LMFDB - Embedded newform 1.68.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1,68,Mod(1,1)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1, base_ring=CyclotomicField(1))

chi = DirichletCharacter(H, H._module([]))

N = Newforms(chi, 68, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1.1");

S:= CuspForms(chi, 68);

N := Newforms(S);

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	1.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.4290351930$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} + \cdots + 20\!\cdots\!00 $ x^5 - x^4 - 939384011925257456x^3 + 31046449413968483513911200x^2 + 156793504704482691874379743265203200*x + 20916736226052669578405116700517591609696000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.06038e8$ of defining polynomial
Character	$\chi$	$=$	1.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34339e10 q^{2} +7.87958e15 q^{3} +3.28965e19 q^{4} -3.01383e23 q^{5} -1.05854e26 q^{6} -4.20742e26 q^{7} +1.54057e30 q^{8} -3.06216e31 q^{9} +O(q^{10})$	\(q-1.34339e10 q^{2} +7.87958e15 q^{3} +3.28965e19 q^{4} -3.01383e23 q^{5} -1.05854e26 q^{6} -4.20742e26 q^{7} +1.54057e30 q^{8} -3.06216e31 q^{9} +4.04876e33 q^{10} +1.62242e34 q^{11} +2.59210e35 q^{12} -3.08856e37 q^{13} +5.65222e36 q^{14} -2.37477e39 q^{15} -2.55506e40 q^{16} -1.59490e41 q^{17} +4.11369e41 q^{18} +1.03902e43 q^{19} -9.91443e42 q^{20} -3.31527e42 q^{21} -2.17954e44 q^{22} -3.59214e45 q^{23} +1.21390e46 q^{24} +2.30691e46 q^{25} +4.14915e47 q^{26} -9.71798e47 q^{27} -1.38409e46 q^{28} +1.54617e49 q^{29} +3.19025e49 q^{30} +1.52983e50 q^{31} +1.15896e50 q^{32} +1.27840e50 q^{33} +2.14258e51 q^{34} +1.26805e50 q^{35} -1.00734e51 q^{36} +1.16261e52 q^{37} -1.39581e53 q^{38} -2.43366e53 q^{39} -4.64301e53 q^{40} +1.50280e54 q^{41} +4.45371e52 q^{42} +1.43610e54 q^{43} +5.33717e53 q^{44} +9.22884e54 q^{45} +4.82565e55 q^{46} +9.41631e55 q^{47} -2.01328e56 q^{48} -4.18201e56 q^{49} -3.09909e56 q^{50} -1.25671e57 q^{51} -1.01603e57 q^{52} +4.49370e57 q^{53} +1.30551e58 q^{54} -4.88968e57 q^{55} -6.48182e56 q^{56} +8.18704e58 q^{57} -2.07711e59 q^{58} +2.81815e59 q^{59} -7.81216e58 q^{60} -4.42593e59 q^{61} -2.05516e60 q^{62} +1.28838e58 q^{63} +2.21365e60 q^{64} +9.30840e60 q^{65} -1.71739e60 q^{66} +9.71985e60 q^{67} -5.24666e60 q^{68} -2.83045e61 q^{69} -1.70348e60 q^{70} -3.66936e61 q^{71} -4.71748e61 q^{72} -1.52158e61 q^{73} -1.56184e62 q^{74} +1.81775e62 q^{75} +3.41801e62 q^{76} -6.82619e60 q^{77} +3.26936e63 q^{78} -2.59876e63 q^{79} +7.70050e63 q^{80} -4.81844e63 q^{81} -2.01885e64 q^{82} -2.25569e64 q^{83} -1.09061e62 q^{84} +4.80676e64 q^{85} -1.92925e64 q^{86} +1.21832e65 q^{87} +2.49944e64 q^{88} -2.92327e64 q^{89} -1.23980e65 q^{90} +1.29949e64 q^{91} -1.18169e65 q^{92} +1.20544e66 q^{93} -1.26498e66 q^{94} -3.13143e66 q^{95} +9.13215e65 q^{96} -8.79797e65 q^{97} +5.61808e66 q^{98} -4.96810e65 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 27\!\cdots\!85 q^{9}+O(q^{10}) $ 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9	$ 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 16\!\cdots\!20 q^{99}+O(q^{100}) $ 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9 - 5287067006725590176865792208438800 * q^10 + 203006981551281631527218766884109060 * q^11 - 2089518810401181456110201520986913024 * q^12 + 177822884499813229570219072356766702 * q^13 + 290753425875732621001638220329334092480 * q^14 - 933024301033068664231979455289562936600 * q^15 + 18266362102387359184851236859301091348480 * q^16 + 75241497376497768433807587544150344102906 * q^17 - 1011568566626978308983328666525487111002968 * q^18 + 3998884102997801557241177083441346393295100 * q^19 - 15074831378020267075078660994426474705846400 * q^20 - 175719863949295325296418063144372762630915040 * q^21 + 1841247880573095918698534631505063585473750432 * q^22 - 419660874260151197180020017656265233426170968 * q^23 - 13019747846573945420584224979196286639928166400 * q^24 + 175279270003539476977813620578789764300129011875 * q^25 + 21416337629300654169124097666599354769371062960 * q^26 + 1886613800166950113417786970194782722207999659320 * q^27 + 14982728306233364463143494158089725946900023694848 * q^28 + 18733547497571939101380180097730717676662894233950 * q^29 + 183881202860964913168534714290880945912096094414400 * q^30 + 365849822471060800690376948081970588059264217288160 * q^31 + 1591165751988992347021834813546084975107380979204096 * q^32 + 2437038903031213839498705428654461423996034595356784 * q^33 + 7817592154883765712044327776251337623941567500782480 * q^34 + 4524513410554564733118320802153390591800073874285200 * q^35 - 38131548457864843751915538710226876689225094472585920 * q^36 - 56389392588768330454829645685931746748224606527380394 * q^37 - 311451303114725973135630000941921262878276604287141280 * q^38 - 717177850582388225418943791249053833224228521090828280 * q^39 - 1886768345048751997029226502271137564956948018338432000 * q^40 + 113555410342193387626817165052765483601006465327103010 * q^41 + 2805540126450529847390961864461873572829871059166350592 * q^42 + 6508304671085340439704487085091235272293711787089861092 * q^43 + 49369449025916528958230735645862721892607029455332199680 * q^44 + 99011086960300194550392336618799141939148872728728967150 * q^45 + 79552980744915855625545425031873854934595203714399466560 * q^46 - 128023594814179612806658270243785955024442659789917751344 * q^47 - 198512903890152994486195174416803543869354744423782432768 * q^48 - 771329931628098140389185999842408514636676231229111640435 * q^49 - 3919481096957516748709549462396523762064894577356951885000 * q^50 - 3397415009313164473667296477133432796692029796730856284840 * q^51 + 672997930141532412373441872733517069231734104886640838016 * q^52 + 1046765637663599376128164919371991945530174947374966490822 * q^53 + 19066352042639687223500180230679796188217876586398799185600 * q^54 + 66942753107037791618627925698973654692865254681839842449400 * q^55 + 164746697572073652525116376573879338548073684495562965299200 * q^56 - 9288557942524759644038647059781434529266663439727382990960 * q^57 - 61137956327686914952144218440883774026433838412764071303120 * q^58 - 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1346157500908637758724100110370785768539984220742218248588800 * q^60 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 - 2197846158587850633685459639420082896328263471188214086352128 * q^62 - 2062780018748959038039139080074136999529046449885010133153768 * q^63 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 + 17054811715744261908524663200762497384422620874171150779332900 * q^65 + 27998862803339055552343423092993933514291039899414979026779520 * q^66 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 + 81678078346863753465296539273887923556029130514802676363013248 * q^68 + 3593002077831059525793506029116145162632074080747869594484320 * q^69 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 - 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 747723581155501189674233444839980291686130221322311676193789440 * q^72 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 + 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 21623812027928499471467875193108522496460255319651250940492500 * q^75 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 + 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 6187393864780941106550100388692932558738785969110711253222726464 * q^78 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 - 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 11195187555119704986426983039082721262159900559267345597680888595 * q^81 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 - 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 52124143836539534782259645807914007868879894236203487647926097920 * q^84 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 + 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 357574798440063707158439877351651513807909832776407769090038489160 * q^87 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 - 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 477054381540878916877701150770415125576892082956059127548575220400 * q^90 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 - 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 1849280210034321990512094428557344731461785100166176155774311127936 * q^93 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 - 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 - 2242556165580376632567217570505477297218451346654758522676359331840 * q^96 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 + 12426885478574475523720488645206508880013087100212747539199689691208 * q^98 + 16175146031966622867876651602334534091105243319832315048256094545620 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.34339e10	−1.10585	−0.552927	−	0.833229i	$-0.686489\pi$
$2$	−1.34339e10	−1.10585	−0.552927	+	0.833229i	$0.686489\pi$
$3$	7.87958e15	0.818354	0.409177	−	0.912455i	$-0.365816\pi$
$3$	7.87958e15	0.818354	0.409177	+	0.912455i	$0.365816\pi$
$4$	3.28965e19	0.222915
$5$	−3.01383e23	−1.15777	−0.578887	−	0.815408i	$-0.696513\pi$
$5$	−3.01383e23	−1.15777	−0.578887	+	0.815408i	$0.696513\pi$
$6$	−1.05854e26	−0.904981
$7$	−4.20742e26	−0.0205699	−0.0102849	−	0.999947i	$-0.503274\pi$
$7$	−4.20742e26	−0.0205699	−0.0102849	+	0.999947i	$0.503274\pi$
$8$	1.54057e30	0.859343
$9$	−3.06216e31	−0.330297
$10$	4.04876e33	1.28033
$11$	1.62242e34	0.210624	0.105312	−	0.994439i	$-0.466416\pi$
$11$	1.62242e34	0.210624	0.105312	+	0.994439i	$0.466416\pi$
$12$	2.59210e35	0.182423
$13$	−3.08856e37	−1.48817	−0.744087	−	0.668082i	$-0.767115\pi$
$13$	−3.08856e37	−1.48817	−0.744087	+	0.668082i	$0.767115\pi$
$14$	5.65222e36	0.0227473
$15$	−2.37477e39	−0.947469
$16$	−2.55506e40	−1.17322
$17$	−1.59490e41	−0.960936	−0.480468	−	0.877012i	$-0.659533\pi$
$17$	−1.59490e41	−0.960936	−0.480468	+	0.877012i	$0.659533\pi$
$18$	4.11369e41	0.365260
$19$	1.03902e43	1.50792	0.753959	−	0.656921i	$-0.228141\pi$
$19$	1.03902e43	1.50792	0.753959	+	0.656921i	$0.228141\pi$
$20$	−9.91443e42	−0.258085
$21$	−3.31527e42	−0.0168334
$22$	−2.17954e44	−0.232919
$23$	−3.59214e45	−0.865905	−0.432953	−	0.901417i	$-0.642528\pi$
$23$	−3.59214e45	−0.865905	−0.432953	+	0.901417i	$0.642528\pi$
$24$	1.21390e46	0.703247
$25$	2.30691e46	0.340440
$26$	4.14915e47	1.64571
$27$	−9.71798e47	−1.08865
$28$	−1.38409e46	−0.00458534
$29$	1.54617e49	1.58097	0.790486	−	0.612480i	$-0.209828\pi$
$29$	1.54617e49	1.58097	0.790486	+	0.612480i	$0.209828\pi$
$30$	3.19025e49	1.04776
$31$	1.52983e50	1.67504	0.837521	−	0.546405i	$-0.184004\pi$
$31$	1.52983e50	1.67504	0.837521	+	0.546405i	$0.184004\pi$
$32$	1.15896e50	0.438072
$33$	1.27840e50	0.172365
$34$	2.14258e51	1.06266
$35$	1.26805e50	0.0238153
$36$	−1.00734e51	−0.0736281
$37$	1.16261e52	0.339372	0.169686	−	0.985498i	$-0.445725\pi$
$37$	1.16261e52	0.339372	0.169686	+	0.985498i	$0.445725\pi$
$38$	−1.39581e53	−1.66754
$39$	−2.43366e53	−1.21785
$40$	−4.64301e53	−0.994925
$41$	1.50280e54	1.40813	0.704064	−	0.710137i	$-0.251367\pi$
$41$	1.50280e54	1.40813	0.704064	+	0.710137i	$0.251367\pi$
$42$	4.45371e52	0.0186153
$43$	1.43610e54	0.272893	0.136447	−	0.990647i	$-0.456432\pi$
$43$	1.43610e54	0.272893	0.136447	+	0.990647i	$0.456432\pi$
$44$	5.33717e53	0.0469512
$45$	9.22884e54	0.382409
$46$	4.82565e55	0.957565
$47$	9.41631e55	0.909081	0.454540	−	0.890726i	$-0.349804\pi$
$47$	9.41631e55	0.909081	0.454540	+	0.890726i	$0.349804\pi$
$48$	−2.01328e56	−0.960112
$49$	−4.18201e56	−0.999577
$50$	−3.09909e56	−0.376477
$51$	−1.25671e57	−0.786386
$52$	−1.01603e57	−0.331737
$53$	4.49370e57	0.775109	0.387554	−	0.921847i	$-0.373320\pi$
$53$	4.49370e57	0.775109	0.387554	+	0.921847i	$0.373320\pi$
$54$	1.30551e58	1.20389
$55$	−4.88968e57	−0.243854
$56$	−6.48182e56	−0.0176766
$57$	8.18704e58	1.23401
$58$	−2.07711e59	−1.74833
$59$	2.81815e59	1.33789	0.668947	−	0.743310i	$-0.266745\pi$
$59$	2.81815e59	1.33789	0.668947	+	0.743310i	$0.266745\pi$
$60$	−7.81216e58	−0.211205
$61$	−4.42593e59	−0.687789	−0.343894	−	0.939008i	$-0.611746\pi$
$61$	−4.42593e59	−0.687789	−0.343894	+	0.939008i	$0.611746\pi$
$62$	−2.05516e60	−1.85235
$63$	1.28838e58	0.00679417
$64$	2.21365e60	0.688780
$65$	9.30840e60	1.72297
$66$	−1.71739e60	−0.190610
$67$	9.71985e60	0.651859	0.325930	−	0.945394i	$-0.394323\pi$
$67$	9.71985e60	0.651859	0.325930	+	0.945394i	$0.394323\pi$
$68$	−5.24666e60	−0.214207
$69$	−2.83045e61	−0.708617
$70$	−1.70348e60	−0.0263362
$71$	−3.66936e61	−0.352725	−0.176362	−	0.984325i	$-0.556433\pi$
$71$	−3.66936e61	−0.352725	−0.176362	+	0.984325i	$0.556433\pi$
$72$	−4.71748e61	−0.283838
$73$	−1.52158e61	−0.0576740	−0.0288370	−	0.999584i	$-0.509180\pi$
$73$	−1.52158e61	−0.0576740	−0.0288370	+	0.999584i	$0.509180\pi$
$74$	−1.56184e62	−0.375296
$75$	1.81775e62	0.278600
$76$	3.41801e62	0.336138
$77$	−6.82619e60	−0.00433250
$78$	3.26936e63	1.34677
$79$	−2.59876e63	−0.698649	−0.349325	−	0.937002i	$-0.613589\pi$
$79$	−2.59876e63	−0.698649	−0.349325	+	0.937002i	$0.613589\pi$
$80$	7.70050e63	1.35833
$81$	−4.81844e63	−0.560607
$82$	−2.01885e64	−1.55719
$83$	−2.25569e64	−1.15921	−0.579607	−	0.814896i	$-0.696794\pi$
$83$	−2.25569e64	−1.15921	−0.579607	+	0.814896i	$0.696794\pi$
$84$	−1.09061e62	−0.00375243
$85$	4.80676e64	1.11255
$86$	−1.92925e64	−0.301780
$87$	1.21832e65	1.29379
$88$	2.49944e64	0.180998
$89$	−2.92327e64	−0.144978	−0.0724892	−	0.997369i	$-0.523094\pi$
$89$	−2.92327e64	−0.144978	−0.0724892	+	0.997369i	$0.523094\pi$
$90$	−1.23980e65	−0.422889
$91$	1.29949e64	0.0306116
$92$	−1.18169e65	−0.193023
$93$	1.20544e66	1.37078
$94$	−1.26498e66	−1.00531
$95$	−3.13143e66	−1.74583
$96$	9.13215e65	0.358498
$97$	−8.79797e65	−0.244078	−0.122039	−	0.992525i	$-0.538943\pi$
$97$	−8.79797e65	−0.244078	−0.122039	+	0.992525i	$0.538943\pi$
$98$	5.61808e66	1.10539
$99$	−4.96810e65	−0.0695683
$100$	7.58891e65	0.0758891
$101$	1.50109e67	1.07558	0.537789	−	0.843080i	$-0.319260\pi$
$101$	1.50109e67	1.07558	0.537789	+	0.843080i	$0.319260\pi$
$102$	1.68826e67	0.869629
$103$	−1.56772e67	−0.582398	−0.291199	−	0.956662i	$-0.594054\pi$
$103$	−1.56772e67	−0.582398	−0.291199	+	0.956662i	$0.594054\pi$
$104$	−4.75814e67	−1.27885
$105$	9.99167e65	0.0194893
$106$	−6.03681e67	−0.857158
$107$	−2.27147e67	−0.235478	−0.117739	−	0.993045i	$-0.537565\pi$
$107$	−2.27147e67	−0.235478	−0.117739	+	0.993045i	$0.537565\pi$
$108$	−3.19687e67	−0.242677
$109$	2.18578e68	1.21848	0.609240	−	0.792986i	$-0.291474\pi$
$109$	2.18578e68	1.21848	0.609240	+	0.792986i	$0.291474\pi$
$110$	6.56877e67	0.269668
$111$	9.16088e67	0.277727
$112$	1.07502e67	0.0241331
$113$	2.20699e68	0.367851	0.183926	−	0.982940i	$-0.441119\pi$
$113$	2.20699e68	0.367851	0.183926	+	0.982940i	$0.441119\pi$
$114$	−1.09984e69	−1.36464
$115$	1.08261e69	1.00252
$116$	5.08635e68	0.352422
$117$	9.45768e68	0.491539
$118$	−3.78588e69	−1.47952
$119$	6.71042e67	0.0197663
$120$	−3.65850e69	−0.814201
$121$	−5.67026e69	−0.955638
$122$	5.94576e69	0.760594
$123$	1.18414e70	1.15235
$124$	5.03259e69	0.373392
$125$	1.34699e70	0.763621
$126$	−1.73080e68	−0.00751336
$127$	1.39984e70	0.466288	0.233144	−	0.972442i	$-0.425099\pi$
$127$	1.39984e70	0.466288	0.233144	+	0.972442i	$0.425099\pi$
$128$	−4.68413e70	−1.19976
$129$	1.13159e70	0.223323
$130$	−1.25048e71	−1.90535
$131$	7.06278e70	0.832506	0.416253	−	0.909249i	$-0.363343\pi$
$131$	7.06278e70	0.832506	0.416253	+	0.909249i	$0.363343\pi$
$132$	4.20547e69	0.0384227
$133$	−4.37159e69	−0.0310177
$134$	−1.30576e71	−0.720862
$135$	2.92883e71	1.26041
$136$	−2.45705e71	−0.825774
$137$	6.38583e71	1.67911	0.839553	−	0.543277i	$-0.182817\pi$
$137$	6.38583e71	1.67911	0.839553	+	0.543277i	$0.182817\pi$
$138$	3.80241e71	0.783627
$139$	−1.22134e72	−1.97625	−0.988126	−	0.153649i	$-0.950898\pi$
$139$	−1.22134e72	−1.97625	−0.988126	+	0.153649i	$0.950898\pi$
$140$	4.17142e69	0.00530878
$141$	7.41966e71	0.743950
$142$	4.92939e71	0.390062
$143$	−5.01093e71	−0.313445
$144$	7.82400e71	0.387512
$145$	−4.65989e72	−1.83041
$146$	2.04409e71	0.0637791
$147$	−3.29525e72	−0.818008
$148$	3.82457e71	0.0756512
$149$	1.00850e73	1.59197	0.795987	−	0.605314i	$-0.206952\pi$
$149$	1.00850e73	1.59197	0.795987	+	0.605314i	$0.206952\pi$
$150$	−2.44195e72	−0.308091
$151$	5.44915e72	0.550302	0.275151	−	0.961401i	$-0.411272\pi$
$151$	5.44915e72	0.550302	0.275151	+	0.961401i	$0.411272\pi$
$152$	1.60068e73	1.29582
$153$	4.88385e72	0.317394
$154$	9.17025e70	0.00479112
$155$	−4.61064e73	−1.93932
$156$	−8.00587e72	−0.271478
$157$	−4.30932e73	−1.17970	−0.589848	−	0.807514i	$-0.700813\pi$
$157$	−4.30932e73	−1.17970	−0.589848	+	0.807514i	$0.700813\pi$
$158$	3.49116e73	0.772605
$159$	3.54085e73	0.634313
$160$	−3.49292e73	−0.507188
$161$	1.51136e72	0.0178116
$162$	6.47306e73	0.619950
$163$	−3.31839e73	−0.258608	−0.129304	−	0.991605i	$-0.541274\pi$
$163$	−3.31839e73	−0.258608	−0.129304	+	0.991605i	$0.541274\pi$
$164$	4.94368e73	0.313893
$165$	−3.85287e73	−0.199559
$166$	3.03028e74	1.28192
$167$	−2.50202e74	−0.865544	−0.432772	−	0.901503i	$-0.642464\pi$
$167$	−2.50202e74	−0.865544	−0.432772	+	0.901503i	$0.642464\pi$
$168$	−5.10741e72	−0.0144657
$169$	5.23192e74	1.21466
$170$	−6.45736e74	−1.23032
$171$	−3.18165e74	−0.498061
$172$	4.72427e73	0.0608320
$173$	3.99627e74	0.423751	0.211876	−	0.977297i	$-0.432043\pi$
$173$	3.99627e74	0.423751	0.211876	+	0.977297i	$0.432043\pi$
$174$	−1.63668e75	−1.43075
$175$	−9.70614e72	−0.00700280
$176$	−4.14536e74	−0.247109
$177$	2.22058e75	1.09487
$178$	3.92710e74	0.160325
$179$	9.62216e74	0.325609	0.162804	−	0.986658i	$-0.447946\pi$
$179$	9.62216e74	0.325609	0.162804	+	0.986658i	$0.447946\pi$
$180$	3.03596e74	0.0852447
$181$	7.00912e75	1.63468	0.817338	−	0.576159i	$-0.195449\pi$
$181$	7.00912e75	1.63468	0.817338	+	0.576159i	$0.195449\pi$
$182$	−1.74572e74	−0.0338520
$183$	−3.48745e75	−0.562855
$184$	−5.53393e75	−0.744110
$185$	−3.50391e75	−0.392916
$186$	−1.61938e76	−1.51588
$187$	−2.58759e75	−0.202396
$188$	3.09763e75	0.202648
$189$	4.08876e74	0.0223935
$190$	4.20674e76	1.93063
$191$	−1.34239e75	−0.0516725	−0.0258362	−	0.999666i	$-0.508225\pi$
$191$	−1.34239e75	−0.0516725	−0.0258362	+	0.999666i	$0.508225\pi$
$192$	1.74427e76	0.563665
$193$	−7.13753e76	−1.93811	−0.969054	−	0.246849i	$-0.920605\pi$
$193$	−7.13753e76	−1.93811	−0.969054	+	0.246849i	$0.920605\pi$
$194$	1.18191e76	0.269915
$195$	7.33463e76	1.41000
$196$	−1.37573e76	−0.222821
$197$	2.98948e75	0.0408297	0.0204149	−	0.999792i	$-0.493501\pi$
$197$	2.98948e75	0.0408297	0.0204149	+	0.999792i	$0.493501\pi$
$198$	6.67411e75	0.0769325
$199$	−6.66765e76	−0.649224	−0.324612	−	0.945847i	$-0.605234\pi$
$199$	−6.66765e76	−0.649224	−0.324612	+	0.945847i	$0.605234\pi$
$200$	3.55395e76	0.292555
$201$	7.65884e76	0.533452
$202$	−2.01656e77	−1.18943
$203$	−6.50539e75	−0.0325204
$204$	−4.13415e76	−0.175297
$205$	−4.52919e77	−1.63029
$206$	2.10606e77	0.644048
$207$	1.09997e77	0.286006
$208$	7.89145e77	1.74596
$209$	1.68572e77	0.317603
$210$	−1.34227e76	−0.0215524
$211$	5.45267e76	0.0746701	0.0373351	−	0.999303i	$-0.488113\pi$
$211$	5.45267e76	0.0746701	0.0373351	+	0.999303i	$0.488113\pi$
$212$	1.47827e77	0.172783
$213$	−2.89130e77	−0.288654
$214$	3.05147e77	0.260404
$215$	−4.32817e77	−0.315948
$216$	−1.49712e78	−0.935527
$217$	−6.43663e76	−0.0344554
$218$	−2.93636e78	−1.34746
$219$	−1.19895e77	−0.0471978
$220$	−1.60853e77	−0.0543588
$221$	4.92595e78	1.43004
$222$	−1.23067e78	−0.307125
$223$	6.27402e78	1.34689	0.673446	−	0.739236i	$-0.264813\pi$
$223$	6.27402e78	1.34689	0.673446	+	0.739236i	$0.264813\pi$
$224$	−4.87625e76	−0.00901109
$225$	−7.06414e77	−0.112446
$226$	−2.96485e78	−0.406790
$227$	3.35983e78	0.397604	0.198802	−	0.980040i	$-0.436295\pi$
$227$	3.35983e78	0.397604	0.198802	+	0.980040i	$0.436295\pi$
$228$	2.69325e78	0.275080
$229$	−5.16052e78	−0.455203	−0.227601	−	0.973754i	$-0.573088\pi$
$229$	−5.16052e78	−0.455203	−0.227601	+	0.973754i	$0.573088\pi$
$230$	−1.45437e79	−1.10864
$231$	−5.37875e76	−0.00354552
$232$	2.38198e79	1.35860
$233$	2.55175e78	0.126013	0.0630065	−	0.998013i	$-0.479931\pi$
$233$	2.55175e78	0.126013	0.0630065	+	0.998013i	$0.479931\pi$
$234$	−1.27054e79	−0.543571
$235$	−2.83791e79	−1.05251
$236$	9.27071e78	0.298237
$237$	−2.04772e79	−0.571742
$238$	−9.01473e77	−0.0218587
$239$	−4.05778e79	−0.854985	−0.427492	−	0.904019i	$-0.640603\pi$
$239$	−4.05778e79	−0.854985	−0.427492	+	0.904019i	$0.640603\pi$
$240$	6.06768e79	1.11159
$241$	4.01993e79	0.640689	0.320345	−	0.947301i	$-0.396201\pi$
$241$	4.01993e79	0.640689	0.320345	+	0.947301i	$0.396201\pi$
$242$	7.61739e79	1.05680
$243$	5.21275e79	0.629879
$244$	−1.45597e79	−0.153318
$245$	1.26039e80	1.15728
$246$	−1.59077e80	−1.27433
$247$	−3.20908e80	−2.24405
$248$	2.35680e80	1.43944
$249$	−1.77739e80	−0.948647
$250$	−1.80953e80	−0.844455
$251$	1.29030e80	0.526771	0.263385	−	0.964691i	$-0.415161\pi$
$251$	1.29030e80	0.526771	0.263385	+	0.964691i	$0.415161\pi$
$252$	4.23832e77	0.00151452
$253$	−5.82794e79	−0.182380
$254$	−1.88054e80	−0.515647
$255$	3.78753e80	0.910457
$256$	3.02586e80	0.637984
$257$	5.23305e80	0.968268	0.484134	−	0.874994i	$-0.339135\pi$
$257$	5.23305e80	0.968268	0.484134	+	0.874994i	$0.339135\pi$
$258$	−1.52017e80	−0.246963
$259$	−4.89159e78	−0.00698084
$260$	3.06213e80	0.384076
$261$	−4.73462e80	−0.522190
$262$	−9.48809e80	−0.920631
$263$	9.85079e80	0.841305	0.420652	−	0.907222i	$-0.361801\pi$
$263$	9.85079e80	0.841305	0.420652	+	0.907222i	$0.361801\pi$
$264$	1.96946e80	0.148120
$265$	−1.35433e81	−0.897400
$266$	5.87277e79	0.0343011
$267$	−2.30341e80	−0.118644
$268$	3.19749e80	0.145309
$269$	−1.08276e81	−0.434343	−0.217171	−	0.976134i	$-0.569683\pi$
$269$	−1.08276e81	−0.434343	−0.217171	+	0.976134i	$0.569683\pi$
$270$	−3.93457e81	−1.39384
$271$	4.87334e81	1.52530	0.762650	−	0.646811i	$-0.223898\pi$
$271$	4.87334e81	1.52530	0.762650	+	0.646811i	$0.223898\pi$
$272$	4.07506e81	1.12739
$273$	1.02394e80	0.0250511
$274$	−8.57868e81	−1.85685
$275$	3.74277e80	0.0717047
$276$	−9.31119e80	−0.157961
$277$	8.18997e81	1.23086	0.615432	−	0.788190i	$-0.288981\pi$
$277$	8.18997e81	1.23086	0.615432	+	0.788190i	$0.288981\pi$
$278$	1.64074e82	2.18545
$279$	−4.68458e81	−0.553261
$280$	1.95351e80	0.0204655
$281$	1.82919e81	0.170058	0.0850289	−	0.996378i	$-0.472902\pi$
$281$	1.82919e81	0.170058	0.0850289	+	0.996378i	$0.472902\pi$
$282$	−9.96751e81	−0.822700
$283$	−1.71714e82	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$283$	−1.71714e82	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$284$	−1.20709e81	−0.0786276
$285$	−2.46743e82	−1.42871
$286$	6.73165e81	0.346625
$287$	−6.32292e80	−0.0289650
$288$	−3.54894e81	−0.144694
$289$	−2.11015e81	−0.0766012
$290$	6.26007e82	2.02417
$291$	−6.93243e81	−0.199742
$292$	−5.00547e80	−0.0128564
$293$	4.86284e82	1.11385	0.556923	−	0.830564i	$-0.311982\pi$
$293$	4.86284e82	1.11385	0.556923	+	0.830564i	$0.311982\pi$
$294$	4.42681e82	0.904598
$295$	−8.49342e82	−1.54898
$296$	1.79108e82	0.291637
$297$	−1.57666e82	−0.229296
$298$	−1.35481e83	−1.76049
$299$	1.10945e83	1.28862
$300$	5.97975e81	0.0621042
$301$	−6.04229e80	−0.00561338
$302$	−7.32035e82	−0.608555
$303$	1.18280e83	0.880203
$304$	−2.65475e83	−1.76913
$305$	1.33390e83	0.796304
$306$	−6.56092e82	−0.350992
$307$	5.34655e81	0.0256412	0.0128206	−	0.999918i	$-0.495919\pi$
$307$	5.34655e81	0.0256412	0.0128206	+	0.999918i	$0.495919\pi$
$308$	−2.24557e80	−0.000965780 0
$309$	−1.23529e83	−0.476608
$310$	6.19390e83	2.14461
$311$	−4.84522e82	−0.150605	−0.0753027	−	0.997161i	$-0.523992\pi$
$311$	−4.84522e82	−0.150605	−0.0753027	+	0.997161i	$0.523992\pi$
$312$	−3.74922e83	−1.04655
$313$	−1.01435e82	−0.0254363	−0.0127182	−	0.999919i	$-0.504048\pi$
$313$	−1.01435e82	−0.0254363	−0.0127182	+	0.999919i	$0.504048\pi$
$314$	5.78911e83	1.30457
$315$	−3.88296e81	−0.00786611
$316$	−8.54901e82	−0.155739
$317$	3.02253e83	0.495319	0.247660	−	0.968847i	$-0.420339\pi$
$317$	3.02253e83	0.495319	0.247660	+	0.968847i	$0.420339\pi$
$318$	−4.75675e83	−0.701458
$319$	2.50853e83	0.332990
$320$	−6.67157e83	−0.797451
$321$	−1.78982e83	−0.192704
$322$	−2.03035e82	−0.0196970
$323$	−1.65713e84	−1.44901
$324$	−1.58510e83	−0.124968
$325$	−7.12503e83	−0.506634
$326$	4.45790e83	0.285983
$327$	1.72230e84	0.997148
$328$	2.31517e84	1.21007
$329$	−3.96184e82	−0.0186997
$330$	5.17591e83	0.220684
$331$	2.27713e84	0.877303	0.438652	−	0.898657i	$-0.355456\pi$
$331$	2.27713e84	0.877303	0.438652	+	0.898657i	$0.355456\pi$
$332$	−7.42042e83	−0.258406
$333$	−3.56010e83	−0.112094
$334$	3.36120e84	0.957166
$335$	−2.92940e84	−0.754705
$336$	8.47071e82	0.0197494
$337$	−7.83609e84	−1.65386	−0.826928	−	0.562308i	$-0.809914\pi$
$337$	−7.83609e84	−1.65386	−0.826928	+	0.562308i	$0.809914\pi$
$338$	−7.02852e84	−1.34324
$339$	1.73901e84	0.301033
$340$	1.58125e84	0.248003
$341$	2.48202e84	0.352803
$342$	4.27420e84	0.550783
$343$	3.51984e83	0.0411311
$344$	2.21242e84	0.234509
$345$	8.53051e84	0.820418
$346$	−5.36856e84	−0.468607
$347$	−1.79418e85	−1.42177	−0.710884	−	0.703309i	$-0.751705\pi$
$347$	−1.79418e85	−1.42177	−0.710884	+	0.703309i	$0.751705\pi$
$348$	4.00783e84	0.288406
$349$	−3.77493e83	−0.0246750	−0.0123375	−	0.999924i	$-0.503927\pi$
$349$	−3.77493e83	−0.0246750	−0.0123375	+	0.999924i	$0.503927\pi$
$350$	1.30392e83	0.00774409
$351$	3.00146e85	1.62011
$352$	1.88032e84	0.0922684
$353$	1.83957e85	0.820848	0.410424	−	0.911895i	$-0.365381\pi$
$353$	1.83957e85	0.820848	0.410424	+	0.911895i	$0.365381\pi$
$354$	−2.98312e85	−1.21077
$355$	1.10588e85	0.408375
$356$	−9.61652e83	−0.0323179
$357$	5.28753e83	0.0161759
$358$	−1.29263e85	−0.360076
$359$	−5.72913e85	−1.45353	−0.726766	−	0.686885i	$-0.758978\pi$
$359$	−5.72913e85	−1.45353	−0.726766	+	0.686885i	$0.758978\pi$
$360$	1.42177e85	0.328621
$361$	6.04782e85	1.27382
$362$	−9.41601e85	−1.80771
$363$	−4.46793e85	−0.782050
$364$	4.27486e83	0.00682378
$365$	4.58580e84	0.0667735
$366$	4.68501e85	0.622435
$367$	9.37834e85	1.13714	0.568568	−	0.822636i	$-0.307497\pi$
$367$	9.37834e85	1.13714	0.568568	+	0.822636i	$0.307497\pi$
$368$	9.17811e85	1.01590
$369$	−4.60182e85	−0.465100
$370$	4.70713e85	0.434508
$371$	−1.89069e84	−0.0159439
$372$	3.96547e85	0.305567
$373$	−8.38977e85	−0.590886	−0.295443	−	0.955360i	$-0.595467\pi$
$373$	−8.38977e85	−0.590886	−0.295443	+	0.955360i	$0.595467\pi$
$374$	3.47615e85	0.223821
$375$	1.06137e86	0.624913
$376$	1.45065e86	0.781212
$377$	−4.77544e86	−2.35276
$378$	−5.49281e84	−0.0247639
$379$	−3.46586e86	−1.43020	−0.715102	−	0.699020i	$-0.753620\pi$
$379$	−3.46586e86	−1.43020	−0.715102	+	0.699020i	$0.753620\pi$
$380$	−1.03013e86	−0.389172
$381$	1.10302e86	0.381588
$382$	1.80335e85	0.0571423
$383$	4.41090e86	1.28047	0.640234	−	0.768180i	$-0.278837\pi$
$383$	4.41090e86	1.28047	0.640234	+	0.768180i	$0.278837\pi$
$384$	−3.69090e86	−0.981830
$385$	2.05730e84	0.00501606
$386$	9.58850e86	2.14327
$387$	−4.39758e85	−0.0901357
$388$	−2.89422e85	−0.0544087
$389$	2.14221e86	0.369446	0.184723	−	0.982791i	$-0.440861\pi$
$389$	2.14221e86	0.369446	0.184723	+	0.982791i	$0.440861\pi$
$390$	−9.85329e86	−1.55925
$391$	5.72910e86	0.832080
$392$	−6.44267e86	−0.858980
$393$	5.56518e86	0.681285
$394$	−4.01604e85	−0.0451518
$395$	7.83223e86	0.808878
$396$	−1.63433e85	−0.0155078
$397$	−1.53266e87	−1.33649	−0.668244	−	0.743942i	$-0.732954\pi$
$397$	−1.53266e87	−1.33649	−0.668244	+	0.743942i	$0.732954\pi$
$398$	8.95727e86	0.717948
$399$	−3.44463e85	−0.0253835
$400$	−5.89428e86	−0.399412
$401$	1.77468e87	1.10607	0.553035	−	0.833158i	$-0.313470\pi$
$401$	1.77468e87	1.10607	0.553035	+	0.833158i	$0.313470\pi$
$402$	−1.02888e87	−0.589920
$403$	−4.72497e87	−2.49276
$404$	4.93807e86	0.239762
$405$	1.45220e87	0.649056
$406$	8.73929e85	0.0359628
$407$	1.88624e86	0.0714798
$408$	−1.93606e87	−0.675776
$409$	6.64558e86	0.213699	0.106849	−	0.994275i	$-0.465924\pi$
$409$	6.64558e86	0.213699	0.106849	+	0.994275i	$0.465924\pi$
$410$	6.08447e87	1.80287
$411$	5.03177e87	1.37410
$412$	−5.15723e86	−0.129825
$413$	−1.18571e86	−0.0275203
$414$	−1.47769e87	−0.316281
$415$	6.79827e87	1.34211
$416$	−3.57953e87	−0.651928
$417$	−9.62368e87	−1.61727
$418$	−2.26459e87	−0.351223
$419$	−7.12912e87	−1.02063	−0.510313	−	0.859989i	$-0.670470\pi$
$419$	−7.12912e87	−1.02063	−0.510313	+	0.859989i	$0.670470\pi$
$420$	3.28691e85	0.00434446
$421$	1.54353e87	0.188393	0.0941965	−	0.995554i	$-0.469972\pi$
$421$	1.54353e87	0.188393	0.0941965	+	0.995554i	$0.469972\pi$
$422$	−7.32508e86	−0.0825744
$423$	−2.88343e87	−0.300266
$424$	6.92286e87	0.666084
$425$	−3.67929e87	−0.327141
$426$	3.88416e87	0.319209
$427$	1.86218e86	0.0141477
$428$	−7.47232e86	−0.0524915
$429$	−3.94840e87	−0.256509
$430$	5.81444e87	0.349393
$431$	2.09290e88	1.16348	0.581742	−	0.813373i	$-0.302371\pi$
$431$	2.09290e88	1.16348	0.581742	+	0.813373i	$0.302371\pi$
$432$	2.48300e88	1.27723
$433$	−1.32958e87	−0.0632951	−0.0316476	−	0.999499i	$-0.510075\pi$
$433$	−1.32958e87	−0.0632951	−0.0316476	+	0.999499i	$0.510075\pi$
$434$	8.64692e86	0.0381027
$435$	−3.67180e88	−1.49792
$436$	7.19044e87	0.271618
$437$	−3.73230e88	−1.30571
$438$	1.61065e87	0.0521939
$439$	3.72682e88	1.11886	0.559432	−	0.828876i	$-0.311019\pi$
$439$	3.72682e88	1.11886	0.559432	+	0.828876i	$0.311019\pi$
$440$	−7.53290e87	−0.209555
$441$	1.28060e88	0.330157
$442$	−6.61748e88	−1.58142
$443$	3.13151e88	0.693793	0.346896	−	0.937904i	$-0.387236\pi$
$443$	3.13151e88	0.693793	0.346896	+	0.937904i	$0.387236\pi$
$444$	3.01361e87	0.0619094
$445$	8.81023e87	0.167852
$446$	−8.42847e88	−1.48947
$447$	7.94656e88	1.30280
$448$	−9.31377e86	−0.0141681
$449$	−5.09792e88	−0.719683	−0.359841	−	0.933014i	$-0.617169\pi$
$449$	−5.09792e88	−0.719683	−0.359841	+	0.933014i	$0.617169\pi$
$450$	9.48991e87	0.124349
$451$	2.43817e88	0.296585
$452$	7.26021e87	0.0819996
$453$	4.29371e88	0.450342
$454$	−4.51357e88	−0.439693
$455$	−3.91644e87	−0.0354413
$456$	1.26127e89	1.06044
$457$	4.22735e88	0.330275	0.165137	−	0.986271i	$-0.447193\pi$
$457$	4.22735e88	0.330275	0.165137	+	0.986271i	$0.447193\pi$
$458$	6.93261e88	0.503388
$459$	1.54992e89	1.04613
$460$	3.56140e88	0.223477
$461$	−2.47810e89	−1.44590	−0.722949	−	0.690901i	$-0.757214\pi$
$461$	−2.47810e89	−1.44590	−0.722949	+	0.690901i	$0.757214\pi$
$462$	7.22577e86	0.00392083
$463$	−8.10015e88	−0.408818	−0.204409	−	0.978886i	$-0.565527\pi$
$463$	−8.10015e88	−0.408818	−0.204409	+	0.978886i	$0.565527\pi$
$464$	−3.95055e89	−1.85483
$465$	−3.63299e89	−1.58705
$466$	−3.42801e88	−0.139352
$467$	3.15046e89	1.19195	0.595977	−	0.803002i	$-0.296765\pi$
$467$	3.15046e89	1.19195	0.595977	+	0.803002i	$0.296765\pi$
$468$	3.11124e88	0.109572
$469$	−4.08955e87	−0.0134087
$470$	3.81243e89	1.16392
$471$	−3.39556e89	−0.965410
$472$	4.34155e89	1.14971
$473$	2.32996e88	0.0574777
$474$	2.75089e89	0.632264
$475$	2.39692e89	0.513356
$476$	2.20749e87	0.00440622
$477$	−1.37605e89	−0.256016
$478$	5.45119e89	0.945489
$479$	−2.89428e89	−0.468059	−0.234029	−	0.972230i	$-0.575191\pi$
$479$	−2.89428e89	−0.468059	−0.234029	+	0.972230i	$0.575191\pi$
$480$	−2.75228e89	−0.415060
$481$	−3.59079e89	−0.505045
$482$	−5.40035e89	−0.708510
$483$	1.19089e88	0.0145762
$484$	−1.86532e89	−0.213026
$485$	2.65156e89	0.282587
$486$	−7.00277e89	−0.696554
$487$	1.75391e90	1.62850	0.814250	−	0.580514i	$-0.197148\pi$
$487$	1.75391e90	1.62850	0.814250	+	0.580514i	$0.197148\pi$
$488$	−6.81845e89	−0.591047
$489$	−2.61475e89	−0.211633
$490$	−1.69319e90	−1.27979
$491$	8.85050e89	0.624795	0.312398	−	0.949951i	$-0.398868\pi$
$491$	8.85050e89	0.624795	0.312398	+	0.949951i	$0.398868\pi$
$492$	3.89541e89	0.256876
$493$	−2.46599e90	−1.51921
$494$	4.31105e90	2.48159
$495$	1.49730e89	0.0805444
$496$	−3.90879e90	−1.96520
$497$	1.54386e88	0.00725550
$498$	2.38773e90	1.04907
$499$	4.21399e90	1.73111	0.865555	−	0.500813i	$-0.166966\pi$
$499$	4.21399e90	1.73111	0.865555	+	0.500813i	$0.166966\pi$
$500$	4.43111e89	0.170223
$501$	−1.97149e90	−0.708321
$502$	−1.73338e90	−0.582532
$503$	2.77704e90	0.873081	0.436540	−	0.899685i	$-0.356204\pi$
$503$	2.77704e90	0.873081	0.436540	+	0.899685i	$0.356204\pi$
$504$	1.98484e88	0.00583852
$505$	−4.52404e90	−1.24528
$506$	7.82921e89	0.201686
$507$	4.12253e90	0.994026
$508$	4.60498e89	0.103943
$509$	−2.09998e90	−0.443782	−0.221891	−	0.975071i	$-0.571223\pi$
$509$	−2.09998e90	−0.443782	−0.221891	+	0.975071i	$0.571223\pi$
$510$	−5.08813e90	−1.00683
$511$	6.40195e87	0.00118635
$512$	2.84765e90	0.494245
$513$	−1.00972e91	−1.64160
$514$	−7.03004e90	−1.07076
$515$	4.72483e90	0.674285
$516$	3.72253e89	0.0497821
$517$	1.52772e90	0.191474
$518$	6.57133e88	0.00771980
$519$	3.14890e90	0.346778
$520$	1.43402e91	1.48062
$521$	3.31742e90	0.321171	0.160586	−	0.987022i	$-0.448662\pi$
$521$	3.31742e90	0.321171	0.160586	+	0.987022i	$0.448662\pi$
$522$	6.36046e90	0.577466
$523$	1.46798e91	1.25001	0.625004	−	0.780622i	$-0.285097\pi$
$523$	1.46798e91	1.25001	0.625004	+	0.780622i	$0.285097\pi$
$524$	2.32340e90	0.185578
$525$	−7.64803e88	−0.00573077
$526$	−1.32335e91	−0.930361
$527$	−2.43992e91	−1.60961
$528$	−3.26637e90	−0.202222
$529$	−4.30593e90	−0.250208
$530$	1.81939e91	0.992395
$531$	−8.62964e90	−0.441902
$532$	−1.43810e89	−0.00691431
$533$	−4.64149e91	−2.09554
$534$	3.09439e90	0.131203
$535$	6.84581e90	0.272630
$536$	1.49741e91	0.560171
$537$	7.58186e90	0.266463
$538$	1.45458e91	0.480320
$539$	−6.78495e90	−0.210535
$540$	9.63482e90	0.280965
$541$	5.22060e91	1.43091	0.715454	−	0.698660i	$-0.246220\pi$
$541$	5.22060e91	1.43091	0.715454	+	0.698660i	$0.246220\pi$
$542$	−6.54681e91	−1.68676
$543$	5.52290e91	1.33774
$544$	−1.84843e91	−0.420960
$545$	−6.58757e91	−1.41072
$546$	−1.37556e90	−0.0277029
$547$	2.90185e91	0.549667	0.274834	−	0.961492i	$-0.411377\pi$
$547$	2.90185e91	0.549667	0.274834	+	0.961492i	$0.411377\pi$
$548$	2.10071e91	0.374298
$549$	1.35529e91	0.227174
$550$	−5.02800e90	−0.0792950
$551$	1.60650e92	2.38398
$552$	−4.36051e91	−0.608945
$553$	1.09341e90	0.0143711
$554$	−1.10024e92	−1.36116
$555$	−2.76093e91	−0.321544
$556$	−4.01779e91	−0.440536
$557$	−5.90102e91	−0.609226	−0.304613	−	0.952476i	$-0.598527\pi$
$557$	−5.90102e91	−0.609226	−0.304613	+	0.952476i	$0.598527\pi$
$558$	6.29323e91	0.611826
$559$	−4.43549e91	−0.406113
$560$	−3.23993e90	−0.0279406
$561$	−2.03891e91	−0.165632
$562$	−2.45732e91	−0.188059
$563$	−8.31074e91	−0.599251	−0.299625	−	0.954057i	$-0.596862\pi$
$563$	−8.31074e91	−0.599251	−0.299625	+	0.954057i	$0.596862\pi$
$564$	2.44080e91	0.165838
$565$	−6.65149e91	−0.425888
$566$	2.30679e92	1.39206
$567$	2.02732e90	0.0115316
$568$	−5.65290e91	−0.303112
$569$	7.52878e90	0.0380596	0.0190298	−	0.999819i	$-0.493942\pi$
$569$	7.52878e90	0.0380596	0.0190298	+	0.999819i	$0.493942\pi$
$570$	3.31473e92	1.57994
$571$	1.98233e92	0.890979	0.445489	−	0.895287i	$-0.353030\pi$
$571$	1.98233e92	0.890979	0.445489	+	0.895287i	$0.353030\pi$
$572$	−1.64842e91	−0.0698716
$573$	−1.05774e91	−0.0422864
$574$	8.49416e90	0.0320311
$575$	−8.28673e91	−0.294789
$576$	−6.77857e91	−0.227502
$577$	−2.56912e91	−0.0813570	−0.0406785	−	0.999172i	$-0.512952\pi$
$577$	−2.56912e91	−0.0813570	−0.0406785	+	0.999172i	$0.512952\pi$
$578$	2.83476e91	0.0847098
$579$	−5.62407e92	−1.58606
$580$	−1.53294e92	−0.408025
$581$	9.49064e90	0.0238449
$582$	9.31298e91	0.220886
$583$	7.29065e91	0.163256
$584$	−2.34411e91	−0.0495618
$585$	−2.85039e92	−0.569091
$586$	−6.53271e92	−1.23175
$587$	−3.55629e92	−0.633317	−0.316658	−	0.948540i	$-0.602561\pi$
$587$	−3.55629e92	−0.633317	−0.316658	+	0.948540i	$0.602561\pi$
$588$	−1.08402e92	−0.182346
$589$	1.58952e93	2.52583
$590$	1.14100e93	1.71294
$591$	2.35558e91	0.0334132
$592$	−2.97053e92	−0.398160
$593$	1.39162e92	0.176274	0.0881368	−	0.996108i	$-0.471909\pi$
$593$	1.39162e92	0.176274	0.0881368	+	0.996108i	$0.471909\pi$
$594$	2.11807e92	0.253568
$595$	−2.02241e91	−0.0228850
$596$	3.31761e92	0.354875
$597$	−5.25383e92	−0.531295
$598$	−1.49043e93	−1.42502
$599$	7.02635e91	0.0635230	0.0317615	−	0.999495i	$-0.489888\pi$
$599$	7.02635e91	0.0635230	0.0317615	+	0.999495i	$0.489888\pi$
$600$	2.80037e92	0.239413
$601$	1.73639e92	0.140395	0.0701977	−	0.997533i	$-0.477637\pi$
$601$	1.73639e92	0.140395	0.0701977	+	0.997533i	$0.477637\pi$
$602$	8.11717e90	0.00620758
$603$	−2.97638e92	−0.215307
$604$	1.79258e92	0.122671
$605$	1.70892e93	1.10641
$606$	−1.58896e93	−0.973377
$607$	8.53328e91	0.0494646	0.0247323	−	0.999694i	$-0.492127\pi$
$607$	8.53328e91	0.0494646	0.0247323	+	0.999694i	$0.492127\pi$
$608$	1.20419e93	0.660577
$609$	−5.12597e91	−0.0266132
$610$	−1.79195e93	−0.880596
$611$	−2.90828e93	−1.35287
$612$	1.60661e92	0.0707520
$613$	1.00190e93	0.417732	0.208866	−	0.977944i	$-0.433023\pi$
$613$	1.00190e93	0.417732	0.208866	+	0.977944i	$0.433023\pi$
$614$	−7.18251e91	−0.0283554
$615$	−3.56881e93	−1.33416
$616$	−1.05162e91	−0.00372311
$617$	5.41310e93	1.81507	0.907534	−	0.419978i	$-0.137962\pi$
$617$	5.41310e93	1.81507	0.907534	+	0.419978i	$0.137962\pi$
$618$	1.65949e93	0.527059
$619$	1.39465e93	0.419595	0.209797	−	0.977745i	$-0.432720\pi$
$619$	1.39465e93	0.419595	0.209797	+	0.977745i	$0.432720\pi$
$620$	−1.51674e93	−0.432303
$621$	3.49083e93	0.942671
$622$	6.50903e92	0.166548
$623$	1.22994e91	0.00298219
$624$	6.21813e93	1.42882
$625$	−5.62281e93	−1.22454
$626$	1.36268e92	0.0281289
$627$	1.32828e93	0.259912
$628$	−1.41761e93	−0.262972
$629$	−1.85425e93	−0.326115
$630$	5.21635e91	0.00869877
$631$	−1.08460e94	−1.71508	−0.857541	−	0.514416i	$-0.828009\pi$
$631$	−1.08460e94	−1.71508	−0.857541	+	0.514416i	$0.828009\pi$
$632$	−4.00358e93	−0.600380
$633$	4.29648e92	0.0611066
$634$	−4.06044e93	−0.547751
$635$	−4.21888e93	−0.539856
$636$	1.16481e93	0.141398
$637$	1.29164e94	1.48755
$638$	−3.36994e93	−0.368239
$639$	1.12362e93	0.116504
$640$	1.41172e94	1.38905
$641$	−1.43305e94	−1.33819	−0.669095	−	0.743177i	$-0.733318\pi$
$641$	−1.43305e94	−1.33819	−0.669095	+	0.743177i	$0.733318\pi$
$642$	2.40443e93	0.213103
$643$	1.53129e94	1.28822	0.644109	−	0.764934i	$-0.277228\pi$
$643$	1.53129e94	1.28822	0.644109	+	0.764934i	$0.277228\pi$
$644$	4.97185e91	0.00397047
$645$	−3.41042e93	−0.258558
$646$	2.22618e94	1.60240
$647$	−1.58467e94	−1.08304	−0.541521	−	0.840687i	$-0.682151\pi$
$647$	−1.58467e94	−1.08304	−0.541521	+	0.840687i	$0.682151\pi$
$648$	−7.42315e93	−0.481754
$649$	4.57221e93	0.281792
$650$	9.57172e93	0.560264
$651$	−5.07180e92	−0.0281967
$652$	−1.09163e93	−0.0576477
$653$	1.67109e94	0.838317	0.419159	−	0.907913i	$-0.362325\pi$
$653$	1.67109e94	0.838317	0.419159	+	0.907913i	$0.362325\pi$
$654$	−2.31373e94	−1.10270
$655$	−2.12860e94	−0.963854
$656$	−3.83974e94	−1.65205
$657$	4.65934e92	0.0190495
$658$	5.32230e92	0.0206791
$659$	2.14450e94	0.791888	0.395944	−	0.918275i	$-0.370417\pi$
$659$	2.14450e94	0.791888	0.395944	+	0.918275i	$0.370417\pi$
$660$	−1.26746e93	−0.0444848
$661$	4.49680e94	1.50022	0.750109	−	0.661314i	$-0.230001\pi$
$661$	4.49680e94	1.50022	0.750109	+	0.661314i	$0.230001\pi$
$662$	−3.05908e94	−0.970170
$663$	3.88144e94	1.17028
$664$	−3.47505e94	−0.996162
$665$	1.31752e93	0.0359115
$666$	4.78262e93	0.123959
$667$	−5.55405e94	−1.36897
$668$	−8.23076e93	−0.192943
$669$	4.94366e94	1.10223
$670$	3.93533e94	0.834595
$671$	−7.18070e93	−0.144865
$672$	−3.84228e92	−0.00737426
$673$	−4.87646e94	−0.890431	−0.445216	−	0.895423i	$-0.646873\pi$
$673$	−4.87646e94	−0.890431	−0.445216	+	0.895423i	$0.646873\pi$
$674$	1.05269e95	1.82892
$675$	−2.24185e94	−0.370621
$676$	1.72112e94	0.270767
$677$	−8.08569e93	−0.121059	−0.0605294	−	0.998166i	$-0.519279\pi$
$677$	−8.08569e93	−0.121059	−0.0605294	+	0.998166i	$0.519279\pi$
$678$	−2.33618e94	−0.332898
$679$	3.70168e92	0.00502066
$680$	7.40514e94	0.956060
$681$	2.64741e94	0.325381
$682$	−3.33432e94	−0.390149
$683$	5.83902e93	0.0650499	0.0325249	−	0.999471i	$-0.489645\pi$
$683$	5.83902e93	0.0650499	0.0325249	+	0.999471i	$0.489645\pi$
$684$	−1.04665e94	−0.111025
$685$	−1.92458e95	−1.94403
$686$	−4.72853e93	−0.0454850
$687$	−4.06628e94	−0.372517
$688$	−3.66932e94	−0.320165
$689$	−1.38791e95	−1.15350
$690$	−1.14598e95	−0.907263
$691$	9.45504e94	0.713098	0.356549	−	0.934277i	$-0.383953\pi$
$691$	9.45504e94	0.713098	0.356549	+	0.934277i	$0.383953\pi$
$692$	1.31463e94	0.0944605
$693$	2.09029e92	0.00143101
$694$	2.41029e95	1.57227
$695$	3.68092e95	2.28805
$696$	1.87690e95	1.11181
$697$	−2.39682e95	−1.35312
$698$	5.07122e93	0.0272870
$699$	2.01067e94	0.103123
$700$	−3.19298e92	−0.00156103
$701$	2.89282e95	1.34824	0.674122	−	0.738620i	$-0.264522\pi$
$701$	2.89282e95	1.34824	0.674122	+	0.738620i	$0.264522\pi$
$702$	−4.03214e95	−1.79160
$703$	1.20797e95	0.511746
$704$	3.59146e94	0.145073
$705$	−2.23616e95	−0.861325
$706$	−2.47126e95	−0.907739
$707$	−6.31573e93	−0.0221245
$708$	7.30493e94	0.244063
$709$	1.32934e95	0.423632	0.211816	−	0.977310i	$-0.432062\pi$
$709$	1.32934e95	0.423632	0.211816	+	0.977310i	$0.432062\pi$
$710$	−1.48564e95	−0.451604
$711$	7.95784e94	0.230762
$712$	−4.50350e94	−0.124586
$713$	−5.49535e95	−1.45043
$714$	−7.10323e93	−0.0178882
$715$	1.51021e95	0.362898
$716$	3.16535e94	0.0725831
$717$	−3.19736e95	−0.699680
$718$	7.69647e95	1.60740
$719$	4.51429e95	0.899853	0.449927	−	0.893065i	$-0.351450\pi$
$719$	4.51429e95	0.899853	0.449927	+	0.893065i	$0.351450\pi$
$720$	−2.35802e95	−0.448651
$721$	6.59604e93	0.0119799
$722$	−8.12460e95	−1.40866
$723$	3.16754e95	0.524311
$724$	2.30575e95	0.364394
$725$	3.56687e95	0.538226
$726$	6.00219e95	0.864834
$727$	−8.14098e95	−1.12014	−0.560072	−	0.828444i	$-0.689227\pi$
$727$	−8.14098e95	−1.12014	−0.560072	+	0.828444i	$0.689227\pi$
$728$	2.00195e94	0.0263059
$729$	8.57458e95	1.07607
$730$	−6.16053e94	−0.0738418
$731$	−2.29044e95	−0.262233
$732$	−1.14725e95	−0.125469
$733$	4.16267e95	0.434899	0.217450	−	0.976072i	$-0.430226\pi$
$733$	4.16267e95	0.434899	0.217450	+	0.976072i	$0.430226\pi$
$734$	−1.25988e96	−1.25751
$735$	9.93132e95	0.947068
$736$	−4.16316e95	−0.379329
$737$	1.57696e95	0.137297
$738$	6.18205e95	0.514333
$739$	−7.08509e95	−0.563322	−0.281661	−	0.959514i	$-0.590885\pi$
$739$	−7.08509e95	−0.563322	−0.281661	+	0.959514i	$0.590885\pi$
$740$	−1.15266e95	−0.0875869
$741$	−2.52862e96	−1.83642
$742$	2.53994e94	0.0176316
$743$	2.43728e95	0.161726	0.0808631	−	0.996725i	$-0.474232\pi$
$743$	2.43728e95	0.161726	0.0808631	+	0.996725i	$0.474232\pi$
$744$	1.85706e96	1.17797
$745$	−3.03945e96	−1.84315
$746$	1.12708e96	0.653435
$747$	6.90729e95	0.382885
$748$	−8.51226e94	−0.0451171
$749$	9.55701e93	0.00484375
$750$	−1.42584e96	−0.691063
$751$	2.39345e96	1.10940	0.554698	−	0.832052i	$-0.312834\pi$
$751$	2.39345e96	1.10940	0.554698	+	0.832052i	$0.312834\pi$
$752$	−2.40592e96	−1.06656
$753$	1.01670e96	0.431085
$754$	6.41529e96	2.60181
$755$	−1.64228e96	−0.637126
$756$	1.34506e94	0.00499184
$757$	3.70408e96	1.31513	0.657565	−	0.753398i	$-0.271586\pi$
$757$	3.70408e96	1.31513	0.657565	+	0.753398i	$0.271586\pi$
$758$	4.65602e96	1.58160
$759$	−4.59217e95	−0.149251
$760$	−4.82418e96	−1.50027
$761$	−3.66225e96	−1.08984	−0.544919	−	0.838489i	$-0.683440\pi$
$761$	−3.66225e96	−1.08984	−0.544919	+	0.838489i	$0.683440\pi$
$762$	−1.48178e96	−0.421981
$763$	−9.19650e94	−0.0250640
$764$	−4.41597e94	−0.0115186
$765$	−1.47191e96	−0.367471
$766$	−5.92558e96	−1.41601
$767$	−8.70403e96	−1.99102
$768$	2.38425e96	0.522096
$769$	5.83109e96	1.22241	0.611206	−	0.791472i	$-0.290685\pi$
$769$	5.83109e96	1.22241	0.611206	+	0.791472i	$0.290685\pi$
$770$	−2.76376e94	−0.00554703
$771$	4.12343e96	0.792386
$772$	−2.34799e96	−0.432033
$773$	2.14263e96	0.377515	0.188757	−	0.982024i	$-0.439554\pi$
$773$	2.14263e96	0.377515	0.188757	+	0.982024i	$0.439554\pi$
$774$	5.90768e95	0.0996770
$775$	3.52917e96	0.570251
$776$	−1.35539e96	−0.209747
$777$	−3.85437e94	−0.00571280
$778$	−2.87783e96	−0.408553
$779$	1.56144e97	2.12334
$780$	2.41283e96	0.314310
$781$	−5.95323e95	−0.0742921
$782$	−7.69643e96	−0.920159
$783$	−1.50256e97	−1.72113
$784$	1.06853e97	1.17273
$785$	1.29876e97	1.36582
$786$	−7.47622e96	−0.753402
$787$	−9.20529e96	−0.888964	−0.444482	−	0.895788i	$-0.646612\pi$
$787$	−9.20529e96	−0.888964	−0.444482	+	0.895788i	$0.646612\pi$
$788$	9.83432e94	0.00910156
$789$	7.76201e96	0.688485
$790$	−1.05218e97	−0.894501
$791$	−9.28573e94	−0.00756666
$792$	−7.65371e95	−0.0597831
$793$	1.36698e97	1.02355
$794$	2.05897e97	1.47796
$795$	−1.06715e97	−0.734391
$796$	−2.19342e96	−0.144722
$797$	−2.70098e97	−1.70871	−0.854355	−	0.519690i	$-0.826047\pi$
$797$	−2.70098e97	−1.70871	−0.854355	+	0.519690i	$0.826047\pi$
$798$	4.62749e95	0.0280704
$799$	−1.50181e97	−0.873569
$800$	2.67363e96	0.149137
$801$	8.95153e95	0.0478859
$802$	−2.38409e97	−1.22315
$803$	−2.46864e95	−0.0121475
$804$	2.51949e96	0.118914
$805$	−4.55499e95	−0.0206218
$806$	6.34749e97	2.75663
$807$	−8.53173e96	−0.355446
$808$	2.31254e97	0.924290
$809$	6.43872e96	0.246902	0.123451	−	0.992351i	$-0.460604\pi$
$809$	6.43872e96	0.246902	0.123451	+	0.992351i	$0.460604\pi$
$810$	−1.95087e97	−0.717762
$811$	−3.69417e97	−1.30412	−0.652061	−	0.758166i	$-0.726095\pi$
$811$	−3.69417e97	−1.30412	−0.652061	+	0.758166i	$0.726095\pi$
$812$	−2.14004e95	−0.00724929
$813$	3.83999e97	1.24824
$814$	−2.53396e96	−0.0790463
$815$	1.00011e97	0.299410
$816$	3.21098e97	0.922607
$817$	1.49214e97	0.411501
$818$	−8.92763e96	−0.236320
$819$	−3.97925e95	−0.0101109
$820$	−1.48994e97	−0.363417
$821$	9.41677e96	0.220499	0.110249	−	0.993904i	$-0.464835\pi$
$821$	9.41677e96	0.220499	0.110249	+	0.993904i	$0.464835\pi$
$822$	−6.75964e97	−1.51956
$823$	−1.85526e97	−0.400415	−0.200208	−	0.979754i	$-0.564162\pi$
$823$	−1.85526e97	−0.400415	−0.200208	+	0.979754i	$0.564162\pi$
$824$	−2.41517e97	−0.500480
$825$	2.94914e96	0.0586798
$826$	1.59288e96	0.0304335
$827$	−8.33671e97	−1.52954	−0.764769	−	0.644304i	$-0.777147\pi$
$827$	−8.33671e97	−1.52954	−0.764769	+	0.644304i	$0.777147\pi$
$828$	3.61852e96	0.0637550
$829$	5.04318e97	0.853351	0.426675	−	0.904405i	$-0.359685\pi$
$829$	5.04318e97	0.853351	0.426675	+	0.904405i	$0.359685\pi$
$830$	−9.13274e97	−1.48418
$831$	6.45336e97	1.00728
$832$	−6.83700e97	−1.02502
$833$	6.66989e97	0.960530
$834$	1.29284e98	1.78847
$835$	7.54066e97	1.00210
$836$	5.54542e96	0.0707986
$837$	−1.48668e98	−1.82354
$838$	9.57721e97	1.12866
$839$	1.80988e97	0.204939	0.102469	−	0.994736i	$-0.467326\pi$
$839$	1.80988e97	0.204939	0.102469	+	0.994736i	$0.467326\pi$
$840$	1.53929e96	0.0167480
$841$	1.43418e98	1.49947
$842$	−2.07357e97	−0.208335
$843$	1.44132e97	0.139167
$844$	1.79374e96	0.0166451
$845$	−1.57681e98	−1.40631
$846$	3.87358e97	0.332051
$847$	2.38572e96	0.0196574
$848$	−1.14817e98	−0.909376
$849$	−1.35303e98	−1.03015
$850$	4.94273e97	0.361770
$851$	−4.17625e97	−0.293864
$852$	−9.51136e96	−0.0643452
$853$	−6.65760e95	−0.00433037	−0.00216519	−	0.999998i	$-0.500689\pi$
$853$	−6.65760e95	−0.00433037	−0.00216519	+	0.999998i	$0.500689\pi$
$854$	−2.50163e96	−0.0156453
$855$	9.58895e97	0.576642
$856$	−3.49935e97	−0.202356
$857$	1.10050e97	0.0611976	0.0305988	−	0.999532i	$-0.490259\pi$
$857$	1.10050e97	0.0611976	0.0305988	+	0.999532i	$0.490259\pi$
$858$	5.30426e97	0.283662
$859$	1.29581e98	0.666457	0.333228	−	0.942846i	$-0.391862\pi$
$859$	1.29581e98	0.666457	0.333228	+	0.942846i	$0.391862\pi$
$860$	−1.42382e97	−0.0704297
$861$	−4.98219e96	−0.0237036
$862$	−2.81159e98	−1.28665
$863$	1.83648e98	0.808398	0.404199	−	0.914671i	$-0.367550\pi$
$863$	1.83648e98	0.808398	0.404199	+	0.914671i	$0.367550\pi$
$864$	−1.12628e98	−0.476909
$865$	−1.20441e98	−0.490608
$866$	1.78615e97	0.0699952
$867$	−1.66271e97	−0.0626869
$868$	−2.11742e96	−0.00768063
$869$	−4.21627e97	−0.147152
$870$	4.93267e98	1.65648
$871$	−3.00204e98	−0.970081
$872$	3.36734e98	1.04709
$873$	2.69408e97	0.0806183
$874$	5.01394e98	1.44393
$875$	−5.66734e96	−0.0157076
$876$	−3.94410e96	−0.0105211
$877$	−8.00025e96	−0.0205408	−0.0102704	−	0.999947i	$-0.503269\pi$
$877$	−8.00025e96	−0.0205408	−0.0102704	+	0.999947i	$0.503269\pi$
$878$	−5.00658e98	−1.23730
$879$	3.83172e98	0.911520
$880$	1.24934e98	0.286096
$881$	−2.46360e98	−0.543096	−0.271548	−	0.962425i	$-0.587536\pi$
$881$	−2.46360e98	−0.543096	−0.271548	+	0.962425i	$0.587536\pi$
$882$	−1.72035e98	−0.365106
$883$	−8.44617e98	−1.72574	−0.862872	−	0.505423i	$-0.831336\pi$
$883$	−8.44617e98	−1.72574	−0.862872	+	0.505423i	$0.831336\pi$
$884$	1.62046e98	0.318778
$885$	−6.69246e98	−1.26761
$886$	−4.20685e98	−0.767234
$887$	4.82704e98	0.847695	0.423848	−	0.905734i	$-0.360679\pi$
$887$	4.82704e98	0.847695	0.423848	+	0.905734i	$0.360679\pi$
$888$	1.41130e98	0.238662
$889$	−5.88972e96	−0.00959148
$890$	−1.18356e98	−0.185620
$891$	−7.81752e97	−0.118077
$892$	2.06393e98	0.300243
$893$	9.78372e98	1.37082
$894$	−1.06753e99	−1.44071
$895$	−2.89996e98	−0.376981
$896$	1.97081e97	0.0246790
$897$	8.74203e98	1.05455
$898$	6.84851e98	0.795864
$899$	2.36537e99	2.64819
$900$	−2.32385e97	−0.0250659
$901$	−7.16701e98	−0.744830
$902$	−3.27542e98	−0.327980
$903$	−4.76108e96	−0.00459373
$904$	3.40002e98	0.316110
$905$	−2.11243e99	−1.89258
$906$	−5.76813e98	−0.498013
$907$	1.79940e99	1.49722	0.748608	−	0.663012i	$-0.230722\pi$
$907$	1.79940e99	1.49722	0.748608	+	0.663012i	$0.230722\pi$
$908$	1.10526e98	0.0886320
$909$	−4.59659e98	−0.355260
$910$	5.26131e97	0.0391929
$911$	1.49982e98	0.107690	0.0538449	−	0.998549i	$-0.482852\pi$
$911$	1.49982e98	0.107690	0.0538449	+	0.998549i	$0.482852\pi$
$912$	−2.09183e99	−1.44777
$913$	−3.65967e98	−0.244158
$914$	−5.67899e98	−0.365236
$915$	1.05106e99	0.651658
$916$	−1.69763e98	−0.101472
$917$	−2.97161e97	−0.0171245
$918$	−2.08215e99	−1.15686
$919$	−2.19761e99	−1.17728	−0.588641	−	0.808394i	$-0.700337\pi$
$919$	−2.19761e99	−1.17728	−0.588641	+	0.808394i	$0.700337\pi$
$920$	1.66783e99	0.861511
$921$	4.21286e97	0.0209836
$922$	3.32906e99	1.59895
$923$	1.13330e99	0.524916
$924$	−1.76942e96	−0.000790350 0
$925$	2.68204e98	0.115536
$926$	1.08817e99	0.452093
$927$	4.80060e98	0.192364
$928$	1.79196e99	0.692580
$929$	−4.06848e99	−1.51672	−0.758361	−	0.651835i	$-0.774000\pi$
$929$	−4.06848e99	−1.51672	−0.758361	+	0.651835i	$0.774000\pi$
$930$	4.88054e99	1.75505
$931$	−4.34519e99	−1.50728
$932$	8.39436e97	0.0280902
$933$	−3.81783e98	−0.123248
$934$	−4.23231e99	−1.31813
$935$	7.79856e98	0.234329
$936$	1.45702e99	0.422401
$937$	4.26633e99	1.19338	0.596689	−	0.802473i	$-0.296483\pi$
$937$	4.26633e99	1.19338	0.596689	+	0.802473i	$0.296483\pi$
$938$	5.49388e97	0.0148280
$939$	−7.99269e97	−0.0208159
$940$	−9.33573e98	−0.234620
$941$	5.86101e99	1.42141	0.710706	−	0.703489i	$-0.248376\pi$
$941$	5.86101e99	1.42141	0.710706	+	0.703489i	$0.248376\pi$
$942$	4.56158e99	1.06760
$943$	−5.39826e99	−1.21931
$944$	−7.20053e99	−1.56965
$945$	−1.23228e98	−0.0259266
$946$	−3.13005e98	−0.0635620
$947$	−4.54626e99	−0.891107	−0.445553	−	0.895255i	$-0.646993\pi$
$947$	−4.54626e99	−0.891107	−0.445553	+	0.895255i	$0.646993\pi$
$948$	−6.73626e98	−0.127450
$949$	4.69951e98	0.0858290
$950$	−3.22001e99	−0.567697
$951$	2.38163e99	0.405347
$952$	1.03379e98	0.0169861
$953$	−5.35590e99	−0.849611	−0.424806	−	0.905285i	$-0.639658\pi$
$953$	−5.35590e99	−0.849611	−0.424806	+	0.905285i	$0.639658\pi$
$954$	1.84857e99	0.283117
$955$	4.04572e98	0.0598250
$956$	−1.33486e99	−0.190589
$957$	1.97662e99	0.272504
$958$	3.88815e99	0.517605
$959$	−2.68679e98	−0.0345390
$960$	−5.25692e99	−0.652597
$961$	1.50624e100	1.80577
$962$	4.82384e99	0.558507
$963$	6.95560e98	0.0777775
$964$	1.32241e99	0.142819
$965$	2.15113e100	2.24389
$966$	−1.59983e98	−0.0161191
$967$	6.14687e99	0.598228	0.299114	−	0.954217i	$-0.403309\pi$
$967$	6.14687e99	0.598228	0.299114	+	0.954217i	$0.403309\pi$
$968$	−8.73543e99	−0.821221
$969$	−1.30575e100	−1.18581
$970$	−3.56208e99	−0.312501
$971$	−1.66038e100	−1.40723	−0.703613	−	0.710583i	$-0.748431\pi$
$971$	−1.66038e100	−1.40723	−0.703613	+	0.710583i	$0.748431\pi$
$972$	1.71481e99	0.140409
$973$	5.13871e98	0.0406512
$974$	−2.35619e100	−1.80089
$975$	−5.61423e99	−0.414606
$976$	1.13085e100	0.806930
$977$	−1.88759e99	−0.130149	−0.0650743	−	0.997880i	$-0.520728\pi$
$977$	−1.88759e99	−0.130149	−0.0650743	+	0.997880i	$0.520728\pi$
$978$	3.51264e99	0.234035
$979$	−4.74276e98	−0.0305359
$980$	4.14622e99	0.257976
$981$	−6.69322e99	−0.402460
$982$	−1.18897e100	−0.690933
$983$	−9.21278e97	−0.00517426	−0.00258713	−	0.999997i	$-0.500824\pi$
$983$	−9.21278e97	−0.00517426	−0.00258713	+	0.999997i	$0.500824\pi$
$984$	1.82426e100	0.990262
$985$	−9.00977e98	−0.0472716
$986$	3.31279e100	1.68003
$987$	−3.12176e98	−0.0153030
$988$	−1.05567e100	−0.500232
$989$	−5.15868e99	−0.236299
$990$	−2.01146e99	−0.0890704
$991$	−1.75044e100	−0.749340	−0.374670	−	0.927158i	$-0.622244\pi$
$991$	−1.75044e100	−0.749340	−0.374670	+	0.927158i	$0.622244\pi$
$992$	1.77302e100	0.733789
$993$	1.79428e100	0.717945
$994$	−2.07400e98	−0.00802353
$995$	2.00952e100	0.751655
$996$	−5.84698e99	−0.211468
$997$	3.70311e100	1.29503	0.647513	−	0.762054i	$-0.275809\pi$
$997$	3.70311e100	1.29503	0.647513	+	0.762054i	$0.275809\pi$
$998$	−5.66104e100	−1.91436
$999$	−1.12982e100	−0.369459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1.68.a.a.1.2	✓	5
3.2	odd	2		9.68.a.a.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.68.a.a.1.2	✓	5	1.1	even	1	trivial
9.68.a.a.1.4		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34339e10 q^{2} +7.87958e15 q^{3} +3.28965e19 q^{4} -3.01383e23 q^{5} -1.05854e26 q^{6} -4.20742e26 q^{7} +1.54057e30 q^{8} -3.06216e31 q^{9} +O(q^{10})\)	\(q-1.34339e10 q^{2} +7.87958e15 q^{3} +3.28965e19 q^{4} -3.01383e23 q^{5} -1.05854e26 q^{6} -4.20742e26 q^{7} +1.54057e30 q^{8} -3.06216e31 q^{9} +4.04876e33 q^{10} +1.62242e34 q^{11} +2.59210e35 q^{12} -3.08856e37 q^{13} +5.65222e36 q^{14} -2.37477e39 q^{15} -2.55506e40 q^{16} -1.59490e41 q^{17} +4.11369e41 q^{18} +1.03902e43 q^{19} -9.91443e42 q^{20} -3.31527e42 q^{21} -2.17954e44 q^{22} -3.59214e45 q^{23} +1.21390e46 q^{24} +2.30691e46 q^{25} +4.14915e47 q^{26} -9.71798e47 q^{27} -1.38409e46 q^{28} +1.54617e49 q^{29} +3.19025e49 q^{30} +1.52983e50 q^{31} +1.15896e50 q^{32} +1.27840e50 q^{33} +2.14258e51 q^{34} +1.26805e50 q^{35} -1.00734e51 q^{36} +1.16261e52 q^{37} -1.39581e53 q^{38} -2.43366e53 q^{39} -4.64301e53 q^{40} +1.50280e54 q^{41} +4.45371e52 q^{42} +1.43610e54 q^{43} +5.33717e53 q^{44} +9.22884e54 q^{45} +4.82565e55 q^{46} +9.41631e55 q^{47} -2.01328e56 q^{48} -4.18201e56 q^{49} -3.09909e56 q^{50} -1.25671e57 q^{51} -1.01603e57 q^{52} +4.49370e57 q^{53} +1.30551e58 q^{54} -4.88968e57 q^{55} -6.48182e56 q^{56} +8.18704e58 q^{57} -2.07711e59 q^{58} +2.81815e59 q^{59} -7.81216e58 q^{60} -4.42593e59 q^{61} -2.05516e60 q^{62} +1.28838e58 q^{63} +2.21365e60 q^{64} +9.30840e60 q^{65} -1.71739e60 q^{66} +9.71985e60 q^{67} -5.24666e60 q^{68} -2.83045e61 q^{69} -1.70348e60 q^{70} -3.66936e61 q^{71} -4.71748e61 q^{72} -1.52158e61 q^{73} -1.56184e62 q^{74} +1.81775e62 q^{75} +3.41801e62 q^{76} -6.82619e60 q^{77} +3.26936e63 q^{78} -2.59876e63 q^{79} +7.70050e63 q^{80} -4.81844e63 q^{81} -2.01885e64 q^{82} -2.25569e64 q^{83} -1.09061e62 q^{84} +4.80676e64 q^{85} -1.92925e64 q^{86} +1.21832e65 q^{87} +2.49944e64 q^{88} -2.92327e64 q^{89} -1.23980e65 q^{90} +1.29949e64 q^{91} -1.18169e65 q^{92} +1.20544e66 q^{93} -1.26498e66 q^{94} -3.13143e66 q^{95} +9.13215e65 q^{96} -8.79797e65 q^{97} +5.61808e66 q^{98} -4.96810e65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 27\!\cdots\!85 q^{9}+O(q^{10}) \) 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9	\( 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 16\!\cdots\!20 q^{99}+O(q^{100}) \) 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9 - 5287067006725590176865792208438800 * q^10 + 203006981551281631527218766884109060 * q^11 - 2089518810401181456110201520986913024 * q^12 + 177822884499813229570219072356766702 * q^13 + 290753425875732621001638220329334092480 * q^14 - 933024301033068664231979455289562936600 * q^15 + 18266362102387359184851236859301091348480 * q^16 + 75241497376497768433807587544150344102906 * q^17 - 1011568566626978308983328666525487111002968 * q^18 + 3998884102997801557241177083441346393295100 * q^19 - 15074831378020267075078660994426474705846400 * q^20 - 175719863949295325296418063144372762630915040 * q^21 + 1841247880573095918698534631505063585473750432 * q^22 - 419660874260151197180020017656265233426170968 * q^23 - 13019747846573945420584224979196286639928166400 * q^24 + 175279270003539476977813620578789764300129011875 * q^25 + 21416337629300654169124097666599354769371062960 * q^26 + 1886613800166950113417786970194782722207999659320 * q^27 + 14982728306233364463143494158089725946900023694848 * q^28 + 18733547497571939101380180097730717676662894233950 * q^29 + 183881202860964913168534714290880945912096094414400 * q^30 + 365849822471060800690376948081970588059264217288160 * q^31 + 1591165751988992347021834813546084975107380979204096 * q^32 + 2437038903031213839498705428654461423996034595356784 * q^33 + 7817592154883765712044327776251337623941567500782480 * q^34 + 4524513410554564733118320802153390591800073874285200 * q^35 - 38131548457864843751915538710226876689225094472585920 * q^36 - 56389392588768330454829645685931746748224606527380394 * q^37 - 311451303114725973135630000941921262878276604287141280 * q^38 - 717177850582388225418943791249053833224228521090828280 * q^39 - 1886768345048751997029226502271137564956948018338432000 * q^40 + 113555410342193387626817165052765483601006465327103010 * q^41 + 2805540126450529847390961864461873572829871059166350592 * q^42 + 6508304671085340439704487085091235272293711787089861092 * q^43 + 49369449025916528958230735645862721892607029455332199680 * q^44 + 99011086960300194550392336618799141939148872728728967150 * q^45 + 79552980744915855625545425031873854934595203714399466560 * q^46 - 128023594814179612806658270243785955024442659789917751344 * q^47 - 198512903890152994486195174416803543869354744423782432768 * q^48 - 771329931628098140389185999842408514636676231229111640435 * q^49 - 3919481096957516748709549462396523762064894577356951885000 * q^50 - 3397415009313164473667296477133432796692029796730856284840 * q^51 + 672997930141532412373441872733517069231734104886640838016 * q^52 + 1046765637663599376128164919371991945530174947374966490822 * q^53 + 19066352042639687223500180230679796188217876586398799185600 * q^54 + 66942753107037791618627925698973654692865254681839842449400 * q^55 + 164746697572073652525116376573879338548073684495562965299200 * q^56 - 9288557942524759644038647059781434529266663439727382990960 * q^57 - 61137956327686914952144218440883774026433838412764071303120 * q^58 - 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1346157500908637758724100110370785768539984220742218248588800 * q^60 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 - 2197846158587850633685459639420082896328263471188214086352128 * q^62 - 2062780018748959038039139080074136999529046449885010133153768 * q^63 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 + 17054811715744261908524663200762497384422620874171150779332900 * q^65 + 27998862803339055552343423092993933514291039899414979026779520 * q^66 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 + 81678078346863753465296539273887923556029130514802676363013248 * q^68 + 3593002077831059525793506029116145162632074080747869594484320 * q^69 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 - 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 747723581155501189674233444839980291686130221322311676193789440 * q^72 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 + 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 21623812027928499471467875193108522496460255319651250940492500 * q^75 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 + 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 6187393864780941106550100388692932558738785969110711253222726464 * q^78 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 - 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 11195187555119704986426983039082721262159900559267345597680888595 * q^81 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 - 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 52124143836539534782259645807914007868879894236203487647926097920 * q^84 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 + 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 357574798440063707158439877351651513807909832776407769090038489160 * q^87 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 - 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 477054381540878916877701150770415125576892082956059127548575220400 * q^90 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 - 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 1849280210034321990512094428557344731461785100166176155774311127936 * q^93 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 - 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 - 2242556165580376632567217570505477297218451346654758522676359331840 * q^96 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 + 12426885478574475523720488645206508880013087100212747539199689691208 * q^98 + 16175146031966622867876651602334534091105243319832315048256094545620 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more