LMFDB - Embedded newform 4.36.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,36,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 36, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	4.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:	$31.0380522535$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 1597028177x + 23572260890640 $ x^3 - x^2 - 1597028177*x + 23572260890640
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-45936.2$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36149e8 q^{3} +1.62199e12 q^{5} -9.39810e14 q^{7} +5.73480e15 q^{9} +O(q^{10})$	$q+2.36149e8 q^{3} +1.62199e12 q^{5} -9.39810e14 q^{7} +5.73480e15 q^{9} -1.97864e18 q^{11} -4.34571e19 q^{13} +3.83032e20 q^{15} +3.66578e20 q^{17} +9.20887e21 q^{19} -2.21935e23 q^{21} +8.36273e23 q^{23} -2.79525e23 q^{25} -1.04606e25 q^{27} -3.77814e25 q^{29} +9.23868e23 q^{31} -4.67253e26 q^{33} -1.52436e27 q^{35} -4.08506e27 q^{37} -1.02623e28 q^{39} +2.27646e28 q^{41} -6.98561e28 q^{43} +9.30181e27 q^{45} +3.00592e29 q^{47} +5.04425e29 q^{49} +8.65670e28 q^{51} +1.51360e30 q^{53} -3.20933e30 q^{55} +2.17466e30 q^{57} -2.89829e30 q^{59} -2.82796e31 q^{61} -5.38963e30 q^{63} -7.04871e31 q^{65} +7.29302e31 q^{67} +1.97485e32 q^{69} +2.68949e32 q^{71} -2.69385e32 q^{73} -6.60096e31 q^{75} +1.85954e33 q^{77} -6.90775e32 q^{79} -2.75719e33 q^{81} -4.54909e33 q^{83} +5.94586e32 q^{85} -8.92203e33 q^{87} -5.48629e33 q^{89} +4.08414e34 q^{91} +2.18170e32 q^{93} +1.49367e34 q^{95} +1.63188e34 q^{97} -1.13471e34 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * 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 $

$2$	0	0
$2$	0	0
$3$	2.36149e8	1.05576	0.527879	−	0.849320i	$-0.322988\pi$
$3$	2.36149e8	1.05576	0.527879	+	0.849320i	$0.322988\pi$
$4$	0	0
$5$	1.62199e12	0.950766	0.475383	−	0.879779i	$-0.342310\pi$
$5$	1.62199e12	0.950766	0.475383	+	0.879779i	$0.342310\pi$
$6$	0	0
$7$	−9.39810e14	−1.52695	−0.763474	−	0.645838i	$-0.776508\pi$
$7$	−9.39810e14	−1.52695	−0.763474	+	0.645838i	$0.776508\pi$
$8$	0	0
$9$	5.73480e15	0.114624
$10$	0	0
$11$	−1.97864e18	−1.18031	−0.590153	−	0.807292i	$-0.700932\pi$
$11$	−1.97864e18	−1.18031	−0.590153	+	0.807292i	$0.700932\pi$
$12$	0	0
$13$	−4.34571e19	−1.39332	−0.696662	−	0.717400i	$-0.745332\pi$
$13$	−4.34571e19	−1.39332	−0.696662	+	0.717400i	$0.745332\pi$
$14$	0	0
$15$	3.83032e20	1.00378
$16$	0	0
$17$	3.66578e20	0.107476	0.0537378	−	0.998555i	$-0.482886\pi$
$17$	3.66578e20	0.107476	0.0537378	+	0.998555i	$0.482886\pi$
$18$	0	0
$19$	9.20887e21	0.385494	0.192747	−	0.981248i	$-0.438260\pi$
$19$	9.20887e21	0.385494	0.192747	+	0.981248i	$0.438260\pi$
$20$	0	0
$21$	−2.21935e23	−1.61209
$22$	0	0
$23$	8.36273e23	1.23626	0.618132	−	0.786074i	$-0.287890\pi$
$23$	8.36273e23	1.23626	0.618132	+	0.786074i	$0.287890\pi$
$24$	0	0
$25$	−2.79525e23	−0.0960441
$26$	0	0
$27$	−1.04606e25	−0.934743
$28$	0	0
$29$	−3.77814e25	−0.966747	−0.483373	−	0.875414i	$-0.660589\pi$
$29$	−3.77814e25	−0.966747	−0.483373	+	0.875414i	$0.660589\pi$
$30$	0	0
$31$	9.23868e23	0.00735835	0.00367917	−	0.999993i	$-0.498829\pi$
$31$	9.23868e23	0.00735835	0.00367917	+	0.999993i	$0.498829\pi$
$32$	0	0
$33$	−4.67253e26	−1.24612
$34$	0	0
$35$	−1.52436e27	−1.45177
$36$	0	0
$37$	−4.08506e27	−1.47119	−0.735593	−	0.677423i	$-0.763097\pi$
$37$	−4.08506e27	−1.47119	−0.735593	+	0.677423i	$0.763097\pi$
$38$	0	0
$39$	−1.02623e28	−1.47101
$40$	0	0
$41$	2.27646e28	1.36001	0.680005	−	0.733207i	$-0.261978\pi$
$41$	2.27646e28	1.36001	0.680005	+	0.733207i	$0.261978\pi$
$42$	0	0
$43$	−6.98561e28	−1.81345	−0.906727	−	0.421719i	$-0.861427\pi$
$43$	−6.98561e28	−1.81345	−0.906727	+	0.421719i	$0.861427\pi$
$44$	0	0
$45$	9.30181e27	0.108980
$46$	0	0
$47$	3.00592e29	1.64538	0.822688	−	0.568493i	$-0.192473\pi$
$47$	3.00592e29	1.64538	0.822688	+	0.568493i	$0.192473\pi$
$48$	0	0
$49$	5.04425e29	1.33157
$50$	0	0
$51$	8.65670e28	0.113468
$52$	0	0
$53$	1.51360e30	1.01201	0.506005	−	0.862531i	$-0.331122\pi$
$53$	1.51360e30	1.01201	0.506005	+	0.862531i	$0.331122\pi$
$54$	0	0
$55$	−3.20933e30	−1.12219
$56$	0	0
$57$	2.17466e30	0.406989
$58$	0	0
$59$	−2.89829e30	−0.296641	−0.148321	−	0.988939i	$-0.547387\pi$
$59$	−2.89829e30	−0.296641	−0.148321	+	0.988939i	$0.547387\pi$
$60$	0	0
$61$	−2.82796e31	−1.61511	−0.807554	−	0.589794i	$-0.799209\pi$
$61$	−2.82796e31	−1.61511	−0.807554	+	0.589794i	$0.799209\pi$
$62$	0	0
$63$	−5.38963e30	−0.175025
$64$	0	0
$65$	−7.04871e31	−1.32472
$66$	0	0
$67$	7.29302e31	0.806489	0.403244	−	0.915092i	$-0.367882\pi$
$67$	7.29302e31	0.806489	0.403244	+	0.915092i	$0.367882\pi$
$68$	0	0
$69$	1.97485e32	1.30519
$70$	0	0
$71$	2.68949e32	1.07807	0.539037	−	0.842282i	$-0.318788\pi$
$71$	2.68949e32	1.07807	0.539037	+	0.842282i	$0.318788\pi$
$72$	0	0
$73$	−2.69385e32	−0.664084	−0.332042	−	0.943265i	$-0.607738\pi$
$73$	−2.69385e32	−0.664084	−0.332042	+	0.943265i	$0.607738\pi$
$74$	0	0
$75$	−6.60096e31	−0.101399
$76$	0	0
$77$	1.85954e33	1.80227
$78$	0	0
$79$	−6.90775e32	−0.427427	−0.213714	−	0.976896i	$-0.568556\pi$
$79$	−6.90775e32	−0.427427	−0.213714	+	0.976896i	$0.568556\pi$
$80$	0	0
$81$	−2.75719e33	−1.10149
$82$	0	0
$83$	−4.54909e33	−1.18593	−0.592964	−	0.805229i	$-0.702042\pi$
$83$	−4.54909e33	−1.18593	−0.592964	+	0.805229i	$0.702042\pi$
$84$	0	0
$85$	5.94586e32	0.102184
$86$	0	0
$87$	−8.92203e33	−1.02065
$88$	0	0
$89$	−5.48629e33	−0.421651	−0.210826	−	0.977524i	$-0.567615\pi$
$89$	−5.48629e33	−0.421651	−0.210826	+	0.977524i	$0.567615\pi$
$90$	0	0
$91$	4.08414e34	2.12753
$92$	0	0
$93$	2.18170e32	0.00776863
$94$	0	0
$95$	1.49367e34	0.366515
$96$	0	0
$97$	1.63188e34	0.278088	0.139044	−	0.990286i	$-0.455597\pi$
$97$	1.63188e34	0.278088	0.139044	+	0.990286i	$0.455597\pi$
$98$	0	0
$99$	−1.13471e34	−0.135291
$100$	0	0
$101$	8.98080e34	0.754555	0.377278	−	0.926100i	$-0.376860\pi$
$101$	8.98080e34	0.754555	0.377278	+	0.926100i	$0.376860\pi$
$102$	0	0
$103$	6.13487e34	0.365724	0.182862	−	0.983139i	$-0.441464\pi$
$103$	6.13487e34	0.365724	0.182862	+	0.983139i	$0.441464\pi$
$104$	0	0
$105$	−3.59977e35	−1.53272
$106$	0	0
$107$	4.08357e34	0.124975	0.0624877	−	0.998046i	$-0.480097\pi$
$107$	4.08357e34	0.124975	0.0624877	+	0.998046i	$0.480097\pi$
$108$	0	0
$109$	−7.14868e35	−1.58220	−0.791101	−	0.611685i	$-0.790492\pi$
$109$	−7.14868e35	−1.58220	−0.791101	+	0.611685i	$0.790492\pi$
$110$	0	0
$111$	−9.64682e35	−1.55322
$112$	0	0
$113$	1.55786e36	1.83508	0.917540	−	0.397644i	$-0.130172\pi$
$113$	1.55786e36	1.83508	0.917540	+	0.397644i	$0.130172\pi$
$114$	0	0
$115$	1.35643e36	1.17540
$116$	0	0
$117$	−2.49218e35	−0.159708
$118$	0	0
$119$	−3.44514e35	−0.164110
$120$	0	0
$121$	1.10476e36	0.393120
$122$	0	0
$123$	5.37583e36	1.43584
$124$	0	0
$125$	−5.17401e36	−1.04208
$126$	0	0
$127$	7.98221e36	1.21775	0.608875	−	0.793266i	$-0.291621\pi$
$127$	7.98221e36	1.21775	0.608875	+	0.793266i	$0.291621\pi$
$128$	0	0
$129$	−1.64964e37	−1.91457
$130$	0	0
$131$	8.29134e36	0.735152	0.367576	−	0.929994i	$-0.380188\pi$
$131$	8.29134e36	0.735152	0.367576	+	0.929994i	$0.380188\pi$
$132$	0	0
$133$	−8.65459e36	−0.588630
$134$	0	0
$135$	−1.69671e37	−0.888721
$136$	0	0
$137$	−3.27013e37	−1.32420	−0.662102	−	0.749413i	$-0.730336\pi$
$137$	−3.27013e37	−1.32420	−0.662102	+	0.749413i	$0.730336\pi$
$138$	0	0
$139$	−1.08784e37	−0.341827	−0.170914	−	0.985286i	$-0.554672\pi$
$139$	−1.08784e37	−0.341827	−0.170914	+	0.985286i	$0.554672\pi$
$140$	0	0
$141$	7.09846e37	1.73712
$142$	0	0
$143$	8.59859e37	1.64455
$144$	0	0
$145$	−6.12811e37	−0.919150
$146$	0	0
$147$	1.19119e38	1.40582
$148$	0	0
$149$	−1.00629e38	−0.937483	−0.468741	−	0.883335i	$-0.655292\pi$
$149$	−1.00629e38	−0.937483	−0.468741	+	0.883335i	$0.655292\pi$
$150$	0	0
$151$	5.84758e37	0.431399	0.215700	−	0.976460i	$-0.430797\pi$
$151$	5.84758e37	0.431399	0.215700	+	0.976460i	$0.430797\pi$
$152$	0	0
$153$	2.10225e36	0.0123193
$154$	0	0
$155$	1.49851e36	0.00699607
$156$	0	0
$157$	−1.78132e38	−0.664504	−0.332252	−	0.943191i	$-0.607808\pi$
$157$	−1.78132e38	−0.664504	−0.332252	+	0.943191i	$0.607808\pi$
$158$	0	0
$159$	3.57436e38	1.06844
$160$	0	0
$161$	−7.85938e38	−1.88771
$162$	0	0
$163$	−3.50626e38	−0.678517	−0.339258	−	0.940693i	$-0.610176\pi$
$163$	−3.50626e38	−0.678517	−0.339258	+	0.940693i	$0.610176\pi$
$164$	0	0
$165$	−7.57881e38	−1.18476
$166$	0	0
$167$	6.88044e38	0.871119	0.435560	−	0.900160i	$-0.356550\pi$
$167$	6.88044e38	0.871119	0.435560	+	0.900160i	$0.356550\pi$
$168$	0	0
$169$	9.15733e38	0.941351
$170$	0	0
$171$	5.28110e37	0.0441868
$172$	0	0
$173$	1.99106e39	1.35918	0.679592	−	0.733590i	$-0.262157\pi$
$173$	1.99106e39	1.35918	0.679592	+	0.733590i	$0.262157\pi$
$174$	0	0
$175$	2.62700e38	0.146654
$176$	0	0
$177$	−6.84429e38	−0.313181
$178$	0	0
$179$	1.41697e39	0.532639	0.266319	−	0.963885i	$-0.414192\pi$
$179$	1.41697e39	0.532639	0.266319	+	0.963885i	$0.414192\pi$
$180$	0	0
$181$	4.22797e39	1.30845	0.654225	−	0.756300i	$-0.272995\pi$
$181$	4.22797e39	1.30845	0.654225	+	0.756300i	$0.272995\pi$
$182$	0	0
$183$	−6.67820e39	−1.70516
$184$	0	0
$185$	−6.62593e39	−1.39875
$186$	0	0
$187$	−7.25325e38	−0.126854
$188$	0	0
$189$	9.83101e39	1.42730
$190$	0	0
$191$	−1.71210e39	−0.206750	−0.103375	−	0.994642i	$-0.532964\pi$
$191$	−1.71210e39	−0.206750	−0.103375	+	0.994642i	$0.532964\pi$
$192$	0	0
$193$	−1.03024e40	−1.03677	−0.518387	−	0.855146i	$-0.673467\pi$
$193$	−1.03024e40	−1.03677	−0.518387	+	0.855146i	$0.673467\pi$
$194$	0	0
$195$	−1.66454e40	−1.39859
$196$	0	0
$197$	6.90833e39	0.485528	0.242764	−	0.970085i	$-0.421946\pi$
$197$	6.90833e39	0.485528	0.242764	+	0.970085i	$0.421946\pi$
$198$	0	0
$199$	−7.95511e39	−0.468508	−0.234254	−	0.972175i	$-0.575265\pi$
$199$	−7.95511e39	−0.468508	−0.234254	+	0.972175i	$0.575265\pi$
$200$	0	0
$201$	1.72224e40	0.851457
$202$	0	0
$203$	3.55073e40	1.47617
$204$	0	0
$205$	3.69240e40	1.29305
$206$	0	0
$207$	4.79586e39	0.141705
$208$	0	0
$209$	−1.82210e40	−0.455001
$210$	0	0
$211$	−2.98582e40	−0.631133	−0.315567	−	0.948903i	$-0.602195\pi$
$211$	−2.98582e40	−0.631133	−0.315567	+	0.948903i	$0.602195\pi$
$212$	0	0
$213$	6.35119e40	1.13818
$214$	0	0
$215$	−1.13306e41	−1.72417
$216$	0	0
$217$	−8.68260e38	−0.0112358
$218$	0	0
$219$	−6.36149e40	−0.701111
$220$	0	0
$221$	−1.59304e40	−0.149748
$222$	0	0
$223$	5.49638e40	0.441308	0.220654	−	0.975352i	$-0.429181\pi$
$223$	5.49638e40	0.441308	0.220654	+	0.975352i	$0.429181\pi$
$224$	0	0
$225$	−1.60302e39	−0.0110089
$226$	0	0
$227$	−1.11217e41	−0.654212	−0.327106	−	0.944988i	$-0.606073\pi$
$227$	−1.11217e41	−0.654212	−0.327106	+	0.944988i	$0.606073\pi$
$228$	0	0
$229$	−2.95870e41	−1.49273	−0.746364	−	0.665538i	$-0.768202\pi$
$229$	−2.95870e41	−1.49273	−0.746364	+	0.665538i	$0.768202\pi$
$230$	0	0
$231$	4.39129e41	1.90276
$232$	0	0
$233$	−1.35414e41	−0.504587	−0.252293	−	0.967651i	$-0.581185\pi$
$233$	−1.35414e41	−0.504587	−0.252293	+	0.967651i	$0.581185\pi$
$234$	0	0
$235$	4.87558e41	1.56437
$236$	0	0
$237$	−1.63126e41	−0.451260
$238$	0	0
$239$	4.07610e41	0.973379	0.486690	−	0.873575i	$-0.338204\pi$
$239$	4.07610e41	0.973379	0.486690	+	0.873575i	$0.338204\pi$
$240$	0	0
$241$	1.77925e41	0.367230	0.183615	−	0.982998i	$-0.441220\pi$
$241$	1.77925e41	0.367230	0.183615	+	0.982998i	$0.441220\pi$
$242$	0	0
$243$	−1.27746e41	−0.228159
$244$	0	0
$245$	8.18173e41	1.26601
$246$	0	0
$247$	−4.00191e41	−0.537119
$248$	0	0
$249$	−1.07426e42	−1.25205
$250$	0	0
$251$	−9.64139e41	−0.976899	−0.488449	−	0.872592i	$-0.662437\pi$
$251$	−9.64139e41	−0.976899	−0.488449	+	0.872592i	$0.662437\pi$
$252$	0	0
$253$	−1.65468e42	−1.45917
$254$	0	0
$255$	1.40411e41	0.107882
$256$	0	0
$257$	−4.03398e41	−0.270336	−0.135168	−	0.990823i	$-0.543157\pi$
$257$	−4.03398e41	−0.270336	−0.135168	+	0.990823i	$0.543157\pi$
$258$	0	0
$259$	3.83918e42	2.24643
$260$	0	0
$261$	−2.16669e41	−0.110812
$262$	0	0
$263$	−3.16142e42	−1.41468	−0.707338	−	0.706876i	$-0.750104\pi$
$263$	−3.16142e42	−1.41468	−0.707338	+	0.706876i	$0.750104\pi$
$264$	0	0
$265$	2.45506e42	0.962185
$266$	0	0
$267$	−1.29558e42	−0.445161
$268$	0	0
$269$	−3.37405e42	−1.01739	−0.508695	−	0.860947i	$-0.669872\pi$
$269$	−3.37405e42	−1.01739	−0.508695	+	0.860947i	$0.669872\pi$
$270$	0	0
$271$	5.44713e42	1.44280	0.721398	−	0.692521i	$-0.243500\pi$
$271$	5.44713e42	1.44280	0.721398	+	0.692521i	$0.243500\pi$
$272$	0	0
$273$	9.64466e42	2.24616
$274$	0	0
$275$	5.53079e41	0.113361
$276$	0	0
$277$	−5.72253e42	−1.03322	−0.516609	−	0.856222i	$-0.672806\pi$
$277$	−5.72253e42	−1.03322	−0.516609	+	0.856222i	$0.672806\pi$
$278$	0	0
$279$	5.29820e39	0.000843441 0
$280$	0	0
$281$	5.67350e42	0.797060	0.398530	−	0.917155i	$-0.369521\pi$
$281$	5.67350e42	0.797060	0.398530	+	0.917155i	$0.369521\pi$
$282$	0	0
$283$	−6.21744e42	−0.771523	−0.385761	−	0.922599i	$-0.626061\pi$
$283$	−6.21744e42	−0.771523	−0.385761	+	0.922599i	$0.626061\pi$
$284$	0	0
$285$	3.52729e42	0.386951
$286$	0	0
$287$	−2.13944e43	−2.07667
$288$	0	0
$289$	−1.14992e43	−0.988449
$290$	0	0
$291$	3.85366e42	0.293593
$292$	0	0
$293$	−1.93797e42	−0.130968	−0.0654838	−	0.997854i	$-0.520859\pi$
$293$	−1.93797e42	−0.130968	−0.0654838	+	0.997854i	$0.520859\pi$
$294$	0	0
$295$	−4.70101e42	−0.282037
$296$	0	0
$297$	2.06978e43	1.10328
$298$	0	0
$299$	−3.63420e43	−1.72252
$300$	0	0
$301$	6.56515e43	2.76905
$302$	0	0
$303$	2.12081e43	0.796627
$304$	0	0
$305$	−4.58693e43	−1.53559
$306$	0	0
$307$	1.65334e43	0.493674	0.246837	−	0.969057i	$-0.420609\pi$
$307$	1.65334e43	0.493674	0.246837	+	0.969057i	$0.420609\pi$
$308$	0	0
$309$	1.44874e43	0.386116
$310$	0	0
$311$	3.30600e43	0.787039	0.393520	−	0.919316i	$-0.371257\pi$
$311$	3.30600e43	0.787039	0.393520	+	0.919316i	$0.371257\pi$
$312$	0	0
$313$	1.28569e42	0.0273596	0.0136798	−	0.999906i	$-0.495645\pi$
$313$	1.28569e42	0.0273596	0.0136798	+	0.999906i	$0.495645\pi$
$314$	0	0
$315$	−8.74193e42	−0.166407
$316$	0	0
$317$	−7.56495e43	−1.28905	−0.644524	−	0.764584i	$-0.722944\pi$
$317$	−7.56495e43	−1.28905	−0.644524	+	0.764584i	$0.722944\pi$
$318$	0	0
$319$	7.47557e43	1.14106
$320$	0	0
$321$	9.64332e42	0.131944
$322$	0	0
$323$	3.37577e42	0.0414313
$324$	0	0
$325$	1.21473e43	0.133820
$326$	0	0
$327$	−1.68815e44	−1.67042
$328$	0	0
$329$	−2.82500e44	−2.51241
$330$	0	0
$331$	4.43305e43	0.354579	0.177289	−	0.984159i	$-0.443267\pi$
$331$	4.43305e43	0.354579	0.177289	+	0.984159i	$0.443267\pi$
$332$	0	0
$333$	−2.34270e43	−0.168633
$334$	0	0
$335$	1.18292e44	0.766782
$336$	0	0
$337$	1.17448e44	0.685995	0.342997	−	0.939336i	$-0.388558\pi$
$337$	1.17448e44	0.685995	0.342997	+	0.939336i	$0.388558\pi$
$338$	0	0
$339$	3.67886e44	1.93740
$340$	0	0
$341$	−1.82800e42	−0.00868509
$342$	0	0
$343$	−1.18046e44	−0.506294
$344$	0	0
$345$	3.20319e44	1.24093
$346$	0	0
$347$	−4.12825e44	−1.44544	−0.722720	−	0.691141i	$-0.757108\pi$
$347$	−4.12825e44	−1.44544	−0.722720	+	0.691141i	$0.757108\pi$
$348$	0	0
$349$	1.81486e44	0.574644	0.287322	−	0.957834i	$-0.407235\pi$
$349$	1.81486e44	0.574644	0.287322	+	0.957834i	$0.407235\pi$
$350$	0	0
$351$	4.54589e44	1.30240
$352$	0	0
$353$	−5.07996e44	−1.31766	−0.658830	−	0.752292i	$-0.728948\pi$
$353$	−5.07996e44	−1.31766	−0.658830	+	0.752292i	$0.728948\pi$
$354$	0	0
$355$	4.36233e44	1.02500
$356$	0	0
$357$	−8.13565e43	−0.173260
$358$	0	0
$359$	−5.34848e44	−1.03295	−0.516473	−	0.856304i	$-0.672755\pi$
$359$	−5.34848e44	−1.03295	−0.516473	+	0.856304i	$0.672755\pi$
$360$	0	0
$361$	−4.85855e44	−0.851394
$362$	0	0
$363$	2.60889e44	0.415040
$364$	0	0
$365$	−4.36940e44	−0.631388
$366$	0	0
$367$	4.12935e44	0.542282	0.271141	−	0.962540i	$-0.412599\pi$
$367$	4.12935e44	0.542282	0.271141	+	0.962540i	$0.412599\pi$
$368$	0	0
$369$	1.30550e44	0.155890
$370$	0	0
$371$	−1.42250e45	−1.54529
$372$	0	0
$373$	1.76736e42	0.00174752	0.000873759	−	1.00000i	$-0.499722\pi$
$373$	1.76736e42	0.00174752	0.000873759		1.00000i	$0.499722\pi$
$374$	0	0
$375$	−1.22184e45	−1.10019
$376$	0	0
$377$	1.64187e45	1.34699
$378$	0	0
$379$	−7.57090e44	−0.566189	−0.283095	−	0.959092i	$-0.591361\pi$
$379$	−7.57090e44	−0.566189	−0.283095	+	0.959092i	$0.591361\pi$
$380$	0	0
$381$	1.88499e45	1.28565
$382$	0	0
$383$	1.04340e45	0.649341	0.324671	−	0.945827i	$-0.394747\pi$
$383$	1.04340e45	0.649341	0.324671	+	0.945827i	$0.394747\pi$
$384$	0	0
$385$	3.01617e45	1.71353
$386$	0	0
$387$	−4.00611e44	−0.207865
$388$	0	0
$389$	−5.06968e44	−0.240360	−0.120180	−	0.992752i	$-0.538347\pi$
$389$	−5.06968e44	−0.240360	−0.120180	+	0.992752i	$0.538347\pi$
$390$	0	0
$391$	3.06559e44	0.132868
$392$	0	0
$393$	1.95799e45	0.776142
$394$	0	0
$395$	−1.12043e45	−0.406383
$396$	0	0
$397$	−3.15468e45	−1.04742	−0.523711	−	0.851896i	$-0.675453\pi$
$397$	−3.15468e45	−1.04742	−0.523711	+	0.851896i	$0.675453\pi$
$398$	0	0
$399$	−2.04377e45	−0.621451
$400$	0	0
$401$	−3.35644e45	−0.935088	−0.467544	−	0.883970i	$-0.654861\pi$
$401$	−3.35644e45	−0.935088	−0.467544	+	0.883970i	$0.654861\pi$
$402$	0	0
$403$	−4.01486e43	−0.0102526
$404$	0	0
$405$	−4.47214e45	−1.04725
$406$	0	0
$407$	8.08284e45	1.73645
$408$	0	0
$409$	3.20017e45	0.630980	0.315490	−	0.948929i	$-0.397831\pi$
$409$	3.20017e45	0.630980	0.315490	+	0.948929i	$0.397831\pi$
$410$	0	0
$411$	−7.72238e45	−1.39804
$412$	0	0
$413$	2.72384e45	0.452956
$414$	0	0
$415$	−7.37859e45	−1.12754
$416$	0	0
$417$	−2.56893e45	−0.360887
$418$	0	0
$419$	1.00345e46	1.29644	0.648219	−	0.761454i	$-0.275514\pi$
$419$	1.00345e46	1.29644	0.648219	+	0.761454i	$0.275514\pi$
$420$	0	0
$421$	−9.90248e45	−1.17708	−0.588542	−	0.808467i	$-0.700298\pi$
$421$	−9.90248e45	−1.17708	−0.588542	+	0.808467i	$0.700298\pi$
$422$	0	0
$423$	1.72384e45	0.188599
$424$	0	0
$425$	−1.02468e44	−0.0103224
$426$	0	0
$427$	2.65775e46	2.46619
$428$	0	0
$429$	2.03055e46	1.73624
$430$	0	0
$431$	8.39933e45	0.662052	0.331026	−	0.943622i	$-0.392605\pi$
$431$	8.39933e45	0.662052	0.331026	+	0.943622i	$0.392605\pi$
$432$	0	0
$433$	−2.57119e46	−1.86894	−0.934471	−	0.356039i	$-0.884127\pi$
$433$	−2.57119e46	−1.86894	−0.934471	+	0.356039i	$0.884127\pi$
$434$	0	0
$435$	−1.44715e46	−0.970400
$436$	0	0
$437$	7.70113e45	0.476573
$438$	0	0
$439$	6.65867e45	0.380416	0.190208	−	0.981744i	$-0.439084\pi$
$439$	6.65867e45	0.380416	0.190208	+	0.981744i	$0.439084\pi$
$440$	0	0
$441$	2.89278e45	0.152630
$442$	0	0
$443$	−6.48769e45	−0.316246	−0.158123	−	0.987419i	$-0.550544\pi$
$443$	−6.48769e45	−0.316246	−0.158123	+	0.987419i	$0.550544\pi$
$444$	0	0
$445$	−8.89871e45	−0.400892
$446$	0	0
$447$	−2.37634e46	−0.989754
$448$	0	0
$449$	−2.80568e46	−1.08076	−0.540378	−	0.841422i	$-0.681719\pi$
$449$	−2.80568e46	−1.08076	−0.540378	+	0.841422i	$0.681719\pi$
$450$	0	0
$451$	−4.50429e46	−1.60523
$452$	0	0
$453$	1.38090e46	0.455453
$454$	0	0
$455$	6.62445e46	2.02279
$456$	0	0
$457$	−3.21337e46	−0.908713	−0.454356	−	0.890820i	$-0.650131\pi$
$457$	−3.21337e46	−0.908713	−0.454356	+	0.890820i	$0.650131\pi$
$458$	0	0
$459$	−3.83464e45	−0.100462
$460$	0	0
$461$	4.02258e46	0.976649	0.488325	−	0.872662i	$-0.337608\pi$
$461$	4.02258e46	0.976649	0.488325	+	0.872662i	$0.337608\pi$
$462$	0	0
$463$	3.16657e46	0.712725	0.356362	−	0.934348i	$-0.384017\pi$
$463$	3.16657e46	0.712725	0.356362	+	0.934348i	$0.384017\pi$
$464$	0	0
$465$	3.53871e44	0.00738615
$466$	0	0
$467$	1.61608e46	0.312908	0.156454	−	0.987685i	$-0.449994\pi$
$467$	1.61608e46	0.312908	0.156454	+	0.987685i	$0.449994\pi$
$468$	0	0
$469$	−6.85405e46	−1.23147
$470$	0	0
$471$	−4.20656e46	−0.701555
$472$	0	0
$473$	1.38220e47	2.14043
$474$	0	0
$475$	−2.57411e45	−0.0370245
$476$	0	0
$477$	8.68023e45	0.116000
$478$	0	0
$479$	3.03202e46	0.376583	0.188291	−	0.982113i	$-0.439705\pi$
$479$	3.03202e46	0.376583	0.188291	+	0.982113i	$0.439705\pi$
$480$	0	0
$481$	1.77525e47	2.04984
$482$	0	0
$483$	−1.85599e47	−1.99297
$484$	0	0
$485$	2.64689e46	0.264396
$486$	0	0
$487$	6.13349e45	0.0570100	0.0285050	−	0.999594i	$-0.490925\pi$
$487$	6.13349e45	0.0570100	0.0285050	+	0.999594i	$0.490925\pi$
$488$	0	0
$489$	−8.27999e46	−0.716349
$490$	0	0
$491$	−1.41212e47	−1.13748	−0.568740	−	0.822517i	$-0.692569\pi$
$491$	−1.41212e47	−1.13748	−0.568740	+	0.822517i	$0.692569\pi$
$492$	0	0
$493$	−1.38498e46	−0.103902
$494$	0	0
$495$	−1.84049e46	−0.128630
$496$	0	0
$497$	−2.52761e47	−1.64616
$498$	0	0
$499$	−1.69443e47	−1.02864	−0.514320	−	0.857598i	$-0.671956\pi$
$499$	−1.69443e47	−1.02864	−0.514320	+	0.857598i	$0.671956\pi$
$500$	0	0
$501$	1.62481e47	0.919691
$502$	0	0
$503$	1.55712e46	0.0822018	0.0411009	−	0.999155i	$-0.486913\pi$
$503$	1.55712e46	0.0822018	0.0411009	+	0.999155i	$0.486913\pi$
$504$	0	0
$505$	1.45668e47	0.717406
$506$	0	0
$507$	2.16249e47	0.993838
$508$	0	0
$509$	9.19658e46	0.394516	0.197258	−	0.980352i	$-0.436796\pi$
$509$	9.19658e46	0.394516	0.197258	+	0.980352i	$0.436796\pi$
$510$	0	0
$511$	2.53171e47	1.01402
$512$	0	0
$513$	−9.63306e46	−0.360338
$514$	0	0
$515$	9.95070e46	0.347718
$516$	0	0
$517$	−5.94763e47	−1.94205
$518$	0	0
$519$	4.70188e47	1.43497
$520$	0	0
$521$	2.04792e47	0.584324	0.292162	−	0.956369i	$-0.405625\pi$
$521$	2.04792e47	0.584324	0.292162	+	0.956369i	$0.405625\pi$
$522$	0	0
$523$	7.20809e46	0.192327	0.0961635	−	0.995366i	$-0.469343\pi$
$523$	7.20809e46	0.192327	0.0961635	+	0.995366i	$0.469343\pi$
$524$	0	0
$525$	6.20365e46	0.154831
$526$	0	0
$527$	3.38669e44	0.000790843 0
$528$	0	0
$529$	2.41766e47	0.528348
$530$	0	0
$531$	−1.66211e46	−0.0340022
$532$	0	0
$533$	−9.89283e47	−1.89493
$534$	0	0
$535$	6.62352e46	0.118822
$536$	0	0
$537$	3.34616e47	0.562338
$538$	0	0
$539$	−9.98074e47	−1.57166
$540$	0	0
$541$	2.56162e47	0.378062	0.189031	−	0.981971i	$-0.439465\pi$
$541$	2.56162e47	0.378062	0.189031	+	0.981971i	$0.439465\pi$
$542$	0	0
$543$	9.98431e47	1.38141
$544$	0	0
$545$	−1.15951e48	−1.50430
$546$	0	0
$547$	−7.45796e47	−0.907489	−0.453745	−	0.891132i	$-0.649912\pi$
$547$	−7.45796e47	−0.907489	−0.453745	+	0.891132i	$0.649912\pi$
$548$	0	0
$549$	−1.62178e47	−0.185130
$550$	0	0
$551$	−3.47924e47	−0.372676
$552$	0	0
$553$	6.49197e47	0.652660
$554$	0	0
$555$	−1.56471e48	−1.47675
$556$	0	0
$557$	1.09114e47	0.0966975	0.0483488	−	0.998831i	$-0.484604\pi$
$557$	1.09114e47	0.0966975	0.0483488	+	0.998831i	$0.484604\pi$
$558$	0	0
$559$	3.03574e48	2.52673
$560$	0	0
$561$	−1.71285e47	−0.133927
$562$	0	0
$563$	1.65566e48	1.21639	0.608197	−	0.793786i	$-0.291893\pi$
$563$	1.65566e48	1.21639	0.608197	+	0.793786i	$0.291893\pi$
$564$	0	0
$565$	2.52683e48	1.74473
$566$	0	0
$567$	2.59123e48	1.68191
$568$	0	0
$569$	−2.88259e48	−1.75921	−0.879606	−	0.475704i	$-0.842194\pi$
$569$	−2.88259e48	−1.75921	−0.879606	+	0.475704i	$0.842194\pi$
$570$	0	0
$571$	−2.99285e48	−1.71772	−0.858862	−	0.512206i	$-0.828828\pi$
$571$	−2.99285e48	−1.71772	−0.858862	+	0.512206i	$0.828828\pi$
$572$	0	0
$573$	−4.04311e47	−0.218278
$574$	0	0
$575$	−2.33759e47	−0.118736
$576$	0	0
$577$	1.37915e48	0.659225	0.329612	−	0.944116i	$-0.393082\pi$
$577$	1.37915e48	0.659225	0.329612	+	0.944116i	$0.393082\pi$
$578$	0	0
$579$	−2.43289e48	−1.09458
$580$	0	0
$581$	4.27528e48	1.81085
$582$	0	0
$583$	−2.99488e48	−1.19448
$584$	0	0
$585$	−4.04230e47	−0.151845
$586$	0	0
$587$	−3.82333e48	−1.35293	−0.676465	−	0.736475i	$-0.736489\pi$
$587$	−3.82333e48	−1.35293	−0.676465	+	0.736475i	$0.736489\pi$
$588$	0	0
$589$	8.50778e45	0.00283660
$590$	0	0
$591$	1.63140e48	0.512600
$592$	0	0
$593$	1.31964e48	0.390839	0.195419	−	0.980720i	$-0.437393\pi$
$593$	1.31964e48	0.390839	0.195419	+	0.980720i	$0.437393\pi$
$594$	0	0
$595$	−5.58798e47	−0.156030
$596$	0	0
$597$	−1.87859e48	−0.494631
$598$	0	0
$599$	−1.62118e48	−0.402590	−0.201295	−	0.979531i	$-0.564515\pi$
$599$	−1.62118e48	−0.402590	−0.201295	+	0.979531i	$0.564515\pi$
$600$	0	0
$601$	−6.04114e48	−1.41519	−0.707596	−	0.706617i	$-0.750220\pi$
$601$	−6.04114e48	−1.41519	−0.707596	+	0.706617i	$0.750220\pi$
$602$	0	0
$603$	4.18240e47	0.0924428
$604$	0	0
$605$	1.79192e48	0.373765
$606$	0	0
$607$	5.93836e48	1.16913	0.584567	−	0.811345i	$-0.301264\pi$
$607$	5.93836e48	1.16913	0.584567	+	0.811345i	$0.301264\pi$
$608$	0	0
$609$	8.38502e48	1.55848
$610$	0	0
$611$	−1.30629e49	−2.29254
$612$	0	0
$613$	3.55399e48	0.589057	0.294529	−	0.955643i	$-0.404837\pi$
$613$	3.55399e48	0.589057	0.294529	+	0.955643i	$0.404837\pi$
$614$	0	0
$615$	8.71956e48	1.36515
$616$	0	0
$617$	1.11694e49	1.65211	0.826055	−	0.563590i	$-0.190580\pi$
$617$	1.11694e49	1.65211	0.826055	+	0.563590i	$0.190580\pi$
$618$	0	0
$619$	7.70707e48	1.07722	0.538608	−	0.842556i	$-0.318950\pi$
$619$	7.70707e48	1.07722	0.538608	+	0.842556i	$0.318950\pi$
$620$	0	0
$621$	−8.74795e48	−1.15559
$622$	0	0
$623$	5.15607e48	0.643840
$624$	0	0
$625$	−7.57867e48	−0.894731
$626$	0	0
$627$	−4.30287e48	−0.480371
$628$	0	0
$629$	−1.49749e48	−0.158117
$630$	0	0
$631$	−1.56273e48	−0.156088	−0.0780442	−	0.996950i	$-0.524868\pi$
$631$	−1.56273e48	−0.156088	−0.0780442	+	0.996950i	$0.524868\pi$
$632$	0	0
$633$	−7.05099e48	−0.666324
$634$	0	0
$635$	1.29471e49	1.15779
$636$	0	0
$637$	−2.19208e49	−1.85531
$638$	0	0
$639$	1.54237e48	0.123573
$640$	0	0
$641$	1.25787e48	0.0954157	0.0477079	−	0.998861i	$-0.484808\pi$
$641$	1.25787e48	0.0954157	0.0477079	+	0.998861i	$0.484808\pi$
$642$	0	0
$643$	−2.12522e49	−1.52655	−0.763276	−	0.646072i	$-0.776410\pi$
$643$	−2.12522e49	−1.52655	−0.763276	+	0.646072i	$0.776410\pi$
$644$	0	0
$645$	−2.67571e49	−1.82030
$646$	0	0
$647$	2.84314e49	1.83220	0.916099	−	0.400951i	$-0.131320\pi$
$647$	2.84314e49	1.83220	0.916099	+	0.400951i	$0.131320\pi$
$648$	0	0
$649$	5.73467e48	0.350127
$650$	0	0
$651$	−2.05039e47	−0.0118623
$652$	0	0
$653$	2.12180e49	1.16338	0.581692	−	0.813409i	$-0.302391\pi$
$653$	2.12180e49	1.16338	0.581692	+	0.813409i	$0.302391\pi$
$654$	0	0
$655$	1.34485e49	0.698957
$656$	0	0
$657$	−1.54487e48	−0.0761198
$658$	0	0
$659$	−2.45276e49	−1.14593	−0.572967	−	0.819578i	$-0.694208\pi$
$659$	−2.45276e49	−1.14593	−0.572967	+	0.819578i	$0.694208\pi$
$660$	0	0
$661$	2.19397e49	0.972086	0.486043	−	0.873935i	$-0.338440\pi$
$661$	2.19397e49	0.972086	0.486043	+	0.873935i	$0.338440\pi$
$662$	0	0
$663$	−3.76195e48	−0.158098
$664$	0	0
$665$	−1.40377e49	−0.559650
$666$	0	0
$667$	−3.15956e49	−1.19515
$668$	0	0
$669$	1.29797e49	0.465914
$670$	0	0
$671$	5.59551e49	1.90632
$672$	0	0
$673$	4.64646e49	1.50265	0.751326	−	0.659931i	$-0.229414\pi$
$673$	4.64646e49	1.50265	0.751326	+	0.659931i	$0.229414\pi$
$674$	0	0
$675$	2.92401e48	0.0897765
$676$	0	0
$677$	2.34704e49	0.684256	0.342128	−	0.939653i	$-0.388852\pi$
$677$	2.34704e49	0.684256	0.342128	+	0.939653i	$0.388852\pi$
$678$	0	0
$679$	−1.53365e49	−0.424626
$680$	0	0
$681$	−2.62638e49	−0.690689
$682$	0	0
$683$	5.70146e49	1.42437	0.712186	−	0.701991i	$-0.247705\pi$
$683$	5.70146e49	1.42437	0.712186	+	0.701991i	$0.247705\pi$
$684$	0	0
$685$	−5.30413e49	−1.25901
$686$	0	0
$687$	−6.98695e49	−1.57596
$688$	0	0
$689$	−6.57769e49	−1.41006
$690$	0	0
$691$	−1.51375e49	−0.308453	−0.154226	−	0.988036i	$-0.549289\pi$
$691$	−1.51375e49	−0.308453	−0.154226	+	0.988036i	$0.549289\pi$
$692$	0	0
$693$	1.06641e49	0.206582
$694$	0	0
$695$	−1.76447e49	−0.324998
$696$	0	0
$697$	8.34499e48	0.146168
$698$	0	0
$699$	−3.19780e49	−0.532721
$700$	0	0
$701$	9.52626e48	0.150958	0.0754792	−	0.997147i	$-0.475951\pi$
$701$	9.52626e48	0.150958	0.0754792	+	0.997147i	$0.475951\pi$
$702$	0	0
$703$	−3.76187e49	−0.567134
$704$	0	0
$705$	1.15136e50	1.65159
$706$	0	0
$707$	−8.44025e49	−1.15217
$708$	0	0
$709$	5.58236e49	0.725286	0.362643	−	0.931928i	$-0.381875\pi$
$709$	5.58236e49	0.725286	0.362643	+	0.931928i	$0.381875\pi$
$710$	0	0
$711$	−3.96146e48	−0.0489933
$712$	0	0
$713$	7.72606e47	0.00909686
$714$	0	0
$715$	1.39468e50	1.56358
$716$	0	0
$717$	9.62568e49	1.02765
$718$	0	0
$719$	−3.27201e49	−0.332705	−0.166352	−	0.986066i	$-0.553199\pi$
$719$	−3.27201e49	−0.332705	−0.166352	+	0.986066i	$0.553199\pi$
$720$	0	0
$721$	−5.76561e49	−0.558442
$722$	0	0
$723$	4.20167e49	0.387706
$724$	0	0
$725$	1.05608e49	0.0928503
$726$	0	0
$727$	−1.29657e50	−1.08629	−0.543144	−	0.839640i	$-0.682766\pi$
$727$	−1.29657e50	−1.08629	−0.543144	+	0.839640i	$0.682766\pi$
$728$	0	0
$729$	1.07779e50	0.860605
$730$	0	0
$731$	−2.56077e49	−0.194902
$732$	0	0
$733$	5.77831e49	0.419258	0.209629	−	0.977781i	$-0.432774\pi$
$733$	5.77831e49	0.419258	0.209629	+	0.977781i	$0.432774\pi$
$734$	0	0
$735$	1.93211e50	1.33660
$736$	0	0
$737$	−1.44302e50	−0.951903
$738$	0	0
$739$	6.88260e49	0.432987	0.216494	−	0.976284i	$-0.430538\pi$
$739$	6.88260e49	0.432987	0.216494	+	0.976284i	$0.430538\pi$
$740$	0	0
$741$	−9.45046e49	−0.567067
$742$	0	0
$743$	−1.77391e50	−1.01538	−0.507688	−	0.861541i	$-0.669500\pi$
$743$	−1.77391e50	−1.01538	−0.507688	+	0.861541i	$0.669500\pi$
$744$	0	0
$745$	−1.63219e50	−0.891327
$746$	0	0
$747$	−2.60881e49	−0.135935
$748$	0	0
$749$	−3.83778e49	−0.190831
$750$	0	0
$751$	2.42348e50	1.15011	0.575057	−	0.818113i	$-0.304980\pi$
$751$	2.42348e50	1.15011	0.575057	+	0.818113i	$0.304980\pi$
$752$	0	0
$753$	−2.27680e50	−1.03137
$754$	0	0
$755$	9.48472e49	0.410160
$756$	0	0
$757$	−2.18483e50	−0.902070	−0.451035	−	0.892506i	$-0.648945\pi$
$757$	−2.18483e50	−0.902070	−0.451035	+	0.892506i	$0.648945\pi$
$758$	0	0
$759$	−3.90752e50	−1.54053
$760$	0	0
$761$	−4.38224e50	−1.64992	−0.824962	−	0.565188i	$-0.808804\pi$
$761$	−4.38224e50	−1.64992	−0.824962	+	0.565188i	$0.808804\pi$
$762$	0	0
$763$	6.71840e50	2.41594
$764$	0	0
$765$	3.40984e48	0.0117127
$766$	0	0
$767$	1.25951e50	0.413318
$768$	0	0
$769$	4.38187e50	1.37388	0.686940	−	0.726715i	$-0.258954\pi$
$769$	4.38187e50	1.37388	0.686940	+	0.726715i	$0.258954\pi$
$770$	0	0
$771$	−9.52621e49	−0.285409
$772$	0	0
$773$	2.98831e50	0.855626	0.427813	−	0.903867i	$-0.359284\pi$
$773$	2.98831e50	0.855626	0.427813	+	0.903867i	$0.359284\pi$
$774$	0	0
$775$	−2.58244e47	−0.000706726 0
$776$	0	0
$777$	9.06618e50	2.37168
$778$	0	0
$779$	2.09636e50	0.524277
$780$	0	0
$781$	−5.32152e50	−1.27246
$782$	0	0
$783$	3.95217e50	0.903659
$784$	0	0
$785$	−2.88928e50	−0.631787
$786$	0	0
$787$	−2.58993e50	−0.541664	−0.270832	−	0.962627i	$-0.587299\pi$
$787$	−2.58993e50	−0.541664	−0.270832	+	0.962627i	$0.587299\pi$
$788$	0	0
$789$	−7.46566e50	−1.49355
$790$	0	0
$791$	−1.46409e51	−2.80207
$792$	0	0
$793$	1.22895e51	2.25037
$794$	0	0
$795$	5.79759e50	1.01583
$796$	0	0
$797$	2.17361e50	0.364470	0.182235	−	0.983255i	$-0.441667\pi$
$797$	2.17361e50	0.364470	0.182235	+	0.983255i	$0.441667\pi$
$798$	0	0
$799$	1.10190e50	0.176838
$800$	0	0
$801$	−3.14628e49	−0.0483312
$802$	0	0
$803$	5.33015e50	0.783822
$804$	0	0
$805$	−1.27479e51	−1.79477
$806$	0	0
$807$	−7.96779e50	−1.07412
$808$	0	0
$809$	−5.88430e50	−0.759619	−0.379809	−	0.925065i	$-0.624010\pi$
$809$	−5.88430e50	−0.759619	−0.379809	+	0.925065i	$0.624010\pi$
$810$	0	0
$811$	−5.27337e50	−0.651963	−0.325982	−	0.945376i	$-0.605695\pi$
$811$	−5.27337e50	−0.651963	−0.325982	+	0.945376i	$0.605695\pi$
$812$	0	0
$813$	1.28633e51	1.52324
$814$	0	0
$815$	−5.68712e50	−0.645111
$816$	0	0
$817$	−6.43295e50	−0.699076
$818$	0	0
$819$	2.34218e50	0.243866
$820$	0	0
$821$	1.35492e51	1.35179	0.675896	−	0.736997i	$-0.263757\pi$
$821$	1.35492e51	1.35179	0.675896	+	0.736997i	$0.263757\pi$
$822$	0	0
$823$	−1.75381e51	−1.67682	−0.838409	−	0.545041i	$-0.816514\pi$
$823$	−1.75381e51	−1.67682	−0.838409	+	0.545041i	$0.816514\pi$
$824$	0	0
$825$	1.30609e50	0.119682
$826$	0	0
$827$	1.88306e51	1.65394	0.826968	−	0.562249i	$-0.190064\pi$
$827$	1.88306e51	1.65394	0.826968	+	0.562249i	$0.190064\pi$
$828$	0	0
$829$	1.25754e51	1.05881	0.529405	−	0.848369i	$-0.322415\pi$
$829$	1.25754e51	1.05881	0.529405	+	0.848369i	$0.322415\pi$
$830$	0	0
$831$	−1.35137e51	−1.09083
$832$	0	0
$833$	1.84911e50	0.143112
$834$	0	0
$835$	1.11600e51	0.828231
$836$	0	0
$837$	−9.66424e48	−0.00687816
$838$	0	0
$839$	2.03970e51	1.39230	0.696149	−	0.717897i	$-0.254895\pi$
$839$	2.03970e51	1.39230	0.696149	+	0.717897i	$0.254895\pi$
$840$	0	0
$841$	−9.98873e49	−0.0654004
$842$	0	0
$843$	1.33979e51	0.841502
$844$	0	0
$845$	1.48531e51	0.895005
$846$	0	0
$847$	−1.03827e51	−0.600275
$848$	0	0
$849$	−1.46824e51	−0.814541
$850$	0	0
$851$	−3.41622e51	−1.81877
$852$	0	0
$853$	2.06862e51	1.05699	0.528497	−	0.848935i	$-0.322756\pi$
$853$	2.06862e51	1.05699	0.528497	+	0.848935i	$0.322756\pi$
$854$	0	0
$855$	8.56591e49	0.0420113
$856$	0	0
$857$	−2.63744e51	−1.24171	−0.620853	−	0.783927i	$-0.713214\pi$
$857$	−2.63744e51	−1.24171	−0.620853	+	0.783927i	$0.713214\pi$
$858$	0	0
$859$	−1.39985e51	−0.632705	−0.316353	−	0.948642i	$-0.602458\pi$
$859$	−1.39985e51	−0.632705	−0.316353	+	0.948642i	$0.602458\pi$
$860$	0	0
$861$	−5.05226e51	−2.19246
$862$	0	0
$863$	2.78961e51	1.16240	0.581198	−	0.813762i	$-0.302584\pi$
$863$	2.78961e51	1.16240	0.581198	+	0.813762i	$0.302584\pi$
$864$	0	0
$865$	3.22949e51	1.29227
$866$	0	0
$867$	−2.71552e51	−1.04356
$868$	0	0
$869$	1.36679e51	0.504495
$870$	0	0
$871$	−3.16933e51	−1.12370
$872$	0	0
$873$	9.35849e49	0.0318755
$874$	0	0
$875$	4.86258e51	1.59120
$876$	0	0
$877$	−3.62148e51	−1.13866	−0.569328	−	0.822110i	$-0.692797\pi$
$877$	−3.62148e51	−1.13866	−0.569328	+	0.822110i	$0.692797\pi$
$878$	0	0
$879$	−4.57650e50	−0.138270
$880$	0	0
$881$	2.65108e51	0.769742	0.384871	−	0.922970i	$-0.374246\pi$
$881$	2.65108e51	0.769742	0.384871	+	0.922970i	$0.374246\pi$
$882$	0	0
$883$	2.11949e51	0.591450	0.295725	−	0.955273i	$-0.404439\pi$
$883$	2.11949e51	0.591450	0.295725	+	0.955273i	$0.404439\pi$
$884$	0	0
$885$	−1.11014e51	−0.297762
$886$	0	0
$887$	3.10956e51	0.801744	0.400872	−	0.916134i	$-0.368707\pi$
$887$	3.10956e51	0.801744	0.400872	+	0.916134i	$0.368707\pi$
$888$	0	0
$889$	−7.50176e51	−1.85944
$890$	0	0
$891$	5.45548e51	1.30009
$892$	0	0
$893$	2.76811e51	0.634283
$894$	0	0
$895$	2.29832e51	0.506415
$896$	0	0
$897$	−8.58213e51	−1.81856
$898$	0	0
$899$	−3.49050e49	−0.00711366
$900$	0	0
$901$	5.54854e50	0.108766
$902$	0	0
$903$	1.55035e52	2.92345
$904$	0	0
$905$	6.85774e51	1.24403
$906$	0	0
$907$	1.02943e51	0.179667	0.0898335	−	0.995957i	$-0.471366\pi$
$907$	1.02943e51	0.179667	0.0898335	+	0.995957i	$0.471366\pi$
$908$	0	0
$909$	5.15031e50	0.0864900
$910$	0	0
$911$	−8.95321e51	−1.44679	−0.723397	−	0.690432i	$-0.757421\pi$
$911$	−8.95321e51	−1.44679	−0.723397	+	0.690432i	$0.757421\pi$
$912$	0	0
$913$	9.00100e51	1.39976
$914$	0	0
$915$	−1.08320e52	−1.62121
$916$	0	0
$917$	−7.79229e51	−1.12254
$918$	0	0
$919$	−1.37824e52	−1.91118	−0.955591	−	0.294696i	$-0.904782\pi$
$919$	−1.37824e52	−1.91118	−0.955591	+	0.294696i	$0.904782\pi$
$920$	0	0
$921$	3.90434e51	0.521200
$922$	0	0
$923$	−1.16877e52	−1.50210
$924$	0	0
$925$	1.14188e51	0.141299
$926$	0	0
$927$	3.51823e50	0.0419207
$928$	0	0
$929$	−2.28048e51	−0.261668	−0.130834	−	0.991404i	$-0.541766\pi$
$929$	−2.28048e51	−0.261668	−0.130834	+	0.991404i	$0.541766\pi$
$930$	0	0
$931$	4.64518e51	0.513314
$932$	0	0
$933$	7.80709e51	0.830922
$934$	0	0
$935$	−1.17647e51	−0.120609
$936$	0	0
$937$	−5.63371e51	−0.556355	−0.278177	−	0.960530i	$-0.589730\pi$
$937$	−5.63371e51	−0.556355	−0.278177	+	0.960530i	$0.589730\pi$
$938$	0	0
$939$	3.03613e50	0.0288851
$940$	0	0
$941$	−1.52748e52	−1.40009	−0.700047	−	0.714097i	$-0.746838\pi$
$941$	−1.52748e52	−1.40009	−0.700047	+	0.714097i	$0.746838\pi$
$942$	0	0
$943$	1.90374e52	1.68133
$944$	0	0
$945$	1.59458e52	1.35703
$946$	0	0
$947$	−9.14668e51	−0.750134	−0.375067	−	0.926998i	$-0.622380\pi$
$947$	−9.14668e51	−0.750134	−0.375067	+	0.926998i	$0.622380\pi$
$948$	0	0
$949$	1.17067e52	0.925284
$950$	0	0
$951$	−1.78646e52	−1.36092
$952$	0	0
$953$	1.27602e52	0.936986	0.468493	−	0.883467i	$-0.344797\pi$
$953$	1.27602e52	0.936986	0.468493	+	0.883467i	$0.344797\pi$
$954$	0	0
$955$	−2.77701e51	−0.196571
$956$	0	0
$957$	1.76535e52	1.20468
$958$	0	0
$959$	3.07330e52	2.02199
$960$	0	0
$961$	−1.57629e52	−0.999946
$962$	0	0
$963$	2.34185e50	0.0143251
$964$	0	0
$965$	−1.67103e52	−0.985729
$966$	0	0
$967$	2.67063e51	0.151932	0.0759662	−	0.997110i	$-0.475796\pi$
$967$	2.67063e51	0.151932	0.0759662	+	0.997110i	$0.475796\pi$
$968$	0	0
$969$	7.97184e50	0.0437414
$970$	0	0
$971$	−2.85873e52	−1.51299	−0.756497	−	0.653997i	$-0.773091\pi$
$971$	−2.85873e52	−1.51299	−0.756497	+	0.653997i	$0.773091\pi$
$972$	0	0
$973$	1.02236e52	0.521953
$974$	0	0
$975$	2.86858e51	0.141282
$976$	0	0
$977$	1.47522e52	0.700975	0.350488	−	0.936567i	$-0.386016\pi$
$977$	1.47522e52	0.700975	0.350488	+	0.936567i	$0.386016\pi$
$978$	0	0
$979$	1.08554e52	0.497677
$980$	0	0
$981$	−4.09963e51	−0.181358
$982$	0	0
$983$	3.06803e52	1.30970	0.654852	−	0.755757i	$-0.272731\pi$
$983$	3.06803e52	1.30970	0.654852	+	0.755757i	$0.272731\pi$
$984$	0	0
$985$	1.12053e52	0.461623
$986$	0	0
$987$	−6.67120e52	−2.65249
$988$	0	0
$989$	−5.84188e52	−2.24191
$990$	0	0
$991$	−2.04429e52	−0.757274	−0.378637	−	0.925545i	$-0.623607\pi$
$991$	−2.04429e52	−0.757274	−0.378637	+	0.925545i	$0.623607\pi$
$992$	0	0
$993$	1.04686e52	0.374349
$994$	0	0
$995$	−1.29031e52	−0.445442
$996$	0	0
$997$	2.06779e52	0.689192	0.344596	−	0.938751i	$-0.388016\pi$
$997$	2.06779e52	0.689192	0.344596	+	0.938751i	$0.388016\pi$
$998$	0	0
$999$	4.27323e52	1.37518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.36.a.a.1.3	✓	3
4.3	odd	2		16.36.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.36.a.a.1.3	✓	3	1.1	even	1	trivial
16.36.a.c.1.1		3	4.3	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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36149e8 q^{3} +1.62199e12 q^{5} -9.39810e14 q^{7} +5.73480e15 q^{9} +O(q^{10})\)	\(q+2.36149e8 q^{3} +1.62199e12 q^{5} -9.39810e14 q^{7} +5.73480e15 q^{9} -1.97864e18 q^{11} -4.34571e19 q^{13} +3.83032e20 q^{15} +3.66578e20 q^{17} +9.20887e21 q^{19} -2.21935e23 q^{21} +8.36273e23 q^{23} -2.79525e23 q^{25} -1.04606e25 q^{27} -3.77814e25 q^{29} +9.23868e23 q^{31} -4.67253e26 q^{33} -1.52436e27 q^{35} -4.08506e27 q^{37} -1.02623e28 q^{39} +2.27646e28 q^{41} -6.98561e28 q^{43} +9.30181e27 q^{45} +3.00592e29 q^{47} +5.04425e29 q^{49} +8.65670e28 q^{51} +1.51360e30 q^{53} -3.20933e30 q^{55} +2.17466e30 q^{57} -2.89829e30 q^{59} -2.82796e31 q^{61} -5.38963e30 q^{63} -7.04871e31 q^{65} +7.29302e31 q^{67} +1.97485e32 q^{69} +2.68949e32 q^{71} -2.69385e32 q^{73} -6.60096e31 q^{75} +1.85954e33 q^{77} -6.90775e32 q^{79} -2.75719e33 q^{81} -4.54909e33 q^{83} +5.94586e32 q^{85} -8.92203e33 q^{87} -5.48629e33 q^{89} +4.08414e34 q^{91} +2.18170e32 q^{93} +1.49367e34 q^{95} +1.63188e34 q^{97} -1.13471e34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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