LMFDB - Embedded newform 15.12.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,12,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("15.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	15.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:	$11.5251477084$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.7191$ of defining polynomial
Character	$\chi$	$=$	15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.7191 q^{2} -243.000 q^{3} -1831.35 q^{4} -3125.00 q^{5} -3576.74 q^{6} +79941.2 q^{7} -57100.5 q^{8} +59049.0 q^{9} +O(q^{10})$	\(q+14.7191 q^{2} -243.000 q^{3} -1831.35 q^{4} -3125.00 q^{5} -3576.74 q^{6} +79941.2 q^{7} -57100.5 q^{8} +59049.0 q^{9} -45997.2 q^{10} +805067. q^{11} +445018. q^{12} -1.19767e6 q^{13} +1.17666e6 q^{14} +759375. q^{15} +2.91013e6 q^{16} +2.63319e6 q^{17} +869148. q^{18} +1.16061e7 q^{19} +5.72296e6 q^{20} -1.94257e7 q^{21} +1.18499e7 q^{22} +1.84216e7 q^{23} +1.38754e7 q^{24} +9.76562e6 q^{25} -1.76286e7 q^{26} -1.43489e7 q^{27} -1.46400e8 q^{28} -1.90527e8 q^{29} +1.11773e7 q^{30} +1.01127e8 q^{31} +1.59776e8 q^{32} -1.95631e8 q^{33} +3.87581e7 q^{34} -2.49816e8 q^{35} -1.08139e8 q^{36} +8.06675e7 q^{37} +1.70831e8 q^{38} +2.91034e8 q^{39} +1.78439e8 q^{40} +2.26316e8 q^{41} -2.85929e8 q^{42} +1.67149e9 q^{43} -1.47436e9 q^{44} -1.84528e8 q^{45} +2.71150e8 q^{46} +8.58507e8 q^{47} -7.07162e8 q^{48} +4.41327e9 q^{49} +1.43741e8 q^{50} -6.39864e8 q^{51} +2.19335e9 q^{52} -3.52750e9 q^{53} -2.11203e8 q^{54} -2.51583e9 q^{55} -4.56468e9 q^{56} -2.82028e9 q^{57} -2.80438e9 q^{58} +4.35760e9 q^{59} -1.39068e9 q^{60} -1.65393e9 q^{61} +1.48849e9 q^{62} +4.72045e9 q^{63} -3.60819e9 q^{64} +3.74272e9 q^{65} -2.87951e9 q^{66} +7.58610e9 q^{67} -4.82228e9 q^{68} -4.47645e9 q^{69} -3.67707e9 q^{70} -2.75809e10 q^{71} -3.37173e9 q^{72} +3.22368e10 q^{73} +1.18735e9 q^{74} -2.37305e9 q^{75} -2.12548e10 q^{76} +6.43580e10 q^{77} +4.28376e9 q^{78} -2.43149e9 q^{79} -9.09417e9 q^{80} +3.48678e9 q^{81} +3.33116e9 q^{82} +1.20729e10 q^{83} +3.55753e10 q^{84} -8.22871e9 q^{85} +2.46028e10 q^{86} +4.62981e10 q^{87} -4.59697e10 q^{88} +4.44073e9 q^{89} -2.71609e9 q^{90} -9.57433e10 q^{91} -3.37364e10 q^{92} -2.45737e10 q^{93} +1.26364e10 q^{94} -3.62690e10 q^{95} -3.88257e10 q^{96} -2.04453e10 q^{97} +6.49594e10 q^{98} +4.75384e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9}+O(q^{10}) $ 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9} + 40625 q^{10} + 295568 q^{11} + 755973 q^{12} + 657492 q^{13} + 3176796 q^{14} + 1518750 q^{15} + 2974065 q^{16} + 8579948 q^{17} - 767637 q^{18} + 17627976 q^{19} + 9721875 q^{20} - 1891512 q^{21} + 25972696 q^{22} - 29841072 q^{23} - 8538777 q^{24} + 19531250 q^{25} - 69052066 q^{26} - 28697814 q^{27} - 54064108 q^{28} - 201881948 q^{29} - 9871875 q^{30} - 71057008 q^{31} - 30902261 q^{32} - 71823024 q^{33} - 126080702 q^{34} - 24325000 q^{35} - 183701439 q^{36} + 705858484 q^{37} + 3909548 q^{38} - 159770556 q^{39} - 109809375 q^{40} - 327655148 q^{41} - 771961428 q^{42} + 3192552120 q^{43} - 822376568 q^{44} - 369056250 q^{45} + 1608947832 q^{46} + 2053064720 q^{47} - 722697795 q^{48} + 7642614770 q^{49} - 126953125 q^{50} - 2084927364 q^{51} - 180610022 q^{52} - 2304299452 q^{53} + 186535791 q^{54} - 923650000 q^{55} - 11220431460 q^{56} - 4283598168 q^{57} - 2489635114 q^{58} - 1478770576 q^{59} - 2362415625 q^{60} - 8264891460 q^{61} + 6261263448 q^{62} + 459637416 q^{63} + 1546316681 q^{64} - 2054662500 q^{65} - 6311365128 q^{66} + 24212177528 q^{67} - 12432065594 q^{68} + 7251380496 q^{69} - 9927487500 q^{70} - 20218888256 q^{71} + 2074922811 q^{72} + 25879583268 q^{73} - 16142374034 q^{74} - 4746093750 q^{75} - 28960704764 q^{76} + 101122045248 q^{77} + 16779652038 q^{78} + 22324995440 q^{79} - 9293953125 q^{80} + 6973568802 q^{81} + 18686740478 q^{82} + 48014508984 q^{83} + 13137578244 q^{84} - 26812337500 q^{85} - 17559698092 q^{86} + 49057313364 q^{87} - 92965609896 q^{88} + 79209683076 q^{89} + 2398865625 q^{90} - 229606712048 q^{91} + 28023036264 q^{92} + 17266852944 q^{93} - 20475612776 q^{94} - 55087425000 q^{95} + 7509249423 q^{96} - 37075227452 q^{97} - 24554976677 q^{98} + 17452994832 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9 + 40625 * q^10 + 295568 * q^11 + 755973 * q^12 + 657492 * q^13 + 3176796 * q^14 + 1518750 * q^15 + 2974065 * q^16 + 8579948 * q^17 - 767637 * q^18 + 17627976 * q^19 + 9721875 * q^20 - 1891512 * q^21 + 25972696 * q^22 - 29841072 * q^23 - 8538777 * q^24 + 19531250 * q^25 - 69052066 * q^26 - 28697814 * q^27 - 54064108 * q^28 - 201881948 * q^29 - 9871875 * q^30 - 71057008 * q^31 - 30902261 * q^32 - 71823024 * q^33 - 126080702 * q^34 - 24325000 * q^35 - 183701439 * q^36 + 705858484 * q^37 + 3909548 * q^38 - 159770556 * q^39 - 109809375 * q^40 - 327655148 * q^41 - 771961428 * q^42 + 3192552120 * q^43 - 822376568 * q^44 - 369056250 * q^45 + 1608947832 * q^46 + 2053064720 * q^47 - 722697795 * q^48 + 7642614770 * q^49 - 126953125 * q^50 - 2084927364 * q^51 - 180610022 * q^52 - 2304299452 * q^53 + 186535791 * q^54 - 923650000 * q^55 - 11220431460 * q^56 - 4283598168 * q^57 - 2489635114 * q^58 - 1478770576 * q^59 - 2362415625 * q^60 - 8264891460 * q^61 + 6261263448 * q^62 + 459637416 * q^63 + 1546316681 * q^64 - 2054662500 * q^65 - 6311365128 * q^66 + 24212177528 * q^67 - 12432065594 * q^68 + 7251380496 * q^69 - 9927487500 * q^70 - 20218888256 * q^71 + 2074922811 * q^72 + 25879583268 * q^73 - 16142374034 * q^74 - 4746093750 * q^75 - 28960704764 * q^76 + 101122045248 * q^77 + 16779652038 * q^78 + 22324995440 * q^79 - 9293953125 * q^80 + 6973568802 * q^81 + 18686740478 * q^82 + 48014508984 * q^83 + 13137578244 * q^84 - 26812337500 * q^85 - 17559698092 * q^86 + 49057313364 * q^87 - 92965609896 * q^88 + 79209683076 * q^89 + 2398865625 * q^90 - 229606712048 * q^91 + 28023036264 * q^92 + 17266852944 * q^93 - 20475612776 * q^94 - 55087425000 * q^95 + 7509249423 * q^96 - 37075227452 * q^97 - 24554976677 * q^98 + 17452994832 * 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 $

$2$	14.7191	0.325249	0.162625	−	0.986688i	$-0.448004\pi$
$2$	14.7191	0.325249	0.162625	+	0.986688i	$0.448004\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	−1831.35	−0.894213
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	−3576.74	−0.187783
$7$	79941.2	1.79776	0.898880	−	0.438195i	$-0.144382\pi$
$7$	79941.2	1.79776	0.898880	+	0.438195i	$0.144382\pi$
$8$	−57100.5	−0.616091
$9$	59049.0	0.333333
$10$	−45997.2	−0.145456
$11$	805067.	1.50720	0.753602	−	0.657331i	$-0.228315\pi$
$11$	805067.	1.50720	0.753602	+	0.657331i	$0.228315\pi$
$12$	445018.	0.516274
$13$	−1.19767e6	−0.894641	−0.447321	−	0.894374i	$-0.647622\pi$
$13$	−1.19767e6	−0.894641	−0.447321	+	0.894374i	$0.647622\pi$
$14$	1.17666e6	0.584720
$15$	759375.	0.258199
$16$	2.91013e6	0.693830
$17$	2.63319e6	0.449793	0.224896	−	0.974383i	$-0.427796\pi$
$17$	2.63319e6	0.449793	0.224896	+	0.974383i	$0.427796\pi$
$18$	869148.	0.108416
$19$	1.16061e7	1.07533	0.537664	−	0.843159i	$-0.319307\pi$
$19$	1.16061e7	1.07533	0.537664	+	0.843159i	$0.319307\pi$
$20$	5.72296e6	0.399904
$21$	−1.94257e7	−1.03794
$22$	1.18499e7	0.490217
$23$	1.84216e7	0.596794	0.298397	−	0.954442i	$-0.403548\pi$
$23$	1.84216e7	0.596794	0.298397	+	0.954442i	$0.403548\pi$
$24$	1.38754e7	0.355700
$25$	9.76562e6	0.200000
$26$	−1.76286e7	−0.290981
$27$	−1.43489e7	−0.192450
$28$	−1.46400e8	−1.60758
$29$	−1.90527e8	−1.72491	−0.862457	−	0.506130i	$-0.831076\pi$
$29$	−1.90527e8	−1.72491	−0.862457	+	0.506130i	$0.831076\pi$
$30$	1.11773e7	0.0839790
$31$	1.01127e8	0.634419	0.317209	−	0.948356i	$-0.397254\pi$
$31$	1.01127e8	0.634419	0.317209	+	0.948356i	$0.397254\pi$
$32$	1.59776e8	0.841759
$33$	−1.95631e8	−0.870185
$34$	3.87581e7	0.146295
$35$	−2.49816e8	−0.803983
$36$	−1.08139e8	−0.298071
$37$	8.06675e7	0.191245	0.0956223	−	0.995418i	$-0.469516\pi$
$37$	8.06675e7	0.191245	0.0956223	+	0.995418i	$0.469516\pi$
$38$	1.70831e8	0.349749
$39$	2.91034e8	0.516522
$40$	1.78439e8	0.275524
$41$	2.26316e8	0.305073	0.152537	−	0.988298i	$-0.451256\pi$
$41$	2.26316e8	0.305073	0.152537	+	0.988298i	$0.451256\pi$
$42$	−2.85929e8	−0.337588
$43$	1.67149e9	1.73391	0.866956	−	0.498385i	$-0.166073\pi$
$43$	1.67149e9	1.73391	0.866956	+	0.498385i	$0.166073\pi$
$44$	−1.47436e9	−1.34776
$45$	−1.84528e8	−0.149071
$46$	2.71150e8	0.194107
$47$	8.58507e8	0.546016	0.273008	−	0.962012i	$-0.411981\pi$
$47$	8.58507e8	0.546016	0.273008	+	0.962012i	$0.411981\pi$
$48$	−7.07162e8	−0.400583
$49$	4.41327e9	2.23194
$50$	1.43741e8	0.0650498
$51$	−6.39864e8	−0.259688
$52$	2.19335e9	0.800000
$53$	−3.52750e9	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$53$	−3.52750e9	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$54$	−2.11203e8	−0.0625942
$55$	−2.51583e9	−0.674042
$56$	−4.56468e9	−1.10758
$57$	−2.82028e9	−0.620841
$58$	−2.80438e9	−0.561027
$59$	4.35760e9	0.793527	0.396763	−	0.917921i	$-0.370133\pi$
$59$	4.35760e9	0.793527	0.396763	+	0.917921i	$0.370133\pi$
$60$	−1.39068e9	−0.230885
$61$	−1.65393e9	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$61$	−1.65393e9	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$62$	1.48849e9	0.206344
$63$	4.72045e9	0.599253
$64$	−3.60819e9	−0.420049
$65$	3.74272e9	0.400096
$66$	−2.87951e9	−0.283027
$67$	7.58610e9	0.686447	0.343224	−	0.939254i	$-0.388481\pi$
$67$	7.58610e9	0.686447	0.343224	+	0.939254i	$0.388481\pi$
$68$	−4.82228e9	−0.402211
$69$	−4.47645e9	−0.344559
$70$	−3.67707e9	−0.261495
$71$	−2.75809e10	−1.81421	−0.907104	−	0.420905i	$-0.861712\pi$
$71$	−2.75809e10	−1.81421	−0.907104	+	0.420905i	$0.861712\pi$
$72$	−3.37173e9	−0.205364
$73$	3.22368e10	1.82002	0.910011	−	0.414584i	$-0.136073\pi$
$73$	3.22368e10	1.82002	0.910011	+	0.414584i	$0.136073\pi$
$74$	1.18735e9	0.0622022
$75$	−2.37305e9	−0.115470
$76$	−2.12548e10	−0.961572
$77$	6.43580e10	2.70959
$78$	4.28376e9	0.167998
$79$	−2.43149e9	−0.0889046	−0.0444523	−	0.999012i	$-0.514154\pi$
$79$	−2.43149e9	−0.0889046	−0.0444523	+	0.999012i	$0.514154\pi$
$80$	−9.09417e9	−0.310290
$81$	3.48678e9	0.111111
$82$	3.33116e9	0.0992247
$83$	1.20729e10	0.336421	0.168210	−	0.985751i	$-0.446201\pi$
$83$	1.20729e10	0.336421	0.168210	+	0.985751i	$0.446201\pi$
$84$	3.55753e10	0.928137
$85$	−8.22871e9	−0.201154
$86$	2.46028e10	0.563953
$87$	4.62981e10	0.995880
$88$	−4.59697e10	−0.928575
$89$	4.44073e9	0.0842964	0.0421482	−	0.999111i	$-0.486580\pi$
$89$	4.44073e9	0.0842964	0.0421482	+	0.999111i	$0.486580\pi$
$90$	−2.71609e9	−0.0484853
$91$	−9.57433e10	−1.60835
$92$	−3.37364e10	−0.533661
$93$	−2.45737e10	−0.366282
$94$	1.26364e10	0.177591
$95$	−3.62690e10	−0.480901
$96$	−3.88257e10	−0.485990
$97$	−2.04453e10	−0.241740	−0.120870	−	0.992668i	$-0.538568\pi$
$97$	−2.04453e10	−0.241740	−0.120870	+	0.992668i	$0.538568\pi$
$98$	6.49594e10	0.725937
$99$	4.75384e10	0.502401
$100$	−1.78843e10	−0.178843
$101$	−1.55947e11	−1.47642	−0.738209	−	0.674572i	$-0.764328\pi$
$101$	−1.55947e11	−1.47642	−0.738209	+	0.674572i	$0.764328\pi$
$102$	−9.41823e9	−0.0844633
$103$	−5.10325e10	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$103$	−5.10325e10	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$104$	6.83876e10	0.551181
$105$	6.07054e10	0.464180
$106$	−5.19217e10	−0.376848
$107$	1.06665e10	0.0735207	0.0367603	−	0.999324i	$-0.488296\pi$
$107$	1.06665e10	0.0735207	0.0367603	+	0.999324i	$0.488296\pi$
$108$	2.62778e10	0.172091
$109$	−3.12513e10	−0.194546	−0.0972729	−	0.995258i	$-0.531012\pi$
$109$	−3.12513e10	−0.194546	−0.0972729	+	0.995258i	$0.531012\pi$
$110$	−3.70308e10	−0.219232
$111$	−1.96022e10	−0.110415
$112$	2.32640e11	1.24734
$113$	−3.96346e10	−0.202369	−0.101184	−	0.994868i	$-0.532263\pi$
$113$	−3.96346e10	−0.202369	−0.101184	+	0.994868i	$0.532263\pi$
$114$	−4.15119e10	−0.201928
$115$	−5.75676e10	−0.266895
$116$	3.48921e11	1.54244
$117$	−7.07213e10	−0.298214
$118$	6.41400e10	0.258094
$119$	2.10500e11	0.808620
$120$	−4.33607e10	−0.159074
$121$	3.62821e11	1.27166
$122$	−2.43443e10	−0.0815489
$123$	−5.49948e10	−0.176134
$124$	−1.85198e11	−0.567305
$125$	−3.05176e10	−0.0894427
$126$	6.94808e10	0.194907
$127$	2.63460e10	0.0707611	0.0353806	−	0.999374i	$-0.488736\pi$
$127$	2.63460e10	0.0707611	0.0353806	+	0.999374i	$0.488736\pi$
$128$	−3.80331e11	−0.978379
$129$	−4.06172e11	−1.00107
$130$	5.50895e10	0.130131
$131$	−2.19917e11	−0.498044	−0.249022	−	0.968498i	$-0.580109\pi$
$131$	−2.19917e11	−0.498044	−0.249022	+	0.968498i	$0.580109\pi$
$132$	3.58269e11	0.778131
$133$	9.27805e11	1.93318
$134$	1.11661e11	0.223266
$135$	4.48403e10	0.0860663
$136$	−1.50356e11	−0.277113
$137$	−5.54041e11	−0.980795	−0.490398	−	0.871499i	$-0.663148\pi$
$137$	−5.54041e11	−0.980795	−0.490398	+	0.871499i	$0.663148\pi$
$138$	−6.58894e10	−0.112068
$139$	−6.17540e11	−1.00945	−0.504723	−	0.863281i	$-0.668406\pi$
$139$	−6.17540e11	−1.00945	−0.504723	+	0.863281i	$0.668406\pi$
$140$	4.57501e11	0.718932
$141$	−2.08617e11	−0.315243
$142$	−4.05966e11	−0.590070
$143$	−9.64205e11	−1.34841
$144$	1.71840e11	0.231277
$145$	5.95397e11	0.771405
$146$	4.74497e11	0.591961
$147$	−1.07243e12	−1.28861
$148$	−1.47730e11	−0.171013
$149$	−6.68742e11	−0.745992	−0.372996	−	0.927833i	$-0.621670\pi$
$149$	−6.68742e11	−0.745992	−0.372996	+	0.927833i	$0.621670\pi$
$150$	−3.49291e10	−0.0375565
$151$	1.38243e12	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$151$	1.38243e12	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$152$	−6.62713e11	−0.662500
$153$	1.55487e11	0.149931
$154$	9.47292e11	0.881292
$155$	−3.16020e11	−0.283721
$156$	−5.32985e11	−0.461880
$157$	−7.93661e11	−0.664029	−0.332014	−	0.943274i	$-0.607728\pi$
$157$	−7.93661e11	−0.664029	−0.332014	+	0.943274i	$0.607728\pi$
$158$	−3.57894e10	−0.0289161
$159$	8.57184e11	0.668944
$160$	−4.99301e11	−0.376446
$161$	1.47265e12	1.07289
$162$	5.13223e10	0.0361388
$163$	−4.67399e11	−0.318168	−0.159084	−	0.987265i	$-0.550854\pi$
$163$	−4.67399e11	−0.318168	−0.159084	+	0.987265i	$0.550854\pi$
$164$	−4.14463e11	−0.272800
$165$	6.11348e11	0.389158
$166$	1.77702e11	0.109421
$167$	2.87482e12	1.71266	0.856328	−	0.516432i	$-0.172740\pi$
$167$	2.87482e12	1.71266	0.856328	+	0.516432i	$0.172740\pi$
$168$	1.10922e12	0.639464
$169$	−3.57745e11	−0.199617
$170$	−1.21119e11	−0.0654250
$171$	6.85328e11	0.358443
$172$	−3.06108e12	−1.55049
$173$	1.53085e12	0.751068	0.375534	−	0.926809i	$-0.377459\pi$
$173$	1.53085e12	0.751068	0.375534	+	0.926809i	$0.377459\pi$
$174$	6.81465e11	0.323909
$175$	7.80676e11	0.359552
$176$	2.34285e12	1.04574
$177$	−1.05890e12	−0.458143
$178$	6.53635e10	0.0274173
$179$	−1.91337e12	−0.778228	−0.389114	−	0.921190i	$-0.627219\pi$
$179$	−1.91337e12	−0.778228	−0.389114	+	0.921190i	$0.627219\pi$
$180$	3.37935e11	0.133301
$181$	−1.70819e12	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$181$	−1.70819e12	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$182$	−1.40925e12	−0.523115
$183$	4.01904e11	0.144758
$184$	−1.05188e12	−0.367680
$185$	−2.52086e11	−0.0855272
$186$	−3.61703e11	−0.119133
$187$	2.11989e12	0.677930
$188$	−1.57223e12	−0.488255
$189$	−1.14707e12	−0.345979
$190$	−5.33847e11	−0.156413
$191$	3.16020e12	0.899562	0.449781	−	0.893139i	$-0.351502\pi$
$191$	3.16020e12	0.899562	0.449781	+	0.893139i	$0.351502\pi$
$192$	8.76790e11	0.242515
$193$	2.77296e12	0.745382	0.372691	−	0.927956i	$-0.378435\pi$
$193$	2.77296e12	0.745382	0.372691	+	0.927956i	$0.378435\pi$
$194$	−3.00936e11	−0.0786256
$195$	−9.09481e11	−0.230995
$196$	−8.08224e12	−1.99583
$197$	3.86504e12	0.928089	0.464044	−	0.885812i	$-0.346398\pi$
$197$	3.86504e12	0.928089	0.464044	+	0.885812i	$0.346398\pi$
$198$	6.99722e11	0.163406
$199$	−1.01839e12	−0.231325	−0.115663	−	0.993289i	$-0.536899\pi$
$199$	−1.01839e12	−0.231325	−0.115663	+	0.993289i	$0.536899\pi$
$200$	−5.57622e11	−0.123218
$201$	−1.84342e12	−0.396321
$202$	−2.29540e12	−0.480204
$203$	−1.52310e13	−3.10098
$204$	1.17181e12	0.232216
$205$	−7.07237e11	−0.136433
$206$	−7.51152e11	−0.141078
$207$	1.08778e12	0.198931
$208$	−3.48538e12	−0.620729
$209$	9.34367e12	1.62074
$210$	8.93528e11	0.150974
$211$	2.05668e12	0.338543	0.169271	−	0.985569i	$-0.445859\pi$
$211$	2.05668e12	0.338543	0.169271	+	0.985569i	$0.445859\pi$
$212$	6.46009e12	1.03608
$213$	6.70216e12	1.04743
$214$	1.57001e11	0.0239125
$215$	−5.22340e12	−0.775429
$216$	8.19330e11	0.118567
$217$	8.08418e12	1.14053
$218$	−4.59990e11	−0.0632759
$219$	−7.83355e12	−1.05079
$220$	4.60737e12	0.602737
$221$	−3.15369e12	−0.402403
$222$	−2.88527e11	−0.0359124
$223$	9.99986e12	1.21428	0.607138	−	0.794597i	$-0.292318\pi$
$223$	9.99986e12	1.21428	0.607138	+	0.794597i	$0.292318\pi$
$224$	1.27727e13	1.51328
$225$	5.76650e11	0.0666667
$226$	−5.83386e11	−0.0658202
$227$	−6.77123e12	−0.745634	−0.372817	−	0.927905i	$-0.621608\pi$
$227$	−6.77123e12	−0.745634	−0.372817	+	0.927905i	$0.621608\pi$
$228$	5.16491e12	0.555164
$229$	1.01933e13	1.06959	0.534796	−	0.844981i	$-0.320389\pi$
$229$	1.01933e13	1.06959	0.534796	+	0.844981i	$0.320389\pi$
$230$	−8.47343e11	−0.0868072
$231$	−1.56390e13	−1.56438
$232$	1.08792e13	1.06270
$233$	6.74446e12	0.643412	0.321706	−	0.946840i	$-0.395744\pi$
$233$	6.74446e12	0.643412	0.321706	+	0.946840i	$0.395744\pi$
$234$	−1.04095e12	−0.0969938
$235$	−2.68283e12	−0.244186
$236$	−7.98029e12	−0.709582
$237$	5.90853e11	0.0513291
$238$	3.09837e12	0.263003
$239$	−1.76501e13	−1.46406	−0.732030	−	0.681272i	$-0.761427\pi$
$239$	−1.76501e13	−1.46406	−0.732030	+	0.681272i	$0.761427\pi$
$240$	2.20988e12	0.179146
$241$	−1.37662e13	−1.09074	−0.545369	−	0.838196i	$-0.683611\pi$
$241$	−1.37662e13	−1.09074	−0.545369	+	0.838196i	$0.683611\pi$
$242$	5.34039e12	0.413608
$243$	−8.47289e11	−0.0641500
$244$	3.02891e12	0.224204
$245$	−1.37915e13	−0.998154
$246$	−8.09473e11	−0.0572874
$247$	−1.39003e13	−0.962033
$248$	−5.77438e12	−0.390860
$249$	−2.93372e12	−0.194233
$250$	−4.49191e11	−0.0290912
$251$	3.02252e13	1.91498	0.957490	−	0.288467i	$-0.0931454\pi$
$251$	3.02252e13	1.91498	0.957490	+	0.288467i	$0.0931454\pi$
$252$	−8.64479e12	−0.535860
$253$	1.48306e13	0.899491
$254$	3.87790e11	0.0230150
$255$	1.99958e12	0.116136
$256$	1.79144e12	0.101832
$257$	6.88263e12	0.382933	0.191466	−	0.981499i	$-0.438676\pi$
$257$	6.88263e12	0.382933	0.191466	+	0.981499i	$0.438676\pi$
$258$	−5.97848e12	−0.325599
$259$	6.44866e12	0.343812
$260$	−6.85423e12	−0.357771
$261$	−1.12504e13	−0.574971
$262$	−3.23698e12	−0.161988
$263$	−3.29948e13	−1.61692	−0.808462	−	0.588549i	$-0.799699\pi$
$263$	−3.29948e13	−1.61692	−0.808462	+	0.588549i	$0.799699\pi$
$264$	1.11706e13	0.536113
$265$	1.10235e13	0.518162
$266$	1.36564e13	0.628765
$267$	−1.07910e12	−0.0486686
$268$	−1.38928e13	−0.613830
$269$	−2.97478e13	−1.28771	−0.643855	−	0.765148i	$-0.722666\pi$
$269$	−2.97478e13	−1.28771	−0.643855	+	0.765148i	$0.722666\pi$
$270$	6.60009e11	0.0279930
$271$	−4.30410e13	−1.78876	−0.894378	−	0.447312i	$-0.852381\pi$
$271$	−4.30410e13	−1.78876	−0.894378	+	0.447312i	$0.852381\pi$
$272$	7.66293e12	0.312080
$273$	2.32656e13	0.928582
$274$	−8.15498e12	−0.319003
$275$	7.86198e12	0.301441
$276$	8.19795e12	0.308110
$277$	1.46706e13	0.540518	0.270259	−	0.962788i	$-0.412891\pi$
$277$	1.46706e13	0.540518	0.270259	+	0.962788i	$0.412891\pi$
$278$	−9.08963e12	−0.328322
$279$	5.97142e12	0.211473
$280$	1.42646e13	0.495327
$281$	4.18378e13	1.42457	0.712286	−	0.701890i	$-0.247660\pi$
$281$	4.18378e13	1.42457	0.712286	+	0.701890i	$0.247660\pi$
$282$	−3.07066e12	−0.102532
$283$	2.34242e13	0.767078	0.383539	−	0.923525i	$-0.374705\pi$
$283$	2.34242e13	0.767078	0.383539	+	0.923525i	$0.374705\pi$
$284$	5.05102e13	1.62229
$285$	8.81337e12	0.277648
$286$	−1.41922e13	−0.438568
$287$	1.80920e13	0.548448
$288$	9.43463e12	0.280586
$289$	−2.73382e13	−0.797686
$290$	8.76370e12	0.250899
$291$	4.96820e12	0.139568
$292$	−5.90369e13	−1.62749
$293$	−7.31258e12	−0.197833	−0.0989166	−	0.995096i	$-0.531538\pi$
$293$	−7.31258e12	−0.197833	−0.0989166	+	0.995096i	$0.531538\pi$
$294$	−1.57851e13	−0.419120
$295$	−1.36175e13	−0.354876
$296$	−4.60616e12	−0.117824
$297$	−1.15518e13	−0.290062
$298$	−9.84328e12	−0.242633
$299$	−2.20630e13	−0.533917
$300$	4.34588e12	0.103255
$301$	1.33621e14	3.11716
$302$	2.03482e13	0.466109
$303$	3.78951e13	0.852410
$304$	3.37752e13	0.746094
$305$	5.16852e12	0.112129
$306$	2.28863e12	0.0487649
$307$	−5.37239e13	−1.12436	−0.562181	−	0.827014i	$-0.690038\pi$
$307$	−5.37239e13	−1.12436	−0.562181	+	0.827014i	$0.690038\pi$
$308$	−1.17862e14	−2.42295
$309$	1.24009e13	0.250427
$310$	−4.65154e12	−0.0922799
$311$	5.41406e13	1.05521	0.527607	−	0.849489i	$-0.323089\pi$
$311$	5.41406e13	1.05521	0.527607	+	0.849489i	$0.323089\pi$
$312$	−1.66182e13	−0.318224
$313$	−4.29721e12	−0.0808524	−0.0404262	−	0.999183i	$-0.512872\pi$
$313$	−4.29721e12	−0.0808524	−0.0404262	+	0.999183i	$0.512872\pi$
$314$	−1.16820e13	−0.215975
$315$	−1.47514e13	−0.267994
$316$	4.45291e12	0.0794996
$317$	−2.81928e13	−0.494666	−0.247333	−	0.968931i	$-0.579554\pi$
$317$	−2.81928e13	−0.494666	−0.247333	+	0.968931i	$0.579554\pi$
$318$	1.26170e13	0.217573
$319$	−1.53387e14	−2.59980
$320$	1.12756e13	0.187851
$321$	−2.59195e12	−0.0424472
$322$	2.16760e13	0.348958
$323$	3.05610e13	0.483675
$324$	−6.38552e12	−0.0993570
$325$	−1.16960e13	−0.178928
$326$	−6.87969e12	−0.103484
$327$	7.59406e12	0.112321
$328$	−1.29227e13	−0.187953
$329$	6.86301e13	0.981606
$330$	8.99848e12	0.126573
$331$	−9.95469e13	−1.37713	−0.688563	−	0.725176i	$-0.741758\pi$
$331$	−9.95469e13	−1.37713	−0.688563	+	0.725176i	$0.741758\pi$
$332$	−2.21097e13	−0.300832
$333$	4.76334e12	0.0637482
$334$	4.23148e13	0.557040
$335$	−2.37066e13	−0.306989
$336$	−5.65314e13	−0.720152
$337$	−9.05175e13	−1.13440	−0.567202	−	0.823579i	$-0.691974\pi$
$337$	−9.05175e13	−1.13440	−0.567202	+	0.823579i	$0.691974\pi$
$338$	−5.26568e12	−0.0649251
$339$	9.63121e12	0.116838
$340$	1.50696e13	0.179874
$341$	8.14136e13	0.956198
$342$	1.00874e13	0.116583
$343$	1.94733e14	2.21473
$344$	−9.54428e13	−1.06825
$345$	1.39889e13	0.154092
$346$	2.25327e13	0.244284
$347$	−6.36892e13	−0.679601	−0.339801	−	0.940498i	$-0.610360\pi$
$347$	−6.36892e13	−0.679601	−0.339801	+	0.940498i	$0.610360\pi$
$348$	−8.47879e13	−0.890529
$349$	7.08598e13	0.732588	0.366294	−	0.930499i	$-0.380626\pi$
$349$	7.08598e13	0.732588	0.366294	+	0.930499i	$0.380626\pi$
$350$	1.14908e13	0.116944
$351$	1.71853e13	0.172174
$352$	1.28631e14	1.26870
$353$	6.62893e13	0.643699	0.321849	−	0.946791i	$-0.395696\pi$
$353$	6.62893e13	0.643699	0.321849	+	0.946791i	$0.395696\pi$
$354$	−1.55860e13	−0.149011
$355$	8.61903e13	0.811339
$356$	−8.13252e12	−0.0753790
$357$	−5.11516e13	−0.466857
$358$	−2.81630e13	−0.253118
$359$	6.80217e13	0.602044	0.301022	−	0.953617i	$-0.402672\pi$
$359$	6.80217e13	0.602044	0.301022	+	0.953617i	$0.402672\pi$
$360$	1.05366e13	0.0918414
$361$	1.82109e13	0.156330
$362$	−2.51430e13	−0.212579
$363$	−8.81654e13	−0.734196
$364$	1.75339e14	1.43821
$365$	−1.00740e14	−0.813939
$366$	5.91566e12	0.0470823
$367$	2.19771e14	1.72309	0.861543	−	0.507685i	$-0.169499\pi$
$367$	2.19771e14	1.72309	0.861543	+	0.507685i	$0.169499\pi$
$368$	5.36094e13	0.414074
$369$	1.33637e13	0.101691
$370$	−3.71048e12	−0.0278177
$371$	−2.81993e14	−2.08296
$372$	4.50031e13	0.327534
$373$	−1.67232e14	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$373$	−1.67232e14	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$374$	3.12029e13	0.220496
$375$	7.41577e12	0.0516398
$376$	−4.90212e13	−0.336396
$377$	2.28189e14	1.54318
$378$	−1.68838e13	−0.112529
$379$	−1.57778e14	−1.03641	−0.518205	−	0.855256i	$-0.673400\pi$
$379$	−1.57778e14	−1.03641	−0.518205	+	0.855256i	$0.673400\pi$
$380$	6.64212e13	0.430028
$381$	−6.40209e12	−0.0408540
$382$	4.65153e13	0.292582
$383$	5.49255e13	0.340550	0.170275	−	0.985397i	$-0.445534\pi$
$383$	5.49255e13	0.340550	0.170275	+	0.985397i	$0.445534\pi$
$384$	9.24205e13	0.564867
$385$	−2.01119e14	−1.21177
$386$	4.08155e13	0.242435
$387$	9.86997e13	0.577971
$388$	3.74424e13	0.216167
$389$	−2.32940e14	−1.32593	−0.662965	−	0.748650i	$-0.730702\pi$
$389$	−2.32940e14	−1.32593	−0.662965	+	0.748650i	$0.730702\pi$
$390$	−1.33867e13	−0.0751311
$391$	4.85076e13	0.268434
$392$	−2.52000e14	−1.37508
$393$	5.34399e13	0.287546
$394$	5.68899e13	0.301860
$395$	7.59842e12	0.0397593
$396$	−8.70593e13	−0.449254
$397$	−4.76134e13	−0.242316	−0.121158	−	0.992633i	$-0.538661\pi$
$397$	−4.76134e13	−0.242316	−0.121158	+	0.992633i	$0.538661\pi$
$398$	−1.49898e13	−0.0752383
$399$	−2.25456e14	−1.11612
$400$	2.84193e13	0.138766
$401$	−2.99074e14	−1.44040	−0.720202	−	0.693764i	$-0.755951\pi$
$401$	−2.99074e14	−1.44040	−0.720202	+	0.693764i	$0.755951\pi$
$402$	−2.71335e13	−0.128903
$403$	−1.21116e14	−0.567577
$404$	2.85593e14	1.32023
$405$	−1.08962e13	−0.0496904
$406$	−2.24186e14	−1.00859
$407$	6.49428e13	0.288245
$408$	3.65366e13	0.159992
$409$	−2.55879e14	−1.10550	−0.552748	−	0.833349i	$-0.686421\pi$
$409$	−2.55879e14	−1.10550	−0.552748	+	0.833349i	$0.686421\pi$
$410$	−1.04099e13	−0.0443747
$411$	1.34632e14	0.566262
$412$	9.34582e13	0.387867
$413$	3.48352e14	1.42657
$414$	1.60111e13	0.0647023
$415$	−3.77279e13	−0.150452
$416$	−1.91359e14	−0.753072
$417$	1.50062e14	0.582804
$418$	1.37530e14	0.527144
$419$	2.21825e14	0.839139	0.419570	−	0.907723i	$-0.362181\pi$
$419$	2.21825e14	0.839139	0.419570	+	0.907723i	$0.362181\pi$
$420$	−1.11173e14	−0.415075
$421$	9.78949e13	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$421$	9.78949e13	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$422$	3.02725e13	0.110111
$423$	5.06940e13	0.182005
$424$	2.01422e14	0.713831
$425$	2.57147e13	0.0899586
$426$	9.86497e13	0.340677
$427$	−1.32217e14	−0.450748
$428$	−1.95340e13	−0.0657432
$429$	2.34302e14	0.778503
$430$	−7.68838e13	−0.252208
$431$	1.11415e14	0.360842	0.180421	−	0.983589i	$-0.442254\pi$
$431$	1.11415e14	0.360842	0.180421	+	0.983589i	$0.442254\pi$
$432$	−4.17572e13	−0.133528
$433$	−3.42045e14	−1.07994	−0.539970	−	0.841684i	$-0.681564\pi$
$433$	−3.42045e14	−1.07994	−0.539970	+	0.841684i	$0.681564\pi$
$434$	1.18992e14	0.370957
$435$	−1.44681e14	−0.445371
$436$	5.72320e13	0.173965
$437$	2.13803e14	0.641750
$438$	−1.15303e14	−0.341769
$439$	1.20947e14	0.354029	0.177015	−	0.984208i	$-0.443356\pi$
$439$	1.20947e14	0.354029	0.177015	+	0.984208i	$0.443356\pi$
$440$	1.43655e14	0.415271
$441$	2.60599e14	0.743980
$442$	−4.64195e13	−0.130881
$443$	−1.27316e14	−0.354539	−0.177270	−	0.984162i	$-0.556726\pi$
$443$	−1.27316e14	−0.354539	−0.177270	+	0.984162i	$0.556726\pi$
$444$	3.58985e13	0.0987347
$445$	−1.38773e13	−0.0376985
$446$	1.47189e14	0.394942
$447$	1.62504e14	0.430699
$448$	−2.88443e14	−0.755147
$449$	−1.61837e14	−0.418526	−0.209263	−	0.977859i	$-0.567106\pi$
$449$	−1.61837e14	−0.418526	−0.209263	+	0.977859i	$0.567106\pi$
$450$	8.48777e12	0.0216833
$451$	1.82199e14	0.459807
$452$	7.25848e13	0.180961
$453$	−3.35931e14	−0.827390
$454$	−9.96664e13	−0.242517
$455$	2.99198e14	0.719276
$456$	1.61039e14	0.382494
$457$	−3.49165e14	−0.819391	−0.409696	−	0.912222i	$-0.634365\pi$
$457$	−3.49165e14	−0.819391	−0.409696	+	0.912222i	$0.634365\pi$
$458$	1.50036e14	0.347884
$459$	−3.77834e13	−0.0865627
$460$	1.05426e14	0.238661
$461$	−4.49619e14	−1.00575	−0.502875	−	0.864359i	$-0.667724\pi$
$461$	−4.49619e14	−1.00575	−0.502875	+	0.864359i	$0.667724\pi$
$462$	−2.30192e14	−0.508814
$463$	4.21098e13	0.0919787	0.0459894	−	0.998942i	$-0.485356\pi$
$463$	4.21098e13	0.0919787	0.0459894	+	0.998942i	$0.485356\pi$
$464$	−5.54459e14	−1.19680
$465$	7.67930e13	0.163806
$466$	9.92723e13	0.209269
$467$	−7.91358e13	−0.164866	−0.0824328	−	0.996597i	$-0.526269\pi$
$467$	−7.91358e13	−0.164866	−0.0824328	+	0.996597i	$0.526269\pi$
$468$	1.29515e14	0.266667
$469$	6.06442e14	1.23407
$470$	−3.94889e13	−0.0794213
$471$	1.92860e14	0.383377
$472$	−2.48821e14	−0.488885
$473$	1.34566e15	2.61336
$474$	8.69682e12	0.0166947
$475$	1.13341e14	0.215066
$476$	−3.85499e14	−0.723078
$477$	−2.08296e14	−0.386215
$478$	−2.59794e14	−0.476184
$479$	−9.62318e14	−1.74371	−0.871853	−	0.489768i	$-0.837081\pi$
$479$	−9.62318e14	−1.74371	−0.871853	+	0.489768i	$0.837081\pi$
$480$	1.21330e14	0.217341
$481$	−9.66132e13	−0.171095
$482$	−2.02626e14	−0.354762
$483$	−3.57853e14	−0.619435
$484$	−6.64451e14	−1.13714
$485$	6.38914e13	0.108109
$486$	−1.24713e13	−0.0208647
$487$	4.57117e14	0.756168	0.378084	−	0.925771i	$-0.376583\pi$
$487$	4.57117e14	0.756168	0.378084	+	0.925771i	$0.376583\pi$
$488$	9.44399e13	0.154471
$489$	1.13578e14	0.183694
$490$	−2.02998e14	−0.324649
$491$	4.40122e14	0.696025	0.348013	−	0.937490i	$-0.386857\pi$
$491$	4.40122e14	0.696025	0.348013	+	0.937490i	$0.386857\pi$
$492$	1.00715e14	0.157501
$493$	−5.01693e14	−0.775854
$494$	−2.04599e14	−0.312900
$495$	−1.48557e14	−0.224681
$496$	2.94292e14	0.440179
$497$	−2.20485e15	−3.26151
$498$	−4.31817e13	−0.0631740
$499$	6.36085e14	0.920369	0.460185	−	0.887823i	$-0.347783\pi$
$499$	6.36085e14	0.920369	0.460185	+	0.887823i	$0.347783\pi$
$500$	5.58883e13	0.0799808
$501$	−6.98581e14	−0.988802
$502$	4.44888e14	0.622846
$503$	−4.81639e12	−0.00666957	−0.00333478	−	0.999994i	$-0.501061\pi$
$503$	−4.81639e12	−0.00666957	−0.00333478	+	0.999994i	$0.501061\pi$
$504$	−2.69540e14	−0.369195
$505$	4.87334e14	0.660274
$506$	2.18294e14	0.292559
$507$	8.69320e13	0.115249
$508$	−4.82488e13	−0.0632755
$509$	−5.73829e14	−0.744449	−0.372224	−	0.928143i	$-0.621405\pi$
$509$	−5.73829e14	−0.744449	−0.372224	+	0.928143i	$0.621405\pi$
$510$	2.94320e13	0.0377731
$511$	2.57705e15	3.27196
$512$	8.05287e14	1.01150
$513$	−1.66535e14	−0.206947
$514$	1.01306e14	0.124548
$515$	1.59476e14	0.193980
$516$	7.43842e14	0.895174
$517$	6.91155e14	0.822958
$518$	9.49185e13	0.111825
$519$	−3.71997e14	−0.433630
$520$	−2.13711e14	−0.246495
$521$	5.14705e14	0.587422	0.293711	−	0.955894i	$-0.405110\pi$
$521$	5.14705e14	0.587422	0.293711	+	0.955894i	$0.405110\pi$
$522$	−1.65596e14	−0.187009
$523$	1.20146e15	1.34261	0.671304	−	0.741182i	$-0.265735\pi$
$523$	1.20146e15	1.34261	0.671304	+	0.741182i	$0.265735\pi$
$524$	4.02745e14	0.445357
$525$	−1.89704e14	−0.207587
$526$	−4.85654e14	−0.525903
$527$	2.66285e14	0.285357
$528$	−5.69313e14	−0.603760
$529$	−6.13454e14	−0.643836
$530$	1.62255e14	0.168532
$531$	2.57312e14	0.264509
$532$	−1.69913e15	−1.72868
$533$	−2.71052e14	−0.272931
$534$	−1.58833e13	−0.0158294
$535$	−3.33327e13	−0.0328795
$536$	−4.33170e14	−0.422914
$537$	4.64948e14	0.449310
$538$	−4.37861e14	−0.418826
$539$	3.55298e15	3.36399
$540$	−8.21183e13	−0.0769616
$541$	−1.89107e15	−1.75438	−0.877190	−	0.480143i	$-0.840585\pi$
$541$	−1.89107e15	−1.75438	−0.877190	+	0.480143i	$0.840585\pi$
$542$	−6.33524e14	−0.581791
$543$	4.15090e14	0.377349
$544$	4.20721e14	0.378617
$545$	9.76602e13	0.0870035
$546$	3.42449e14	0.302020
$547$	1.37400e14	0.119966	0.0599830	−	0.998199i	$-0.480895\pi$
$547$	1.37400e14	0.119966	0.0599830	+	0.998199i	$0.480895\pi$
$548$	1.01464e15	0.877040
$549$	−9.76626e13	−0.0835758
$550$	1.15721e14	0.0980434
$551$	−2.21127e15	−1.85485
$552$	2.55608e14	0.212280
$553$	−1.94377e14	−0.159829
$554$	2.15939e14	0.175803
$555$	6.12569e13	0.0493792
$556$	1.13093e15	0.902661
$557$	−1.18622e15	−0.937483	−0.468742	−	0.883335i	$-0.655292\pi$
$557$	−1.18622e15	−0.937483	−0.468742	+	0.883335i	$0.655292\pi$
$558$	8.78939e13	0.0687814
$559$	−2.00189e15	−1.55123
$560$	−7.26999e14	−0.557827
$561$	−5.15134e14	−0.391403
$562$	6.15815e14	0.463340
$563$	−2.58208e15	−1.92386	−0.961929	−	0.273298i	$-0.911885\pi$
$563$	−2.58208e15	−1.92386	−0.961929	+	0.273298i	$0.911885\pi$
$564$	3.82051e14	0.281894
$565$	1.23858e14	0.0905020
$566$	3.44783e14	0.249491
$567$	2.78738e14	0.199751
$568$	1.57488e15	1.11772
$569$	6.86455e13	0.0482497	0.0241249	−	0.999709i	$-0.492320\pi$
$569$	6.86455e13	0.0482497	0.0241249	+	0.999709i	$0.492320\pi$
$570$	1.29725e14	0.0903049
$571$	−1.48521e15	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$571$	−1.48521e15	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$572$	1.76579e15	1.20576
$573$	−7.67929e14	−0.519363
$574$	2.66297e14	0.178382
$575$	1.79899e14	0.119359
$576$	−2.13060e14	−0.140016
$577$	−1.11682e14	−0.0726966	−0.0363483	−	0.999339i	$-0.511573\pi$
$577$	−1.11682e14	−0.0726966	−0.0363483	+	0.999339i	$0.511573\pi$
$578$	−4.02394e14	−0.259447
$579$	−6.73830e14	−0.430346
$580$	−1.09038e15	−0.689800
$581$	9.65124e14	0.604803
$582$	7.31274e13	0.0453945
$583$	−2.83988e15	−1.74631
$584$	−1.84074e15	−1.12130
$585$	2.21004e14	0.133365
$586$	−1.07635e14	−0.0643450
$587$	1.03228e15	0.611347	0.305673	−	0.952136i	$-0.401118\pi$
$587$	1.03228e15	0.611347	0.305673	+	0.952136i	$0.401118\pi$
$588$	1.96398e15	1.15229
$589$	1.17368e15	0.682208
$590$	−2.00437e14	−0.115423
$591$	−9.39204e14	−0.535832
$592$	2.34753e14	0.132691
$593$	5.05946e14	0.283337	0.141668	−	0.989914i	$-0.454753\pi$
$593$	5.05946e14	0.283337	0.141668	+	0.989914i	$0.454753\pi$
$594$	−1.70032e14	−0.0943423
$595$	−6.57813e14	−0.361626
$596$	1.22470e15	0.667076
$597$	2.47469e14	0.133556
$598$	−3.24748e14	−0.173656
$599$	−1.86439e14	−0.0987846	−0.0493923	−	0.998779i	$-0.515728\pi$
$599$	−1.86439e14	−0.0987846	−0.0493923	+	0.998779i	$0.515728\pi$
$600$	1.35502e14	0.0711401
$601$	−6.39018e14	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$601$	−6.39018e14	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$602$	1.96678e15	1.01385
$603$	4.47952e14	0.228816
$604$	−2.53172e15	−1.28148
$605$	−1.13381e15	−0.568706
$606$	5.57782e14	0.277246
$607$	3.07440e15	1.51434	0.757169	−	0.653219i	$-0.226582\pi$
$607$	3.07440e15	1.51434	0.757169	+	0.653219i	$0.226582\pi$
$608$	1.85438e15	0.905166
$609$	3.70112e15	1.79035
$610$	7.60759e13	0.0364698
$611$	−1.02821e15	−0.488489
$612$	−2.84751e14	−0.134070
$613$	1.45823e15	0.680447	0.340223	−	0.940345i	$-0.389497\pi$
$613$	1.45823e15	0.680447	0.340223	+	0.940345i	$0.389497\pi$
$614$	−7.90767e14	−0.365698
$615$	1.71859e14	0.0787695
$616$	−3.67488e15	−1.66935
$617$	2.52038e15	1.13474	0.567372	−	0.823462i	$-0.307960\pi$
$617$	2.52038e15	1.13474	0.567372	+	0.823462i	$0.307960\pi$
$618$	1.82530e14	0.0814512
$619$	2.76117e15	1.22122	0.610612	−	0.791930i	$-0.290924\pi$
$619$	2.76117e15	1.22122	0.610612	+	0.791930i	$0.290924\pi$
$620$	5.78743e14	0.253707
$621$	−2.64330e14	−0.114853
$622$	7.96900e14	0.343207
$623$	3.54997e14	0.151545
$624$	8.46948e14	0.358378
$625$	9.53674e13	0.0400000
$626$	−6.32511e13	−0.0262972
$627$	−2.27051e15	−0.935734
$628$	1.45347e15	0.593783
$629$	2.12413e14	0.0860205
$630$	−2.17127e14	−0.0871649
$631$	−5.89126e14	−0.234448	−0.117224	−	0.993105i	$-0.537400\pi$
$631$	−5.89126e14	−0.234448	−0.117224	+	0.993105i	$0.537400\pi$
$632$	1.38839e14	0.0547733
$633$	−4.99774e14	−0.195458
$634$	−4.14972e14	−0.160890
$635$	−8.23314e13	−0.0316453
$636$	−1.56980e15	−0.598178
$637$	−5.28565e15	−1.99679
$638$	−2.25772e15	−0.845582
$639$	−1.62862e15	−0.604736
$640$	1.18854e15	0.437544
$641$	1.59787e15	0.583208	0.291604	−	0.956539i	$-0.405811\pi$
$641$	1.59787e15	0.583208	0.291604	+	0.956539i	$0.405811\pi$
$642$	−3.81512e13	−0.0138059
$643$	4.44445e15	1.59462	0.797310	−	0.603569i	$-0.206255\pi$
$643$	4.44445e15	1.59462	0.797310	+	0.603569i	$0.206255\pi$
$644$	−2.69693e15	−0.959395
$645$	1.26929e15	0.447694
$646$	4.49830e14	0.157315
$647$	−8.54247e14	−0.296217	−0.148108	−	0.988971i	$-0.547318\pi$
$647$	−8.54247e14	−0.296217	−0.148108	+	0.988971i	$0.547318\pi$
$648$	−1.99097e14	−0.0684546
$649$	3.50816e15	1.19601
$650$	−1.72155e14	−0.0581963
$651$	−1.96446e15	−0.658486
$652$	8.55971e14	0.284510
$653$	−2.86428e15	−0.944045	−0.472022	−	0.881587i	$-0.656476\pi$
$653$	−2.86428e15	−0.944045	−0.472022	+	0.881587i	$0.656476\pi$
$654$	1.11778e14	0.0365323
$655$	6.87242e14	0.222732
$656$	6.58609e14	0.211669
$657$	1.90355e15	0.606674
$658$	1.01017e15	0.319267
$659$	4.78135e15	1.49858	0.749291	−	0.662241i	$-0.230394\pi$
$659$	4.78135e15	1.49858	0.749291	+	0.662241i	$0.230394\pi$
$660$	−1.11959e15	−0.347991
$661$	−5.52275e14	−0.170234	−0.0851172	−	0.996371i	$-0.527126\pi$
$661$	−5.52275e14	−0.170234	−0.0851172	+	0.996371i	$0.527126\pi$
$662$	−1.46524e15	−0.447909
$663$	7.66347e14	0.232328
$664$	−6.89369e14	−0.207266
$665$	−2.89939e15	−0.864545
$666$	7.01120e13	0.0207341
$667$	−3.50982e15	−1.02942
$668$	−5.26480e15	−1.53148
$669$	−2.42997e15	−0.701062
$670$	−3.48939e14	−0.0998478
$671$	−1.33152e15	−0.377897
$672$	−3.10377e15	−0.873693
$673$	−3.31175e15	−0.924645	−0.462323	−	0.886712i	$-0.652984\pi$
$673$	−3.31175e15	−0.924645	−0.462323	+	0.886712i	$0.652984\pi$
$674$	−1.33234e15	−0.368964
$675$	−1.40126e14	−0.0384900
$676$	6.55156e14	0.178500
$677$	6.69181e15	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$677$	6.69181e15	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$678$	1.41763e14	0.0380013
$679$	−1.63442e15	−0.434590
$680$	4.69863e14	0.123929
$681$	1.64541e15	0.430492
$682$	1.19833e15	0.311003
$683$	−7.20810e15	−1.85570	−0.927848	−	0.372959i	$-0.878343\pi$
$683$	−7.20810e15	−1.85570	−0.927848	+	0.372959i	$0.878343\pi$
$684$	−1.25507e15	−0.320524
$685$	1.73138e15	0.438625
$686$	2.86629e15	0.720340
$687$	−2.47696e15	−0.617529
$688$	4.86426e15	1.20304
$689$	4.22479e15	1.03657
$690$	2.05904e14	0.0501182
$691$	7.79467e15	1.88221	0.941106	−	0.338112i	$-0.109788\pi$
$691$	7.79467e15	1.88221	0.941106	+	0.338112i	$0.109788\pi$
$692$	−2.80352e15	−0.671615
$693$	3.80028e15	0.903197
$694$	−9.37448e14	−0.221040
$695$	1.92981e15	0.451438
$696$	−2.64364e15	−0.613553
$697$	5.95932e14	0.137220
$698$	1.04299e15	0.238274
$699$	−1.63890e15	−0.371474
$700$	−1.42969e15	−0.321516
$701$	6.71290e15	1.49782	0.748912	−	0.662669i	$-0.230577\pi$
$701$	6.71290e15	1.49782	0.748912	+	0.662669i	$0.230577\pi$
$702$	2.52952e14	0.0559994
$703$	9.36234e14	0.205651
$704$	−2.90483e15	−0.633099
$705$	6.51929e14	0.140981
$706$	9.75719e14	0.209362
$707$	−1.24666e16	−2.65424
$708$	1.93921e15	0.409677
$709$	−4.10214e15	−0.859917	−0.429959	−	0.902849i	$-0.641472\pi$
$709$	−4.10214e15	−0.859917	−0.429959	+	0.902849i	$0.641472\pi$
$710$	1.26864e15	0.263887
$711$	−1.43577e14	−0.0296349
$712$	−2.53568e14	−0.0519343
$713$	1.86291e15	0.378617
$714$	−7.52905e14	−0.151845
$715$	3.01314e15	0.603026
$716$	3.50404e15	0.695902
$717$	4.28898e15	0.845276
$718$	1.00122e15	0.195814
$719$	−3.31818e15	−0.644008	−0.322004	−	0.946738i	$-0.604356\pi$
$719$	−3.31818e15	−0.644008	−0.322004	+	0.946738i	$0.604356\pi$
$720$	−5.37001e14	−0.103430
$721$	−4.07960e15	−0.779783
$722$	2.68047e14	0.0508460
$723$	3.34519e15	0.629738
$724$	3.12829e15	0.584446
$725$	−1.86061e15	−0.344983
$726$	−1.29772e15	−0.238797
$727$	3.34273e15	0.610467	0.305233	−	0.952278i	$-0.401266\pi$
$727$	3.34273e15	0.610467	0.305233	+	0.952278i	$0.401266\pi$
$728$	5.46699e15	0.990890
$729$	2.05891e14	0.0370370
$730$	−1.48280e15	−0.264733
$731$	4.40134e15	0.779901
$732$	−7.36026e14	−0.129444
$733$	−6.14966e15	−1.07344	−0.536722	−	0.843759i	$-0.680338\pi$
$733$	−6.14966e15	−1.07344	−0.536722	+	0.843759i	$0.680338\pi$
$734$	3.23483e15	0.560432
$735$	3.35133e15	0.576284
$736$	2.94334e15	0.502357
$737$	6.10732e15	1.03462
$738$	1.96702e14	0.0330749
$739$	−4.15701e15	−0.693804	−0.346902	−	0.937901i	$-0.612766\pi$
$739$	−4.15701e15	−0.693804	−0.346902	+	0.937901i	$0.612766\pi$
$740$	4.61657e14	0.0764795
$741$	3.37776e15	0.555430
$742$	−4.15068e15	−0.677482
$743$	−4.90364e15	−0.794474	−0.397237	−	0.917716i	$-0.630031\pi$
$743$	−4.90364e15	−0.794474	−0.397237	+	0.917716i	$0.630031\pi$
$744$	1.40317e15	0.225663
$745$	2.08982e15	0.333618
$746$	−2.46150e15	−0.390064
$747$	7.12893e14	0.112140
$748$	−3.88226e15	−0.606214
$749$	8.52690e14	0.132173
$750$	1.09153e14	0.0167958
$751$	−2.00580e15	−0.306385	−0.153192	−	0.988196i	$-0.548955\pi$
$751$	−2.00580e15	−0.306385	−0.153192	+	0.988196i	$0.548955\pi$
$752$	2.49837e15	0.378842
$753$	−7.34473e15	−1.10561
$754$	3.35873e15	0.501918
$755$	−4.32010e15	−0.640894
$756$	2.10068e15	0.309379
$757$	2.85016e15	0.416718	0.208359	−	0.978052i	$-0.433188\pi$
$757$	2.85016e15	0.416718	0.208359	+	0.978052i	$0.433188\pi$
$758$	−2.32235e15	−0.337091
$759$	−3.60384e15	−0.519321
$760$	2.07098e15	0.296279
$761$	8.65962e15	1.22994	0.614969	−	0.788551i	$-0.289168\pi$
$761$	8.65962e15	1.22994	0.614969	+	0.788551i	$0.289168\pi$
$762$	−9.42329e13	−0.0132877
$763$	−2.49827e15	−0.349747
$764$	−5.78743e15	−0.804400
$765$	−4.85897e14	−0.0670512
$766$	8.08453e14	0.110763
$767$	−5.21897e15	−0.709922
$768$	−4.35320e14	−0.0587925
$769$	−7.49727e15	−1.00533	−0.502665	−	0.864481i	$-0.667647\pi$
$769$	−7.49727e15	−1.00533	−0.502665	+	0.864481i	$0.667647\pi$
$770$	−2.96029e15	−0.394126
$771$	−1.67248e15	−0.221086
$772$	−5.07826e15	−0.666530
$773$	2.07867e15	0.270894	0.135447	−	0.990785i	$-0.456753\pi$
$773$	2.07867e15	0.270894	0.135447	+	0.990785i	$0.456753\pi$
$774$	1.45277e15	0.187984
$775$	9.87564e14	0.126884
$776$	1.16743e15	0.148934
$777$	−1.56703e15	−0.198500
$778$	−3.42866e15	−0.431258
$779$	2.62664e15	0.328053
$780$	1.66558e15	0.206559
$781$	−2.22045e16	−2.73438
$782$	7.13988e14	0.0873079
$783$	2.73385e15	0.331960
$784$	1.28432e16	1.54859
$785$	2.48019e15	0.296963
$786$	7.86587e14	0.0935239
$787$	−1.58421e16	−1.87047	−0.935235	−	0.354026i	$-0.884812\pi$
$787$	−1.58421e16	−1.87047	−0.935235	+	0.354026i	$0.884812\pi$
$788$	−7.07823e15	−0.829909
$789$	8.01774e15	0.933531
$790$	1.11842e14	0.0129317
$791$	−3.16844e15	−0.363810
$792$	−2.71447e15	−0.309525
$793$	1.98086e15	0.224311
$794$	−7.00826e14	−0.0788129
$795$	−2.67870e15	−0.299161
$796$	1.86503e15	0.206854
$797$	−4.58469e15	−0.504997	−0.252499	−	0.967597i	$-0.581252\pi$
$797$	−4.58469e15	−0.504997	−0.252499	+	0.967597i	$0.581252\pi$
$798$	−3.31852e15	−0.363018
$799$	2.26061e15	0.245594
$800$	1.56032e15	0.168352
$801$	2.62221e14	0.0280988
$802$	−4.40210e15	−0.468490
$803$	2.59528e16	2.74315
$804$	3.37595e15	0.354395
$805$	−4.60202e15	−0.479812
$806$	−1.78272e15	−0.184604
$807$	7.22872e15	0.743459
$808$	8.90465e15	0.909608
$809$	−1.72813e15	−0.175331	−0.0876656	−	0.996150i	$-0.527941\pi$
$809$	−1.72813e15	−0.175331	−0.0876656	+	0.996150i	$0.527941\pi$
$810$	−1.60382e14	−0.0161618
$811$	4.24638e14	0.0425015	0.0212507	−	0.999774i	$-0.493235\pi$
$811$	4.24638e14	0.0425015	0.0212507	+	0.999774i	$0.493235\pi$
$812$	2.78932e16	2.77294
$813$	1.04590e16	1.03274
$814$	9.55899e14	0.0937514
$815$	1.46062e15	0.142289
$816$	−1.86209e15	−0.180179
$817$	1.93994e16	1.86452
$818$	−3.76631e15	−0.359561
$819$	−5.65355e15	−0.536117
$820$	1.29520e15	0.122000
$821$	−1.96517e16	−1.83871	−0.919354	−	0.393431i	$-0.871288\pi$
$821$	−1.96517e16	−1.83871	−0.919354	+	0.393431i	$0.871288\pi$
$822$	1.98166e15	0.184176
$823$	5.95701e14	0.0549957	0.0274979	−	0.999622i	$-0.491246\pi$
$823$	5.95701e14	0.0549957	0.0274979	+	0.999622i	$0.491246\pi$
$824$	2.91398e15	0.267231
$825$	−1.91046e15	−0.174037
$826$	5.12743e15	0.463991
$827$	−1.40975e16	−1.26725	−0.633626	−	0.773639i	$-0.718434\pi$
$827$	−1.40975e16	−1.26725	−0.633626	+	0.773639i	$0.718434\pi$
$828$	−1.99210e15	−0.177887
$829$	−7.87187e15	−0.698277	−0.349139	−	0.937071i	$-0.613526\pi$
$829$	−7.87187e15	−0.698277	−0.349139	+	0.937071i	$0.613526\pi$
$830$	−5.55320e14	−0.0489343
$831$	−3.56497e15	−0.312068
$832$	4.32142e15	0.375793
$833$	1.16210e16	1.00391
$834$	2.20878e15	0.189557
$835$	−8.98381e15	−0.765923
$836$	−1.71115e16	−1.44929
$837$	−1.45106e15	−0.122094
$838$	3.26507e15	0.272929
$839$	6.12117e15	0.508327	0.254164	−	0.967161i	$-0.418200\pi$
$839$	6.12117e15	0.508327	0.254164	+	0.967161i	$0.418200\pi$
$840$	−3.46631e15	−0.285977
$841$	2.41000e16	1.97533
$842$	1.44092e15	0.117334
$843$	−1.01666e16	−0.822476
$844$	−3.76650e15	−0.302729
$845$	1.11795e15	0.0892713
$846$	7.46170e14	0.0591971
$847$	2.90043e16	2.28615
$848$	−1.02655e16	−0.803902
$849$	−5.69208e15	−0.442873
$850$	3.78497e14	0.0292590
$851$	1.48603e15	0.114134
$852$	−1.22740e16	−0.936629
$853$	2.83032e15	0.214594	0.107297	−	0.994227i	$-0.465781\pi$
$853$	2.83032e15	0.214594	0.107297	+	0.994227i	$0.465781\pi$
$854$	−1.94611e15	−0.146605
$855$	−2.14165e15	−0.160300
$856$	−6.09060e14	−0.0452954
$857$	−4.83193e14	−0.0357048	−0.0178524	−	0.999841i	$-0.505683\pi$
$857$	−4.83193e14	−0.0357048	−0.0178524	+	0.999841i	$0.505683\pi$
$858$	3.44871e15	0.253208
$859$	1.23886e16	0.903775	0.451888	−	0.892075i	$-0.350751\pi$
$859$	1.23886e16	0.903775	0.451888	+	0.892075i	$0.350751\pi$
$860$	9.56587e15	0.693399
$861$	−4.39635e15	−0.316647
$862$	1.63992e15	0.117364
$863$	−2.26458e16	−1.61038	−0.805191	−	0.593016i	$-0.797937\pi$
$863$	−2.26458e16	−1.61038	−0.805191	+	0.593016i	$0.797937\pi$
$864$	−2.29262e15	−0.161997
$865$	−4.78391e15	−0.335888
$866$	−5.03459e15	−0.351249
$867$	6.64319e15	0.460544
$868$	−1.48050e16	−1.01988
$869$	−1.95751e15	−0.133997
$870$	−2.12958e15	−0.144857
$871$	−9.08565e15	−0.614124
$872$	1.78446e15	0.119858
$873$	−1.20727e15	−0.0805799
$874$	3.14698e15	0.208728
$875$	−2.43961e15	−0.160797
$876$	1.43460e16	0.939630
$877$	−8.89622e15	−0.579039	−0.289519	−	0.957172i	$-0.593495\pi$
$877$	−8.89622e15	−0.579039	−0.289519	+	0.957172i	$0.593495\pi$
$878$	1.78023e15	0.115148
$879$	1.77696e15	0.114219
$880$	−7.32141e15	−0.467671
$881$	−1.82720e16	−1.15989	−0.579946	−	0.814655i	$-0.696927\pi$
$881$	−1.82720e16	−1.15989	−0.579946	+	0.814655i	$0.696927\pi$
$882$	3.83579e15	0.241979
$883$	8.75758e15	0.549035	0.274518	−	0.961582i	$-0.411482\pi$
$883$	8.75758e15	0.549035	0.274518	+	0.961582i	$0.411482\pi$
$884$	5.77551e15	0.359834
$885$	3.30905e15	0.204888
$886$	−1.87398e15	−0.115314
$887$	8.52159e15	0.521124	0.260562	−	0.965457i	$-0.416092\pi$
$887$	8.52159e15	0.521124	0.260562	+	0.965457i	$0.416092\pi$
$888$	1.11930e15	0.0680258
$889$	2.10613e15	0.127212
$890$	−2.04261e14	−0.0122614
$891$	2.80709e15	0.167467
$892$	−1.83132e16	−1.08582
$893$	9.96390e15	0.587146
$894$	2.39192e15	0.140084
$895$	5.97928e15	0.348034
$896$	−3.04042e16	−1.75889
$897$	5.36132e15	0.308257
$898$	−2.38209e15	−0.136125
$899$	−1.92673e16	−1.09432
$900$	−1.05605e15	−0.0596142
$901$	−9.28858e15	−0.521150
$902$	2.68181e15	0.149552
$903$	−3.24699e16	−1.79969
$904$	2.26316e15	0.124677
$905$	5.33809e15	0.292293
$906$	−4.94461e15	−0.269108
$907$	2.10550e16	1.13898	0.569490	−	0.821998i	$-0.307141\pi$
$907$	2.10550e16	1.13898	0.569490	+	0.821998i	$0.307141\pi$
$908$	1.24005e16	0.666756
$909$	−9.20851e15	−0.492139
$910$	4.40392e15	0.233944
$911$	3.04479e16	1.60771	0.803854	−	0.594827i	$-0.202780\pi$
$911$	3.04479e16	1.60771	0.803854	+	0.594827i	$0.202780\pi$
$912$	−8.20739e15	−0.430758
$913$	9.71950e15	0.507055
$914$	−5.13939e15	−0.266506
$915$	−1.25595e15	−0.0647375
$916$	−1.86674e16	−0.956443
$917$	−1.75805e16	−0.895363
$918$	−5.56137e14	−0.0281544
$919$	−4.47695e15	−0.225293	−0.112646	−	0.993635i	$-0.535933\pi$
$919$	−4.47695e15	−0.225293	−0.112646	+	0.993635i	$0.535933\pi$
$920$	3.28714e15	0.164431
$921$	1.30549e16	0.649151
$922$	−6.61799e15	−0.327119
$923$	3.30328e16	1.62307
$924$	2.86405e16	1.39889
$925$	7.87769e14	0.0382489
$926$	6.19818e14	0.0299160
$927$	−3.01342e15	−0.144584
$928$	−3.04417e16	−1.45196
$929$	2.63310e16	1.24848	0.624238	−	0.781234i	$-0.285409\pi$
$929$	2.63310e16	1.24848	0.624238	+	0.781234i	$0.285409\pi$
$930$	1.13032e15	0.0532778
$931$	5.12208e16	2.40007
$932$	−1.23515e16	−0.575348
$933$	−1.31562e16	−0.609228
$934$	−1.16481e15	−0.0536224
$935$	−6.62466e15	−0.303179
$936$	4.03822e15	0.183727
$937$	1.18259e16	0.534894	0.267447	−	0.963573i	$-0.413820\pi$
$937$	1.18259e16	0.534894	0.267447	+	0.963573i	$0.413820\pi$
$938$	8.92628e15	0.401379
$939$	1.04422e15	0.0466802
$940$	4.91320e15	0.218354
$941$	6.96003e15	0.307517	0.153758	−	0.988108i	$-0.450862\pi$
$941$	6.96003e15	0.307517	0.153758	+	0.988108i	$0.450862\pi$
$942$	2.83872e15	0.124693
$943$	4.16911e15	0.182066
$944$	1.26812e16	0.550573
$945$	3.58459e15	0.154727
$946$	1.98069e16	0.849993
$947$	−3.38501e16	−1.44423	−0.722113	−	0.691775i	$-0.756829\pi$
$947$	−3.38501e16	−1.44423	−0.722113	+	0.691775i	$0.756829\pi$
$948$	−1.08206e15	−0.0458991
$949$	−3.86091e16	−1.62827
$950$	1.66827e15	0.0699499
$951$	6.85084e15	0.285595
$952$	−1.20197e16	−0.498183
$953$	4.50062e16	1.85465	0.927323	−	0.374262i	$-0.122104\pi$
$953$	4.50062e16	1.85465	0.927323	+	0.374262i	$0.122104\pi$
$954$	−3.06592e15	−0.125616
$955$	−9.87563e15	−0.402297
$956$	3.23235e16	1.30918
$957$	3.72730e16	1.50099
$958$	−1.41644e16	−0.567139
$959$	−4.42907e16	−1.76323
$960$	−2.73997e15	−0.108456
$961$	−1.51819e16	−0.597513
$962$	−1.42206e15	−0.0556486
$963$	6.29844e14	0.0245069
$964$	2.52107e16	0.975353
$965$	−8.66551e15	−0.333345
$966$	−5.26728e15	−0.201471
$967$	−1.48359e16	−0.564246	−0.282123	−	0.959378i	$-0.591039\pi$
$967$	−1.48359e16	−0.564246	−0.282123	+	0.959378i	$0.591039\pi$
$968$	−2.07172e16	−0.783461
$969$	−7.42632e15	−0.279250
$970$	9.40424e14	0.0351624
$971$	1.14414e16	0.425378	0.212689	−	0.977120i	$-0.431778\pi$
$971$	1.14414e16	0.425378	0.212689	+	0.977120i	$0.431778\pi$
$972$	1.55168e15	0.0573638
$973$	−4.93669e16	−1.81474
$974$	6.72835e15	0.245943
$975$	2.84213e15	0.103304
$976$	−4.81314e15	−0.173962
$977$	−2.64437e16	−0.950390	−0.475195	−	0.879881i	$-0.657622\pi$
$977$	−2.64437e16	−0.950390	−0.475195	+	0.879881i	$0.657622\pi$
$978$	1.67177e15	0.0597464
$979$	3.57508e15	0.127052
$980$	2.52570e16	0.892562
$981$	−1.84536e15	−0.0648486
$982$	6.47820e15	0.226382
$983$	3.63656e16	1.26371	0.631854	−	0.775088i	$-0.282294\pi$
$983$	3.63656e16	1.26371	0.631854	+	0.775088i	$0.282294\pi$
$984$	3.14023e15	0.108515
$985$	−1.20782e16	−0.415054
$986$	−7.38447e15	−0.252346
$987$	−1.66771e16	−0.566731
$988$	2.54562e16	0.860262
$989$	3.07915e16	1.03479
$990$	−2.18663e15	−0.0730772
$991$	5.09067e16	1.69188	0.845941	−	0.533276i	$-0.179039\pi$
$991$	5.09067e16	1.69188	0.845941	+	0.533276i	$0.179039\pi$
$992$	1.61576e16	0.534027
$993$	2.41899e16	0.795084
$994$	−3.24534e16	−1.06080
$995$	3.18247e15	0.103452
$996$	5.37266e15	0.173685
$997$	−4.28291e16	−1.37694	−0.688471	−	0.725264i	$-0.741718\pi$
$997$	−4.28291e16	−1.37694	−0.688471	+	0.725264i	$0.741718\pi$
$998$	9.36259e15	0.299349
$999$	−1.15749e15	−0.0368051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.a.c.1.2	✓	2
3.2	odd	2		45.12.a.c.1.1		2
4.3	odd	2		240.12.a.m.1.1		2
5.2	odd	4		75.12.b.d.49.3		4
5.3	odd	4		75.12.b.d.49.2		4
5.4	even	2		75.12.a.c.1.1		2
15.2	even	4		225.12.b.i.199.2		4
15.8	even	4		225.12.b.i.199.3		4
15.14	odd	2		225.12.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.2	✓	2	1.1	even	1	trivial
45.12.a.c.1.1		2	3.2	odd	2
75.12.a.c.1.1		2	5.4	even	2
75.12.b.d.49.2		4	5.3	odd	4
75.12.b.d.49.3		4	5.2	odd	4
225.12.a.i.1.2		2	15.14	odd	2
225.12.b.i.199.2		4	15.2	even	4
225.12.b.i.199.3		4	15.8	even	4
240.12.a.m.1.1		2	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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)

\(f(q)\)	\(=\)	\(q+14.7191 q^{2} -243.000 q^{3} -1831.35 q^{4} -3125.00 q^{5} -3576.74 q^{6} +79941.2 q^{7} -57100.5 q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q+14.7191 q^{2} -243.000 q^{3} -1831.35 q^{4} -3125.00 q^{5} -3576.74 q^{6} +79941.2 q^{7} -57100.5 q^{8} +59049.0 q^{9} -45997.2 q^{10} +805067. q^{11} +445018. q^{12} -1.19767e6 q^{13} +1.17666e6 q^{14} +759375. q^{15} +2.91013e6 q^{16} +2.63319e6 q^{17} +869148. q^{18} +1.16061e7 q^{19} +5.72296e6 q^{20} -1.94257e7 q^{21} +1.18499e7 q^{22} +1.84216e7 q^{23} +1.38754e7 q^{24} +9.76562e6 q^{25} -1.76286e7 q^{26} -1.43489e7 q^{27} -1.46400e8 q^{28} -1.90527e8 q^{29} +1.11773e7 q^{30} +1.01127e8 q^{31} +1.59776e8 q^{32} -1.95631e8 q^{33} +3.87581e7 q^{34} -2.49816e8 q^{35} -1.08139e8 q^{36} +8.06675e7 q^{37} +1.70831e8 q^{38} +2.91034e8 q^{39} +1.78439e8 q^{40} +2.26316e8 q^{41} -2.85929e8 q^{42} +1.67149e9 q^{43} -1.47436e9 q^{44} -1.84528e8 q^{45} +2.71150e8 q^{46} +8.58507e8 q^{47} -7.07162e8 q^{48} +4.41327e9 q^{49} +1.43741e8 q^{50} -6.39864e8 q^{51} +2.19335e9 q^{52} -3.52750e9 q^{53} -2.11203e8 q^{54} -2.51583e9 q^{55} -4.56468e9 q^{56} -2.82028e9 q^{57} -2.80438e9 q^{58} +4.35760e9 q^{59} -1.39068e9 q^{60} -1.65393e9 q^{61} +1.48849e9 q^{62} +4.72045e9 q^{63} -3.60819e9 q^{64} +3.74272e9 q^{65} -2.87951e9 q^{66} +7.58610e9 q^{67} -4.82228e9 q^{68} -4.47645e9 q^{69} -3.67707e9 q^{70} -2.75809e10 q^{71} -3.37173e9 q^{72} +3.22368e10 q^{73} +1.18735e9 q^{74} -2.37305e9 q^{75} -2.12548e10 q^{76} +6.43580e10 q^{77} +4.28376e9 q^{78} -2.43149e9 q^{79} -9.09417e9 q^{80} +3.48678e9 q^{81} +3.33116e9 q^{82} +1.20729e10 q^{83} +3.55753e10 q^{84} -8.22871e9 q^{85} +2.46028e10 q^{86} +4.62981e10 q^{87} -4.59697e10 q^{88} +4.44073e9 q^{89} -2.71609e9 q^{90} -9.57433e10 q^{91} -3.37364e10 q^{92} -2.45737e10 q^{93} +1.26364e10 q^{94} -3.62690e10 q^{95} -3.88257e10 q^{96} -2.04453e10 q^{97} +6.49594e10 q^{98} +4.75384e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9}+O(q^{10}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9} + 40625 q^{10} + 295568 q^{11} + 755973 q^{12} + 657492 q^{13} + 3176796 q^{14} + 1518750 q^{15} + 2974065 q^{16} + 8579948 q^{17} - 767637 q^{18} + 17627976 q^{19} + 9721875 q^{20} - 1891512 q^{21} + 25972696 q^{22} - 29841072 q^{23} - 8538777 q^{24} + 19531250 q^{25} - 69052066 q^{26} - 28697814 q^{27} - 54064108 q^{28} - 201881948 q^{29} - 9871875 q^{30} - 71057008 q^{31} - 30902261 q^{32} - 71823024 q^{33} - 126080702 q^{34} - 24325000 q^{35} - 183701439 q^{36} + 705858484 q^{37} + 3909548 q^{38} - 159770556 q^{39} - 109809375 q^{40} - 327655148 q^{41} - 771961428 q^{42} + 3192552120 q^{43} - 822376568 q^{44} - 369056250 q^{45} + 1608947832 q^{46} + 2053064720 q^{47} - 722697795 q^{48} + 7642614770 q^{49} - 126953125 q^{50} - 2084927364 q^{51} - 180610022 q^{52} - 2304299452 q^{53} + 186535791 q^{54} - 923650000 q^{55} - 11220431460 q^{56} - 4283598168 q^{57} - 2489635114 q^{58} - 1478770576 q^{59} - 2362415625 q^{60} - 8264891460 q^{61} + 6261263448 q^{62} + 459637416 q^{63} + 1546316681 q^{64} - 2054662500 q^{65} - 6311365128 q^{66} + 24212177528 q^{67} - 12432065594 q^{68} + 7251380496 q^{69} - 9927487500 q^{70} - 20218888256 q^{71} + 2074922811 q^{72} + 25879583268 q^{73} - 16142374034 q^{74} - 4746093750 q^{75} - 28960704764 q^{76} + 101122045248 q^{77} + 16779652038 q^{78} + 22324995440 q^{79} - 9293953125 q^{80} + 6973568802 q^{81} + 18686740478 q^{82} + 48014508984 q^{83} + 13137578244 q^{84} - 26812337500 q^{85} - 17559698092 q^{86} + 49057313364 q^{87} - 92965609896 q^{88} + 79209683076 q^{89} + 2398865625 q^{90} - 229606712048 q^{91} + 28023036264 q^{92} + 17266852944 q^{93} - 20475612776 q^{94} - 55087425000 q^{95} + 7509249423 q^{96} - 37075227452 q^{97} - 24554976677 q^{98} + 17452994832 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9 + 40625 * q^10 + 295568 * q^11 + 755973 * q^12 + 657492 * q^13 + 3176796 * q^14 + 1518750 * q^15 + 2974065 * q^16 + 8579948 * q^17 - 767637 * q^18 + 17627976 * q^19 + 9721875 * q^20 - 1891512 * q^21 + 25972696 * q^22 - 29841072 * q^23 - 8538777 * q^24 + 19531250 * q^25 - 69052066 * q^26 - 28697814 * q^27 - 54064108 * q^28 - 201881948 * q^29 - 9871875 * q^30 - 71057008 * q^31 - 30902261 * q^32 - 71823024 * q^33 - 126080702 * q^34 - 24325000 * q^35 - 183701439 * q^36 + 705858484 * q^37 + 3909548 * q^38 - 159770556 * q^39 - 109809375 * q^40 - 327655148 * q^41 - 771961428 * q^42 + 3192552120 * q^43 - 822376568 * q^44 - 369056250 * q^45 + 1608947832 * q^46 + 2053064720 * q^47 - 722697795 * q^48 + 7642614770 * q^49 - 126953125 * q^50 - 2084927364 * q^51 - 180610022 * q^52 - 2304299452 * q^53 + 186535791 * q^54 - 923650000 * q^55 - 11220431460 * q^56 - 4283598168 * q^57 - 2489635114 * q^58 - 1478770576 * q^59 - 2362415625 * q^60 - 8264891460 * q^61 + 6261263448 * q^62 + 459637416 * q^63 + 1546316681 * q^64 - 2054662500 * q^65 - 6311365128 * q^66 + 24212177528 * q^67 - 12432065594 * q^68 + 7251380496 * q^69 - 9927487500 * q^70 - 20218888256 * q^71 + 2074922811 * q^72 + 25879583268 * q^73 - 16142374034 * q^74 - 4746093750 * q^75 - 28960704764 * q^76 + 101122045248 * q^77 + 16779652038 * q^78 + 22324995440 * q^79 - 9293953125 * q^80 + 6973568802 * q^81 + 18686740478 * q^82 + 48014508984 * q^83 + 13137578244 * q^84 - 26812337500 * q^85 - 17559698092 * q^86 + 49057313364 * q^87 - 92965609896 * q^88 + 79209683076 * q^89 + 2398865625 * q^90 - 229606712048 * q^91 + 28023036264 * q^92 + 17266852944 * q^93 - 20475612776 * q^94 - 55087425000 * q^95 + 7509249423 * q^96 - 37075227452 * q^97 - 24554976677 * q^98 + 17452994832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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