LMFDB - Embedded newform 625.2.b.a.624.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(624,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("625.624");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			624.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	625.624
Dual form			625.2.b.a.624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -0.618034i q^{7} +2.23607i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -0.618034i q^{7} +2.23607i q^{8} +2.00000 q^{9} -5.23607 q^{11} -0.618034i q^{12} +1.85410i q^{13} +1.00000 q^{14} -4.85410 q^{16} +5.23607i q^{17} +3.23607i q^{18} -0.854102 q^{19} +0.618034 q^{21} -8.47214i q^{22} +3.76393i q^{23} -2.23607 q^{24} -3.00000 q^{26} +5.00000i q^{27} +0.381966i q^{28} +3.61803 q^{29} -3.00000 q^{31} -3.38197i q^{32} -5.23607i q^{33} -8.47214 q^{34} -1.23607 q^{36} +0.236068i q^{37} -1.38197i q^{38} -1.85410 q^{39} -0.763932 q^{41} +1.00000i q^{42} -4.85410i q^{43} +3.23607 q^{44} -6.09017 q^{46} -0.618034i q^{47} -4.85410i q^{48} +6.61803 q^{49} -5.23607 q^{51} -1.14590i q^{52} -3.47214i q^{53} -8.09017 q^{54} +1.38197 q^{56} -0.854102i q^{57} +5.85410i q^{58} +10.8541 q^{59} +8.70820 q^{61} -4.85410i q^{62} -1.23607i q^{63} -4.23607 q^{64} +8.47214 q^{66} -4.76393i q^{67} -3.23607i q^{68} -3.76393 q^{69} -6.61803 q^{71} +4.47214i q^{72} -9.00000i q^{73} -0.381966 q^{74} +0.527864 q^{76} +3.23607i q^{77} -3.00000i q^{78} +8.09017 q^{79} +1.00000 q^{81} -1.23607i q^{82} -6.23607i q^{83} -0.381966 q^{84} +7.85410 q^{86} +3.61803i q^{87} -11.7082i q^{88} +8.94427 q^{89} +1.14590 q^{91} -2.32624i q^{92} -3.00000i q^{93} +1.00000 q^{94} +3.38197 q^{96} +3.85410i q^{97} +10.7082i q^{98} -10.4721 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^6 + 8 * q^9	$ 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 + 8 * q^9 - 12 * q^11 + 4 * q^14 - 6 * q^16 + 10 * q^19 - 2 * q^21 - 12 * q^26 + 10 * q^29 - 12 * q^31 - 16 * q^34 + 4 * q^36 + 6 * q^39 - 12 * q^41 + 4 * q^44 - 2 * q^46 + 22 * q^49 - 12 * q^51 - 10 * q^54 + 10 * q^56 + 30 * q^59 + 8 * q^61 - 8 * q^64 + 16 * q^66 - 24 * q^69 - 22 * q^71 - 6 * q^74 + 20 * q^76 + 10 * q^79 + 4 * q^81 - 6 * q^84 + 18 * q^86 + 18 * q^91 + 4 * q^94 + 18 * q^96 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	1.61803i	1.14412i	0.820211	+	0.572061i	$0.193856\pi$
$2$	1.61803i	1.14412i	−0.820211	+	0.572061i	$0.806144\pi$
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	− 0.618034i	− 0.233595i	−0.993156	−	0.116797i	$-0.962737\pi$
$7$	− 0.618034i	− 0.233595i	0.993156	−	0.116797i	$-0.0372628\pi$
$8$	2.23607i	0.790569i
$9$	2.00000	0.666667
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	− 0.618034i	− 0.178411i
$13$	1.85410i	0.514235i	0.966380	+	0.257118i	$0.0827728\pi$
$13$	1.85410i	0.514235i	−0.966380	+	0.257118i	$0.917227\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.23607i	1.26993i	0.772540	+	0.634967i	$0.218986\pi$
$17$	5.23607i	1.26993i	−0.772540	+	0.634967i	$0.781014\pi$
$18$	3.23607i	0.762749i
$19$	−0.854102	−0.195944	−0.0979722	−	0.995189i	$-0.531236\pi$
$19$	−0.854102	−0.195944	−0.0979722	+	0.995189i	$0.531236\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	− 8.47214i	− 1.80627i
$23$	3.76393i	0.784834i	0.919787	+	0.392417i	$0.128361\pi$
$23$	3.76393i	0.784834i	−0.919787	+	0.392417i	$0.871639\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−3.00000	−0.588348
$27$	5.00000i	0.962250i
$28$	0.381966i	0.0721848i
$29$	3.61803	0.671852	0.335926	−	0.941888i	$-0.390951\pi$
$29$	3.61803	0.671852	0.335926	+	0.941888i	$0.390951\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 3.38197i	− 0.597853i
$33$	− 5.23607i	− 0.911482i
$34$	−8.47214	−1.45296
$35$	0	0
$36$	−1.23607	−0.206011
$37$	0.236068i	0.0388093i	0.999812	+	0.0194047i	$0.00617709\pi$
$37$	0.236068i	0.0388093i	−0.999812	+	0.0194047i	$0.993823\pi$
$38$	− 1.38197i	− 0.224184i
$39$	−1.85410	−0.296894
$40$	0	0
$41$	−0.763932	−0.119306	−0.0596531	−	0.998219i	$-0.518999\pi$
$41$	−0.763932	−0.119306	−0.0596531	+	0.998219i	$0.518999\pi$
$42$	1.00000i	0.154303i
$43$	− 4.85410i	− 0.740244i	−0.928983	−	0.370122i	$-0.879316\pi$
$43$	− 4.85410i	− 0.740244i	0.928983	−	0.370122i	$-0.120684\pi$
$44$	3.23607	0.487856
$45$	0	0
$46$	−6.09017	−0.897947
$47$	− 0.618034i	− 0.0901495i	−0.998984	−	0.0450748i	$-0.985647\pi$
$47$	− 0.618034i	− 0.0901495i	0.998984	−	0.0450748i	$-0.0143526\pi$
$48$	− 4.85410i	− 0.700629i
$49$	6.61803	0.945433
$50$	0	0
$51$	−5.23607	−0.733196
$52$	− 1.14590i	− 0.158907i
$53$	− 3.47214i	− 0.476935i	−0.971151	−	0.238467i	$-0.923355\pi$
$53$	− 3.47214i	− 0.476935i	0.971151	−	0.238467i	$-0.0766450\pi$
$54$	−8.09017	−1.10093
$55$	0	0
$56$	1.38197	0.184673
$57$	− 0.854102i	− 0.113129i
$58$	5.85410i	0.768681i
$59$	10.8541	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$59$	10.8541	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$60$	0	0
$61$	8.70820	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$61$	8.70820	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$62$	− 4.85410i	− 0.616472i
$63$	− 1.23607i	− 0.155730i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	8.47214	1.04285
$67$	− 4.76393i	− 0.582007i	−0.956722	−	0.291003i	$-0.906011\pi$
$67$	− 4.76393i	− 0.582007i	0.956722	−	0.291003i	$-0.0939891\pi$
$68$	− 3.23607i	− 0.392431i
$69$	−3.76393	−0.453124
$70$	0	0
$71$	−6.61803	−0.785416	−0.392708	−	0.919663i	$-0.628462\pi$
$71$	−6.61803	−0.785416	−0.392708	+	0.919663i	$0.628462\pi$
$72$	4.47214i	0.527046i
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	−0.381966	−0.0444026
$75$	0	0
$76$	0.527864	0.0605502
$77$	3.23607i	0.368784i
$78$	− 3.00000i	− 0.339683i
$79$	8.09017	0.910215	0.455108	−	0.890436i	$-0.349601\pi$
$79$	8.09017	0.910215	0.455108	+	0.890436i	$0.349601\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 1.23607i	− 0.136501i
$83$	− 6.23607i	− 0.684497i	−0.939609	−	0.342249i	$-0.888811\pi$
$83$	− 6.23607i	− 0.684497i	0.939609	−	0.342249i	$-0.111189\pi$
$84$	−0.381966	−0.0416759
$85$	0	0
$86$	7.85410	0.846930
$87$	3.61803i	0.387894i
$88$	− 11.7082i	− 1.24810i
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	0	0
$91$	1.14590	0.120123
$92$	− 2.32624i	− 0.242527i
$93$	− 3.00000i	− 0.311086i
$94$	1.00000	0.103142
$95$	0	0
$96$	3.38197	0.345170
$97$	3.85410i	0.391325i	0.980671	+	0.195662i	$0.0626857\pi$
$97$	3.85410i	0.391325i	−0.980671	+	0.195662i	$0.937314\pi$
$98$	10.7082i	1.08169i
$99$	−10.4721	−1.05249
$100$	0	0
$101$	1.47214	0.146483	0.0732415	−	0.997314i	$-0.476666\pi$
$101$	1.47214	0.146483	0.0732415	+	0.997314i	$0.476666\pi$
$102$	− 8.47214i	− 0.838866i
$103$	8.56231i	0.843669i	0.906673	+	0.421835i	$0.138614\pi$
$103$	8.56231i	0.843669i	−0.906673	+	0.421835i	$0.861386\pi$
$104$	−4.14590	−0.406539
$105$	0	0
$106$	5.61803	0.545672
$107$	16.4164i	1.58703i	0.608548	+	0.793517i	$0.291752\pi$
$107$	16.4164i	1.58703i	−0.608548	+	0.793517i	$0.708248\pi$
$108$	− 3.09017i	− 0.297352i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	3.00000i	0.283473i
$113$	16.8541i	1.58550i	0.609547	+	0.792750i	$0.291352\pi$
$113$	16.8541i	1.58550i	−0.609547	+	0.792750i	$0.708648\pi$
$114$	1.38197	0.129433
$115$	0	0
$116$	−2.23607	−0.207614
$117$	3.70820i	0.342824i
$118$	17.5623i	1.61674i
$119$	3.23607	0.296650
$120$	0	0
$121$	16.4164	1.49240
$122$	14.0902i	1.27566i
$123$	− 0.763932i	− 0.0688814i
$124$	1.85410	0.166503
$125$	0	0
$126$	2.00000	0.178174
$127$	− 19.8885i	− 1.76482i	−0.470479	−	0.882411i	$-0.655919\pi$
$127$	− 19.8885i	− 1.76482i	0.470479	−	0.882411i	$-0.344081\pi$
$128$	− 13.6180i	− 1.20368i
$129$	4.85410	0.427380
$130$	0	0
$131$	6.79837	0.593977	0.296988	−	0.954881i	$-0.404018\pi$
$131$	6.79837	0.593977	0.296988	+	0.954881i	$0.404018\pi$
$132$	3.23607i	0.281664i
$133$	0.527864i	0.0457716i
$134$	7.70820	0.665887
$135$	0	0
$136$	−11.7082	−1.00397
$137$	11.9443i	1.02047i	0.860036	+	0.510234i	$0.170441\pi$
$137$	11.9443i	1.02047i	−0.860036	+	0.510234i	$0.829559\pi$
$138$	− 6.09017i	− 0.518430i
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0.618034	0.0520479
$142$	− 10.7082i	− 0.898613i
$143$	− 9.70820i	− 0.811841i
$144$	−9.70820	−0.809017
$145$	0	0
$146$	14.5623	1.20519
$147$	6.61803i	0.545846i
$148$	− 0.145898i	− 0.0119927i
$149$	3.94427	0.323127	0.161564	−	0.986862i	$-0.448346\pi$
$149$	3.94427	0.323127	0.161564	+	0.986862i	$0.448346\pi$
$150$	0	0
$151$	14.5623	1.18506	0.592532	−	0.805547i	$-0.298128\pi$
$151$	14.5623	1.18506	0.592532	+	0.805547i	$0.298128\pi$
$152$	− 1.90983i	− 0.154908i
$153$	10.4721i	0.846622i
$154$	−5.23607	−0.421934
$155$	0	0
$156$	1.14590	0.0917453
$157$	− 13.1803i	− 1.05191i	−0.850514	−	0.525953i	$-0.823709\pi$
$157$	− 13.1803i	− 1.05191i	0.850514	−	0.525953i	$-0.176291\pi$
$158$	13.0902i	1.04140i
$159$	3.47214	0.275358
$160$	0	0
$161$	2.32624	0.183333
$162$	1.61803i	0.127125i
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	0.472136	0.0368676
$165$	0	0
$166$	10.0902	0.783149
$167$	− 14.5623i	− 1.12687i	−0.826162	−	0.563433i	$-0.809480\pi$
$167$	− 14.5623i	− 1.12687i	0.826162	−	0.563433i	$-0.190520\pi$
$168$	1.38197i	0.106621i
$169$	9.56231	0.735562
$170$	0	0
$171$	−1.70820	−0.130630
$172$	3.00000i	0.228748i
$173$	18.8885i	1.43607i	0.696007	+	0.718035i	$0.254958\pi$
$173$	18.8885i	1.43607i	−0.696007	+	0.718035i	$0.745042\pi$
$174$	−5.85410	−0.443798
$175$	0	0
$176$	25.4164	1.91583
$177$	10.8541i	0.815844i
$178$	14.4721i	1.08473i
$179$	0.527864	0.0394544	0.0197272	−	0.999805i	$-0.493720\pi$
$179$	0.527864	0.0394544	0.0197272	+	0.999805i	$0.493720\pi$
$180$	0	0
$181$	0.291796	0.0216890	0.0108445	−	0.999941i	$-0.496548\pi$
$181$	0.291796	0.0216890	0.0108445	+	0.999941i	$0.496548\pi$
$182$	1.85410i	0.137435i
$183$	8.70820i	0.643729i
$184$	−8.41641	−0.620466
$185$	0	0
$186$	4.85410	0.355920
$187$	− 27.4164i	− 2.00489i
$188$	0.381966i	0.0278577i
$189$	3.09017	0.224777
$190$	0	0
$191$	−1.81966	−0.131666	−0.0658330	−	0.997831i	$-0.520970\pi$
$191$	−1.81966	−0.131666	−0.0658330	+	0.997831i	$0.520970\pi$
$192$	− 4.23607i	− 0.305712i
$193$	7.70820i	0.554849i	0.960747	+	0.277424i	$0.0894808\pi$
$193$	7.70820i	0.554849i	−0.960747	+	0.277424i	$0.910519\pi$
$194$	−6.23607	−0.447724
$195$	0	0
$196$	−4.09017	−0.292155
$197$	− 3.70820i	− 0.264199i	−0.991236	−	0.132099i	$-0.957828\pi$
$197$	− 3.70820i	− 0.264199i	0.991236	−	0.132099i	$-0.0421718\pi$
$198$	− 16.9443i	− 1.20418i
$199$	17.5623	1.24496	0.622479	−	0.782636i	$-0.286125\pi$
$199$	17.5623	1.24496	0.622479	+	0.782636i	$0.286125\pi$
$200$	0	0
$201$	4.76393	0.336022
$202$	2.38197i	0.167595i
$203$	− 2.23607i	− 0.156941i
$204$	3.23607	0.226570
$205$	0	0
$206$	−13.8541	−0.965261
$207$	7.52786i	0.523223i
$208$	− 9.00000i	− 0.624038i
$209$	4.47214	0.309344
$210$	0	0
$211$	−9.18034	−0.632001	−0.316000	−	0.948759i	$-0.602340\pi$
$211$	−9.18034	−0.632001	−0.316000	+	0.948759i	$0.602340\pi$
$212$	2.14590i	0.147381i
$213$	− 6.61803i	− 0.453460i
$214$	−26.5623	−1.81576
$215$	0	0
$216$	−11.1803	−0.760726
$217$	1.85410i	0.125865i
$218$	− 16.1803i	− 1.09587i
$219$	9.00000	0.608164
$220$	0	0
$221$	−9.70820	−0.653044
$222$	− 0.381966i	− 0.0256359i
$223$	− 0.180340i	− 0.0120765i	−0.999982	−	0.00603823i	$-0.998078\pi$
$223$	− 0.180340i	− 0.0120765i	0.999982	−	0.00603823i	$-0.00192204\pi$
$224$	−2.09017	−0.139655
$225$	0	0
$226$	−27.2705	−1.81401
$227$	− 14.7639i	− 0.979917i	−0.871746	−	0.489958i	$-0.837012\pi$
$227$	− 14.7639i	− 0.979917i	0.871746	−	0.489958i	$-0.162988\pi$
$228$	0.527864i	0.0349587i
$229$	−21.7082	−1.43452	−0.717259	−	0.696806i	$-0.754604\pi$
$229$	−21.7082	−1.43452	−0.717259	+	0.696806i	$0.754604\pi$
$230$	0	0
$231$	−3.23607	−0.212918
$232$	8.09017i	0.531146i
$233$	− 2.94427i	− 0.192886i	−0.995339	−	0.0964428i	$-0.969254\pi$
$233$	− 2.94427i	− 0.192886i	0.995339	−	0.0964428i	$-0.0307465\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−6.70820	−0.436667
$237$	8.09017i	0.525513i
$238$	5.23607i	0.339404i
$239$	20.5279	1.32784	0.663919	−	0.747805i	$-0.268892\pi$
$239$	20.5279	1.32784	0.663919	+	0.747805i	$0.268892\pi$
$240$	0	0
$241$	2.52786	0.162834	0.0814170	−	0.996680i	$-0.474055\pi$
$241$	2.52786	0.162834	0.0814170	+	0.996680i	$0.474055\pi$
$242$	26.5623i	1.70749i
$243$	16.0000i	1.02640i
$244$	−5.38197	−0.344545
$245$	0	0
$246$	1.23607	0.0788088
$247$	− 1.58359i	− 0.100762i
$248$	− 6.70820i	− 0.425971i
$249$	6.23607	0.395195
$250$	0	0
$251$	−29.1803	−1.84185	−0.920923	−	0.389744i	$-0.872564\pi$
$251$	−29.1803	−1.84185	−0.920923	+	0.389744i	$0.872564\pi$
$252$	0.763932i	0.0481232i
$253$	− 19.7082i	− 1.23904i
$254$	32.1803	2.01917
$255$	0	0
$256$	13.5623	0.847644
$257$	− 22.8541i	− 1.42560i	−0.701367	−	0.712800i	$-0.747427\pi$
$257$	− 22.8541i	− 1.42560i	0.701367	−	0.712800i	$-0.252573\pi$
$258$	7.85410i	0.488975i
$259$	0.145898	0.00906566
$260$	0	0
$261$	7.23607	0.447901
$262$	11.0000i	0.679582i
$263$	− 10.9098i	− 0.672729i	−0.941732	−	0.336364i	$-0.890803\pi$
$263$	− 10.9098i	− 0.672729i	0.941732	−	0.336364i	$-0.109197\pi$
$264$	11.7082	0.720590
$265$	0	0
$266$	−0.854102	−0.0523684
$267$	8.94427i	0.547381i
$268$	2.94427i	0.179850i
$269$	12.7639	0.778231	0.389115	−	0.921189i	$-0.372781\pi$
$269$	12.7639	0.778231	0.389115	+	0.921189i	$0.372781\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 25.4164i	− 1.54110i
$273$	1.14590i	0.0693529i
$274$	−19.3262	−1.16754
$275$	0	0
$276$	2.32624	0.140023
$277$	24.7082i	1.48457i	0.670083	+	0.742286i	$0.266258\pi$
$277$	24.7082i	1.48457i	−0.670083	+	0.742286i	$0.733742\pi$
$278$	− 8.09017i	− 0.485216i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	10.0902	0.601929	0.300965	−	0.953635i	$-0.402691\pi$
$281$	10.0902	0.601929	0.300965	+	0.953635i	$0.402691\pi$
$282$	1.00000i	0.0595491i
$283$	− 29.8541i	− 1.77464i	−0.461152	−	0.887321i	$-0.652564\pi$
$283$	− 29.8541i	− 1.77464i	0.461152	−	0.887321i	$-0.347436\pi$
$284$	4.09017	0.242707
$285$	0	0
$286$	15.7082	0.928846
$287$	0.472136i	0.0278693i
$288$	− 6.76393i	− 0.398569i
$289$	−10.4164	−0.612730
$290$	0	0
$291$	−3.85410	−0.225931
$292$	5.56231i	0.325509i
$293$	− 19.5279i	− 1.14083i	−0.821357	−	0.570415i	$-0.806782\pi$
$293$	− 19.5279i	− 1.14083i	0.821357	−	0.570415i	$-0.193218\pi$
$294$	−10.7082	−0.624515
$295$	0	0
$296$	−0.527864	−0.0306815
$297$	− 26.1803i	− 1.51914i
$298$	6.38197i	0.369697i
$299$	−6.97871	−0.403589
$300$	0	0
$301$	−3.00000	−0.172917
$302$	23.5623i	1.35586i
$303$	1.47214i	0.0845720i
$304$	4.14590	0.237784
$305$	0	0
$306$	−16.9443	−0.968640
$307$	− 9.23607i	− 0.527130i	−0.964642	−	0.263565i	$-0.915102\pi$
$307$	− 9.23607i	− 0.527130i	0.964642	−	0.263565i	$-0.0848984\pi$
$308$	− 2.00000i	− 0.113961i
$309$	−8.56231	−0.487093
$310$	0	0
$311$	8.50658	0.482364	0.241182	−	0.970480i	$-0.422465\pi$
$311$	8.50658	0.482364	0.241182	+	0.970480i	$0.422465\pi$
$312$	− 4.14590i	− 0.234715i
$313$	− 16.7639i	− 0.947553i	−0.880645	−	0.473777i	$-0.842890\pi$
$313$	− 16.7639i	− 0.947553i	0.880645	−	0.473777i	$-0.157110\pi$
$314$	21.3262	1.20351
$315$	0	0
$316$	−5.00000	−0.281272
$317$	− 7.65248i	− 0.429806i	−0.976635	−	0.214903i	$-0.931056\pi$
$317$	− 7.65248i	− 0.429806i	0.976635	−	0.214903i	$-0.0689435\pi$
$318$	5.61803i	0.315044i
$319$	−18.9443	−1.06068
$320$	0	0
$321$	−16.4164	−0.916275
$322$	3.76393i	0.209756i
$323$	− 4.47214i	− 0.248836i
$324$	−0.618034	−0.0343352
$325$	0	0
$326$	−17.7984	−0.985761
$327$	− 10.0000i	− 0.553001i
$328$	− 1.70820i	− 0.0943198i
$329$	−0.381966	−0.0210585
$330$	0	0
$331$	−23.1246	−1.27104	−0.635522	−	0.772083i	$-0.719215\pi$
$331$	−23.1246	−1.27104	−0.635522	+	0.772083i	$0.719215\pi$
$332$	3.85410i	0.211521i
$333$	0.472136i	0.0258729i
$334$	23.5623	1.28927
$335$	0	0
$336$	−3.00000	−0.163663
$337$	− 7.85410i	− 0.427840i	−0.976851	−	0.213920i	$-0.931377\pi$
$337$	− 7.85410i	− 0.427840i	0.976851	−	0.213920i	$-0.0686232\pi$
$338$	15.4721i	0.841573i
$339$	−16.8541	−0.915389
$340$	0	0
$341$	15.7082	0.850647
$342$	− 2.76393i	− 0.149456i
$343$	− 8.41641i	− 0.454443i
$344$	10.8541	0.585214
$345$	0	0
$346$	−30.5623	−1.64304
$347$	19.9098i	1.06882i	0.845227	+	0.534408i	$0.179465\pi$
$347$	19.9098i	1.06882i	−0.845227	+	0.534408i	$0.820535\pi$
$348$	− 2.23607i	− 0.119866i
$349$	−21.7082	−1.16201	−0.581007	−	0.813899i	$-0.697341\pi$
$349$	−21.7082	−1.16201	−0.581007	+	0.813899i	$0.697341\pi$
$350$	0	0
$351$	−9.27051	−0.494823
$352$	17.7082i	0.943850i
$353$	12.9098i	0.687121i	0.939131	+	0.343560i	$0.111633\pi$
$353$	12.9098i	0.687121i	−0.939131	+	0.343560i	$0.888367\pi$
$354$	−17.5623	−0.933426
$355$	0	0
$356$	−5.52786	−0.292976
$357$	3.23607i	0.171271i
$358$	0.854102i	0.0451407i
$359$	−13.7426	−0.725309	−0.362655	−	0.931924i	$-0.618130\pi$
$359$	−13.7426	−0.725309	−0.362655	+	0.931924i	$0.618130\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0.472136i	0.0248149i
$363$	16.4164i	0.861638i
$364$	−0.708204	−0.0371200
$365$	0	0
$366$	−14.0902	−0.736505
$367$	25.5623i	1.33434i	0.744905	+	0.667171i	$0.232495\pi$
$367$	25.5623i	1.33434i	−0.744905	+	0.667171i	$0.767505\pi$
$368$	− 18.2705i	− 0.952416i
$369$	−1.52786	−0.0795374
$370$	0	0
$371$	−2.14590	−0.111409
$372$	1.85410i	0.0961307i
$373$	− 28.2705i	− 1.46379i	−0.681417	−	0.731896i	$-0.738636\pi$
$373$	− 28.2705i	− 1.46379i	0.681417	−	0.731896i	$-0.261364\pi$
$374$	44.3607	2.29384
$375$	0	0
$376$	1.38197	0.0712695
$377$	6.70820i	0.345490i
$378$	5.00000i	0.257172i
$379$	−14.5967	−0.749785	−0.374892	−	0.927068i	$-0.622320\pi$
$379$	−14.5967	−0.749785	−0.374892	+	0.927068i	$0.622320\pi$
$380$	0	0
$381$	19.8885	1.01892
$382$	− 2.94427i	− 0.150642i
$383$	33.3607i	1.70465i	0.523012	+	0.852326i	$0.324808\pi$
$383$	33.3607i	1.70465i	−0.523012	+	0.852326i	$0.675192\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	−12.4721	−0.634815
$387$	− 9.70820i	− 0.493496i
$388$	− 2.38197i	− 0.120926i
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	−19.7082	−0.996687
$392$	14.7984i	0.747431i
$393$	6.79837i	0.342933i
$394$	6.00000	0.302276
$395$	0	0
$396$	6.47214	0.325237
$397$	− 29.0344i	− 1.45720i	−0.684941	−	0.728598i	$-0.740172\pi$
$397$	− 29.0344i	− 1.45720i	0.684941	−	0.728598i	$-0.259828\pi$
$398$	28.4164i	1.42439i
$399$	−0.527864	−0.0264263
$400$	0	0
$401$	26.5967	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$401$	26.5967	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$402$	7.70820i	0.384450i
$403$	− 5.56231i	− 0.277078i
$404$	−0.909830	−0.0452657
$405$	0	0
$406$	3.61803	0.179560
$407$	− 1.23607i	− 0.0612696i
$408$	− 11.7082i	− 0.579642i
$409$	−1.58359	−0.0783036	−0.0391518	−	0.999233i	$-0.512466\pi$
$409$	−1.58359	−0.0783036	−0.0391518	+	0.999233i	$0.512466\pi$
$410$	0	0
$411$	−11.9443	−0.589167
$412$	− 5.29180i	− 0.260708i
$413$	− 6.70820i	− 0.330089i
$414$	−12.1803	−0.598631
$415$	0	0
$416$	6.27051	0.307437
$417$	− 5.00000i	− 0.244851i
$418$	7.23607i	0.353928i
$419$	9.47214	0.462744	0.231372	−	0.972865i	$-0.425679\pi$
$419$	9.47214	0.462744	0.231372	+	0.972865i	$0.425679\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	− 14.8541i	− 0.723086i
$423$	− 1.23607i	− 0.0600997i
$424$	7.76393	0.377050
$425$	0	0
$426$	10.7082	0.518814
$427$	− 5.38197i	− 0.260452i
$428$	− 10.1459i	− 0.490420i
$429$	9.70820	0.468717
$430$	0	0
$431$	−29.8328	−1.43700	−0.718498	−	0.695529i	$-0.755170\pi$
$431$	−29.8328	−1.43700	−0.718498	+	0.695529i	$0.755170\pi$
$432$	− 24.2705i	− 1.16772i
$433$	26.8541i	1.29053i	0.763961	+	0.645263i	$0.223252\pi$
$433$	26.8541i	1.29053i	−0.763961	+	0.645263i	$0.776748\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	6.18034	0.295985
$437$	− 3.21478i	− 0.153784i
$438$	14.5623i	0.695814i
$439$	40.9787	1.95581	0.977904	−	0.209056i	$-0.0670391\pi$
$439$	40.9787	1.95581	0.977904	+	0.209056i	$0.0670391\pi$
$440$	0	0
$441$	13.2361	0.630289
$442$	− 15.7082i	− 0.747163i
$443$	29.9443i	1.42270i	0.702840	+	0.711348i	$0.251915\pi$
$443$	29.9443i	1.42270i	−0.702840	+	0.711348i	$0.748085\pi$
$444$	0.145898	0.00692401
$445$	0	0
$446$	0.291796	0.0138169
$447$	3.94427i	0.186558i
$448$	2.61803i	0.123690i
$449$	−4.67376	−0.220568	−0.110284	−	0.993900i	$-0.535176\pi$
$449$	−4.67376	−0.220568	−0.110284	+	0.993900i	$0.535176\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	− 10.4164i	− 0.489947i
$453$	14.5623i	0.684197i
$454$	23.8885	1.12114
$455$	0	0
$456$	1.90983	0.0894360
$457$	21.4164i	1.00182i	0.865500	+	0.500909i	$0.167001\pi$
$457$	21.4164i	1.00182i	−0.865500	+	0.500909i	$0.832999\pi$
$458$	− 35.1246i	− 1.64127i
$459$	−26.1803	−1.22199
$460$	0	0
$461$	0.819660	0.0381754	0.0190877	−	0.999818i	$-0.493924\pi$
$461$	0.819660	0.0381754	0.0190877	+	0.999818i	$0.493924\pi$
$462$	− 5.23607i	− 0.243604i
$463$	− 24.1246i	− 1.12117i	−0.828098	−	0.560583i	$-0.810577\pi$
$463$	− 24.1246i	− 1.12117i	0.828098	−	0.560583i	$-0.189423\pi$
$464$	−17.5623	−0.815310
$465$	0	0
$466$	4.76393	0.220685
$467$	− 27.4508i	− 1.27027i	−0.772400	−	0.635137i	$-0.780944\pi$
$467$	− 27.4508i	− 1.27027i	0.772400	−	0.635137i	$-0.219056\pi$
$468$	− 2.29180i	− 0.105938i
$469$	−2.94427	−0.135954
$470$	0	0
$471$	13.1803	0.607318
$472$	24.2705i	1.11714i
$473$	25.4164i	1.16865i
$474$	−13.0902	−0.601251
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 6.94427i	− 0.317956i
$478$	33.2148i	1.51921i
$479$	10.8541	0.495937	0.247968	−	0.968768i	$-0.420237\pi$
$479$	10.8541	0.495937	0.247968	+	0.968768i	$0.420237\pi$
$480$	0	0
$481$	−0.437694	−0.0199571
$482$	4.09017i	0.186302i
$483$	2.32624i	0.105847i
$484$	−10.1459	−0.461177
$485$	0	0
$486$	−25.8885	−1.17433
$487$	36.4164i	1.65018i	0.564998	+	0.825092i	$0.308877\pi$
$487$	36.4164i	1.65018i	−0.564998	+	0.825092i	$0.691123\pi$
$488$	19.4721i	0.881462i
$489$	−11.0000	−0.497437
$490$	0	0
$491$	−43.2492	−1.95181	−0.975905	−	0.218196i	$-0.929983\pi$
$491$	−43.2492	−1.95181	−0.975905	+	0.218196i	$0.929983\pi$
$492$	0.472136i	0.0212855i
$493$	18.9443i	0.853207i
$494$	2.56231	0.115284
$495$	0	0
$496$	14.5623	0.653867
$497$	4.09017i	0.183469i
$498$	10.0902i	0.452151i
$499$	−7.56231	−0.338535	−0.169268	−	0.985570i	$-0.554140\pi$
$499$	−7.56231	−0.338535	−0.169268	+	0.985570i	$0.554140\pi$
$500$	0	0
$501$	14.5623	0.650596
$502$	− 47.2148i	− 2.10730i
$503$	− 37.4164i	− 1.66832i	−0.551526	−	0.834158i	$-0.685954\pi$
$503$	− 37.4164i	− 1.66832i	0.551526	−	0.834158i	$-0.314046\pi$
$504$	2.76393	0.123115
$505$	0	0
$506$	31.8885	1.41762
$507$	9.56231i	0.424677i
$508$	12.2918i	0.545360i
$509$	20.3262	0.900945	0.450472	−	0.892790i	$-0.351256\pi$
$509$	20.3262	0.900945	0.450472	+	0.892790i	$0.351256\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	− 5.29180i	− 0.233867i
$513$	− 4.27051i	− 0.188548i
$514$	36.9787	1.63106
$515$	0	0
$516$	−3.00000	−0.132068
$517$	3.23607i	0.142322i
$518$	0.236068i	0.0103722i
$519$	−18.8885	−0.829115
$520$	0	0
$521$	29.3607	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$521$	29.3607	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$522$	11.7082i	0.512454i
$523$	− 13.1459i	− 0.574830i	−0.957806	−	0.287415i	$-0.907204\pi$
$523$	− 13.1459i	− 0.574830i	0.957806	−	0.287415i	$-0.0927959\pi$
$524$	−4.20163	−0.183549
$525$	0	0
$526$	17.6525	0.769685
$527$	− 15.7082i	− 0.684260i
$528$	25.4164i	1.10611i
$529$	8.83282	0.384035
$530$	0	0
$531$	21.7082	0.942056
$532$	− 0.326238i	− 0.0141442i
$533$	− 1.41641i	− 0.0613514i
$534$	−14.4721	−0.626271
$535$	0	0
$536$	10.6525	0.460117
$537$	0.527864i	0.0227790i
$538$	20.6525i	0.890391i
$539$	−34.6525	−1.49259
$540$	0	0
$541$	27.1246	1.16618	0.583089	−	0.812408i	$-0.301844\pi$
$541$	27.1246	1.16618	0.583089	+	0.812408i	$0.301844\pi$
$542$	− 12.9443i	− 0.556004i
$543$	0.291796i	0.0125222i
$544$	17.7082	0.759233
$545$	0	0
$546$	−1.85410	−0.0793482
$547$	21.2918i	0.910371i	0.890397	+	0.455186i	$0.150427\pi$
$547$	21.2918i	0.910371i	−0.890397	+	0.455186i	$0.849573\pi$
$548$	− 7.38197i	− 0.315342i
$549$	17.4164	0.743314
$550$	0	0
$551$	−3.09017	−0.131646
$552$	− 8.41641i	− 0.358226i
$553$	− 5.00000i	− 0.212622i
$554$	−39.9787	−1.69853
$555$	0	0
$556$	3.09017	0.131052
$557$	− 4.76393i	− 0.201854i	−0.994894	−	0.100927i	$-0.967819\pi$
$557$	− 4.76393i	− 0.201854i	0.994894	−	0.100927i	$-0.0321809\pi$
$558$	− 9.70820i	− 0.410981i
$559$	9.00000	0.380659
$560$	0	0
$561$	27.4164	1.15752
$562$	16.3262i	0.688681i
$563$	7.38197i	0.311113i	0.987827	+	0.155556i	$0.0497170\pi$
$563$	7.38197i	0.311113i	−0.987827	+	0.155556i	$0.950283\pi$
$564$	−0.381966	−0.0160837
$565$	0	0
$566$	48.3050	2.03041
$567$	− 0.618034i	− 0.0259550i
$568$	− 14.7984i	− 0.620926i
$569$	−20.5279	−0.860573	−0.430286	−	0.902692i	$-0.641587\pi$
$569$	−20.5279	−0.860573	−0.430286	+	0.902692i	$0.641587\pi$
$570$	0	0
$571$	−8.12461	−0.340004	−0.170002	−	0.985444i	$-0.554377\pi$
$571$	−8.12461	−0.340004	−0.170002	+	0.985444i	$0.554377\pi$
$572$	6.00000i	0.250873i
$573$	− 1.81966i	− 0.0760174i
$574$	−0.763932	−0.0318859
$575$	0	0
$576$	−8.47214	−0.353006
$577$	33.7771i	1.40616i	0.711111	+	0.703079i	$0.248192\pi$
$577$	33.7771i	1.40616i	−0.711111	+	0.703079i	$0.751808\pi$
$578$	− 16.8541i	− 0.701038i
$579$	−7.70820	−0.320342
$580$	0	0
$581$	−3.85410	−0.159895
$582$	− 6.23607i	− 0.258493i
$583$	18.1803i	0.752953i
$584$	20.1246	0.832762
$585$	0	0
$586$	31.5967	1.30525
$587$	− 5.29180i	− 0.218416i	−0.994019	−	0.109208i	$-0.965169\pi$
$587$	− 5.29180i	− 0.218416i	0.994019	−	0.109208i	$-0.0348314\pi$
$588$	− 4.09017i	− 0.168676i
$589$	2.56231	0.105578
$590$	0	0
$591$	3.70820	0.152535
$592$	− 1.14590i	− 0.0470961i
$593$	− 10.9098i	− 0.448013i	−0.974588	−	0.224007i	$-0.928086\pi$
$593$	− 10.9098i	− 0.448013i	0.974588	−	0.224007i	$-0.0719137\pi$
$594$	42.3607	1.73808
$595$	0	0
$596$	−2.43769	−0.0998518
$597$	17.5623i	0.718777i
$598$	− 11.2918i	− 0.461756i
$599$	9.47214	0.387021	0.193510	−	0.981098i	$-0.438013\pi$
$599$	9.47214	0.387021	0.193510	+	0.981098i	$0.438013\pi$
$600$	0	0
$601$	2.72949	0.111338	0.0556691	−	0.998449i	$-0.482271\pi$
$601$	2.72949	0.111338	0.0556691	+	0.998449i	$0.482271\pi$
$602$	− 4.85410i	− 0.197838i
$603$	− 9.52786i	− 0.388005i
$604$	−9.00000	−0.366205
$605$	0	0
$606$	−2.38197	−0.0967608
$607$	35.5623i	1.44343i	0.692191	+	0.721715i	$0.256646\pi$
$607$	35.5623i	1.44343i	−0.692191	+	0.721715i	$0.743354\pi$
$608$	2.88854i	0.117146i
$609$	2.23607	0.0906100
$610$	0	0
$611$	1.14590	0.0463581
$612$	− 6.47214i	− 0.261621i
$613$	− 14.9787i	− 0.604985i	−0.953152	−	0.302492i	$-0.902181\pi$
$613$	− 14.9787i	− 0.604985i	0.953152	−	0.302492i	$-0.0978186\pi$
$614$	14.9443	0.603102
$615$	0	0
$616$	−7.23607	−0.291549
$617$	− 14.2361i	− 0.573123i	−0.958062	−	0.286561i	$-0.907488\pi$
$617$	− 14.2361i	− 0.573123i	0.958062	−	0.286561i	$-0.0925122\pi$
$618$	− 13.8541i	− 0.557294i
$619$	−30.5279	−1.22702	−0.613509	−	0.789688i	$-0.710243\pi$
$619$	−30.5279	−1.22702	−0.613509	+	0.789688i	$0.710243\pi$
$620$	0	0
$621$	−18.8197	−0.755207
$622$	13.7639i	0.551883i
$623$	− 5.52786i	− 0.221469i
$624$	9.00000	0.360288
$625$	0	0
$626$	27.1246	1.08412
$627$	4.47214i	0.178600i
$628$	8.14590i	0.325057i
$629$	−1.23607	−0.0492853
$630$	0	0
$631$	−10.2361	−0.407491	−0.203746	−	0.979024i	$-0.565312\pi$
$631$	−10.2361	−0.407491	−0.203746	+	0.979024i	$0.565312\pi$
$632$	18.0902i	0.719588i
$633$	− 9.18034i	− 0.364886i
$634$	12.3820	0.491751
$635$	0	0
$636$	−2.14590	−0.0850904
$637$	12.2705i	0.486175i
$638$	− 30.6525i	− 1.21354i
$639$	−13.2361	−0.523611
$640$	0	0
$641$	−1.09017	−0.0430591	−0.0215296	−	0.999768i	$-0.506854\pi$
$641$	−1.09017	−0.0430591	−0.0215296	+	0.999768i	$0.506854\pi$
$642$	− 26.5623i	− 1.04833i
$643$	− 30.8328i	− 1.21593i	−0.793965	−	0.607964i	$-0.791987\pi$
$643$	− 30.8328i	− 1.21593i	0.793965	−	0.607964i	$-0.208013\pi$
$644$	−1.43769	−0.0566531
$645$	0	0
$646$	7.23607	0.284699
$647$	36.5410i	1.43658i	0.695746	+	0.718288i	$0.255074\pi$
$647$	36.5410i	1.43658i	−0.695746	+	0.718288i	$0.744926\pi$
$648$	2.23607i	0.0878410i
$649$	−56.8328	−2.23088
$650$	0	0
$651$	−1.85410	−0.0726680
$652$	− 6.79837i	− 0.266245i
$653$	19.0902i	0.747056i	0.927619	+	0.373528i	$0.121852\pi$
$653$	19.0902i	0.747056i	−0.927619	+	0.373528i	$0.878148\pi$
$654$	16.1803	0.632701
$655$	0	0
$656$	3.70820	0.144781
$657$	− 18.0000i	− 0.702247i
$658$	− 0.618034i	− 0.0240935i
$659$	−15.5279	−0.604880	−0.302440	−	0.953168i	$-0.597801\pi$
$659$	−15.5279	−0.604880	−0.302440	+	0.953168i	$0.597801\pi$
$660$	0	0
$661$	19.6869	0.765732	0.382866	−	0.923804i	$-0.374937\pi$
$661$	19.6869	0.765732	0.382866	+	0.923804i	$0.374937\pi$
$662$	− 37.4164i	− 1.45423i
$663$	− 9.70820i	− 0.377035i
$664$	13.9443	0.541143
$665$	0	0
$666$	−0.763932	−0.0296018
$667$	13.6180i	0.527292i
$668$	9.00000i	0.348220i
$669$	0.180340	0.00697234
$670$	0	0
$671$	−45.5967	−1.76024
$672$	− 2.09017i	− 0.0806301i
$673$	12.1803i	0.469518i	0.972054	+	0.234759i	$0.0754300\pi$
$673$	12.1803i	0.469518i	−0.972054	+	0.234759i	$0.924570\pi$
$674$	12.7082	0.489502
$675$	0	0
$676$	−5.90983	−0.227301
$677$	− 10.6180i	− 0.408084i	−0.978962	−	0.204042i	$-0.934592\pi$
$677$	− 10.6180i	− 0.408084i	0.978962	−	0.204042i	$-0.0654079\pi$
$678$	− 27.2705i	− 1.04732i
$679$	2.38197	0.0914115
$680$	0	0
$681$	14.7639	0.565755
$682$	25.4164i	0.973245i
$683$	− 13.4721i	− 0.515497i	−0.966212	−	0.257748i	$-0.917019\pi$
$683$	− 13.4721i	− 0.515497i	0.966212	−	0.257748i	$-0.0829806\pi$
$684$	1.05573	0.0403668
$685$	0	0
$686$	13.6180	0.519939
$687$	− 21.7082i	− 0.828220i
$688$	23.5623i	0.898304i
$689$	6.43769	0.245257
$690$	0	0
$691$	36.2705	1.37980	0.689898	−	0.723907i	$-0.257656\pi$
$691$	36.2705	1.37980	0.689898	+	0.723907i	$0.257656\pi$
$692$	− 11.6738i	− 0.443770i
$693$	6.47214i	0.245856i
$694$	−32.2148	−1.22286
$695$	0	0
$696$	−8.09017	−0.306657
$697$	− 4.00000i	− 0.151511i
$698$	− 35.1246i	− 1.32949i
$699$	2.94427	0.111363
$700$	0	0
$701$	−41.0132	−1.54905	−0.774523	−	0.632546i	$-0.782010\pi$
$701$	−41.0132	−1.54905	−0.774523	+	0.632546i	$0.782010\pi$
$702$	− 15.0000i	− 0.566139i
$703$	− 0.201626i	− 0.00760447i
$704$	22.1803	0.835953
$705$	0	0
$706$	−20.8885	−0.786151
$707$	− 0.909830i	− 0.0342177i
$708$	− 6.70820i	− 0.252110i
$709$	33.5410	1.25966	0.629830	−	0.776733i	$-0.283125\pi$
$709$	33.5410	1.25966	0.629830	+	0.776733i	$0.283125\pi$
$710$	0	0
$711$	16.1803	0.606810
$712$	20.0000i	0.749532i
$713$	− 11.2918i	− 0.422881i
$714$	−5.23607	−0.195955
$715$	0	0
$716$	−0.326238	−0.0121921
$717$	20.5279i	0.766627i
$718$	− 22.2361i	− 0.829843i
$719$	23.2918	0.868637	0.434319	−	0.900759i	$-0.356989\pi$
$719$	23.2918	0.868637	0.434319	+	0.900759i	$0.356989\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	− 29.5623i	− 1.10020i
$723$	2.52786i	0.0940123i
$724$	−0.180340	−0.00670228
$725$	0	0
$726$	−26.5623	−0.985820
$727$	− 24.5623i	− 0.910966i	−0.890245	−	0.455483i	$-0.849467\pi$
$727$	− 24.5623i	− 0.910966i	0.890245	−	0.455483i	$-0.150533\pi$
$728$	2.56231i	0.0949654i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	25.4164	0.940060
$732$	− 5.38197i	− 0.198923i
$733$	− 19.9787i	− 0.737931i	−0.929443	−	0.368965i	$-0.879712\pi$
$733$	− 19.9787i	− 0.737931i	0.929443	−	0.368965i	$-0.120288\pi$
$734$	−41.3607	−1.52665
$735$	0	0
$736$	12.7295	0.469215
$737$	24.9443i	0.918834i
$738$	− 2.47214i	− 0.0910006i
$739$	−15.9787	−0.587786	−0.293893	−	0.955838i	$-0.594951\pi$
$739$	−15.9787	−0.587786	−0.293893	+	0.955838i	$0.594951\pi$
$740$	0	0
$741$	1.58359	0.0581747
$742$	− 3.47214i	− 0.127466i
$743$	28.3607i	1.04045i	0.854029	+	0.520226i	$0.174152\pi$
$743$	28.3607i	1.04045i	−0.854029	+	0.520226i	$0.825848\pi$
$744$	6.70820	0.245935
$745$	0	0
$746$	45.7426	1.67476
$747$	− 12.4721i	− 0.456332i
$748$	16.9443i	0.619544i
$749$	10.1459	0.370723
$750$	0	0
$751$	−5.11146	−0.186520	−0.0932598	−	0.995642i	$-0.529729\pi$
$751$	−5.11146	−0.186520	−0.0932598	+	0.995642i	$0.529729\pi$
$752$	3.00000i	0.109399i
$753$	− 29.1803i	− 1.06339i
$754$	−10.8541	−0.395283
$755$	0	0
$756$	−1.90983	−0.0694598
$757$	− 30.4164i	− 1.10550i	−0.833346	−	0.552752i	$-0.813578\pi$
$757$	− 30.4164i	− 1.10550i	0.833346	−	0.552752i	$-0.186422\pi$
$758$	− 23.6180i	− 0.857846i
$759$	19.7082	0.715362
$760$	0	0
$761$	−18.4508	−0.668843	−0.334421	−	0.942424i	$-0.608541\pi$
$761$	−18.4508	−0.668843	−0.334421	+	0.942424i	$0.608541\pi$
$762$	32.1803i	1.16577i
$763$	6.18034i	0.223743i
$764$	1.12461	0.0406870
$765$	0	0
$766$	−53.9787	−1.95033
$767$	20.1246i	0.726658i
$768$	13.5623i	0.489388i
$769$	−13.4164	−0.483808	−0.241904	−	0.970300i	$-0.577772\pi$
$769$	−13.4164	−0.483808	−0.241904	+	0.970300i	$0.577772\pi$
$770$	0	0
$771$	22.8541	0.823070
$772$	− 4.76393i	− 0.171458i
$773$	− 36.1591i	− 1.30055i	−0.759699	−	0.650275i	$-0.774654\pi$
$773$	− 36.1591i	− 1.30055i	0.759699	−	0.650275i	$-0.225346\pi$
$774$	15.7082	0.564620
$775$	0	0
$776$	−8.61803	−0.309369
$777$	0.145898i	0.00523406i
$778$	− 24.2705i	− 0.870140i
$779$	0.652476	0.0233774
$780$	0	0
$781$	34.6525	1.23996
$782$	− 31.8885i	− 1.14033i
$783$	18.0902i	0.646490i
$784$	−32.1246	−1.14731
$785$	0	0
$786$	−11.0000	−0.392357
$787$	11.8197i	0.421325i	0.977559	+	0.210663i	$0.0675622\pi$
$787$	11.8197i	0.421325i	−0.977559	+	0.210663i	$0.932438\pi$
$788$	2.29180i	0.0816419i
$789$	10.9098	0.388400
$790$	0	0
$791$	10.4164	0.370365
$792$	− 23.4164i	− 0.832066i
$793$	16.1459i	0.573358i
$794$	46.9787	1.66721
$795$	0	0
$796$	−10.8541	−0.384713
$797$	− 9.76393i	− 0.345856i	−0.984934	−	0.172928i	$-0.944677\pi$
$797$	− 9.76393i	− 0.345856i	0.984934	−	0.172928i	$-0.0553228\pi$
$798$	− 0.854102i	− 0.0302349i
$799$	3.23607	0.114484
$800$	0	0
$801$	17.8885	0.632061
$802$	43.0344i	1.51960i
$803$	47.1246i	1.66299i
$804$	−2.94427	−0.103836
$805$	0	0
$806$	9.00000	0.317011
$807$	12.7639i	0.449312i
$808$	3.29180i	0.115805i
$809$	−30.9787	−1.08915	−0.544577	−	0.838711i	$-0.683310\pi$
$809$	−30.9787	−1.08915	−0.544577	+	0.838711i	$0.683310\pi$
$810$	0	0
$811$	−14.7082	−0.516475	−0.258237	−	0.966081i	$-0.583142\pi$
$811$	−14.7082	−0.516475	−0.258237	+	0.966081i	$0.583142\pi$
$812$	1.38197i	0.0484975i
$813$	− 8.00000i	− 0.280572i
$814$	2.00000	0.0701000
$815$	0	0
$816$	25.4164	0.889752
$817$	4.14590i	0.145047i
$818$	− 2.56231i	− 0.0895889i
$819$	2.29180	0.0800818
$820$	0	0
$821$	−40.6869	−1.41998	−0.709992	−	0.704210i	$-0.751301\pi$
$821$	−40.6869	−1.41998	−0.709992	+	0.704210i	$0.751301\pi$
$822$	− 19.3262i	− 0.674080i
$823$	47.7082i	1.66300i	0.555522	+	0.831502i	$0.312518\pi$
$823$	47.7082i	1.66300i	−0.555522	+	0.831502i	$0.687482\pi$
$824$	−19.1459	−0.666979
$825$	0	0
$826$	10.8541	0.377663
$827$	0.965558i	0.0335757i	0.999859	+	0.0167879i	$0.00534400\pi$
$827$	0.965558i	0.0335757i	−0.999859	+	0.0167879i	$0.994656\pi$
$828$	− 4.65248i	− 0.161685i
$829$	35.8541	1.24526	0.622632	−	0.782515i	$-0.286063\pi$
$829$	35.8541	1.24526	0.622632	+	0.782515i	$0.286063\pi$
$830$	0	0
$831$	−24.7082	−0.857118
$832$	− 7.85410i	− 0.272292i
$833$	34.6525i	1.20064i
$834$	8.09017	0.280140
$835$	0	0
$836$	−2.76393	−0.0955926
$837$	− 15.0000i	− 0.518476i
$838$	15.3262i	0.529436i
$839$	−10.8541	−0.374725	−0.187363	−	0.982291i	$-0.559994\pi$
$839$	−10.8541	−0.374725	−0.187363	+	0.982291i	$0.559994\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	51.7771i	1.78436i
$843$	10.0902i	0.347524i
$844$	5.67376	0.195299
$845$	0	0
$846$	2.00000	0.0687614
$847$	− 10.1459i	− 0.348617i
$848$	16.8541i	0.578772i
$849$	29.8541	1.02459
$850$	0	0
$851$	−0.888544	−0.0304589
$852$	4.09017i	0.140127i
$853$	− 15.3050i	− 0.524032i	−0.965064	−	0.262016i	$-0.915613\pi$
$853$	− 15.3050i	− 0.524032i	0.965064	−	0.262016i	$-0.0843873\pi$
$854$	8.70820	0.297989
$855$	0	0
$856$	−36.7082	−1.25466
$857$	− 19.6869i	− 0.672492i	−0.941774	−	0.336246i	$-0.890843\pi$
$857$	− 19.6869i	− 0.672492i	0.941774	−	0.336246i	$-0.109157\pi$
$858$	15.7082i	0.536269i
$859$	1.58359	0.0540315	0.0270157	−	0.999635i	$-0.491400\pi$
$859$	1.58359	0.0540315	0.0270157	+	0.999635i	$0.491400\pi$
$860$	0	0
$861$	−0.472136	−0.0160904
$862$	− 48.2705i	− 1.64410i
$863$	− 21.4377i	− 0.729748i	−0.931057	−	0.364874i	$-0.881112\pi$
$863$	− 21.4377i	− 0.729748i	0.931057	−	0.364874i	$-0.118888\pi$
$864$	16.9098	0.575284
$865$	0	0
$866$	−43.4508	−1.47652
$867$	− 10.4164i	− 0.353760i
$868$	− 1.14590i	− 0.0388943i
$869$	−42.3607	−1.43699
$870$	0	0
$871$	8.83282	0.299289
$872$	− 22.3607i	− 0.757228i
$873$	7.70820i	0.260883i
$874$	5.20163	0.175948
$875$	0	0
$876$	−5.56231	−0.187933
$877$	36.5410i	1.23390i	0.787001	+	0.616951i	$0.211632\pi$
$877$	36.5410i	1.23390i	−0.787001	+	0.616951i	$0.788368\pi$
$878$	66.3050i	2.23768i
$879$	19.5279	0.658659
$880$	0	0
$881$	−40.3607	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$881$	−40.3607	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$882$	21.4164i	0.721128i
$883$	− 20.5836i	− 0.692693i	−0.938107	−	0.346347i	$-0.887422\pi$
$883$	− 20.5836i	− 0.692693i	0.938107	−	0.346347i	$-0.112578\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−48.4508	−1.62774
$887$	− 29.8885i	− 1.00356i	−0.864996	−	0.501780i	$-0.832679\pi$
$887$	− 29.8885i	− 1.00356i	0.864996	−	0.501780i	$-0.167321\pi$
$888$	− 0.527864i	− 0.0177140i
$889$	−12.2918	−0.412254
$890$	0	0
$891$	−5.23607	−0.175415
$892$	0.111456i	0.00373183i
$893$	0.527864i	0.0176643i
$894$	−6.38197	−0.213445
$895$	0	0
$896$	−8.41641	−0.281172
$897$	− 6.97871i	− 0.233012i
$898$	− 7.56231i	− 0.252357i
$899$	−10.8541	−0.362005
$900$	0	0
$901$	18.1803	0.605675
$902$	6.47214i	0.215499i
$903$	− 3.00000i	− 0.0998337i
$904$	−37.6869	−1.25345
$905$	0	0
$906$	−23.5623	−0.782805
$907$	33.2492i	1.10402i	0.833837	+	0.552011i	$0.186139\pi$
$907$	33.2492i	1.10402i	−0.833837	+	0.552011i	$0.813861\pi$
$908$	9.12461i	0.302811i
$909$	2.94427	0.0976553
$910$	0	0
$911$	−40.2361	−1.33308	−0.666540	−	0.745469i	$-0.732226\pi$
$911$	−40.2361	−1.33308	−0.666540	+	0.745469i	$0.732226\pi$
$912$	4.14590i	0.137284i
$913$	32.6525i	1.08064i
$914$	−34.6525	−1.14620
$915$	0	0
$916$	13.4164	0.443291
$917$	− 4.20163i	− 0.138750i
$918$	− 42.3607i	− 1.39811i
$919$	53.2148	1.75539	0.877697	−	0.479216i	$-0.159079\pi$
$919$	53.2148	1.75539	0.877697	+	0.479216i	$0.159079\pi$
$920$	0	0
$921$	9.23607	0.304339
$922$	1.32624i	0.0436773i
$923$	− 12.2705i	− 0.403889i
$924$	2.00000	0.0657952
$925$	0	0
$926$	39.0344	1.28275
$927$	17.1246i	0.562446i
$928$	− 12.2361i	− 0.401669i
$929$	−41.6312	−1.36588	−0.682938	−	0.730477i	$-0.739298\pi$
$929$	−41.6312	−1.36588	−0.682938	+	0.730477i	$0.739298\pi$
$930$	0	0
$931$	−5.65248	−0.185252
$932$	1.81966i	0.0596049i
$933$	8.50658i	0.278493i
$934$	44.4164	1.45335
$935$	0	0
$936$	−8.29180	−0.271026
$937$	− 17.7295i	− 0.579197i	−0.957148	−	0.289599i	$-0.906478\pi$
$937$	− 17.7295i	− 0.579197i	0.957148	−	0.289599i	$-0.0935218\pi$
$938$	− 4.76393i	− 0.155548i
$939$	16.7639	0.547070
$940$	0	0
$941$	−46.4164	−1.51313	−0.756566	−	0.653918i	$-0.773124\pi$
$941$	−46.4164	−1.51313	−0.756566	+	0.653918i	$0.773124\pi$
$942$	21.3262i	0.694846i
$943$	− 2.87539i	− 0.0936355i
$944$	−52.6869	−1.71481
$945$	0	0
$946$	−41.1246	−1.33708
$947$	− 2.65248i	− 0.0861939i	−0.999071	−	0.0430969i	$-0.986278\pi$
$947$	− 2.65248i	− 0.0861939i	0.999071	−	0.0430969i	$-0.0137224\pi$
$948$	− 5.00000i	− 0.162392i
$949$	16.6869	0.541680
$950$	0	0
$951$	7.65248	0.248149
$952$	7.23607i	0.234522i
$953$	− 7.74265i	− 0.250809i	−0.992106	−	0.125404i	$-0.959977\pi$
$953$	− 7.74265i	− 0.250809i	0.992106	−	0.125404i	$-0.0400228\pi$
$954$	11.2361	0.363781
$955$	0	0
$956$	−12.6869	−0.410324
$957$	− 18.9443i	− 0.612381i
$958$	17.5623i	0.567412i
$959$	7.38197	0.238376
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 0.708204i	− 0.0228334i
$963$	32.8328i	1.05802i
$964$	−1.56231	−0.0503185
$965$	0	0
$966$	−3.76393	−0.121103
$967$	− 4.11146i	− 0.132216i	−0.997812	−	0.0661078i	$-0.978942\pi$
$967$	− 4.11146i	− 0.132216i	0.997812	−	0.0661078i	$-0.0210581\pi$
$968$	36.7082i	1.17985i
$969$	4.47214	0.143666
$970$	0	0
$971$	5.61803	0.180291	0.0901456	−	0.995929i	$-0.471267\pi$
$971$	5.61803	0.180291	0.0901456	+	0.995929i	$0.471267\pi$
$972$	− 9.88854i	− 0.317175i
$973$	3.09017i	0.0990663i
$974$	−58.9230	−1.88801
$975$	0	0
$976$	−42.2705	−1.35305
$977$	2.34752i	0.0751040i	0.999295	+	0.0375520i	$0.0119560\pi$
$977$	2.34752i	0.0751040i	−0.999295	+	0.0375520i	$0.988044\pi$
$978$	− 17.7984i	− 0.569129i
$979$	−46.8328	−1.49678
$980$	0	0
$981$	−20.0000	−0.638551
$982$	− 69.9787i	− 2.23311i
$983$	9.61803i	0.306768i	0.988167	+	0.153384i	$0.0490171\pi$
$983$	9.61803i	0.306768i	−0.988167	+	0.153384i	$0.950983\pi$
$984$	1.70820	0.0544556
$985$	0	0
$986$	−30.6525	−0.976174
$987$	− 0.381966i	− 0.0121581i
$988$	0.978714i	0.0311370i
$989$	18.2705	0.580968
$990$	0	0
$991$	−15.3607	−0.487948	−0.243974	−	0.969782i	$-0.578451\pi$
$991$	−15.3607	−0.487948	−0.243974	+	0.969782i	$0.578451\pi$
$992$	10.1459i	0.322133i
$993$	− 23.1246i	− 0.733837i
$994$	−6.61803	−0.209911
$995$	0	0
$996$	−3.85410	−0.122122
$997$	− 24.8885i	− 0.788228i	−0.919062	−	0.394114i	$-0.871051\pi$
$997$	− 24.8885i	− 0.788228i	0.919062	−	0.394114i	$-0.128949\pi$
$998$	− 12.2361i	− 0.387326i
$999$	−1.18034	−0.0373443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.b.a.624.4		4
5.2	odd	4		625.2.a.b.1.1		2
5.3	odd	4		625.2.a.c.1.2		2
5.4	even	2	inner	625.2.b.a.624.1		4
15.2	even	4		5625.2.a.f.1.2		2
15.8	even	4		5625.2.a.d.1.1		2
20.3	even	4		10000.2.a.l.1.2		2
20.7	even	4		10000.2.a.c.1.1		2
25.2	odd	20		25.2.d.a.21.1	yes	4
25.3	odd	20		625.2.d.b.376.1		4
25.4	even	10		625.2.e.c.249.1		8
25.6	even	5		625.2.e.c.374.1		8
25.8	odd	20		625.2.d.b.251.1		4
25.9	even	10		125.2.e.a.99.2		8
25.11	even	5		125.2.e.a.24.2		8
25.12	odd	20		25.2.d.a.6.1	✓	4
25.13	odd	20		125.2.d.a.26.1		4
25.14	even	10		125.2.e.a.24.1		8
25.16	even	5		125.2.e.a.99.1		8
25.17	odd	20		625.2.d.h.251.1		4
25.19	even	10		625.2.e.c.374.2		8
25.21	even	5		625.2.e.c.249.2		8
25.22	odd	20		625.2.d.h.376.1		4
25.23	odd	20		125.2.d.a.101.1		4
75.2	even	20		225.2.h.b.46.1		4
75.62	even	20		225.2.h.b.181.1		4
100.27	even	20		400.2.u.b.321.1		4
100.87	even	20		400.2.u.b.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.6.1	✓	4	25.12	odd	20
25.2.d.a.21.1	yes	4	25.2	odd	20
125.2.d.a.26.1		4	25.13	odd	20
125.2.d.a.101.1		4	25.23	odd	20
125.2.e.a.24.1		8	25.14	even	10
125.2.e.a.24.2		8	25.11	even	5
125.2.e.a.99.1		8	25.16	even	5
125.2.e.a.99.2		8	25.9	even	10
225.2.h.b.46.1		4	75.2	even	20
225.2.h.b.181.1		4	75.62	even	20
400.2.u.b.81.1		4	100.87	even	20
400.2.u.b.321.1		4	100.27	even	20
625.2.a.b.1.1		2	5.2	odd	4
625.2.a.c.1.2		2	5.3	odd	4
625.2.b.a.624.1		4	5.4	even	2	inner
625.2.b.a.624.4		4	1.1	even	1	trivial
625.2.d.b.251.1		4	25.8	odd	20
625.2.d.b.376.1		4	25.3	odd	20
625.2.d.h.251.1		4	25.17	odd	20
625.2.d.h.376.1		4	25.22	odd	20
625.2.e.c.249.1		8	25.4	even	10
625.2.e.c.249.2		8	25.21	even	5
625.2.e.c.374.1		8	25.6	even	5
625.2.e.c.374.2		8	25.19	even	10
5625.2.a.d.1.1		2	15.8	even	4
5625.2.a.f.1.2		2	15.2	even	4
10000.2.a.c.1.1		2	20.7	even	4
10000.2.a.l.1.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -0.618034i q^{7} +2.23607i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -0.618034i q^{7} +2.23607i q^{8} +2.00000 q^{9} -5.23607 q^{11} -0.618034i q^{12} +1.85410i q^{13} +1.00000 q^{14} -4.85410 q^{16} +5.23607i q^{17} +3.23607i q^{18} -0.854102 q^{19} +0.618034 q^{21} -8.47214i q^{22} +3.76393i q^{23} -2.23607 q^{24} -3.00000 q^{26} +5.00000i q^{27} +0.381966i q^{28} +3.61803 q^{29} -3.00000 q^{31} -3.38197i q^{32} -5.23607i q^{33} -8.47214 q^{34} -1.23607 q^{36} +0.236068i q^{37} -1.38197i q^{38} -1.85410 q^{39} -0.763932 q^{41} +1.00000i q^{42} -4.85410i q^{43} +3.23607 q^{44} -6.09017 q^{46} -0.618034i q^{47} -4.85410i q^{48} +6.61803 q^{49} -5.23607 q^{51} -1.14590i q^{52} -3.47214i q^{53} -8.09017 q^{54} +1.38197 q^{56} -0.854102i q^{57} +5.85410i q^{58} +10.8541 q^{59} +8.70820 q^{61} -4.85410i q^{62} -1.23607i q^{63} -4.23607 q^{64} +8.47214 q^{66} -4.76393i q^{67} -3.23607i q^{68} -3.76393 q^{69} -6.61803 q^{71} +4.47214i q^{72} -9.00000i q^{73} -0.381966 q^{74} +0.527864 q^{76} +3.23607i q^{77} -3.00000i q^{78} +8.09017 q^{79} +1.00000 q^{81} -1.23607i q^{82} -6.23607i q^{83} -0.381966 q^{84} +7.85410 q^{86} +3.61803i q^{87} -11.7082i q^{88} +8.94427 q^{89} +1.14590 q^{91} -2.32624i q^{92} -3.00000i q^{93} +1.00000 q^{94} +3.38197 q^{96} +3.85410i q^{97} +10.7082i q^{98} -10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^6 + 8 * q^9	\( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 + 8 * q^9 - 12 * q^11 + 4 * q^14 - 6 * q^16 + 10 * q^19 - 2 * q^21 - 12 * q^26 + 10 * q^29 - 12 * q^31 - 16 * q^34 + 4 * q^36 + 6 * q^39 - 12 * q^41 + 4 * q^44 - 2 * q^46 + 22 * q^49 - 12 * q^51 - 10 * q^54 + 10 * q^56 + 30 * q^59 + 8 * q^61 - 8 * q^64 + 16 * q^66 - 24 * q^69 - 22 * q^71 - 6 * q^74 + 20 * q^76 + 10 * q^79 + 4 * q^81 - 6 * q^84 + 18 * q^86 + 18 * q^91 + 4 * q^94 + 18 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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