LMFDB - Embedded newform 625.2.b.b.624.3

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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			624.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	625.624
Dual form			625.2.b.b.624.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} +0.381966i q^{3} +1.61803 q^{4} -0.236068 q^{6} +2.85410i q^{7} +2.23607i q^{8} +2.85410 q^{9} +O(q^{10})$	\(q+0.618034i q^{2} +0.381966i q^{3} +1.61803 q^{4} -0.236068 q^{6} +2.85410i q^{7} +2.23607i q^{8} +2.85410 q^{9} -1.61803 q^{11} +0.618034i q^{12} +3.47214i q^{13} -1.76393 q^{14} +1.85410 q^{16} -7.47214i q^{17} +1.76393i q^{18} -5.85410 q^{19} -1.09017 q^{21} -1.00000i q^{22} +4.85410i q^{23} -0.854102 q^{24} -2.14590 q^{26} +2.23607i q^{27} +4.61803i q^{28} +7.23607 q^{29} +2.00000 q^{31} +5.61803i q^{32} -0.618034i q^{33} +4.61803 q^{34} +4.61803 q^{36} -3.00000i q^{37} -3.61803i q^{38} -1.32624 q^{39} -2.47214 q^{41} -0.673762i q^{42} +8.47214i q^{43} -2.61803 q^{44} -3.00000 q^{46} -9.38197i q^{47} +0.708204i q^{48} -1.14590 q^{49} +2.85410 q^{51} +5.61803i q^{52} -0.472136i q^{53} -1.38197 q^{54} -6.38197 q^{56} -2.23607i q^{57} +4.47214i q^{58} -3.94427 q^{59} -2.14590 q^{61} +1.23607i q^{62} +8.14590i q^{63} +0.236068 q^{64} +0.381966 q^{66} +2.52786i q^{67} -12.0902i q^{68} -1.85410 q^{69} +12.3262 q^{71} +6.38197i q^{72} -6.85410i q^{73} +1.85410 q^{74} -9.47214 q^{76} -4.61803i q^{77} -0.819660i q^{78} -6.70820 q^{79} +7.70820 q^{81} -1.52786i q^{82} -4.94427i q^{83} -1.76393 q^{84} -5.23607 q^{86} +2.76393i q^{87} -3.61803i q^{88} -1.38197 q^{89} -9.90983 q^{91} +7.85410i q^{92} +0.763932i q^{93} +5.79837 q^{94} -2.14590 q^{96} +8.38197i q^{97} -0.708204i q^{98} -4.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9	$ 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9} - 2 q^{11} - 16 q^{14} - 6 q^{16} - 10 q^{19} + 18 q^{21} + 10 q^{24} - 22 q^{26} + 20 q^{29} + 8 q^{31} + 14 q^{34} + 14 q^{36} + 26 q^{39} + 8 q^{41} - 6 q^{44} - 12 q^{46} - 18 q^{49} - 2 q^{51} - 10 q^{54} - 30 q^{56} + 20 q^{59} - 22 q^{61} - 8 q^{64} + 6 q^{66} + 6 q^{69} + 18 q^{71} - 6 q^{74} - 20 q^{76} + 4 q^{81} - 16 q^{84} - 12 q^{86} - 10 q^{89} - 62 q^{91} - 26 q^{94} - 22 q^{96} - 14 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 - 2 * q^11 - 16 * q^14 - 6 * q^16 - 10 * q^19 + 18 * q^21 + 10 * q^24 - 22 * q^26 + 20 * q^29 + 8 * q^31 + 14 * q^34 + 14 * q^36 + 26 * q^39 + 8 * q^41 - 6 * q^44 - 12 * q^46 - 18 * q^49 - 2 * q^51 - 10 * q^54 - 30 * q^56 + 20 * q^59 - 22 * q^61 - 8 * q^64 + 6 * q^66 + 6 * q^69 + 18 * q^71 - 6 * q^74 - 20 * q^76 + 4 * q^81 - 16 * q^84 - 12 * q^86 - 10 * q^89 - 62 * q^91 - 26 * q^94 - 22 * q^96 - 14 * 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 $

$2$	0.618034i	0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$	0.618034i	0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$	0.381966i	0.220528i	0.993902	+	0.110264i	$0.0351697\pi$
$3$	0.381966i	0.220528i	−0.993902	+	0.110264i	$0.964830\pi$
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	−0.236068	−0.0963743
$7$	2.85410i	1.07875i	0.842066	+	0.539375i	$0.181339\pi$
$7$	2.85410i	1.07875i	−0.842066	+	0.539375i	$0.818661\pi$
$8$	2.23607i	0.790569i
$9$	2.85410	0.951367
$10$	0	0
$11$	−1.61803	−0.487856	−0.243928	−	0.969793i	$-0.578436\pi$
$11$	−1.61803	−0.487856	−0.243928	+	0.969793i	$0.578436\pi$
$12$	0.618034i	0.178411i
$13$	3.47214i	0.962997i	0.876447	+	0.481499i	$0.159907\pi$
$13$	3.47214i	0.962997i	−0.876447	+	0.481499i	$0.840093\pi$
$14$	−1.76393	−0.471431
$15$	0	0
$16$	1.85410	0.463525
$17$	− 7.47214i	− 1.81226i	−0.423000	−	0.906130i	$-0.639023\pi$
$17$	− 7.47214i	− 1.81226i	0.423000	−	0.906130i	$-0.360977\pi$
$18$	1.76393i	0.415763i
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	0	0
$21$	−1.09017	−0.237895
$22$	− 1.00000i	− 0.213201i
$23$	4.85410i	1.01215i	0.862489	+	0.506075i	$0.168904\pi$
$23$	4.85410i	1.01215i	−0.862489	+	0.506075i	$0.831096\pi$
$24$	−0.854102	−0.174343
$25$	0	0
$26$	−2.14590	−0.420845
$27$	2.23607i	0.430331i
$28$	4.61803i	0.872726i
$29$	7.23607	1.34370	0.671852	−	0.740685i	$-0.265499\pi$
$29$	7.23607	1.34370	0.671852	+	0.740685i	$0.265499\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	5.61803i	0.993137i
$33$	− 0.618034i	− 0.107586i
$34$	4.61803	0.791986
$35$	0	0
$36$	4.61803	0.769672
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	− 3.61803i	− 0.586923i
$39$	−1.32624	−0.212368
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	− 0.673762i	− 0.103964i
$43$	8.47214i	1.29199i	0.763342	+	0.645994i	$0.223557\pi$
$43$	8.47214i	1.29199i	−0.763342	+	0.645994i	$0.776443\pi$
$44$	−2.61803	−0.394683
$45$	0	0
$46$	−3.00000	−0.442326
$47$	− 9.38197i	− 1.36850i	−0.729247	−	0.684250i	$-0.760130\pi$
$47$	− 9.38197i	− 1.36850i	0.729247	−	0.684250i	$-0.239870\pi$
$48$	0.708204i	0.102220i
$49$	−1.14590	−0.163700
$50$	0	0
$51$	2.85410	0.399654
$52$	5.61803i	0.779081i
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	−1.38197	−0.188062
$55$	0	0
$56$	−6.38197	−0.852826
$57$	− 2.23607i	− 0.296174i
$58$	4.47214i	0.587220i
$59$	−3.94427	−0.513500	−0.256750	−	0.966478i	$-0.582652\pi$
$59$	−3.94427	−0.513500	−0.256750	+	0.966478i	$0.582652\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	1.23607i	0.156981i
$63$	8.14590i	1.02629i
$64$	0.236068	0.0295085
$65$	0	0
$66$	0.381966	0.0470168
$67$	2.52786i	0.308828i	0.988006	+	0.154414i	$0.0493489\pi$
$67$	2.52786i	0.308828i	−0.988006	+	0.154414i	$0.950651\pi$
$68$	− 12.0902i	− 1.46615i
$69$	−1.85410	−0.223208
$70$	0	0
$71$	12.3262	1.46286	0.731428	−	0.681919i	$-0.238854\pi$
$71$	12.3262	1.46286	0.731428	+	0.681919i	$0.238854\pi$
$72$	6.38197i	0.752122i
$73$	− 6.85410i	− 0.802212i	−0.916032	−	0.401106i	$-0.868626\pi$
$73$	− 6.85410i	− 0.802212i	0.916032	−	0.401106i	$-0.131374\pi$
$74$	1.85410	0.215535
$75$	0	0
$76$	−9.47214	−1.08653
$77$	− 4.61803i	− 0.526274i
$78$	− 0.819660i	− 0.0928082i
$79$	−6.70820	−0.754732	−0.377366	−	0.926064i	$-0.623170\pi$
$79$	−6.70820	−0.754732	−0.377366	+	0.926064i	$0.623170\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	− 1.52786i	− 0.168724i
$83$	− 4.94427i	− 0.542704i	−0.962480	−	0.271352i	$-0.912529\pi$
$83$	− 4.94427i	− 0.542704i	0.962480	−	0.271352i	$-0.0874708\pi$
$84$	−1.76393	−0.192461
$85$	0	0
$86$	−5.23607	−0.564620
$87$	2.76393i	0.296325i
$88$	− 3.61803i	− 0.385684i
$89$	−1.38197	−0.146488	−0.0732441	−	0.997314i	$-0.523335\pi$
$89$	−1.38197	−0.146488	−0.0732441	+	0.997314i	$0.523335\pi$
$90$	0	0
$91$	−9.90983	−1.03883
$92$	7.85410i	0.818847i
$93$	0.763932i	0.0792161i
$94$	5.79837	0.598057
$95$	0	0
$96$	−2.14590	−0.219015
$97$	8.38197i	0.851060i	0.904944	+	0.425530i	$0.139912\pi$
$97$	8.38197i	0.851060i	−0.904944	+	0.425530i	$0.860088\pi$
$98$	− 0.708204i	− 0.0715394i
$99$	−4.61803	−0.464130
$100$	0	0
$101$	14.2361	1.41654	0.708271	−	0.705941i	$-0.249476\pi$
$101$	14.2361	1.41654	0.708271	+	0.705941i	$0.249476\pi$
$102$	1.76393i	0.174655i
$103$	− 11.8541i	− 1.16802i	−0.811747	−	0.584010i	$-0.801483\pi$
$103$	− 11.8541i	− 1.16802i	0.811747	−	0.584010i	$-0.198517\pi$
$104$	−7.76393	−0.761316
$105$	0	0
$106$	0.291796	0.0283417
$107$	− 12.7984i	− 1.23727i	−0.785680	−	0.618633i	$-0.787687\pi$
$107$	− 12.7984i	− 1.23727i	0.785680	−	0.618633i	$-0.212313\pi$
$108$	3.61803i	0.348145i
$109$	−9.14590	−0.876018	−0.438009	−	0.898971i	$-0.644316\pi$
$109$	−9.14590	−0.876018	−0.438009	+	0.898971i	$0.644316\pi$
$110$	0	0
$111$	1.14590	0.108764
$112$	5.29180i	0.500028i
$113$	− 12.7082i	− 1.19549i	−0.801687	−	0.597744i	$-0.796064\pi$
$113$	− 12.7082i	− 1.19549i	0.801687	−	0.597744i	$-0.203936\pi$
$114$	1.38197	0.129433
$115$	0	0
$116$	11.7082	1.08708
$117$	9.90983i	0.916164i
$118$	− 2.43769i	− 0.224408i
$119$	21.3262	1.95497
$120$	0	0
$121$	−8.38197	−0.761997
$122$	− 1.32624i	− 0.120072i
$123$	− 0.944272i	− 0.0851421i
$124$	3.23607	0.290607
$125$	0	0
$126$	−5.03444	−0.448504
$127$	− 13.8541i	− 1.22935i	−0.788779	−	0.614676i	$-0.789287\pi$
$127$	− 13.8541i	− 1.22935i	0.788779	−	0.614676i	$-0.210713\pi$
$128$	11.3820i	1.00603i
$129$	−3.23607	−0.284920
$130$	0	0
$131$	−5.23607	−0.457477	−0.228739	−	0.973488i	$-0.573460\pi$
$131$	−5.23607	−0.457477	−0.228739	+	0.973488i	$0.573460\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	− 16.7082i	− 1.44879i
$134$	−1.56231	−0.134963
$135$	0	0
$136$	16.7082	1.43272
$137$	− 15.7639i	− 1.34680i	−0.739277	−	0.673402i	$-0.764832\pi$
$137$	− 15.7639i	− 1.34680i	0.739277	−	0.673402i	$-0.235168\pi$
$138$	− 1.14590i	− 0.0975453i
$139$	18.4164	1.56206	0.781030	−	0.624494i	$-0.214695\pi$
$139$	18.4164	1.56206	0.781030	+	0.624494i	$0.214695\pi$
$140$	0	0
$141$	3.58359	0.301793
$142$	7.61803i	0.639291i
$143$	− 5.61803i	− 0.469804i
$144$	5.29180	0.440983
$145$	0	0
$146$	4.23607	0.350579
$147$	− 0.437694i	− 0.0361004i
$148$	− 4.85410i	− 0.399005i
$149$	11.3820	0.932447	0.466223	−	0.884667i	$-0.345614\pi$
$149$	11.3820	0.932447	0.466223	+	0.884667i	$0.345614\pi$
$150$	0	0
$151$	−1.94427	−0.158223	−0.0791113	−	0.996866i	$-0.525208\pi$
$151$	−1.94427	−0.158223	−0.0791113	+	0.996866i	$0.525208\pi$
$152$	− 13.0902i	− 1.06175i
$153$	− 21.3262i	− 1.72412i
$154$	2.85410	0.229990
$155$	0	0
$156$	−2.14590	−0.171809
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	− 4.14590i	− 0.329830i
$159$	0.180340	0.0143019
$160$	0	0
$161$	−13.8541	−1.09186
$162$	4.76393i	0.374290i
$163$	4.85410i	0.380203i	0.981764	+	0.190101i	$0.0608816\pi$
$163$	4.85410i	0.380203i	−0.981764	+	0.190101i	$0.939118\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	3.05573	0.237170
$167$	− 8.52786i	− 0.659906i	−0.943997	−	0.329953i	$-0.892967\pi$
$167$	− 8.52786i	− 0.659906i	0.943997	−	0.329953i	$-0.107033\pi$
$168$	− 2.43769i	− 0.188072i
$169$	0.944272	0.0726363
$170$	0	0
$171$	−16.7082	−1.27771
$172$	13.7082i	1.04524i
$173$	2.29180i	0.174242i	0.996198	+	0.0871210i	$0.0277667\pi$
$173$	2.29180i	0.174242i	−0.996198	+	0.0871210i	$0.972233\pi$
$174$	−1.70820	−0.129499
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 1.50658i	− 0.113241i
$178$	− 0.854102i	− 0.0640176i
$179$	−20.1246	−1.50418	−0.752092	−	0.659058i	$-0.770955\pi$
$179$	−20.1246	−1.50418	−0.752092	+	0.659058i	$0.770955\pi$
$180$	0	0
$181$	2.32624	0.172908	0.0864540	−	0.996256i	$-0.472446\pi$
$181$	2.32624	0.172908	0.0864540	+	0.996256i	$0.472446\pi$
$182$	− 6.12461i	− 0.453986i
$183$	− 0.819660i	− 0.0605910i
$184$	−10.8541	−0.800175
$185$	0	0
$186$	−0.472136	−0.0346187
$187$	12.0902i	0.884121i
$188$	− 15.1803i	− 1.10714i
$189$	−6.38197	−0.464220
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0.0901699i	0.00650746i
$193$	5.05573i	0.363919i	0.983306	+	0.181960i	$0.0582440\pi$
$193$	5.05573i	0.363919i	−0.983306	+	0.181960i	$0.941756\pi$
$194$	−5.18034	−0.371927
$195$	0	0
$196$	−1.85410	−0.132436
$197$	7.52786i	0.536338i	0.963372	+	0.268169i	$0.0864186\pi$
$197$	7.52786i	0.536338i	−0.963372	+	0.268169i	$0.913581\pi$
$198$	− 2.85410i	− 0.202832i
$199$	4.14590	0.293895	0.146947	−	0.989144i	$-0.453055\pi$
$199$	4.14590	0.293895	0.146947	+	0.989144i	$0.453055\pi$
$200$	0	0
$201$	−0.965558	−0.0681052
$202$	8.79837i	0.619051i
$203$	20.6525i	1.44952i
$204$	4.61803	0.323327
$205$	0	0
$206$	7.32624	0.510443
$207$	13.8541i	0.962927i
$208$	6.43769i	0.446374i
$209$	9.47214	0.655201
$210$	0	0
$211$	−9.18034	−0.632001	−0.316000	−	0.948759i	$-0.602340\pi$
$211$	−9.18034	−0.632001	−0.316000	+	0.948759i	$0.602340\pi$
$212$	− 0.763932i	− 0.0524671i
$213$	4.70820i	0.322601i
$214$	7.90983	0.540705
$215$	0	0
$216$	−5.00000	−0.340207
$217$	5.70820i	0.387498i
$218$	− 5.65248i	− 0.382834i
$219$	2.61803	0.176910
$220$	0	0
$221$	25.9443	1.74520
$222$	0.708204i	0.0475315i
$223$	− 6.85410i	− 0.458985i	−0.973310	−	0.229492i	$-0.926293\pi$
$223$	− 6.85410i	− 0.458985i	0.973310	−	0.229492i	$-0.0737066\pi$
$224$	−16.0344	−1.07135
$225$	0	0
$226$	7.85410	0.522447
$227$	− 5.23607i	− 0.347530i	−0.984787	−	0.173765i	$-0.944407\pi$
$227$	− 5.23607i	− 0.347530i	0.984787	−	0.173765i	$-0.0555933\pi$
$228$	− 3.61803i	− 0.239610i
$229$	16.7082	1.10411	0.552055	−	0.833808i	$-0.313844\pi$
$229$	16.7082	1.10411	0.552055	+	0.833808i	$0.313844\pi$
$230$	0	0
$231$	1.76393	0.116058
$232$	16.1803i	1.06229i
$233$	10.9098i	0.714727i	0.933965	+	0.357363i	$0.116324\pi$
$233$	10.9098i	0.714727i	−0.933965	+	0.357363i	$0.883676\pi$
$234$	−6.12461	−0.400378
$235$	0	0
$236$	−6.38197	−0.415431
$237$	− 2.56231i	− 0.166440i
$238$	13.1803i	0.854355i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−7.14590	−0.460308	−0.230154	−	0.973154i	$-0.573923\pi$
$241$	−7.14590	−0.460308	−0.230154	+	0.973154i	$0.573923\pi$
$242$	− 5.18034i	− 0.333005i
$243$	9.65248i	0.619207i
$244$	−3.47214	−0.222281
$245$	0	0
$246$	0.583592	0.0372085
$247$	− 20.3262i	− 1.29333i
$248$	4.47214i	0.283981i
$249$	1.88854	0.119682
$250$	0	0
$251$	13.1803	0.831936	0.415968	−	0.909379i	$-0.363443\pi$
$251$	13.1803	0.831936	0.415968	+	0.909379i	$0.363443\pi$
$252$	13.1803i	0.830283i
$253$	− 7.85410i	− 0.493783i
$254$	8.56231	0.537247
$255$	0	0
$256$	−6.56231	−0.410144
$257$	− 9.70820i	− 0.605581i	−0.953057	−	0.302791i	$-0.902082\pi$
$257$	− 9.70820i	− 0.605581i	0.953057	−	0.302791i	$-0.0979183\pi$
$258$	− 2.00000i	− 0.124515i
$259$	8.56231	0.532036
$260$	0	0
$261$	20.6525	1.27836
$262$	− 3.23607i	− 0.199925i
$263$	3.67376i	0.226534i	0.993565	+	0.113267i	$0.0361315\pi$
$263$	3.67376i	0.226534i	−0.993565	+	0.113267i	$0.963868\pi$
$264$	1.38197	0.0850541
$265$	0	0
$266$	10.3262	0.633142
$267$	− 0.527864i	− 0.0323048i
$268$	4.09017i	0.249847i
$269$	−26.1803	−1.59624	−0.798122	−	0.602496i	$-0.794173\pi$
$269$	−26.1803	−1.59624	−0.798122	+	0.602496i	$0.794173\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	− 13.8541i	− 0.840028i
$273$	− 3.78522i	− 0.229092i
$274$	9.74265	0.588575
$275$	0	0
$276$	−3.00000	−0.180579
$277$	28.7082i	1.72491i	0.506135	+	0.862454i	$0.331074\pi$
$277$	28.7082i	1.72491i	−0.506135	+	0.862454i	$0.668926\pi$
$278$	11.3820i	0.682645i
$279$	5.70820	0.341741
$280$	0	0
$281$	−25.3607	−1.51289	−0.756446	−	0.654057i	$-0.773066\pi$
$281$	−25.3607	−1.51289	−0.756446	+	0.654057i	$0.773066\pi$
$282$	2.21478i	0.131888i
$283$	21.3607i	1.26976i	0.772611	+	0.634880i	$0.218951\pi$
$283$	21.3607i	1.26976i	−0.772611	+	0.634880i	$0.781049\pi$
$284$	19.9443	1.18347
$285$	0	0
$286$	3.47214	0.205312
$287$	− 7.05573i	− 0.416486i
$288$	16.0344i	0.944839i
$289$	−38.8328	−2.28428
$290$	0	0
$291$	−3.20163	−0.187683
$292$	− 11.0902i	− 0.649003i
$293$	− 1.32624i	− 0.0774796i	−0.999249	−	0.0387398i	$-0.987666\pi$
$293$	− 1.32624i	− 0.0774796i	0.999249	−	0.0387398i	$-0.0123344\pi$
$294$	0.270510	0.0157765
$295$	0	0
$296$	6.70820	0.389906
$297$	− 3.61803i	− 0.209940i
$298$	7.03444i	0.407494i
$299$	−16.8541	−0.974698
$300$	0	0
$301$	−24.1803	−1.39373
$302$	− 1.20163i	− 0.0691458i
$303$	5.43769i	0.312387i
$304$	−10.8541	−0.622525
$305$	0	0
$306$	13.1803	0.753470
$307$	32.9787i	1.88219i	0.338136	+	0.941097i	$0.390204\pi$
$307$	32.9787i	1.88219i	−0.338136	+	0.941097i	$0.609796\pi$
$308$	− 7.47214i	− 0.425764i
$309$	4.52786	0.257581
$310$	0	0
$311$	−5.56231	−0.315409	−0.157705	−	0.987486i	$-0.550409\pi$
$311$	−5.56231	−0.315409	−0.157705	+	0.987486i	$0.550409\pi$
$312$	− 2.96556i	− 0.167892i
$313$	− 8.03444i	− 0.454134i	−0.973879	−	0.227067i	$-0.927086\pi$
$313$	− 8.03444i	− 0.454134i	0.973879	−	0.227067i	$-0.0729136\pi$
$314$	8.03444	0.453410
$315$	0	0
$316$	−10.8541	−0.610591
$317$	− 17.7984i	− 0.999656i	−0.866125	−	0.499828i	$-0.833396\pi$
$317$	− 17.7984i	− 0.999656i	0.866125	−	0.499828i	$-0.166604\pi$
$318$	0.111456i	0.00625015i
$319$	−11.7082	−0.655534
$320$	0	0
$321$	4.88854	0.272852
$322$	− 8.56231i	− 0.477159i
$323$	43.7426i	2.43391i
$324$	12.4721	0.692896
$325$	0	0
$326$	−3.00000	−0.166155
$327$	− 3.49342i	− 0.193187i
$328$	− 5.52786i	− 0.305225i
$329$	26.7771	1.47627
$330$	0	0
$331$	28.7082	1.57795	0.788973	−	0.614428i	$-0.210613\pi$
$331$	28.7082	1.57795	0.788973	+	0.614428i	$0.210613\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 8.56231i	− 0.469211i
$334$	5.27051	0.288389
$335$	0	0
$336$	−2.02129	−0.110270
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	0.583592i	0.0317432i
$339$	4.85410	0.263639
$340$	0	0
$341$	−3.23607	−0.175243
$342$	− 10.3262i	− 0.558379i
$343$	16.7082i	0.902158i
$344$	−18.9443	−1.02141
$345$	0	0
$346$	−1.41641	−0.0761466
$347$	− 10.2361i	− 0.549501i	−0.961516	−	0.274750i	$-0.911405\pi$
$347$	− 10.2361i	− 0.549501i	0.961516	−	0.274750i	$-0.0885952\pi$
$348$	4.47214i	0.239732i
$349$	−20.1246	−1.07725	−0.538623	−	0.842547i	$-0.681055\pi$
$349$	−20.1246	−1.07725	−0.538623	+	0.842547i	$0.681055\pi$
$350$	0	0
$351$	−7.76393	−0.414408
$352$	− 9.09017i	− 0.484508i
$353$	24.6525i	1.31212i	0.754709	+	0.656059i	$0.227778\pi$
$353$	24.6525i	1.31212i	−0.754709	+	0.656059i	$0.772222\pi$
$354$	0.931116	0.0494883
$355$	0	0
$356$	−2.23607	−0.118511
$357$	8.14590i	0.431127i
$358$	− 12.4377i	− 0.657353i
$359$	−5.65248	−0.298326	−0.149163	−	0.988813i	$-0.547658\pi$
$359$	−5.65248	−0.298326	−0.149163	+	0.988813i	$0.547658\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	1.43769i	0.0755635i
$363$	− 3.20163i	− 0.168042i
$364$	−16.0344	−0.840433
$365$	0	0
$366$	0.506578	0.0264792
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	9.00000i	0.469157i
$369$	−7.05573	−0.367307
$370$	0	0
$371$	1.34752	0.0699600
$372$	1.23607i	0.0640871i
$373$	− 11.8541i	− 0.613782i	−0.951745	−	0.306891i	$-0.900711\pi$
$373$	− 11.8541i	− 0.613782i	0.951745	−	0.306891i	$-0.0992887\pi$
$374$	−7.47214	−0.386375
$375$	0	0
$376$	20.9787	1.08189
$377$	25.1246i	1.29398i
$378$	− 3.94427i	− 0.202871i
$379$	−7.23607	−0.371692	−0.185846	−	0.982579i	$-0.559503\pi$
$379$	−7.23607	−0.371692	−0.185846	+	0.982579i	$0.559503\pi$
$380$	0	0
$381$	5.29180	0.271107
$382$	− 1.85410i	− 0.0948641i
$383$	27.6180i	1.41122i	0.708603	+	0.705608i	$0.249326\pi$
$383$	27.6180i	1.41122i	−0.708603	+	0.705608i	$0.750674\pi$
$384$	−4.34752	−0.221859
$385$	0	0
$386$	−3.12461	−0.159039
$387$	24.1803i	1.22916i
$388$	13.5623i	0.688522i
$389$	5.32624	0.270051	0.135025	−	0.990842i	$-0.456888\pi$
$389$	5.32624	0.270051	0.135025	+	0.990842i	$0.456888\pi$
$390$	0	0
$391$	36.2705	1.83428
$392$	− 2.56231i	− 0.129416i
$393$	− 2.00000i	− 0.100887i
$394$	−4.65248	−0.234388
$395$	0	0
$396$	−7.47214	−0.375489
$397$	15.4164i	0.773727i	0.922137	+	0.386864i	$0.126442\pi$
$397$	15.4164i	0.773727i	−0.922137	+	0.386864i	$0.873558\pi$
$398$	2.56231i	0.128437i
$399$	6.38197	0.319498
$400$	0	0
$401$	−5.56231	−0.277768	−0.138884	−	0.990309i	$-0.544352\pi$
$401$	−5.56231	−0.277768	−0.138884	+	0.990309i	$0.544352\pi$
$402$	− 0.596748i	− 0.0297631i
$403$	6.94427i	0.345919i
$404$	23.0344	1.14601
$405$	0	0
$406$	−12.7639	−0.633463
$407$	4.85410i	0.240609i
$408$	6.38197i	0.315954i
$409$	18.2918	0.904471	0.452236	−	0.891899i	$-0.350627\pi$
$409$	18.2918	0.904471	0.452236	+	0.891899i	$0.350627\pi$
$410$	0	0
$411$	6.02129	0.297008
$412$	− 19.1803i	− 0.944948i
$413$	− 11.2574i	− 0.553938i
$414$	−8.56231	−0.420814
$415$	0	0
$416$	−19.5066	−0.956389
$417$	7.03444i	0.344478i
$418$	5.85410i	0.286333i
$419$	27.7639	1.35636	0.678178	−	0.734897i	$-0.262770\pi$
$419$	27.7639	1.35636	0.678178	+	0.734897i	$0.262770\pi$
$420$	0	0
$421$	−9.18034	−0.447422	−0.223711	−	0.974655i	$-0.571817\pi$
$421$	−9.18034	−0.447422	−0.223711	+	0.974655i	$0.571817\pi$
$422$	− 5.67376i	− 0.276194i
$423$	− 26.7771i	− 1.30195i
$424$	1.05573	0.0512707
$425$	0	0
$426$	−2.90983	−0.140982
$427$	− 6.12461i	− 0.296391i
$428$	− 20.7082i	− 1.00097i
$429$	2.14590	0.103605
$430$	0	0
$431$	−31.4164	−1.51328	−0.756638	−	0.653835i	$-0.773159\pi$
$431$	−31.4164	−1.51328	−0.756638	+	0.653835i	$0.773159\pi$
$432$	4.14590i	0.199470i
$433$	− 6.85410i	− 0.329387i	−0.986345	−	0.164694i	$-0.947336\pi$
$433$	− 6.85410i	− 0.329387i	0.986345	−	0.164694i	$-0.0526635\pi$
$434$	−3.52786	−0.169343
$435$	0	0
$436$	−14.7984	−0.708714
$437$	− 28.4164i	− 1.35934i
$438$	1.61803i	0.0773127i
$439$	24.2705	1.15837	0.579184	−	0.815197i	$-0.303371\pi$
$439$	24.2705	1.15837	0.579184	+	0.815197i	$0.303371\pi$
$440$	0	0
$441$	−3.27051	−0.155739
$442$	16.0344i	0.762681i
$443$	− 14.2918i	− 0.679024i	−0.940602	−	0.339512i	$-0.889738\pi$
$443$	− 14.2918i	− 0.679024i	0.940602	−	0.339512i	$-0.110262\pi$
$444$	1.85410	0.0879918
$445$	0	0
$446$	4.23607	0.200584
$447$	4.34752i	0.205631i
$448$	0.673762i	0.0318323i
$449$	29.0689	1.37185	0.685923	−	0.727674i	$-0.259399\pi$
$449$	29.0689	1.37185	0.685923	+	0.727674i	$0.259399\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	− 20.5623i	− 0.967170i
$453$	− 0.742646i	− 0.0348925i
$454$	3.23607	0.151876
$455$	0	0
$456$	5.00000	0.234146
$457$	− 0.888544i	− 0.0415643i	−0.999784	−	0.0207822i	$-0.993384\pi$
$457$	− 0.888544i	− 0.0415643i	0.999784	−	0.0207822i	$-0.00661564\pi$
$458$	10.3262i	0.482513i
$459$	16.7082	0.779872
$460$	0	0
$461$	−26.0902	−1.21514	−0.607570	−	0.794266i	$-0.707856\pi$
$461$	−26.0902	−1.21514	−0.607570	+	0.794266i	$0.707856\pi$
$462$	1.09017i	0.0507193i
$463$	19.3262i	0.898166i	0.893490	+	0.449083i	$0.148249\pi$
$463$	19.3262i	0.898166i	−0.893490	+	0.449083i	$0.851751\pi$
$464$	13.4164	0.622841
$465$	0	0
$466$	−6.74265	−0.312347
$467$	12.3262i	0.570390i	0.958469	+	0.285195i	$0.0920584\pi$
$467$	12.3262i	0.570390i	−0.958469	+	0.285195i	$0.907942\pi$
$468$	16.0344i	0.741192i
$469$	−7.21478	−0.333148
$470$	0	0
$471$	4.96556	0.228801
$472$	− 8.81966i	− 0.405958i
$473$	− 13.7082i	− 0.630304i
$474$	1.58359	0.0727368
$475$	0	0
$476$	34.5066	1.58161
$477$	− 1.34752i	− 0.0616989i
$478$	0	0
$479$	−27.2361	−1.24445	−0.622224	−	0.782839i	$-0.713771\pi$
$479$	−27.2361	−1.24445	−0.622224	+	0.782839i	$0.713771\pi$
$480$	0	0
$481$	10.4164	0.474947
$482$	− 4.41641i	− 0.201162i
$483$	− 5.29180i	− 0.240785i
$484$	−13.5623	−0.616468
$485$	0	0
$486$	−5.96556	−0.270603
$487$	39.3607i	1.78360i	0.452427	+	0.891801i	$0.350558\pi$
$487$	39.3607i	1.78360i	−0.452427	+	0.891801i	$0.649442\pi$
$488$	− 4.79837i	− 0.217212i
$489$	−1.85410	−0.0838454
$490$	0	0
$491$	13.0557	0.589197	0.294598	−	0.955621i	$-0.404814\pi$
$491$	13.0557	0.589197	0.294598	+	0.955621i	$0.404814\pi$
$492$	− 1.52786i	− 0.0688814i
$493$	− 54.0689i	− 2.43514i
$494$	12.5623	0.565205
$495$	0	0
$496$	3.70820	0.166503
$497$	35.1803i	1.57805i
$498$	1.16718i	0.0523028i
$499$	1.58359	0.0708913	0.0354457	−	0.999372i	$-0.488715\pi$
$499$	1.58359	0.0708913	0.0354457	+	0.999372i	$0.488715\pi$
$500$	0	0
$501$	3.25735	0.145528
$502$	8.14590i	0.363569i
$503$	9.00000i	0.401290i	0.979664	+	0.200645i	$0.0643038\pi$
$503$	9.00000i	0.401290i	−0.979664	+	0.200645i	$0.935696\pi$
$504$	−18.2148	−0.811351
$505$	0	0
$506$	4.85410	0.215791
$507$	0.360680i	0.0160184i
$508$	− 22.4164i	− 0.994567i
$509$	−33.6180	−1.49009	−0.745047	−	0.667012i	$-0.767573\pi$
$509$	−33.6180	−1.49009	−0.745047	+	0.667012i	$0.767573\pi$
$510$	0	0
$511$	19.5623	0.865385
$512$	18.7082i	0.826794i
$513$	− 13.0902i	− 0.577945i
$514$	6.00000	0.264649
$515$	0	0
$516$	−5.23607	−0.230505
$517$	15.1803i	0.667631i
$518$	5.29180i	0.232508i
$519$	−0.875388	−0.0384253
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	12.7639i	0.558662i
$523$	19.1246i	0.836261i	0.908387	+	0.418130i	$0.137315\pi$
$523$	19.1246i	0.836261i	−0.908387	+	0.418130i	$0.862685\pi$
$524$	−8.47214	−0.370107
$525$	0	0
$526$	−2.27051	−0.0989989
$527$	− 14.9443i	− 0.650983i
$528$	− 1.14590i	− 0.0498688i
$529$	−0.562306	−0.0244481
$530$	0	0
$531$	−11.2574	−0.488528
$532$	− 27.0344i	− 1.17209i
$533$	− 8.58359i	− 0.371797i
$534$	0.326238	0.0141177
$535$	0	0
$536$	−5.65248	−0.244150
$537$	− 7.68692i	− 0.331715i
$538$	− 16.1803i	− 0.697584i
$539$	1.85410	0.0798618
$540$	0	0
$541$	10.2918	0.442479	0.221239	−	0.975220i	$-0.428990\pi$
$541$	10.2918	0.442479	0.221239	+	0.975220i	$0.428990\pi$
$542$	− 8.03444i	− 0.345109i
$543$	0.888544i	0.0381311i
$544$	41.9787	1.79982
$545$	0	0
$546$	2.33939	0.100117
$547$	18.7082i	0.799905i	0.916536	+	0.399953i	$0.130973\pi$
$547$	18.7082i	0.799905i	−0.916536	+	0.399953i	$0.869027\pi$
$548$	− 25.5066i	− 1.08959i
$549$	−6.12461	−0.261392
$550$	0	0
$551$	−42.3607	−1.80463
$552$	− 4.14590i	− 0.176461i
$553$	− 19.1459i	− 0.814166i
$554$	−17.7426	−0.753813
$555$	0	0
$556$	29.7984	1.26373
$557$	− 16.6180i	− 0.704129i	−0.935976	−	0.352064i	$-0.885480\pi$
$557$	− 16.6180i	− 0.704129i	0.935976	−	0.352064i	$-0.114520\pi$
$558$	3.52786i	0.149346i
$559$	−29.4164	−1.24418
$560$	0	0
$561$	−4.61803	−0.194974
$562$	− 15.6738i	− 0.661158i
$563$	− 17.8328i	− 0.751564i	−0.926708	−	0.375782i	$-0.877374\pi$
$563$	− 17.8328i	− 0.751564i	0.926708	−	0.375782i	$-0.122626\pi$
$564$	5.79837	0.244156
$565$	0	0
$566$	−13.2016	−0.554906
$567$	22.0000i	0.923913i
$568$	27.5623i	1.15649i
$569$	7.76393	0.325481	0.162740	−	0.986669i	$-0.447967\pi$
$569$	7.76393	0.325481	0.162740	+	0.986669i	$0.447967\pi$
$570$	0	0
$571$	28.7082	1.20140	0.600700	−	0.799474i	$-0.294888\pi$
$571$	28.7082	1.20140	0.600700	+	0.799474i	$0.294888\pi$
$572$	− 9.09017i	− 0.380079i
$573$	− 1.14590i	− 0.0478706i
$574$	4.36068	0.182011
$575$	0	0
$576$	0.673762	0.0280734
$577$	− 32.5967i	− 1.35702i	−0.734591	−	0.678510i	$-0.762626\pi$
$577$	− 32.5967i	− 1.35702i	0.734591	−	0.678510i	$-0.237374\pi$
$578$	− 24.0000i	− 0.998268i
$579$	−1.93112	−0.0802545
$580$	0	0
$581$	14.1115	0.585442
$582$	− 1.97871i	− 0.0820203i
$583$	0.763932i	0.0316388i
$584$	15.3262	0.634204
$585$	0	0
$586$	0.819660	0.0338598
$587$	7.85410i	0.324173i	0.986777	+	0.162087i	$0.0518224\pi$
$587$	7.85410i	0.324173i	−0.986777	+	0.162087i	$0.948178\pi$
$588$	− 0.708204i	− 0.0292058i
$589$	−11.7082	−0.482428
$590$	0	0
$591$	−2.87539	−0.118278
$592$	− 5.56231i	− 0.228609i
$593$	− 21.0000i	− 0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$	− 21.0000i	− 0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$	2.23607	0.0917470
$595$	0	0
$596$	18.4164	0.754365
$597$	1.58359i	0.0648121i
$598$	− 10.4164i	− 0.425959i
$599$	−2.76393	−0.112931	−0.0564656	−	0.998405i	$-0.517983\pi$
$599$	−2.76393	−0.112931	−0.0564656	+	0.998405i	$0.517983\pi$
$600$	0	0
$601$	−1.81966	−0.0742255	−0.0371127	−	0.999311i	$-0.511816\pi$
$601$	−1.81966	−0.0742255	−0.0371127	+	0.999311i	$0.511816\pi$
$602$	− 14.9443i	− 0.609083i
$603$	7.21478i	0.293809i
$604$	−3.14590	−0.128005
$605$	0	0
$606$	−3.36068	−0.136518
$607$	− 31.4164i	− 1.27515i	−0.770387	−	0.637576i	$-0.779937\pi$
$607$	− 31.4164i	− 1.27515i	0.770387	−	0.637576i	$-0.220063\pi$
$608$	− 32.8885i	− 1.33381i
$609$	−7.88854	−0.319660
$610$	0	0
$611$	32.5755	1.31786
$612$	− 34.5066i	− 1.39485i
$613$	− 14.2918i	− 0.577240i	−0.957444	−	0.288620i	$-0.906804\pi$
$613$	− 14.2918i	− 0.577240i	0.957444	−	0.288620i	$-0.0931965\pi$
$614$	−20.3820	−0.822549
$615$	0	0
$616$	10.3262	0.416056
$617$	26.6738i	1.07385i	0.843631	+	0.536923i	$0.180413\pi$
$617$	26.6738i	1.07385i	−0.843631	+	0.536923i	$0.819587\pi$
$618$	2.79837i	0.112567i
$619$	36.3050	1.45922	0.729610	−	0.683864i	$-0.239702\pi$
$619$	36.3050	1.45922	0.729610	+	0.683864i	$0.239702\pi$
$620$	0	0
$621$	−10.8541	−0.435560
$622$	− 3.43769i	− 0.137839i
$623$	− 3.94427i	− 0.158024i
$624$	−2.45898	−0.0984380
$625$	0	0
$626$	4.96556	0.198464
$627$	3.61803i	0.144490i
$628$	− 21.0344i	− 0.839366i
$629$	−22.4164	−0.893801
$630$	0	0
$631$	−38.5279	−1.53377	−0.766885	−	0.641785i	$-0.778194\pi$
$631$	−38.5279	−1.53377	−0.766885	+	0.641785i	$0.778194\pi$
$632$	− 15.0000i	− 0.596668i
$633$	− 3.50658i	− 0.139374i
$634$	11.0000	0.436866
$635$	0	0
$636$	0.291796	0.0115705
$637$	− 3.97871i	− 0.157642i
$638$	− 7.23607i	− 0.286479i
$639$	35.1803	1.39171
$640$	0	0
$641$	30.5410	1.20630	0.603149	−	0.797629i	$-0.293912\pi$
$641$	30.5410	1.20630	0.603149	+	0.797629i	$0.293912\pi$
$642$	3.02129i	0.119241i
$643$	− 13.7639i	− 0.542796i	−0.962467	−	0.271398i	$-0.912514\pi$
$643$	− 13.7639i	− 0.542796i	0.962467	−	0.271398i	$-0.0874860\pi$
$644$	−22.4164	−0.883330
$645$	0	0
$646$	−27.0344	−1.06366
$647$	− 32.5967i	− 1.28151i	−0.767745	−	0.640755i	$-0.778621\pi$
$647$	− 32.5967i	− 1.28151i	0.767745	−	0.640755i	$-0.221379\pi$
$648$	17.2361i	0.677097i
$649$	6.38197	0.250514
$650$	0	0
$651$	−2.18034	−0.0854543
$652$	7.85410i	0.307590i
$653$	7.74265i	0.302993i	0.988458	+	0.151497i	$0.0484093\pi$
$653$	7.74265i	0.302993i	−0.988458	+	0.151497i	$0.951591\pi$
$654$	2.15905	0.0844257
$655$	0	0
$656$	−4.58359	−0.178959
$657$	− 19.5623i	− 0.763198i
$658$	16.5492i	0.645153i
$659$	−35.4508	−1.38097	−0.690485	−	0.723347i	$-0.742603\pi$
$659$	−35.4508	−1.38097	−0.690485	+	0.723347i	$0.742603\pi$
$660$	0	0
$661$	33.7082	1.31110	0.655549	−	0.755153i	$-0.272437\pi$
$661$	33.7082	1.31110	0.655549	+	0.755153i	$0.272437\pi$
$662$	17.7426i	0.689588i
$663$	9.90983i	0.384866i
$664$	11.0557	0.429045
$665$	0	0
$666$	5.29180	0.205053
$667$	35.1246i	1.36003i
$668$	− 13.7984i	− 0.533875i
$669$	2.61803	0.101219
$670$	0	0
$671$	3.47214	0.134040
$672$	− 6.12461i	− 0.236262i
$673$	12.4164i	0.478617i	0.970944	+	0.239309i	$0.0769208\pi$
$673$	12.4164i	0.478617i	−0.970944	+	0.239309i	$0.923079\pi$
$674$	−1.23607	−0.0476116
$675$	0	0
$676$	1.52786	0.0587640
$677$	23.5066i	0.903431i	0.892162	+	0.451716i	$0.149188\pi$
$677$	23.5066i	0.903431i	−0.892162	+	0.451716i	$0.850812\pi$
$678$	3.00000i	0.115214i
$679$	−23.9230	−0.918080
$680$	0	0
$681$	2.00000	0.0766402
$682$	− 2.00000i	− 0.0765840i
$683$	− 26.3262i	− 1.00735i	−0.863895	−	0.503673i	$-0.831982\pi$
$683$	− 26.3262i	− 1.00735i	0.863895	−	0.503673i	$-0.168018\pi$
$684$	−27.0344	−1.03369
$685$	0	0
$686$	−10.3262	−0.394258
$687$	6.38197i	0.243487i
$688$	15.7082i	0.598870i
$689$	1.63932	0.0624531
$690$	0	0
$691$	−51.5410	−1.96071	−0.980356	−	0.197234i	$-0.936804\pi$
$691$	−51.5410	−1.96071	−0.980356	+	0.197234i	$0.936804\pi$
$692$	3.70820i	0.140965i
$693$	− 13.1803i	− 0.500680i
$694$	6.32624	0.240141
$695$	0	0
$696$	−6.18034	−0.234265
$697$	18.4721i	0.699682i
$698$	− 12.4377i	− 0.470774i
$699$	−4.16718	−0.157617
$700$	0	0
$701$	5.94427	0.224512	0.112256	−	0.993679i	$-0.464192\pi$
$701$	5.94427	0.224512	0.112256	+	0.993679i	$0.464192\pi$
$702$	− 4.79837i	− 0.181103i
$703$	17.5623i	0.662375i
$704$	−0.381966	−0.0143959
$705$	0	0
$706$	−15.2361	−0.573417
$707$	40.6312i	1.52809i
$708$	− 2.43769i	− 0.0916142i
$709$	40.1246	1.50691	0.753456	−	0.657499i	$-0.228385\pi$
$709$	40.1246	1.50691	0.753456	+	0.657499i	$0.228385\pi$
$710$	0	0
$711$	−19.1459	−0.718027
$712$	− 3.09017i	− 0.115809i
$713$	9.70820i	0.363575i
$714$	−5.03444	−0.188409
$715$	0	0
$716$	−32.5623	−1.21691
$717$	0	0
$718$	− 3.49342i	− 0.130373i
$719$	8.21478	0.306360	0.153180	−	0.988198i	$-0.451049\pi$
$719$	8.21478	0.306360	0.153180	+	0.988198i	$0.451049\pi$
$720$	0	0
$721$	33.8328	1.26000
$722$	9.43769i	0.351235i
$723$	− 2.72949i	− 0.101511i
$724$	3.76393	0.139885
$725$	0	0
$726$	1.97871	0.0734370
$727$	17.9787i	0.666794i	0.942787	+	0.333397i	$0.108195\pi$
$727$	17.9787i	0.666794i	−0.942787	+	0.333397i	$0.891805\pi$
$728$	− 22.1591i	− 0.821269i
$729$	19.4377	0.719915
$730$	0	0
$731$	63.3050	2.34142
$732$	− 1.32624i	− 0.0490192i
$733$	25.8328i	0.954157i	0.878861	+	0.477078i	$0.158304\pi$
$733$	25.8328i	0.954157i	−0.878861	+	0.477078i	$0.841696\pi$
$734$	11.1246	0.410617
$735$	0	0
$736$	−27.2705	−1.00520
$737$	− 4.09017i	− 0.150663i
$738$	− 4.36068i	− 0.160519i
$739$	−20.5279	−0.755130	−0.377565	−	0.925983i	$-0.623238\pi$
$739$	−20.5279	−0.755130	−0.377565	+	0.925983i	$0.623238\pi$
$740$	0	0
$741$	7.76393	0.285215
$742$	0.832816i	0.0305736i
$743$	37.0902i	1.36071i	0.732884	+	0.680353i	$0.238174\pi$
$743$	37.0902i	1.36071i	−0.732884	+	0.680353i	$0.761826\pi$
$744$	−1.70820	−0.0626258
$745$	0	0
$746$	7.32624	0.268233
$747$	− 14.1115i	− 0.516311i
$748$	19.5623i	0.715269i
$749$	36.5279	1.33470
$750$	0	0
$751$	9.03444	0.329671	0.164836	−	0.986321i	$-0.447291\pi$
$751$	9.03444	0.329671	0.164836	+	0.986321i	$0.447291\pi$
$752$	− 17.3951i	− 0.634335i
$753$	5.03444i	0.183465i
$754$	−15.5279	−0.565491
$755$	0	0
$756$	−10.3262	−0.375562
$757$	0.167184i	0.00607642i	0.999995	+	0.00303821i	$0.000967093\pi$
$757$	0.167184i	0.00607642i	−0.999995	+	0.00303821i	$0.999033\pi$
$758$	− 4.47214i	− 0.162435i
$759$	3.00000	0.108893
$760$	0	0
$761$	6.27051	0.227306	0.113653	−	0.993521i	$-0.463745\pi$
$761$	6.27051	0.227306	0.113653	+	0.993521i	$0.463745\pi$
$762$	3.27051i	0.118478i
$763$	− 26.1033i	− 0.945004i
$764$	−4.85410	−0.175615
$765$	0	0
$766$	−17.0689	−0.616724
$767$	− 13.6950i	− 0.494500i
$768$	− 2.50658i	− 0.0904483i
$769$	−33.5410	−1.20952	−0.604760	−	0.796408i	$-0.706731\pi$
$769$	−33.5410	−1.20952	−0.604760	+	0.796408i	$0.706731\pi$
$770$	0	0
$771$	3.70820	0.133548
$772$	8.18034i	0.294417i
$773$	− 45.7214i	− 1.64448i	−0.569139	−	0.822242i	$-0.692723\pi$
$773$	− 45.7214i	− 1.64448i	0.569139	−	0.822242i	$-0.307277\pi$
$774$	−14.9443	−0.537161
$775$	0	0
$776$	−18.7426	−0.672822
$777$	3.27051i	0.117329i
$778$	3.29180i	0.118017i
$779$	14.4721	0.518518
$780$	0	0
$781$	−19.9443	−0.713662
$782$	22.4164i	0.801609i
$783$	16.1803i	0.578238i
$784$	−2.12461	−0.0758790
$785$	0	0
$786$	1.23607	0.0440891
$787$	4.88854i	0.174258i	0.996197	+	0.0871289i	$0.0277692\pi$
$787$	4.88854i	0.174258i	−0.996197	+	0.0871289i	$0.972231\pi$
$788$	12.1803i	0.433907i
$789$	−1.40325	−0.0499571
$790$	0	0
$791$	36.2705	1.28963
$792$	− 10.3262i	− 0.366927i
$793$	− 7.45085i	− 0.264587i
$794$	−9.52786	−0.338131
$795$	0	0
$796$	6.70820	0.237766
$797$	19.6869i	0.697346i	0.937244	+	0.348673i	$0.113368\pi$
$797$	19.6869i	0.697346i	−0.937244	+	0.348673i	$0.886632\pi$
$798$	3.94427i	0.139626i
$799$	−70.1033	−2.48008
$800$	0	0
$801$	−3.94427	−0.139364
$802$	− 3.43769i	− 0.121389i
$803$	11.0902i	0.391364i
$804$	−1.56231	−0.0550983
$805$	0	0
$806$	−4.29180	−0.151172
$807$	− 10.0000i	− 0.352017i
$808$	31.8328i	1.11987i
$809$	−20.4508	−0.719014	−0.359507	−	0.933142i	$-0.617055\pi$
$809$	−20.4508	−0.719014	−0.359507	+	0.933142i	$0.617055\pi$
$810$	0	0
$811$	−38.1246	−1.33874	−0.669368	−	0.742931i	$-0.733435\pi$
$811$	−38.1246	−1.33874	−0.669368	+	0.742931i	$0.733435\pi$
$812$	33.4164i	1.17269i
$813$	− 4.96556i	− 0.174150i
$814$	−3.00000	−0.105150
$815$	0	0
$816$	5.29180	0.185250
$817$	− 49.5967i	− 1.73517i
$818$	11.3050i	0.395268i
$819$	−28.2837	−0.988311
$820$	0	0
$821$	−13.5279	−0.472126	−0.236063	−	0.971738i	$-0.575857\pi$
$821$	−13.5279	−0.472126	−0.236063	+	0.971738i	$0.575857\pi$
$822$	3.72136i	0.129797i
$823$	− 12.7082i	− 0.442980i	−0.975163	−	0.221490i	$-0.928908\pi$
$823$	− 12.7082i	− 0.442980i	0.975163	−	0.221490i	$-0.0710920\pi$
$824$	26.5066	0.923400
$825$	0	0
$826$	6.95743	0.242080
$827$	47.1246i	1.63868i	0.573306	+	0.819342i	$0.305661\pi$
$827$	47.1246i	1.63868i	−0.573306	+	0.819342i	$0.694339\pi$
$828$	22.4164i	0.779024i
$829$	55.1246	1.91456	0.957278	−	0.289168i	$-0.0933785\pi$
$829$	55.1246	1.91456	0.957278	+	0.289168i	$0.0933785\pi$
$830$	0	0
$831$	−10.9656	−0.380391
$832$	0.819660i	0.0284166i
$833$	8.56231i	0.296666i
$834$	−4.34752	−0.150542
$835$	0	0
$836$	15.3262	0.530069
$837$	4.47214i	0.154580i
$838$	17.1591i	0.592750i
$839$	−11.5066	−0.397251	−0.198626	−	0.980075i	$-0.563648\pi$
$839$	−11.5066	−0.397251	−0.198626	+	0.980075i	$0.563648\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	− 5.67376i	− 0.195531i
$843$	− 9.68692i	− 0.333635i
$844$	−14.8541	−0.511299
$845$	0	0
$846$	16.5492	0.568972
$847$	− 23.9230i	− 0.822004i
$848$	− 0.875388i	− 0.0300610i
$849$	−8.15905	−0.280018
$850$	0	0
$851$	14.5623	0.499189
$852$	7.61803i	0.260990i
$853$	− 8.68692i	− 0.297434i	−0.988880	−	0.148717i	$-0.952486\pi$
$853$	− 8.68692i	− 0.297434i	0.988880	−	0.148717i	$-0.0475144\pi$
$854$	3.78522	0.129528
$855$	0	0
$856$	28.6180	0.978144
$857$	36.0689i	1.23209i	0.787711	+	0.616045i	$0.211266\pi$
$857$	36.0689i	1.23209i	−0.787711	+	0.616045i	$0.788734\pi$
$858$	1.32624i	0.0452770i
$859$	12.2361	0.417489	0.208745	−	0.977970i	$-0.433062\pi$
$859$	12.2361	0.417489	0.208745	+	0.977970i	$0.433062\pi$
$860$	0	0
$861$	2.69505	0.0918470
$862$	− 19.4164i	− 0.661325i
$863$	49.2492i	1.67646i	0.545314	+	0.838232i	$0.316410\pi$
$863$	49.2492i	1.67646i	−0.545314	+	0.838232i	$0.683590\pi$
$864$	−12.5623	−0.427378
$865$	0	0
$866$	4.23607	0.143947
$867$	− 14.8328i	− 0.503749i
$868$	9.23607i	0.313493i
$869$	10.8541	0.368200
$870$	0	0
$871$	−8.77709	−0.297400
$872$	− 20.4508i	− 0.692553i
$873$	23.9230i	0.809670i
$874$	17.5623	0.594054
$875$	0	0
$876$	4.23607	0.143123
$877$	− 28.9787i	− 0.978542i	−0.872132	−	0.489271i	$-0.837263\pi$
$877$	− 28.9787i	− 0.978542i	0.872132	−	0.489271i	$-0.162737\pi$
$878$	15.0000i	0.506225i
$879$	0.506578	0.0170864
$880$	0	0
$881$	−23.4508	−0.790079	−0.395040	−	0.918664i	$-0.629269\pi$
$881$	−23.4508	−0.790079	−0.395040	+	0.918664i	$0.629269\pi$
$882$	− 2.02129i	− 0.0680602i
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	41.9787	1.41190
$885$	0	0
$886$	8.83282	0.296744
$887$	− 42.0689i	− 1.41253i	−0.707945	−	0.706267i	$-0.750378\pi$
$887$	− 42.0689i	− 1.41253i	0.707945	−	0.706267i	$-0.249622\pi$
$888$	2.56231i	0.0859854i
$889$	39.5410	1.32616
$890$	0	0
$891$	−12.4721	−0.417832
$892$	− 11.0902i	− 0.371326i
$893$	54.9230i	1.83793i
$894$	−2.68692	−0.0898640
$895$	0	0
$896$	−32.4853	−1.08526
$897$	− 6.43769i	− 0.214948i
$898$	17.9656i	0.599518i
$899$	14.4721	0.482673
$900$	0	0
$901$	−3.52786	−0.117530
$902$	2.47214i	0.0823131i
$903$	− 9.23607i	− 0.307357i
$904$	28.4164	0.945116
$905$	0	0
$906$	0.458980	0.0152486
$907$	46.1459i	1.53225i	0.642692	+	0.766125i	$0.277818\pi$
$907$	46.1459i	1.53225i	−0.642692	+	0.766125i	$0.722182\pi$
$908$	− 8.47214i	− 0.281158i
$909$	40.6312	1.34765
$910$	0	0
$911$	41.4721	1.37403	0.687017	−	0.726642i	$-0.258920\pi$
$911$	41.4721	1.37403	0.687017	+	0.726642i	$0.258920\pi$
$912$	− 4.14590i	− 0.137284i
$913$	8.00000i	0.264761i
$914$	0.549150	0.0181643
$915$	0	0
$916$	27.0344	0.893243
$917$	− 14.9443i	− 0.493503i
$918$	10.3262i	0.340817i
$919$	−12.6393	−0.416933	−0.208466	−	0.978030i	$-0.566847\pi$
$919$	−12.6393	−0.416933	−0.208466	+	0.978030i	$0.566847\pi$
$920$	0	0
$921$	−12.5967	−0.415077
$922$	− 16.1246i	− 0.531036i
$923$	42.7984i	1.40873i
$924$	2.85410	0.0938931
$925$	0	0
$926$	−11.9443	−0.392513
$927$	− 33.8328i	− 1.11122i
$928$	40.6525i	1.33448i
$929$	15.6525	0.513541	0.256771	−	0.966472i	$-0.417342\pi$
$929$	15.6525	0.513541	0.256771	+	0.966472i	$0.417342\pi$
$930$	0	0
$931$	6.70820	0.219853
$932$	17.6525i	0.578226i
$933$	− 2.12461i	− 0.0695567i
$934$	−7.61803	−0.249270
$935$	0	0
$936$	−22.1591	−0.724291
$937$	8.70820i	0.284485i	0.989832	+	0.142242i	$0.0454312\pi$
$937$	8.70820i	0.284485i	−0.989832	+	0.142242i	$0.954569\pi$
$938$	− 4.45898i	− 0.145591i
$939$	3.06888	0.100149
$940$	0	0
$941$	−57.0689	−1.86039	−0.930196	−	0.367063i	$-0.880363\pi$
$941$	−57.0689	−1.86039	−0.930196	+	0.367063i	$0.880363\pi$
$942$	3.06888i	0.0999896i
$943$	− 12.0000i	− 0.390774i
$944$	−7.31308	−0.238021
$945$	0	0
$946$	8.47214	0.275453
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	− 4.14590i	− 0.134653i
$949$	23.7984	0.772528
$950$	0	0
$951$	6.79837	0.220452
$952$	47.6869i	1.54554i
$953$	− 20.9230i	− 0.677762i	−0.940829	−	0.338881i	$-0.889952\pi$
$953$	− 20.9230i	− 0.677762i	0.940829	−	0.338881i	$-0.110048\pi$
$954$	0.832816	0.0269634
$955$	0	0
$956$	0	0
$957$	− 4.47214i	− 0.144564i
$958$	− 16.8328i	− 0.543844i
$959$	44.9919	1.45286
$960$	0	0
$961$	−27.0000	−0.870968
$962$	6.43769i	0.207560i
$963$	− 36.5279i	− 1.17709i
$964$	−11.5623	−0.372397
$965$	0	0
$966$	3.27051	0.105227
$967$	− 27.0689i	− 0.870477i	−0.900315	−	0.435238i	$-0.856664\pi$
$967$	− 27.0689i	− 0.870477i	0.900315	−	0.435238i	$-0.143336\pi$
$968$	− 18.7426i	− 0.602411i
$969$	−16.7082	−0.536745
$970$	0	0
$971$	−55.6869	−1.78708	−0.893539	−	0.448985i	$-0.851786\pi$
$971$	−55.6869	−1.78708	−0.893539	+	0.448985i	$0.851786\pi$
$972$	15.6180i	0.500949i
$973$	52.5623i	1.68507i
$974$	−24.3262	−0.779463
$975$	0	0
$976$	−3.97871	−0.127356
$977$	31.0689i	0.993982i	0.867756	+	0.496991i	$0.165562\pi$
$977$	31.0689i	0.993982i	−0.867756	+	0.496991i	$0.834438\pi$
$978$	− 1.14590i	− 0.0366418i
$979$	2.23607	0.0714650
$980$	0	0
$981$	−26.1033	−0.833415
$982$	8.06888i	0.257488i
$983$	52.9443i	1.68866i	0.535824	+	0.844330i	$0.320001\pi$
$983$	52.9443i	1.68866i	−0.535824	+	0.844330i	$0.679999\pi$
$984$	2.11146	0.0673108
$985$	0	0
$986$	33.4164	1.06420
$987$	10.2279i	0.325559i
$988$	− 32.8885i	− 1.04632i
$989$	−41.1246	−1.30769
$990$	0	0
$991$	30.5410	0.970167	0.485084	−	0.874468i	$-0.338789\pi$
$991$	30.5410	0.970167	0.485084	+	0.874468i	$0.338789\pi$
$992$	11.2361i	0.356746i
$993$	10.9656i	0.347981i
$994$	−21.7426	−0.689635
$995$	0	0
$996$	3.05573	0.0968244
$997$	− 25.5623i	− 0.809566i	−0.914413	−	0.404783i	$-0.867347\pi$
$997$	− 25.5623i	− 0.809566i	0.914413	−	0.404783i	$-0.132653\pi$
$998$	0.978714i	0.0309806i
$999$	6.70820	0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.b.b.624.3		4
5.2	odd	4		625.2.a.d.1.1	yes	2
5.3	odd	4		625.2.a.a.1.2	✓	2
5.4	even	2	inner	625.2.b.b.624.2		4
15.2	even	4		5625.2.a.c.1.2		2
15.8	even	4		5625.2.a.e.1.1		2
20.3	even	4		10000.2.a.m.1.1		2
20.7	even	4		10000.2.a.b.1.2		2
25.2	odd	20		625.2.d.c.501.1		4
25.3	odd	20		625.2.d.e.376.1		4
25.4	even	10		625.2.e.f.249.1		8
25.6	even	5		625.2.e.f.374.1		8
25.8	odd	20		625.2.d.e.251.1		4
25.9	even	10		625.2.e.e.499.2		8
25.11	even	5		625.2.e.e.124.2		8
25.12	odd	20		625.2.d.c.126.1		4
25.13	odd	20		625.2.d.i.126.1		4
25.14	even	10		625.2.e.e.124.1		8
25.16	even	5		625.2.e.e.499.1		8
25.17	odd	20		625.2.d.f.251.1		4
25.19	even	10		625.2.e.f.374.2		8
25.21	even	5		625.2.e.f.249.2		8
25.22	odd	20		625.2.d.f.376.1		4
25.23	odd	20		625.2.d.i.501.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.a.1.2	✓	2	5.3	odd	4
625.2.a.d.1.1	yes	2	5.2	odd	4
625.2.b.b.624.2		4	5.4	even	2	inner
625.2.b.b.624.3		4	1.1	even	1	trivial
625.2.d.c.126.1		4	25.12	odd	20
625.2.d.c.501.1		4	25.2	odd	20
625.2.d.e.251.1		4	25.8	odd	20
625.2.d.e.376.1		4	25.3	odd	20
625.2.d.f.251.1		4	25.17	odd	20
625.2.d.f.376.1		4	25.22	odd	20
625.2.d.i.126.1		4	25.13	odd	20
625.2.d.i.501.1		4	25.23	odd	20
625.2.e.e.124.1		8	25.14	even	10
625.2.e.e.124.2		8	25.11	even	5
625.2.e.e.499.1		8	25.16	even	5
625.2.e.e.499.2		8	25.9	even	10
625.2.e.f.249.1		8	25.4	even	10
625.2.e.f.249.2		8	25.21	even	5
625.2.e.f.374.1		8	25.6	even	5
625.2.e.f.374.2		8	25.19	even	10
5625.2.a.c.1.2		2	15.2	even	4
5625.2.a.e.1.1		2	15.8	even	4
10000.2.a.b.1.2		2	20.7	even	4
10000.2.a.m.1.1		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)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +0.381966i q^{3} +1.61803 q^{4} -0.236068 q^{6} +2.85410i q^{7} +2.23607i q^{8} +2.85410 q^{9} +O(q^{10})\)	\(q+0.618034i q^{2} +0.381966i q^{3} +1.61803 q^{4} -0.236068 q^{6} +2.85410i q^{7} +2.23607i q^{8} +2.85410 q^{9} -1.61803 q^{11} +0.618034i q^{12} +3.47214i q^{13} -1.76393 q^{14} +1.85410 q^{16} -7.47214i q^{17} +1.76393i q^{18} -5.85410 q^{19} -1.09017 q^{21} -1.00000i q^{22} +4.85410i q^{23} -0.854102 q^{24} -2.14590 q^{26} +2.23607i q^{27} +4.61803i q^{28} +7.23607 q^{29} +2.00000 q^{31} +5.61803i q^{32} -0.618034i q^{33} +4.61803 q^{34} +4.61803 q^{36} -3.00000i q^{37} -3.61803i q^{38} -1.32624 q^{39} -2.47214 q^{41} -0.673762i q^{42} +8.47214i q^{43} -2.61803 q^{44} -3.00000 q^{46} -9.38197i q^{47} +0.708204i q^{48} -1.14590 q^{49} +2.85410 q^{51} +5.61803i q^{52} -0.472136i q^{53} -1.38197 q^{54} -6.38197 q^{56} -2.23607i q^{57} +4.47214i q^{58} -3.94427 q^{59} -2.14590 q^{61} +1.23607i q^{62} +8.14590i q^{63} +0.236068 q^{64} +0.381966 q^{66} +2.52786i q^{67} -12.0902i q^{68} -1.85410 q^{69} +12.3262 q^{71} +6.38197i q^{72} -6.85410i q^{73} +1.85410 q^{74} -9.47214 q^{76} -4.61803i q^{77} -0.819660i q^{78} -6.70820 q^{79} +7.70820 q^{81} -1.52786i q^{82} -4.94427i q^{83} -1.76393 q^{84} -5.23607 q^{86} +2.76393i q^{87} -3.61803i q^{88} -1.38197 q^{89} -9.90983 q^{91} +7.85410i q^{92} +0.763932i q^{93} +5.79837 q^{94} -2.14590 q^{96} +8.38197i q^{97} -0.708204i q^{98} -4.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9	\( 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9} - 2 q^{11} - 16 q^{14} - 6 q^{16} - 10 q^{19} + 18 q^{21} + 10 q^{24} - 22 q^{26} + 20 q^{29} + 8 q^{31} + 14 q^{34} + 14 q^{36} + 26 q^{39} + 8 q^{41} - 6 q^{44} - 12 q^{46} - 18 q^{49} - 2 q^{51} - 10 q^{54} - 30 q^{56} + 20 q^{59} - 22 q^{61} - 8 q^{64} + 6 q^{66} + 6 q^{69} + 18 q^{71} - 6 q^{74} - 20 q^{76} + 4 q^{81} - 16 q^{84} - 12 q^{86} - 10 q^{89} - 62 q^{91} - 26 q^{94} - 22 q^{96} - 14 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 - 2 * q^11 - 16 * q^14 - 6 * q^16 - 10 * q^19 + 18 * q^21 + 10 * q^24 - 22 * q^26 + 20 * q^29 + 8 * q^31 + 14 * q^34 + 14 * q^36 + 26 * q^39 + 8 * q^41 - 6 * q^44 - 12 * q^46 - 18 * q^49 - 2 * q^51 - 10 * q^54 - 30 * q^56 + 20 * q^59 - 22 * q^61 - 8 * q^64 + 6 * q^66 + 6 * q^69 + 18 * q^71 - 6 * q^74 - 20 * q^76 + 4 * q^81 - 16 * q^84 - 12 * q^86 - 10 * q^89 - 62 * q^91 - 26 * q^94 - 22 * q^96 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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