LMFDB - Embedded newform 845.2.c.c.506.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(506,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("845.506");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.c (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:	$6.74735897080$
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 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.c.506.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{2} +2.23607 q^{3} +1.61803 q^{4} +1.00000i q^{5} -1.38197i q^{6} -4.23607i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-0.618034i q^{2} +2.23607 q^{3} +1.61803 q^{4} +1.00000i q^{5} -1.38197i q^{6} -4.23607i q^{7} -2.23607i q^{8} +2.00000 q^{9} +0.618034 q^{10} +0.236068i q^{11} +3.61803 q^{12} -2.61803 q^{14} +2.23607i q^{15} +1.85410 q^{16} +3.47214 q^{17} -1.23607i q^{18} +4.23607i q^{19} +1.61803i q^{20} -9.47214i q^{21} +0.145898 q^{22} -3.76393 q^{23} -5.00000i q^{24} -1.00000 q^{25} -2.23607 q^{27} -6.85410i q^{28} -7.47214 q^{29} +1.38197 q^{30} -5.61803i q^{32} +0.527864i q^{33} -2.14590i q^{34} +4.23607 q^{35} +3.23607 q^{36} -3.00000i q^{37} +2.61803 q^{38} +2.23607 q^{40} +11.9443i q^{41} -5.85410 q^{42} +6.23607 q^{43} +0.381966i q^{44} +2.00000i q^{45} +2.32624i q^{46} +4.94427i q^{47} +4.14590 q^{48} -10.9443 q^{49} +0.618034i q^{50} +7.76393 q^{51} +6.00000 q^{53} +1.38197i q^{54} -0.236068 q^{55} -9.47214 q^{56} +9.47214i q^{57} +4.61803i q^{58} +0.708204i q^{59} +3.61803i q^{60} +14.4164 q^{61} -8.47214i q^{63} +0.236068 q^{64} +0.326238 q^{66} -2.70820i q^{67} +5.61803 q^{68} -8.41641 q^{69} -2.61803i q^{70} -6.23607i q^{71} -4.47214i q^{72} +6.00000i q^{73} -1.85410 q^{74} -2.23607 q^{75} +6.85410i q^{76} +1.00000 q^{77} +1.85410i q^{80} -11.0000 q^{81} +7.38197 q^{82} +8.94427i q^{83} -15.3262i q^{84} +3.47214i q^{85} -3.85410i q^{86} -16.7082 q^{87} +0.527864 q^{88} +9.00000i q^{89} +1.23607 q^{90} -6.09017 q^{92} +3.05573 q^{94} -4.23607 q^{95} -12.5623i q^{96} -3.47214i q^{97} +6.76393i q^{98} +0.472136i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 8 * q^9	$ 4 q + 2 q^{4} + 8 q^{9} - 2 q^{10} + 10 q^{12} - 6 q^{14} - 6 q^{16} - 4 q^{17} + 14 q^{22} - 24 q^{23} - 4 q^{25} - 12 q^{29} + 10 q^{30} + 8 q^{35} + 4 q^{36} + 6 q^{38} - 10 q^{42} + 16 q^{43} + 30 q^{48} - 8 q^{49} + 40 q^{51} + 24 q^{53} + 8 q^{55} - 20 q^{56} + 4 q^{61} - 8 q^{64} - 30 q^{66} + 18 q^{68} + 20 q^{69} + 6 q^{74} + 4 q^{77} - 44 q^{81} + 34 q^{82} - 40 q^{87} + 20 q^{88} - 4 q^{90} - 2 q^{92} + 48 q^{94} - 8 q^{95}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^9 - 2 * q^10 + 10 * q^12 - 6 * q^14 - 6 * q^16 - 4 * q^17 + 14 * q^22 - 24 * q^23 - 4 * q^25 - 12 * q^29 + 10 * q^30 + 8 * q^35 + 4 * q^36 + 6 * q^38 - 10 * q^42 + 16 * q^43 + 30 * q^48 - 8 * q^49 + 40 * q^51 + 24 * q^53 + 8 * q^55 - 20 * q^56 + 4 * q^61 - 8 * q^64 - 30 * q^66 + 18 * q^68 + 20 * q^69 + 6 * q^74 + 4 * q^77 - 44 * q^81 + 34 * q^82 - 40 * q^87 + 20 * q^88 - 4 * q^90 - 2 * q^92 + 48 * q^94 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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	$-0.929881\pi$
$2$	− 0.618034i	− 0.437016i	0.975835	−	0.218508i	$-0.0701190\pi$
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	1.61803	0.809017
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.38197i	− 0.564185i
$7$	− 4.23607i	− 1.60108i	−0.599277	−	0.800542i	$-0.704545\pi$
$7$	− 4.23607i	− 1.60108i	0.599277	−	0.800542i	$-0.295455\pi$
$8$	− 2.23607i	− 0.790569i
$9$	2.00000	0.666667
$10$	0.618034	0.195440
$11$	0.236068i	0.0711772i	0.999367	+	0.0355886i	$0.0113306\pi$
$11$	0.236068i	0.0711772i	−0.999367	+	0.0355886i	$0.988669\pi$
$12$	3.61803	1.04444
$13$	0	0
$13$	0	0
$14$	−2.61803	−0.699699
$15$	2.23607i	0.577350i
$16$	1.85410	0.463525
$17$	3.47214	0.842117	0.421058	−	0.907034i	$-0.361659\pi$
$17$	3.47214	0.842117	0.421058	+	0.907034i	$0.361659\pi$
$18$	− 1.23607i	− 0.291344i
$19$	4.23607i	0.971821i	0.874009	+	0.485910i	$0.161512\pi$
$19$	4.23607i	0.971821i	−0.874009	+	0.485910i	$0.838488\pi$
$20$	1.61803i	0.361803i
$21$	− 9.47214i	− 2.06699i
$22$	0.145898	0.0311056
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	− 5.00000i	− 1.02062i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−2.23607	−0.430331
$28$	− 6.85410i	− 1.29530i
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	1.38197	0.252311
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 5.61803i	− 0.993137i
$33$	0.527864i	0.0918893i
$34$	− 2.14590i	− 0.368018i
$35$	4.23607	0.716026
$36$	3.23607	0.539345
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	2.61803	0.424701
$39$	0	0
$40$	2.23607	0.353553
$41$	11.9443i	1.86538i	0.360677	+	0.932691i	$0.382546\pi$
$41$	11.9443i	1.86538i	−0.360677	+	0.932691i	$0.617454\pi$
$42$	−5.85410	−0.903308
$43$	6.23607	0.950991	0.475496	−	0.879718i	$-0.342269\pi$
$43$	6.23607	0.950991	0.475496	+	0.879718i	$0.342269\pi$
$44$	0.381966i	0.0575835i
$45$	2.00000i	0.298142i
$46$	2.32624i	0.342985i
$47$	4.94427i	0.721196i	0.932721	+	0.360598i	$0.117427\pi$
$47$	4.94427i	0.721196i	−0.932721	+	0.360598i	$0.882573\pi$
$48$	4.14590	0.598409
$49$	−10.9443	−1.56347
$50$	0.618034i	0.0874032i
$51$	7.76393	1.08717
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.38197i	0.188062i
$55$	−0.236068	−0.0318314
$56$	−9.47214	−1.26577
$57$	9.47214i	1.25462i
$58$	4.61803i	0.606378i
$59$	0.708204i	0.0922003i	0.998937	+	0.0461001i	$0.0146793\pi$
$59$	0.708204i	0.0922003i	−0.998937	+	0.0461001i	$0.985321\pi$
$60$	3.61803i	0.467086i
$61$	14.4164	1.84583	0.922916	−	0.385002i	$-0.125799\pi$
$61$	14.4164	1.84583	0.922916	+	0.385002i	$0.125799\pi$
$62$	0	0
$63$	− 8.47214i	− 1.06739i
$64$	0.236068	0.0295085
$65$	0	0
$66$	0.326238	0.0401571
$67$	− 2.70820i	− 0.330860i	−0.986222	−	0.165430i	$-0.947099\pi$
$67$	− 2.70820i	− 0.330860i	0.986222	−	0.165430i	$-0.0529012\pi$
$68$	5.61803	0.681287
$69$	−8.41641	−1.01322
$70$	− 2.61803i	− 0.312915i
$71$	− 6.23607i	− 0.740085i	−0.929015	−	0.370043i	$-0.879343\pi$
$71$	− 6.23607i	− 0.740085i	0.929015	−	0.370043i	$-0.120657\pi$
$72$	− 4.47214i	− 0.527046i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−1.85410	−0.215535
$75$	−2.23607	−0.258199
$76$	6.85410i	0.786219i
$77$	1.00000	0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.85410i	0.207295i
$81$	−11.0000	−1.22222
$82$	7.38197	0.815202
$83$	8.94427i	0.981761i	0.871227	+	0.490881i	$0.163325\pi$
$83$	8.94427i	0.981761i	−0.871227	+	0.490881i	$0.836675\pi$
$84$	− 15.3262i	− 1.67223i
$85$	3.47214i	0.376606i
$86$	− 3.85410i	− 0.415599i
$87$	−16.7082	−1.79131
$88$	0.527864	0.0562705
$89$	9.00000i	0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$	9.00000i	0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	1.23607	0.130293
$91$	0	0
$92$	−6.09017	−0.634944
$93$	0	0
$94$	3.05573	0.315174
$95$	−4.23607	−0.434611
$96$	− 12.5623i	− 1.28213i
$97$	− 3.47214i	− 0.352542i	−0.984342	−	0.176271i	$-0.943596\pi$
$97$	− 3.47214i	− 0.352542i	0.984342	−	0.176271i	$-0.0564035\pi$
$98$	6.76393i	0.683260i
$99$	0.472136i	0.0474514i
$100$	−1.61803	−0.161803
$101$	−0.527864	−0.0525244	−0.0262622	−	0.999655i	$-0.508360\pi$
$101$	−0.527864	−0.0525244	−0.0262622	+	0.999655i	$0.508360\pi$
$102$	− 4.79837i	− 0.475110i
$103$	−12.9443	−1.27544	−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443	−1.27544	−0.637719	+	0.770270i	$0.720122\pi$
$104$	0	0
$105$	9.47214	0.924386
$106$	− 3.70820i	− 0.360173i
$107$	−5.76393	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$107$	−5.76393	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$108$	−3.61803	−0.348145
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0.145898i	0.0139108i
$111$	− 6.70820i	− 0.636715i
$112$	− 7.85410i	− 0.742143i
$113$	1.47214	0.138487	0.0692435	−	0.997600i	$-0.477941\pi$
$113$	1.47214	0.138487	0.0692435	+	0.997600i	$0.477941\pi$
$114$	5.85410	0.548287
$115$	− 3.76393i	− 0.350988i
$116$	−12.0902	−1.12254
$117$	0	0
$118$	0.437694	0.0402930
$119$	− 14.7082i	− 1.34830i
$120$	5.00000	0.456435
$121$	10.9443	0.994934
$122$	− 8.90983i	− 0.806658i
$123$	26.7082i	2.40820i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	−5.23607	−0.466466
$127$	4.23607	0.375890	0.187945	−	0.982180i	$-0.439817\pi$
$127$	4.23607	0.375890	0.187945	+	0.982180i	$0.439817\pi$
$128$	− 11.3820i	− 1.00603i
$129$	13.9443	1.22772
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0.854102i	0.0743400i
$133$	17.9443	1.55597
$134$	−1.67376	−0.144591
$135$	− 2.23607i	− 0.192450i
$136$	− 7.76393i	− 0.665752i
$137$	− 1.47214i	− 0.125773i	−0.998021	−	0.0628865i	$-0.979969\pi$
$137$	− 1.47214i	− 0.125773i	0.998021	−	0.0628865i	$-0.0200306\pi$
$138$	5.20163i	0.442792i
$139$	−16.7082	−1.41717	−0.708586	−	0.705625i	$-0.750666\pi$
$139$	−16.7082	−1.41717	−0.708586	+	0.705625i	$0.750666\pi$
$140$	6.85410	0.579277
$141$	11.0557i	0.931060i
$142$	−3.85410	−0.323429
$143$	0	0
$144$	3.70820	0.309017
$145$	− 7.47214i	− 0.620527i
$146$	3.70820	0.306893
$147$	−24.4721	−2.01843
$148$	− 4.85410i	− 0.399005i
$149$	4.52786i	0.370937i	0.982650	+	0.185469i	$0.0593803\pi$
$149$	4.52786i	0.370937i	−0.982650	+	0.185469i	$0.940620\pi$
$150$	1.38197i	0.112837i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	9.47214	0.768292
$153$	6.94427	0.561411
$154$	− 0.618034i	− 0.0498026i
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	13.4164	1.06399
$160$	5.61803	0.444145
$161$	15.9443i	1.25658i
$162$	6.79837i	0.534131i
$163$	− 14.7082i	− 1.15204i	−0.817437	−	0.576018i	$-0.804606\pi$
$163$	− 14.7082i	− 1.15204i	0.817437	−	0.576018i	$-0.195394\pi$
$164$	19.3262i	1.50913i
$165$	−0.527864	−0.0410942
$166$	5.52786	0.429045
$167$	17.1803i	1.32945i	0.747086	+	0.664727i	$0.231452\pi$
$167$	17.1803i	1.32945i	−0.747086	+	0.664727i	$0.768548\pi$
$168$	−21.1803	−1.63410
$169$	0	0
$170$	2.14590	0.164583
$171$	8.47214i	0.647880i
$172$	10.0902	0.769368
$173$	18.8885	1.43607	0.718035	−	0.696007i	$-0.245042\pi$
$173$	18.8885	1.43607	0.718035	+	0.696007i	$0.245042\pi$
$174$	10.3262i	0.782830i
$175$	4.23607i	0.320217i
$176$	0.437694i	0.0329924i
$177$	1.58359i	0.119030i
$178$	5.56231	0.416912
$179$	8.23607	0.615593	0.307796	−	0.951452i	$-0.400408\pi$
$179$	8.23607	0.615593	0.307796	+	0.951452i	$0.400408\pi$
$180$	3.23607i	0.241202i
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	32.2361	2.38296
$184$	8.41641i	0.620466i
$185$	3.00000	0.220564
$186$	0	0
$187$	0.819660i	0.0599395i
$188$	8.00000i	0.583460i
$189$	9.47214i	0.688997i
$190$	2.61803i	0.189932i
$191$	−27.1803	−1.96670	−0.983350	−	0.181721i	$-0.941833\pi$
$191$	−27.1803	−1.96670	−0.983350	+	0.181721i	$0.941833\pi$
$192$	0.527864	0.0380953
$193$	3.47214i	0.249930i	0.992161	+	0.124965i	$0.0398818\pi$
$193$	3.47214i	0.249930i	−0.992161	+	0.124965i	$0.960118\pi$
$194$	−2.14590	−0.154067
$195$	0	0
$196$	−17.7082	−1.26487
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0.291796	0.0207370
$199$	1.29180	0.0915730	0.0457865	−	0.998951i	$-0.485421\pi$
$199$	1.29180	0.0915730	0.0457865	+	0.998951i	$0.485421\pi$
$200$	2.23607i	0.158114i
$201$	− 6.05573i	− 0.427138i
$202$	0.326238i	0.0229540i
$203$	31.6525i	2.22157i
$204$	12.5623	0.879537
$205$	−11.9443	−0.834224
$206$	8.00000i	0.557386i
$207$	−7.52786	−0.523223
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	− 5.85410i	− 0.403971i
$211$	−13.1803	−0.907372	−0.453686	−	0.891162i	$-0.649891\pi$
$211$	−13.1803	−0.907372	−0.453686	+	0.891162i	$0.649891\pi$
$212$	9.70820	0.666762
$213$	− 13.9443i	− 0.955446i
$214$	3.56231i	0.243514i
$215$	6.23607i	0.425296i
$216$	5.00000i	0.340207i
$217$	0	0
$218$	−1.23607	−0.0837171
$219$	13.4164i	0.906597i
$220$	−0.381966	−0.0257521
$221$	0	0
$222$	−4.14590	−0.278254
$223$	0.708204i	0.0474248i	0.999719	+	0.0237124i	$0.00754861\pi$
$223$	0.708204i	0.0474248i	−0.999719	+	0.0237124i	$0.992451\pi$
$224$	−23.7984	−1.59010
$225$	−2.00000	−0.133333
$226$	− 0.909830i	− 0.0605210i
$227$	− 6.23607i	− 0.413902i	−0.978351	−	0.206951i	$-0.933646\pi$
$227$	− 6.23607i	− 0.413902i	0.978351	−	0.206951i	$-0.0663541\pi$
$228$	15.3262i	1.01500i
$229$	− 15.8885i	− 1.04994i	−0.851119	−	0.524972i	$-0.824076\pi$
$229$	− 15.8885i	− 1.04994i	0.851119	−	0.524972i	$-0.175924\pi$
$230$	−2.32624	−0.153388
$231$	2.23607	0.147122
$232$	16.7082i	1.09695i
$233$	15.8885	1.04089	0.520447	−	0.853894i	$-0.325765\pi$
$233$	15.8885	1.04089	0.520447	+	0.853894i	$0.325765\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	1.14590i	0.0745916i
$237$	0	0
$238$	−9.09017	−0.589228
$239$	− 25.8885i	− 1.67459i	−0.546751	−	0.837295i	$-0.684136\pi$
$239$	− 25.8885i	− 1.67459i	0.546751	−	0.837295i	$-0.315864\pi$
$240$	4.14590i	0.267617i
$241$	− 15.0000i	− 0.966235i	−0.875556	−	0.483117i	$-0.839504\pi$
$241$	− 15.0000i	− 0.966235i	0.875556	−	0.483117i	$-0.160496\pi$
$242$	− 6.76393i	− 0.434802i
$243$	−17.8885	−1.14755
$244$	23.3262	1.49331
$245$	− 10.9443i	− 0.699204i
$246$	16.5066	1.05242
$247$	0	0
$248$	0	0
$249$	20.0000i	1.26745i
$250$	−0.618034	−0.0390879
$251$	−20.2361	−1.27729	−0.638645	−	0.769502i	$-0.720505\pi$
$251$	−20.2361	−1.27729	−0.638645	+	0.769502i	$0.720505\pi$
$252$	− 13.7082i	− 0.863536i
$253$	− 0.888544i	− 0.0558623i
$254$	− 2.61803i	− 0.164270i
$255$	7.76393i	0.486196i
$256$	−6.56231	−0.410144
$257$	−17.4721	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$257$	−17.4721	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$258$	− 8.61803i	− 0.536535i
$259$	−12.7082	−0.789649
$260$	0	0
$261$	−14.9443	−0.925027
$262$	7.41641i	0.458187i
$263$	3.76393	0.232094	0.116047	−	0.993244i	$-0.462978\pi$
$263$	3.76393	0.232094	0.116047	+	0.993244i	$0.462978\pi$
$264$	1.18034	0.0726449
$265$	6.00000i	0.368577i
$266$	− 11.0902i	− 0.679982i
$267$	20.1246i	1.23161i
$268$	− 4.38197i	− 0.267671i
$269$	−6.52786	−0.398011	−0.199005	−	0.979998i	$-0.563771\pi$
$269$	−6.52786	−0.398011	−0.199005	+	0.979998i	$0.563771\pi$
$270$	−1.38197	−0.0841038
$271$	− 1.29180i	− 0.0784710i	−0.999230	−	0.0392355i	$-0.987508\pi$
$271$	− 1.29180i	− 0.0784710i	0.999230	−	0.0392355i	$-0.0124923\pi$
$272$	6.43769	0.390343
$273$	0	0
$274$	−0.909830	−0.0549648
$275$	− 0.236068i	− 0.0142354i
$276$	−13.6180	−0.819709
$277$	−16.8885	−1.01473	−0.507367	−	0.861730i	$-0.669381\pi$
$277$	−16.8885	−1.01473	−0.507367	+	0.861730i	$0.669381\pi$
$278$	10.3262i	0.619327i
$279$	0	0
$280$	− 9.47214i	− 0.566068i
$281$	15.8885i	0.947831i	0.880570	+	0.473916i	$0.157160\pi$
$281$	15.8885i	0.947831i	−0.880570	+	0.473916i	$0.842840\pi$
$282$	6.83282	0.406888
$283$	10.7082	0.636537	0.318268	−	0.948001i	$-0.396899\pi$
$283$	10.7082	0.636537	0.318268	+	0.948001i	$0.396899\pi$
$284$	− 10.0902i	− 0.598741i
$285$	−9.47214	−0.561081
$286$	0	0
$287$	50.5967	2.98663
$288$	− 11.2361i	− 0.662092i
$289$	−4.94427	−0.290840
$290$	−4.61803	−0.271180
$291$	− 7.76393i	− 0.455130i
$292$	9.70820i	0.568130i
$293$	29.9443i	1.74936i	0.484698	+	0.874682i	$0.338929\pi$
$293$	29.9443i	1.74936i	−0.484698	+	0.874682i	$0.661071\pi$
$294$	15.1246i	0.882085i
$295$	−0.708204	−0.0412332
$296$	−6.70820	−0.389906
$297$	− 0.527864i	− 0.0306298i
$298$	2.79837	0.162105
$299$	0	0
$300$	−3.61803	−0.208887
$301$	− 26.4164i	− 1.52262i
$302$	−4.94427	−0.284511
$303$	−1.18034	−0.0678088
$304$	7.85410i	0.450464i
$305$	14.4164i	0.825481i
$306$	− 4.29180i	− 0.245346i
$307$	− 24.9443i	− 1.42364i	−0.702360	−	0.711822i	$-0.747870\pi$
$307$	− 24.9443i	− 1.42364i	0.702360	−	0.711822i	$-0.252130\pi$
$308$	1.61803	0.0921960
$309$	−28.9443	−1.64658
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	3.88854	0.219793	0.109897	−	0.993943i	$-0.464948\pi$
$313$	3.88854	0.219793	0.109897	+	0.993943i	$0.464948\pi$
$314$	11.1246i	0.627798i
$315$	8.47214	0.477351
$316$	0	0
$317$	− 11.8885i	− 0.667727i	−0.942621	−	0.333864i	$-0.891648\pi$
$317$	− 11.8885i	− 0.667727i	0.942621	−	0.333864i	$-0.108352\pi$
$318$	− 8.29180i	− 0.464981i
$319$	− 1.76393i	− 0.0987612i
$320$	0.236068i	0.0131966i
$321$	−12.8885	−0.719368
$322$	9.85410	0.549148
$323$	14.7082i	0.818386i
$324$	−17.7984	−0.988799
$325$	0	0
$326$	−9.09017	−0.503458
$327$	− 4.47214i	− 0.247310i
$328$	26.7082	1.47471
$329$	20.9443	1.15470
$330$	0.326238i	0.0179588i
$331$	− 5.65248i	− 0.310688i	−0.987860	−	0.155344i	$-0.950351\pi$
$331$	− 5.65248i	− 0.310688i	0.987860	−	0.155344i	$-0.0496486\pi$
$332$	14.4721i	0.794262i
$333$	− 6.00000i	− 0.328798i
$334$	10.6180	0.580993
$335$	2.70820	0.147965
$336$	− 17.5623i	− 0.958102i
$337$	7.88854	0.429716	0.214858	−	0.976645i	$-0.431071\pi$
$337$	7.88854	0.429716	0.214858	+	0.976645i	$0.431071\pi$
$338$	0	0
$339$	3.29180	0.178786
$340$	5.61803i	0.304681i
$341$	0	0
$342$	5.23607	0.283134
$343$	16.7082i	0.902158i
$344$	− 13.9443i	− 0.751825i
$345$	− 8.41641i	− 0.453124i
$346$	− 11.6738i	− 0.627585i
$347$	30.7082	1.64850	0.824251	−	0.566224i	$-0.191596\pi$
$347$	30.7082	1.64850	0.824251	+	0.566224i	$0.191596\pi$
$348$	−27.0344	−1.44920
$349$	− 2.41641i	− 0.129347i	−0.997906	−	0.0646737i	$-0.979399\pi$
$349$	− 2.41641i	− 0.129347i	0.997906	−	0.0646737i	$-0.0206006\pi$
$350$	2.61803	0.139940
$351$	0	0
$352$	1.32624	0.0706887
$353$	− 19.4721i	− 1.03640i	−0.855260	−	0.518199i	$-0.826603\pi$
$353$	− 19.4721i	− 1.03640i	0.855260	−	0.518199i	$-0.173397\pi$
$354$	0.978714	0.0520180
$355$	6.23607	0.330976
$356$	14.5623i	0.771801i
$357$	− 32.8885i	− 1.74065i
$358$	− 5.09017i	− 0.269024i
$359$	− 17.8885i	− 0.944121i	−0.881566	−	0.472061i	$-0.843510\pi$
$359$	− 17.8885i	− 0.944121i	0.881566	−	0.472061i	$-0.156490\pi$
$360$	4.47214	0.235702
$361$	1.05573	0.0555646
$362$	3.70820i	0.194899i
$363$	24.4721	1.28445
$364$	0	0
$365$	−6.00000	−0.314054
$366$	− 19.9230i	− 1.04139i
$367$	5.65248	0.295057	0.147528	−	0.989058i	$-0.452868\pi$
$367$	5.65248	0.295057	0.147528	+	0.989058i	$0.452868\pi$
$368$	−6.97871	−0.363791
$369$	23.8885i	1.24359i
$370$	− 1.85410i	− 0.0963902i
$371$	− 25.4164i	− 1.31955i
$372$	0	0
$373$	27.9443	1.44690	0.723450	−	0.690377i	$-0.242555\pi$
$373$	27.9443	1.44690	0.723450	+	0.690377i	$0.242555\pi$
$374$	0.506578	0.0261945
$375$	− 2.23607i	− 0.115470i
$376$	11.0557	0.570156
$377$	0	0
$378$	5.85410	0.301103
$379$	10.8197i	0.555769i	0.960615	+	0.277884i	$0.0896332\pi$
$379$	10.8197i	0.555769i	−0.960615	+	0.277884i	$0.910367\pi$
$380$	−6.85410	−0.351608
$381$	9.47214	0.485272
$382$	16.7984i	0.859480i
$383$	4.23607i	0.216453i	0.994126	+	0.108226i	$0.0345172\pi$
$383$	4.23607i	0.216453i	−0.994126	+	0.108226i	$0.965483\pi$
$384$	− 25.4508i	− 1.29878i
$385$	1.00000i	0.0509647i
$386$	2.14590	0.109223
$387$	12.4721	0.633994
$388$	− 5.61803i	− 0.285212i
$389$	0.111456	0.00565105	0.00282553	−	0.999996i	$-0.499101\pi$
$389$	0.111456	0.00565105	0.00282553	+	0.999996i	$0.499101\pi$
$390$	0	0
$391$	−13.0689	−0.660922
$392$	24.4721i	1.23603i
$393$	−26.8328	−1.35354
$394$	1.85410	0.0934083
$395$	0	0
$396$	0.763932i	0.0383890i
$397$	− 27.9443i	− 1.40248i	−0.712924	−	0.701241i	$-0.752630\pi$
$397$	− 27.9443i	− 1.40248i	0.712924	−	0.701241i	$-0.247370\pi$
$398$	− 0.798374i	− 0.0400189i
$399$	40.1246	2.00874
$400$	−1.85410	−0.0927051
$401$	− 8.88854i	− 0.443873i	−0.975061	−	0.221936i	$-0.928762\pi$
$401$	− 8.88854i	− 0.443873i	0.975061	−	0.221936i	$-0.0712377\pi$
$402$	−3.74265	−0.186666
$403$	0	0
$404$	−0.854102	−0.0424932
$405$	− 11.0000i	− 0.546594i
$406$	19.5623	0.970861
$407$	0.708204	0.0351044
$408$	− 17.3607i	− 0.859482i
$409$	24.8885i	1.23066i	0.788270	+	0.615330i	$0.210977\pi$
$409$	24.8885i	1.23066i	−0.788270	+	0.615330i	$0.789023\pi$
$410$	7.38197i	0.364569i
$411$	− 3.29180i	− 0.162372i
$412$	−20.9443	−1.03185
$413$	3.00000	0.147620
$414$	4.65248i	0.228657i
$415$	−8.94427	−0.439057
$416$	0	0
$417$	−37.3607	−1.82956
$418$	0.618034i	0.0302290i
$419$	17.6525	0.862380	0.431190	−	0.902261i	$-0.358094\pi$
$419$	17.6525	0.862380	0.431190	+	0.902261i	$0.358094\pi$
$420$	15.3262	0.747844
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	8.14590i	0.396536i
$423$	9.88854i	0.480797i
$424$	− 13.4164i	− 0.651558i
$425$	−3.47214	−0.168423
$426$	−8.61803	−0.417545
$427$	− 61.0689i	− 2.95533i
$428$	−9.32624	−0.450801
$429$	0	0
$430$	3.85410	0.185861
$431$	27.1803i	1.30923i	0.755962	+	0.654615i	$0.227169\pi$
$431$	27.1803i	1.30923i	−0.755962	+	0.654615i	$0.772831\pi$
$432$	−4.14590	−0.199470
$433$	14.5279	0.698165	0.349082	−	0.937092i	$-0.386493\pi$
$433$	14.5279	0.698165	0.349082	+	0.937092i	$0.386493\pi$
$434$	0	0
$435$	− 16.7082i	− 0.801097i
$436$	− 3.23607i	− 0.154980i
$437$	− 15.9443i	− 0.762718i
$438$	8.29180	0.396197
$439$	22.7082	1.08380	0.541902	−	0.840442i	$-0.317704\pi$
$439$	22.7082	1.08380	0.541902	+	0.840442i	$0.317704\pi$
$440$	0.527864i	0.0251649i
$441$	−21.8885	−1.04231
$442$	0	0
$443$	0.944272	0.0448637	0.0224319	−	0.999748i	$-0.492859\pi$
$443$	0.944272	0.0448637	0.0224319	+	0.999748i	$0.492859\pi$
$444$	− 10.8541i	− 0.515113i
$445$	−9.00000	−0.426641
$446$	0.437694	0.0207254
$447$	10.1246i	0.478878i
$448$	− 1.00000i	− 0.0472456i
$449$	− 3.94427i	− 0.186142i	−0.995659	−	0.0930709i	$-0.970332\pi$
$449$	− 3.94427i	− 0.186142i	0.995659	−	0.0930709i	$-0.0296683\pi$
$450$	1.23607i	0.0582688i
$451$	−2.81966	−0.132773
$452$	2.38197	0.112038
$453$	− 17.8885i	− 0.840477i
$454$	−3.85410	−0.180882
$455$	0	0
$456$	21.1803	0.991860
$457$	− 1.58359i	− 0.0740773i	−0.999314	−	0.0370387i	$-0.988208\pi$
$457$	− 1.58359i	− 0.0740773i	0.999314	−	0.0370387i	$-0.0117925\pi$
$458$	−9.81966	−0.458843
$459$	−7.76393	−0.362389
$460$	− 6.09017i	− 0.283956i
$461$	− 1.58359i	− 0.0737552i	−0.999320	−	0.0368776i	$-0.988259\pi$
$461$	− 1.58359i	− 0.0737552i	0.999320	−	0.0368776i	$-0.0117412\pi$
$462$	− 1.38197i	− 0.0642949i
$463$	− 11.0557i	− 0.513803i	−0.966438	−	0.256902i	$-0.917298\pi$
$463$	− 11.0557i	− 0.513803i	0.966438	−	0.256902i	$-0.0827016\pi$
$464$	−13.8541	−0.643161
$465$	0	0
$466$	− 9.81966i	− 0.454887i
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−11.4721	−0.529734
$470$	3.05573i	0.140950i
$471$	−40.2492	−1.85459
$472$	1.58359	0.0728907
$473$	1.47214i	0.0676889i
$474$	0	0
$475$	− 4.23607i	− 0.194364i
$476$	− 23.7984i	− 1.09080i
$477$	12.0000	0.549442
$478$	−16.0000	−0.731823
$479$	− 32.1246i	− 1.46781i	−0.679252	−	0.733905i	$-0.737695\pi$
$479$	− 32.1246i	− 1.46781i	0.679252	−	0.733905i	$-0.262305\pi$
$480$	12.5623	0.573388
$481$	0	0
$482$	−9.27051	−0.422260
$483$	35.6525i	1.62224i
$484$	17.7082	0.804918
$485$	3.47214	0.157662
$486$	11.0557i	0.501498i
$487$	14.1246i	0.640047i	0.947410	+	0.320024i	$0.103691\pi$
$487$	14.1246i	0.640047i	−0.947410	+	0.320024i	$0.896309\pi$
$488$	− 32.2361i	− 1.45926i
$489$	− 32.8885i	− 1.48727i
$490$	−6.76393	−0.305563
$491$	0.236068	0.0106536	0.00532680	−	0.999986i	$-0.498304\pi$
$491$	0.236068	0.0106536	0.00532680	+	0.999986i	$0.498304\pi$
$492$	43.2148i	1.94827i
$493$	−25.9443	−1.16847
$494$	0	0
$495$	−0.472136	−0.0212209
$496$	0	0
$497$	−26.4164	−1.18494
$498$	12.3607	0.553895
$499$	− 21.8885i	− 0.979866i	−0.871760	−	0.489933i	$-0.837021\pi$
$499$	− 21.8885i	− 0.979866i	0.871760	−	0.489933i	$-0.162979\pi$
$500$	− 1.61803i	− 0.0723607i
$501$	38.4164i	1.71632i
$502$	12.5066i	0.558196i
$503$	9.18034	0.409331	0.204666	−	0.978832i	$-0.434389\pi$
$503$	9.18034	0.409331	0.204666	+	0.978832i	$0.434389\pi$
$504$	−18.9443	−0.843845
$505$	− 0.527864i	− 0.0234896i
$506$	−0.549150	−0.0244127
$507$	0	0
$508$	6.85410	0.304102
$509$	− 33.3607i	− 1.47869i	−0.673329	−	0.739343i	$-0.735136\pi$
$509$	− 33.3607i	− 1.47869i	0.673329	−	0.739343i	$-0.264864\pi$
$510$	4.79837	0.212476
$511$	25.4164	1.12436
$512$	− 18.7082i	− 0.826794i
$513$	− 9.47214i	− 0.418205i
$514$	10.7984i	0.476296i
$515$	− 12.9443i	− 0.570393i
$516$	22.5623	0.993250
$517$	−1.16718	−0.0513327
$518$	7.85410i	0.345089i
$519$	42.2361	1.85396
$520$	0	0
$521$	29.7771	1.30456	0.652279	−	0.757979i	$-0.273813\pi$
$521$	29.7771	1.30456	0.652279	+	0.757979i	$0.273813\pi$
$522$	9.23607i	0.404252i
$523$	5.29180	0.231394	0.115697	−	0.993285i	$-0.463090\pi$
$523$	5.29180	0.231394	0.115697	+	0.993285i	$0.463090\pi$
$524$	−19.4164	−0.848210
$525$	9.47214i	0.413398i
$526$	− 2.32624i	− 0.101429i
$527$	0	0
$528$	0.978714i	0.0425930i
$529$	−8.83282	−0.384035
$530$	3.70820	0.161074
$531$	1.41641i	0.0614669i
$532$	29.0344	1.25880
$533$	0	0
$534$	12.4377	0.538232
$535$	− 5.76393i	− 0.249197i
$536$	−6.05573	−0.261568
$537$	18.4164	0.794727
$538$	4.03444i	0.173937i
$539$	− 2.58359i	− 0.111283i
$540$	− 3.61803i	− 0.155695i
$541$	27.8885i	1.19902i	0.800366	+	0.599511i	$0.204638\pi$
$541$	27.8885i	1.19902i	−0.800366	+	0.599511i	$0.795362\pi$
$542$	−0.798374	−0.0342931
$543$	−13.4164	−0.575753
$544$	− 19.5066i	− 0.836338i
$545$	2.00000	0.0856706
$546$	0	0
$547$	18.8328	0.805233	0.402617	−	0.915369i	$-0.368101\pi$
$547$	18.8328	0.805233	0.402617	+	0.915369i	$0.368101\pi$
$548$	− 2.38197i	− 0.101753i
$549$	28.8328	1.23055
$550$	−0.145898	−0.00622111
$551$	− 31.6525i	− 1.34844i
$552$	18.8197i	0.801018i
$553$	0	0
$554$	10.4377i	0.443455i
$555$	6.70820	0.284747
$556$	−27.0344	−1.14652
$557$	9.94427i	0.421352i	0.977556	+	0.210676i	$0.0675666\pi$
$557$	9.94427i	0.421352i	−0.977556	+	0.210676i	$0.932433\pi$
$558$	0	0
$559$	0	0
$560$	7.85410	0.331896
$561$	1.83282i	0.0773815i
$562$	9.81966	0.414217
$563$	−30.7082	−1.29420	−0.647098	−	0.762407i	$-0.724018\pi$
$563$	−30.7082	−1.29420	−0.647098	+	0.762407i	$0.724018\pi$
$564$	17.8885i	0.753244i
$565$	1.47214i	0.0619332i
$566$	− 6.61803i	− 0.278177i
$567$	46.5967i	1.95688i
$568$	−13.9443	−0.585089
$569$	−16.8885	−0.708005	−0.354002	−	0.935245i	$-0.615179\pi$
$569$	−16.8885	−0.708005	−0.354002	+	0.935245i	$0.615179\pi$
$570$	5.85410i	0.245201i
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−60.7771	−2.53900
$574$	− 31.2705i	− 1.30521i
$575$	3.76393	0.156967
$576$	0.472136	0.0196723
$577$	27.8885i	1.16102i	0.814255	+	0.580508i	$0.197146\pi$
$577$	27.8885i	1.16102i	−0.814255	+	0.580508i	$0.802854\pi$
$578$	3.05573i	0.127102i
$579$	7.76393i	0.322658i
$580$	− 12.0902i	− 0.502017i
$581$	37.8885	1.57188
$582$	−4.79837	−0.198899
$583$	1.41641i	0.0586616i
$584$	13.4164	0.555175
$585$	0	0
$586$	18.5066	0.764500
$587$	10.2361i	0.422488i	0.977433	+	0.211244i	$0.0677514\pi$
$587$	10.2361i	0.422488i	−0.977433	+	0.211244i	$0.932249\pi$
$588$	−39.5967	−1.63294
$589$	0	0
$590$	0.437694i	0.0180196i
$591$	6.70820i	0.275939i
$592$	− 5.56231i	− 0.228609i
$593$	7.88854i	0.323944i	0.986795	+	0.161972i	$0.0517854\pi$
$593$	7.88854i	0.323944i	−0.986795	+	0.161972i	$0.948215\pi$
$594$	−0.326238	−0.0133857
$595$	14.7082	0.602978
$596$	7.32624i	0.300094i
$597$	2.88854	0.118220
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	5.00000i	0.204124i
$601$	43.9443	1.79252	0.896262	−	0.443525i	$-0.146272\pi$
$601$	43.9443	1.79252	0.896262	+	0.443525i	$0.146272\pi$
$602$	−16.3262	−0.665408
$603$	− 5.41641i	− 0.220573i
$604$	− 12.9443i	− 0.526695i
$605$	10.9443i	0.444948i
$606$	0.729490i	0.0296335i
$607$	−0.708204	−0.0287451	−0.0143726	−	0.999897i	$-0.504575\pi$
$607$	−0.708204	−0.0287451	−0.0143726	+	0.999897i	$0.504575\pi$
$608$	23.7984	0.965152
$609$	70.7771i	2.86803i
$610$	8.90983	0.360748
$611$	0	0
$612$	11.2361	0.454191
$613$	39.9443i	1.61333i	0.591006	+	0.806667i	$0.298731\pi$
$613$	39.9443i	1.61333i	−0.591006	+	0.806667i	$0.701269\pi$
$614$	−15.4164	−0.622156
$615$	−26.7082	−1.07698
$616$	− 2.23607i	− 0.0900937i
$617$	− 40.4164i	− 1.62710i	−0.581493	−	0.813552i	$-0.697531\pi$
$617$	− 40.4164i	− 1.62710i	0.581493	−	0.813552i	$-0.302469\pi$
$618$	17.8885i	0.719583i
$619$	− 12.0000i	− 0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$	− 12.0000i	− 0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	0	0
$621$	8.41641	0.337739
$622$	− 14.8328i	− 0.594742i
$623$	38.1246	1.52743
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 2.40325i	− 0.0960533i
$627$	−2.23607	−0.0893000
$628$	−29.1246	−1.16220
$629$	− 10.4164i	− 0.415329i
$630$	− 5.23607i	− 0.208610i
$631$	16.1246i	0.641911i	0.947094	+	0.320955i	$0.104004\pi$
$631$	16.1246i	0.641911i	−0.947094	+	0.320955i	$0.895996\pi$
$632$	0	0
$633$	−29.4721	−1.17141
$634$	−7.34752	−0.291807
$635$	4.23607i	0.168103i
$636$	21.7082	0.860786
$637$	0	0
$638$	−1.09017	−0.0431602
$639$	− 12.4721i	− 0.493390i
$640$	11.3820	0.449912
$641$	−8.88854	−0.351076	−0.175538	−	0.984473i	$-0.556167\pi$
$641$	−8.88854	−0.351076	−0.175538	+	0.984473i	$0.556167\pi$
$642$	7.96556i	0.314376i
$643$	− 17.2918i	− 0.681922i	−0.940077	−	0.340961i	$-0.889248\pi$
$643$	− 17.2918i	− 0.681922i	0.940077	−	0.340961i	$-0.110752\pi$
$644$	25.7984i	1.01660i
$645$	13.9443i	0.549055i
$646$	9.09017	0.357648
$647$	−2.59675	−0.102089	−0.0510443	−	0.998696i	$-0.516255\pi$
$647$	−2.59675	−0.102089	−0.0510443	+	0.998696i	$0.516255\pi$
$648$	24.5967i	0.966252i
$649$	−0.167184	−0.00656256
$650$	0	0
$651$	0	0
$652$	− 23.7984i	− 0.932016i
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−2.76393	−0.108078
$655$	− 12.0000i	− 0.468879i
$656$	22.1459i	0.864652i
$657$	12.0000i	0.468165i
$658$	− 12.9443i	− 0.504620i
$659$	−43.7639	−1.70480	−0.852400	−	0.522890i	$-0.824854\pi$
$659$	−43.7639	−1.70480	−0.852400	+	0.522890i	$0.824854\pi$
$660$	−0.854102	−0.0332459
$661$	41.3607i	1.60874i	0.594126	+	0.804372i	$0.297498\pi$
$661$	41.3607i	1.60874i	−0.594126	+	0.804372i	$0.702502\pi$
$662$	−3.49342	−0.135776
$663$	0	0
$664$	20.0000	0.776151
$665$	17.9443i	0.695849i
$666$	−3.70820	−0.143690
$667$	28.1246	1.08899
$668$	27.7984i	1.07555i
$669$	1.58359i	0.0612252i
$670$	− 1.67376i	− 0.0646631i
$671$	3.40325i	0.131381i
$672$	−53.2148	−2.05280
$673$	9.58359	0.369420	0.184710	−	0.982793i	$-0.440865\pi$
$673$	9.58359	0.369420	0.184710	+	0.982793i	$0.440865\pi$
$674$	− 4.87539i	− 0.187793i
$675$	2.23607	0.0860663
$676$	0	0
$677$	12.1115	0.465481	0.232741	−	0.972539i	$-0.425231\pi$
$677$	12.1115	0.465481	0.232741	+	0.972539i	$0.425231\pi$
$678$	− 2.03444i	− 0.0781323i
$679$	−14.7082	−0.564449
$680$	7.76393	0.297733
$681$	− 13.9443i	− 0.534346i
$682$	0	0
$683$	− 25.7639i	− 0.985829i	−0.870078	−	0.492915i	$-0.835931\pi$
$683$	− 25.7639i	− 0.985829i	0.870078	−	0.492915i	$-0.164069\pi$
$684$	13.7082i	0.524146i
$685$	1.47214	0.0562474
$686$	10.3262	0.394258
$687$	− 35.5279i	− 1.35547i
$688$	11.5623	0.440809
$689$	0	0
$690$	−5.20163	−0.198023
$691$	38.5967i	1.46829i	0.678993	+	0.734145i	$0.262417\pi$
$691$	38.5967i	1.46829i	−0.678993	+	0.734145i	$0.737583\pi$
$692$	30.5623	1.16180
$693$	2.00000	0.0759737
$694$	− 18.9787i	− 0.720422i
$695$	− 16.7082i	− 0.633778i
$696$	37.3607i	1.41615i
$697$	41.4721i	1.57087i
$698$	−1.49342	−0.0565269
$699$	35.5279	1.34379
$700$	6.85410i	0.259061i
$701$	−7.88854	−0.297946	−0.148973	−	0.988841i	$-0.547597\pi$
$701$	−7.88854	−0.297946	−0.148973	+	0.988841i	$0.547597\pi$
$702$	0	0
$703$	12.7082	0.479299
$704$	0.0557281i	0.00210033i
$705$	−11.0557	−0.416383
$706$	−12.0344	−0.452922
$707$	2.23607i	0.0840960i
$708$	2.56231i	0.0962974i
$709$	− 24.3050i	− 0.912792i	−0.889777	−	0.456396i	$-0.849140\pi$
$709$	− 24.3050i	− 0.912792i	0.889777	−	0.456396i	$-0.150860\pi$
$710$	− 3.85410i	− 0.144642i
$711$	0	0
$712$	20.1246	0.754202
$713$	0	0
$714$	−20.3262	−0.760690
$715$	0	0
$716$	13.3262	0.498025
$717$	− 57.8885i	− 2.16189i
$718$	−11.0557	−0.412596
$719$	32.1246	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$719$	32.1246	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$720$	3.70820i	0.138197i
$721$	54.8328i	2.04208i
$722$	− 0.652476i	− 0.0242826i
$723$	− 33.5410i	− 1.24740i
$724$	−9.70820	−0.360803
$725$	7.47214	0.277508
$726$	− 15.1246i	− 0.561327i
$727$	−28.9443	−1.07348	−0.536742	−	0.843747i	$-0.680345\pi$
$727$	−28.9443	−1.07348	−0.536742	+	0.843747i	$0.680345\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	3.70820i	0.137247i
$731$	21.6525	0.800846
$732$	52.1591	1.92785
$733$	− 43.8885i	− 1.62106i	−0.585697	−	0.810530i	$-0.699179\pi$
$733$	− 43.8885i	− 1.62106i	0.585697	−	0.810530i	$-0.300821\pi$
$734$	− 3.49342i	− 0.128945i
$735$	− 24.4721i	− 0.902668i
$736$	21.1459i	0.779448i
$737$	0.639320	0.0235497
$738$	14.7639	0.543468
$739$	− 27.5410i	− 1.01311i	−0.862207	−	0.506556i	$-0.830918\pi$
$739$	− 27.5410i	− 1.01311i	0.862207	−	0.506556i	$-0.169082\pi$
$740$	4.85410	0.178440
$741$	0	0
$742$	−15.7082	−0.576666
$743$	− 38.1246i	− 1.39866i	−0.714801	−	0.699328i	$-0.753483\pi$
$743$	− 38.1246i	− 1.39866i	0.714801	−	0.699328i	$-0.246517\pi$
$744$	0	0
$745$	−4.52786	−0.165888
$746$	− 17.2705i	− 0.632318i
$747$	17.8885i	0.654508i
$748$	1.32624i	0.0484921i
$749$	24.4164i	0.892156i
$750$	−1.38197	−0.0504623
$751$	30.7082	1.12056	0.560279	−	0.828304i	$-0.310694\pi$
$751$	30.7082	1.12056	0.560279	+	0.828304i	$0.310694\pi$
$752$	9.16718i	0.334293i
$753$	−45.2492	−1.64897
$754$	0	0
$755$	8.00000	0.291150
$756$	15.3262i	0.557410i
$757$	−6.88854	−0.250368	−0.125184	−	0.992134i	$-0.539952\pi$
$757$	−6.88854	−0.250368	−0.125184	+	0.992134i	$0.539952\pi$
$758$	6.68692	0.242880
$759$	− 1.98684i	− 0.0721179i
$760$	9.47214i	0.343590i
$761$	37.9443i	1.37548i	0.725958	+	0.687739i	$0.241397\pi$
$761$	37.9443i	1.37548i	−0.725958	+	0.687739i	$0.758603\pi$
$762$	− 5.85410i	− 0.212072i
$763$	−8.47214	−0.306712
$764$	−43.9787	−1.59109
$765$	6.94427i	0.251071i
$766$	2.61803	0.0945934
$767$	0	0
$768$	−14.6738	−0.529494
$769$	29.8328i	1.07580i	0.843009	+	0.537899i	$0.180782\pi$
$769$	29.8328i	1.07580i	−0.843009	+	0.537899i	$0.819218\pi$
$770$	0.618034	0.0222724
$771$	−39.0689	−1.40703
$772$	5.61803i	0.202197i
$773$	− 20.0557i	− 0.721354i	−0.932691	−	0.360677i	$-0.882546\pi$
$773$	− 20.0557i	− 0.721354i	0.932691	−	0.360677i	$-0.117454\pi$
$774$	− 7.70820i	− 0.277066i
$775$	0	0
$776$	−7.76393	−0.278709
$777$	−28.4164	−1.01943
$778$	− 0.0688837i	− 0.00246960i
$779$	−50.5967	−1.81282
$780$	0	0
$781$	1.47214	0.0526772
$782$	8.07701i	0.288833i
$783$	16.7082	0.597102
$784$	−20.2918	−0.724707
$785$	− 18.0000i	− 0.642448i
$786$	16.5836i	0.591517i
$787$	− 29.5410i	− 1.05302i	−0.850168	−	0.526512i	$-0.823500\pi$
$787$	− 29.5410i	− 1.05302i	0.850168	−	0.526512i	$-0.176500\pi$
$788$	4.85410i	0.172920i
$789$	8.41641	0.299632
$790$	0	0
$791$	− 6.23607i	− 0.221729i
$792$	1.05573	0.0375137
$793$	0	0
$794$	−17.2705	−0.612907
$795$	13.4164i	0.475831i
$796$	2.09017	0.0740841
$797$	−53.8328	−1.90686	−0.953428	−	0.301620i	$-0.902472\pi$
$797$	−53.8328	−1.90686	−0.953428	+	0.301620i	$0.902472\pi$
$798$	− 24.7984i	− 0.877853i
$799$	17.1672i	0.607331i
$800$	5.61803i	0.198627i
$801$	18.0000i	0.635999i
$802$	−5.49342	−0.193979
$803$	−1.41641	−0.0499839
$804$	− 9.79837i	− 0.345562i
$805$	−15.9443	−0.561962
$806$	0	0
$807$	−14.5967	−0.513830
$808$	1.18034i	0.0415242i
$809$	−2.88854	−0.101556	−0.0507779	−	0.998710i	$-0.516170\pi$
$809$	−2.88854	−0.101556	−0.0507779	+	0.998710i	$0.516170\pi$
$810$	−6.79837	−0.238871
$811$	31.7771i	1.11584i	0.829893	+	0.557922i	$0.188401\pi$
$811$	31.7771i	1.11584i	−0.829893	+	0.557922i	$0.811599\pi$
$812$	51.2148i	1.79729i
$813$	− 2.88854i	− 0.101306i
$814$	− 0.437694i	− 0.0153412i
$815$	14.7082	0.515206
$816$	14.3951	0.503930
$817$	26.4164i	0.924193i
$818$	15.3820	0.537818
$819$	0	0
$820$	−19.3262	−0.674902
$821$	− 25.3607i	− 0.885094i	−0.896745	−	0.442547i	$-0.854075\pi$
$821$	− 25.3607i	− 0.885094i	0.896745	−	0.442547i	$-0.145925\pi$
$822$	−2.03444	−0.0709593
$823$	26.5967	0.927104	0.463552	−	0.886070i	$-0.346575\pi$
$823$	26.5967	0.927104	0.463552	+	0.886070i	$0.346575\pi$
$824$	28.9443i	1.00832i
$825$	− 0.527864i	− 0.0183779i
$826$	− 1.85410i	− 0.0645125i
$827$	8.94427i	0.311023i	0.987834	+	0.155511i	$0.0497025\pi$
$827$	8.94427i	0.311023i	−0.987834	+	0.155511i	$0.950297\pi$
$828$	−12.1803	−0.423296
$829$	43.2492	1.50211	0.751054	−	0.660241i	$-0.229546\pi$
$829$	43.2492	1.50211	0.751054	+	0.660241i	$0.229546\pi$
$830$	5.52786i	0.191875i
$831$	−37.7639	−1.31002
$832$	0	0
$833$	−38.0000	−1.31662
$834$	23.0902i	0.799547i
$835$	−17.1803	−0.594550
$836$	−1.61803	−0.0559609
$837$	0	0
$838$	− 10.9098i	− 0.376874i
$839$	− 30.7082i	− 1.06016i	−0.847946	−	0.530082i	$-0.822161\pi$
$839$	− 30.7082i	− 1.06016i	0.847946	−	0.530082i	$-0.177839\pi$
$840$	− 21.1803i	− 0.730791i
$841$	26.8328	0.925270
$842$	3.70820	0.127793
$843$	35.5279i	1.22364i
$844$	−21.3262	−0.734079
$845$	0	0
$846$	6.11146	0.210116
$847$	− 46.3607i	− 1.59297i
$848$	11.1246	0.382021
$849$	23.9443	0.821765
$850$	2.14590i	0.0736037i
$851$	11.2918i	0.387078i
$852$	− 22.5623i	− 0.772972i
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	−37.7426	−1.29153
$855$	−8.47214	−0.289741
$856$	12.8885i	0.440521i
$857$	−35.8885	−1.22593	−0.612965	−	0.790110i	$-0.710023\pi$
$857$	−35.8885	−1.22593	−0.612965	+	0.790110i	$0.710023\pi$
$858$	0	0
$859$	21.8885	0.746827	0.373414	−	0.927665i	$-0.378187\pi$
$859$	21.8885	0.746827	0.373414	+	0.927665i	$0.378187\pi$
$860$	10.0902i	0.344072i
$861$	113.138	3.85572
$862$	16.7984	0.572155
$863$	14.8328i	0.504915i	0.967608	+	0.252457i	$0.0812388\pi$
$863$	14.8328i	0.504915i	−0.967608	+	0.252457i	$0.918761\pi$
$864$	12.5623i	0.427378i
$865$	18.8885i	0.642230i
$866$	− 8.97871i	− 0.305109i
$867$	−11.0557	−0.375472
$868$	0	0
$869$	0	0
$870$	−10.3262	−0.350092
$871$	0	0
$872$	−4.47214	−0.151446
$873$	− 6.94427i	− 0.235028i
$874$	−9.85410	−0.333320
$875$	−4.23607	−0.143205
$876$	21.7082i	0.733452i
$877$	− 33.7214i	− 1.13869i	−0.822099	−	0.569345i	$-0.807197\pi$
$877$	− 33.7214i	− 1.13869i	0.822099	−	0.569345i	$-0.192803\pi$
$878$	− 14.0344i	− 0.473639i
$879$	66.9574i	2.25842i
$880$	−0.437694	−0.0147547
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	13.5279i	0.455507i
$883$	42.8328	1.44144	0.720720	−	0.693227i	$-0.243812\pi$
$883$	42.8328	1.44144	0.720720	+	0.693227i	$0.243812\pi$
$884$	0	0
$885$	−1.58359	−0.0532319
$886$	− 0.583592i	− 0.0196062i
$887$	44.4853	1.49367	0.746835	−	0.665009i	$-0.231572\pi$
$887$	44.4853	1.49367	0.746835	+	0.665009i	$0.231572\pi$
$888$	−15.0000	−0.503367
$889$	− 17.9443i	− 0.601832i
$890$	5.56231i	0.186449i
$891$	− 2.59675i	− 0.0869943i
$892$	1.14590i	0.0383675i
$893$	−20.9443	−0.700873
$894$	6.25735	0.209277
$895$	8.23607i	0.275301i
$896$	−48.2148	−1.61074
$897$	0	0
$898$	−2.43769	−0.0813469
$899$	0	0
$900$	−3.23607	−0.107869
$901$	20.8328	0.694042
$902$	1.74265i	0.0580238i
$903$	− 59.0689i	− 1.96569i
$904$	− 3.29180i	− 0.109484i
$905$	− 6.00000i	− 0.199447i
$906$	−11.0557	−0.367302
$907$	−20.1246	−0.668227	−0.334113	−	0.942533i	$-0.608437\pi$
$907$	−20.1246	−0.668227	−0.334113	+	0.942533i	$0.608437\pi$
$908$	− 10.0902i	− 0.334854i
$909$	−1.05573	−0.0350163
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	17.5623i	0.581546i
$913$	−2.11146	−0.0698790
$914$	−0.978714	−0.0323730
$915$	32.2361i	1.06569i
$916$	− 25.7082i	− 0.849423i
$917$	50.8328i	1.67865i
$918$	4.79837i	0.158370i
$919$	46.7082	1.54076	0.770381	−	0.637584i	$-0.220066\pi$
$919$	46.7082	1.54076	0.770381	+	0.637584i	$0.220066\pi$
$920$	−8.41641	−0.277481
$921$	− 55.7771i	− 1.83792i
$922$	−0.978714	−0.0322322
$923$	0	0
$924$	3.61803	0.119025
$925$	3.00000i	0.0986394i
$926$	−6.83282	−0.224540
$927$	−25.8885	−0.850291
$928$	41.9787i	1.37802i
$929$	19.9443i	0.654350i	0.944964	+	0.327175i	$0.106097\pi$
$929$	19.9443i	0.654350i	−0.944964	+	0.327175i	$0.893903\pi$
$930$	0	0
$931$	− 46.3607i	− 1.51941i
$932$	25.7082	0.842100
$933$	53.6656	1.75693
$934$	5.52786i	0.180877i
$935$	−0.819660	−0.0268058
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	7.09017i	0.231502i
$939$	8.69505	0.283752
$940$	−8.00000	−0.260931
$941$	− 11.8885i	− 0.387555i	−0.981045	−	0.193778i	$-0.937926\pi$
$941$	− 11.8885i	− 0.387555i	0.981045	−	0.193778i	$-0.0620741\pi$
$942$	24.8754i	0.810484i
$943$	− 44.9574i	− 1.46402i
$944$	1.31308i	0.0427372i
$945$	−9.47214	−0.308129
$946$	0.909830	0.0295811
$947$	− 39.1803i	− 1.27319i	−0.771198	−	0.636595i	$-0.780342\pi$
$947$	− 39.1803i	− 1.27319i	0.771198	−	0.636595i	$-0.219658\pi$
$948$	0	0
$949$	0	0
$950$	−2.61803	−0.0849402
$951$	− 26.5836i	− 0.862032i
$952$	−32.8885	−1.06592
$953$	−42.1935	−1.36678	−0.683391	−	0.730053i	$-0.739495\pi$
$953$	−42.1935	−1.36678	−0.683391	+	0.730053i	$0.739495\pi$
$954$	− 7.41641i	− 0.240115i
$955$	− 27.1803i	− 0.879535i
$956$	− 41.8885i	− 1.35477i
$957$	− 3.94427i	− 0.127500i
$958$	−19.8541	−0.641457
$959$	−6.23607	−0.201373
$960$	0.527864i	0.0170367i
$961$	31.0000	1.00000
$962$	0	0
$963$	−11.5279	−0.371480
$964$	− 24.2705i	− 0.781700i
$965$	−3.47214	−0.111772
$966$	22.0344	0.708947
$967$	− 54.8328i	− 1.76330i	−0.471900	−	0.881652i	$-0.656432\pi$
$967$	− 54.8328i	− 1.76330i	0.471900	−	0.881652i	$-0.343568\pi$
$968$	− 24.4721i	− 0.786564i
$969$	32.8885i	1.05653i
$970$	− 2.14590i	− 0.0689006i
$971$	−11.7639	−0.377523	−0.188761	−	0.982023i	$-0.560447\pi$
$971$	−11.7639	−0.377523	−0.188761	+	0.982023i	$0.560447\pi$
$972$	−28.9443	−0.928388
$973$	70.7771i	2.26901i
$974$	8.72949	0.279711
$975$	0	0
$976$	26.7295	0.855590
$977$	7.58359i	0.242621i	0.992615	+	0.121310i	$0.0387096\pi$
$977$	7.58359i	0.242621i	−0.992615	+	0.121310i	$0.961290\pi$
$978$	−20.3262	−0.649961
$979$	−2.12461	−0.0679029
$980$	− 17.7082i	− 0.565668i
$981$	− 4.00000i	− 0.127710i
$982$	− 0.145898i	− 0.00465579i
$983$	− 38.8328i	− 1.23857i	−0.785165	−	0.619287i	$-0.787422\pi$
$983$	− 38.8328i	− 1.23857i	0.785165	−	0.619287i	$-0.212578\pi$
$984$	59.7214	1.90385
$985$	−3.00000	−0.0955879
$986$	16.0344i	0.510641i
$987$	46.8328	1.49070
$988$	0	0
$989$	−23.4721	−0.746371
$990$	0.291796i	0.00927389i
$991$	−46.7082	−1.48373	−0.741867	−	0.670547i	$-0.766060\pi$
$991$	−46.7082	−1.48373	−0.741867	+	0.670547i	$0.766060\pi$
$992$	0	0
$993$	− 12.6393i	− 0.401097i
$994$	16.3262i	0.517837i
$995$	1.29180i	0.0409527i
$996$	32.3607i	1.02539i
$997$	−18.8885	−0.598206	−0.299103	−	0.954221i	$-0.596687\pi$
$997$	−18.8885	−0.598206	−0.299103	+	0.954221i	$0.596687\pi$
$998$	−13.5279	−0.428217
$999$	6.70820i	0.212238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.c.506.2		4
13.2	odd	12		65.2.e.a.61.2	yes	4
13.3	even	3		845.2.m.e.316.3		8
13.4	even	6		845.2.m.e.361.3		8
13.5	odd	4		845.2.a.e.1.1		2
13.6	odd	12		65.2.e.a.16.2	✓	4
13.7	odd	12		845.2.e.g.146.1		4
13.8	odd	4		845.2.a.b.1.2		2
13.9	even	3		845.2.m.e.361.2		8
13.10	even	6		845.2.m.e.316.2		8
13.11	odd	12		845.2.e.g.191.1		4
13.12	even	2	inner	845.2.c.c.506.3		4
39.2	even	12		585.2.j.e.451.1		4
39.5	even	4		7605.2.a.ba.1.2		2
39.8	even	4		7605.2.a.bf.1.1		2
39.32	even	12		585.2.j.e.406.1		4
52.15	even	12		1040.2.q.n.321.2		4
52.19	even	12		1040.2.q.n.81.2		4
65.2	even	12		325.2.o.a.74.3		8
65.19	odd	12		325.2.e.b.276.1		4
65.28	even	12		325.2.o.a.74.2		8
65.32	even	12		325.2.o.a.224.2		8
65.34	odd	4		4225.2.a.y.1.1		2
65.44	odd	4		4225.2.a.u.1.2		2
65.54	odd	12		325.2.e.b.126.1		4
65.58	even	12		325.2.o.a.224.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.a.16.2	✓	4	13.6	odd	12
65.2.e.a.61.2	yes	4	13.2	odd	12
325.2.e.b.126.1		4	65.54	odd	12
325.2.e.b.276.1		4	65.19	odd	12
325.2.o.a.74.2		8	65.28	even	12
325.2.o.a.74.3		8	65.2	even	12
325.2.o.a.224.2		8	65.32	even	12
325.2.o.a.224.3		8	65.58	even	12
585.2.j.e.406.1		4	39.32	even	12
585.2.j.e.451.1		4	39.2	even	12
845.2.a.b.1.2		2	13.8	odd	4
845.2.a.e.1.1		2	13.5	odd	4
845.2.c.c.506.2		4	1.1	even	1	trivial
845.2.c.c.506.3		4	13.12	even	2	inner
845.2.e.g.146.1		4	13.7	odd	12
845.2.e.g.191.1		4	13.11	odd	12
845.2.m.e.316.2		8	13.10	even	6
845.2.m.e.316.3		8	13.3	even	3
845.2.m.e.361.2		8	13.9	even	3
845.2.m.e.361.3		8	13.4	even	6
1040.2.q.n.81.2		4	52.19	even	12
1040.2.q.n.321.2		4	52.15	even	12
4225.2.a.u.1.2		2	65.44	odd	4
4225.2.a.y.1.1		2	65.34	odd	4
7605.2.a.ba.1.2		2	39.5	even	4
7605.2.a.bf.1.1		2	39.8	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{2} +2.23607 q^{3} +1.61803 q^{4} +1.00000i q^{5} -1.38197i q^{6} -4.23607i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-0.618034i q^{2} +2.23607 q^{3} +1.61803 q^{4} +1.00000i q^{5} -1.38197i q^{6} -4.23607i q^{7} -2.23607i q^{8} +2.00000 q^{9} +0.618034 q^{10} +0.236068i q^{11} +3.61803 q^{12} -2.61803 q^{14} +2.23607i q^{15} +1.85410 q^{16} +3.47214 q^{17} -1.23607i q^{18} +4.23607i q^{19} +1.61803i q^{20} -9.47214i q^{21} +0.145898 q^{22} -3.76393 q^{23} -5.00000i q^{24} -1.00000 q^{25} -2.23607 q^{27} -6.85410i q^{28} -7.47214 q^{29} +1.38197 q^{30} -5.61803i q^{32} +0.527864i q^{33} -2.14590i q^{34} +4.23607 q^{35} +3.23607 q^{36} -3.00000i q^{37} +2.61803 q^{38} +2.23607 q^{40} +11.9443i q^{41} -5.85410 q^{42} +6.23607 q^{43} +0.381966i q^{44} +2.00000i q^{45} +2.32624i q^{46} +4.94427i q^{47} +4.14590 q^{48} -10.9443 q^{49} +0.618034i q^{50} +7.76393 q^{51} +6.00000 q^{53} +1.38197i q^{54} -0.236068 q^{55} -9.47214 q^{56} +9.47214i q^{57} +4.61803i q^{58} +0.708204i q^{59} +3.61803i q^{60} +14.4164 q^{61} -8.47214i q^{63} +0.236068 q^{64} +0.326238 q^{66} -2.70820i q^{67} +5.61803 q^{68} -8.41641 q^{69} -2.61803i q^{70} -6.23607i q^{71} -4.47214i q^{72} +6.00000i q^{73} -1.85410 q^{74} -2.23607 q^{75} +6.85410i q^{76} +1.00000 q^{77} +1.85410i q^{80} -11.0000 q^{81} +7.38197 q^{82} +8.94427i q^{83} -15.3262i q^{84} +3.47214i q^{85} -3.85410i q^{86} -16.7082 q^{87} +0.527864 q^{88} +9.00000i q^{89} +1.23607 q^{90} -6.09017 q^{92} +3.05573 q^{94} -4.23607 q^{95} -12.5623i q^{96} -3.47214i q^{97} +6.76393i q^{98} +0.472136i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 8 * q^9	\( 4 q + 2 q^{4} + 8 q^{9} - 2 q^{10} + 10 q^{12} - 6 q^{14} - 6 q^{16} - 4 q^{17} + 14 q^{22} - 24 q^{23} - 4 q^{25} - 12 q^{29} + 10 q^{30} + 8 q^{35} + 4 q^{36} + 6 q^{38} - 10 q^{42} + 16 q^{43} + 30 q^{48} - 8 q^{49} + 40 q^{51} + 24 q^{53} + 8 q^{55} - 20 q^{56} + 4 q^{61} - 8 q^{64} - 30 q^{66} + 18 q^{68} + 20 q^{69} + 6 q^{74} + 4 q^{77} - 44 q^{81} + 34 q^{82} - 40 q^{87} + 20 q^{88} - 4 q^{90} - 2 q^{92} + 48 q^{94} - 8 q^{95}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^9 - 2 * q^10 + 10 * q^12 - 6 * q^14 - 6 * q^16 - 4 * q^17 + 14 * q^22 - 24 * q^23 - 4 * q^25 - 12 * q^29 + 10 * q^30 + 8 * q^35 + 4 * q^36 + 6 * q^38 - 10 * q^42 + 16 * q^43 + 30 * q^48 - 8 * q^49 + 40 * q^51 + 24 * q^53 + 8 * q^55 - 20 * q^56 + 4 * q^61 - 8 * q^64 - 30 * q^66 + 18 * q^68 + 20 * q^69 + 6 * q^74 + 4 * q^77 - 44 * q^81 + 34 * q^82 - 40 * q^87 + 20 * q^88 - 4 * q^90 - 2 * q^92 + 48 * q^94 - 8 * q^95

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