LMFDB - Embedded newform 845.2.c.e.506.4

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:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.e.506.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205i q^{2} +2.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.00000i q^{7} +1.73205i q^{8} +4.46410 q^{9} -1.73205 q^{10} -4.73205i q^{11} -2.73205 q^{12} -3.46410 q^{14} +2.73205i q^{15} -5.00000 q^{16} +3.46410 q^{17} +7.73205i q^{18} +6.19615i q^{19} -1.00000i q^{20} +5.46410i q^{21} +8.19615 q^{22} -1.26795 q^{23} +4.73205i q^{24} -1.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -2.53590 q^{29} -4.73205 q^{30} -10.1962i q^{31} -5.19615i q^{32} -12.9282i q^{33} +6.00000i q^{34} -2.00000 q^{35} -4.46410 q^{36} -4.00000i q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410i q^{41} -9.46410 q^{42} +0.196152 q^{43} +4.73205i q^{44} +4.46410i q^{45} -2.19615i q^{46} +6.00000i q^{47} -13.6603 q^{48} +3.00000 q^{49} -1.73205i q^{50} +9.46410 q^{51} +10.3923 q^{53} +6.92820i q^{54} +4.73205 q^{55} -3.46410 q^{56} +16.9282i q^{57} -4.39230i q^{58} +9.12436i q^{59} -2.73205i q^{60} -8.39230 q^{61} +17.6603 q^{62} +8.92820i q^{63} -1.00000 q^{64} +22.3923 q^{66} -6.39230i q^{67} -3.46410 q^{68} -3.46410 q^{69} -3.46410i q^{70} -4.73205i q^{71} +7.73205i q^{72} -4.00000i q^{73} +6.92820 q^{74} -2.73205 q^{75} -6.19615i q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000i q^{80} -2.46410 q^{81} +6.00000 q^{82} +6.00000i q^{83} -5.46410i q^{84} +3.46410i q^{85} +0.339746i q^{86} -6.92820 q^{87} +8.19615 q^{88} -12.9282i q^{89} -7.73205 q^{90} +1.26795 q^{92} -27.8564i q^{93} -10.3923 q^{94} -6.19615 q^{95} -14.1962i q^{96} -2.00000i q^{97} +5.19615i q^{98} -21.1244i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49}+ \cdots - 4 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^12 - 20 * q^16 + 12 * q^22 - 12 * q^23 - 4 * q^25 + 16 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^35 - 4 * q^36 - 36 * q^38 - 24 * q^42 - 20 * q^43 - 20 * q^48 + 12 * q^49 + 24 * q^51 + 12 * q^55 + 8 * q^61 + 36 * q^62 - 4 * q^64 + 48 * q^66 - 4 * q^75 + 24 * q^77 + 8 * q^79 + 4 * q^81 + 24 * q^82 + 12 * q^88 - 24 * q^90 + 12 * q^92 - 4 * q^95

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	4.73205i	1.93185i
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.73205i	0.612372i
$9$	4.46410	1.48803
$10$	−1.73205	−0.547723
$11$	− 4.73205i	− 1.42677i	−0.700774	−	0.713384i	$-0.747162\pi$
$11$	− 4.73205i	− 1.42677i	0.700774	−	0.713384i	$-0.252838\pi$
$12$	−2.73205	−0.788675
$13$	0	0
$13$	0	0
$14$	−3.46410	−0.925820
$15$	2.73205i	0.705412i
$16$	−5.00000	−1.25000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	7.73205i	1.82246i
$19$	6.19615i	1.42149i	0.703447	+	0.710747i	$0.251643\pi$
$19$	6.19615i	1.42149i	−0.703447	+	0.710747i	$0.748357\pi$
$20$	− 1.00000i	− 0.223607i
$21$	5.46410i	1.19236i
$22$	8.19615	1.74743
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	4.73205i	0.965926i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	− 2.00000i	− 0.377964i
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	−4.73205	−0.863950
$31$	− 10.1962i	− 1.83128i	−0.401996	−	0.915642i	$-0.631683\pi$
$31$	− 10.1962i	− 1.83128i	0.401996	−	0.915642i	$-0.368317\pi$
$32$	− 5.19615i	− 0.918559i
$33$	− 12.9282i	− 2.25051i
$34$	6.00000i	1.02899i
$35$	−2.00000	−0.338062
$36$	−4.46410	−0.744017
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	−10.7321	−1.74097
$39$	0	0
$40$	−1.73205	−0.273861
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	−9.46410	−1.46034
$43$	0.196152	0.0299130	0.0149565	−	0.999888i	$-0.495239\pi$
$43$	0.196152	0.0299130	0.0149565	+	0.999888i	$0.495239\pi$
$44$	4.73205i	0.713384i
$45$	4.46410i	0.665469i
$46$	− 2.19615i	− 0.323805i
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	−13.6603	−1.97169
$49$	3.00000	0.428571
$50$	− 1.73205i	− 0.244949i
$51$	9.46410	1.32524
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	6.92820i	0.942809i
$55$	4.73205	0.638070
$56$	−3.46410	−0.462910
$57$	16.9282i	2.24220i
$58$	− 4.39230i	− 0.576738i
$59$	9.12436i	1.18789i	0.804506	+	0.593945i	$0.202430\pi$
$59$	9.12436i	1.18789i	−0.804506	+	0.593945i	$0.797570\pi$
$60$	− 2.73205i	− 0.352706i
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	17.6603	2.24285
$63$	8.92820i	1.12485i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	22.3923	2.75630
$67$	− 6.39230i	− 0.780944i	−0.920615	−	0.390472i	$-0.872312\pi$
$67$	− 6.39230i	− 0.780944i	0.920615	−	0.390472i	$-0.127688\pi$
$68$	−3.46410	−0.420084
$69$	−3.46410	−0.417029
$70$	− 3.46410i	− 0.414039i
$71$	− 4.73205i	− 0.561591i	−0.959768	−	0.280796i	$-0.909402\pi$
$71$	− 4.73205i	− 0.561591i	0.959768	−	0.280796i	$-0.0905983\pi$
$72$	7.73205i	0.911231i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	6.92820	0.805387
$75$	−2.73205	−0.315470
$76$	− 6.19615i	− 0.710747i
$77$	9.46410	1.07853
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	− 5.00000i	− 0.559017i
$81$	−2.46410	−0.273789
$82$	6.00000	0.662589
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	− 5.46410i	− 0.596182i
$85$	3.46410i	0.375735i
$86$	0.339746i	0.0366357i
$87$	−6.92820	−0.742781
$88$	8.19615	0.873713
$89$	− 12.9282i	− 1.37039i	−0.728361	−	0.685193i	$-0.759718\pi$
$89$	− 12.9282i	− 1.37039i	0.728361	−	0.685193i	$-0.240282\pi$
$90$	−7.73205	−0.815030
$91$	0	0
$92$	1.26795	0.132193
$93$	− 27.8564i	− 2.88857i
$94$	−10.3923	−1.07188
$95$	−6.19615	−0.635712
$96$	− 14.1962i	− 1.44889i
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	5.19615i	0.524891i
$99$	− 21.1244i	− 2.12308i
$100$	1.00000	0.100000
$101$	0.928203	0.0923597	0.0461798	−	0.998933i	$-0.485295\pi$
$101$	0.928203	0.0923597	0.0461798	+	0.998933i	$0.485295\pi$
$102$	16.3923i	1.62308i
$103$	0.196152	0.0193275	0.00966374	−	0.999953i	$-0.496924\pi$
$103$	0.196152	0.0193275	0.00966374	+	0.999953i	$0.496924\pi$
$104$	0	0
$105$	−5.46410	−0.533242
$106$	18.0000i	1.74831i
$107$	17.6603	1.70728	0.853641	−	0.520862i	$-0.174390\pi$
$107$	17.6603	1.70728	0.853641	+	0.520862i	$0.174390\pi$
$108$	−4.00000	−0.384900
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	8.19615i	0.781472i
$111$	− 10.9282i	− 1.03726i
$112$	− 10.0000i	− 0.944911i
$113$	8.53590	0.802990	0.401495	−	0.915861i	$-0.368491\pi$
$113$	8.53590	0.802990	0.401495	+	0.915861i	$0.368491\pi$
$114$	−29.3205	−2.74612
$115$	− 1.26795i	− 0.118237i
$116$	2.53590	0.235452
$117$	0	0
$118$	−15.8038	−1.45486
$119$	6.92820i	0.635107i
$120$	−4.73205	−0.431975
$121$	−11.3923	−1.03566
$122$	− 14.5359i	− 1.31602i
$123$	− 9.46410i	− 0.853349i
$124$	10.1962i	0.915642i
$125$	− 1.00000i	− 0.0894427i
$126$	−15.4641	−1.37765
$127$	−16.1962	−1.43718	−0.718588	−	0.695436i	$-0.755211\pi$
$127$	−16.1962	−1.43718	−0.718588	+	0.695436i	$0.755211\pi$
$128$	− 12.1244i	− 1.07165i
$129$	0.535898	0.0471832
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	12.9282i	1.12526i
$133$	−12.3923	−1.07455
$134$	11.0718	0.956458
$135$	4.00000i	0.344265i
$136$	6.00000i	0.514496i
$137$	0.928203i	0.0793018i	0.999214	+	0.0396509i	$0.0126246\pi$
$137$	0.928203i	0.0793018i	−0.999214	+	0.0396509i	$0.987375\pi$
$138$	− 6.00000i	− 0.510754i
$139$	12.3923	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$139$	12.3923	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$140$	2.00000	0.169031
$141$	16.3923i	1.38048i
$142$	8.19615	0.687806
$143$	0	0
$144$	−22.3205	−1.86004
$145$	− 2.53590i	− 0.210595i
$146$	6.92820	0.573382
$147$	8.19615	0.676007
$148$	4.00000i	0.328798i
$149$	7.85641i	0.643622i	0.946804	+	0.321811i	$0.104292\pi$
$149$	7.85641i	0.643622i	−0.946804	+	0.321811i	$0.895708\pi$
$150$	− 4.73205i	− 0.386370i
$151$	− 1.80385i	− 0.146795i	−0.997303	−	0.0733975i	$-0.976616\pi$
$151$	− 1.80385i	− 0.146795i	0.997303	−	0.0733975i	$-0.0233842\pi$
$152$	−10.7321	−0.870484
$153$	15.4641	1.25020
$154$	16.3923i	1.32093i
$155$	10.1962	0.818975
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 14.5359i	− 1.15641i
$159$	28.3923	2.25166
$160$	5.19615	0.410792
$161$	− 2.53590i	− 0.199857i
$162$	− 4.26795i	− 0.335322i
$163$	− 14.3923i	− 1.12729i	−0.826016	−	0.563646i	$-0.809398\pi$
$163$	− 14.3923i	− 1.12729i	0.826016	−	0.563646i	$-0.190602\pi$
$164$	3.46410i	0.270501i
$165$	12.9282	1.00646
$166$	−10.3923	−0.806599
$167$	− 0.928203i	− 0.0718265i	−0.999355	−	0.0359133i	$-0.988566\pi$
$167$	− 0.928203i	− 0.0718265i	0.999355	−	0.0359133i	$-0.0114340\pi$
$168$	−9.46410	−0.730171
$169$	0	0
$170$	−6.00000	−0.460179
$171$	27.6603i	2.11523i
$172$	−0.196152	−0.0149565
$173$	−8.53590	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$173$	−8.53590	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$174$	− 12.0000i	− 0.909718i
$175$	− 2.00000i	− 0.151186i
$176$	23.6603i	1.78346i
$177$	24.9282i	1.87372i
$178$	22.3923	1.67837
$179$	18.9282	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$179$	18.9282	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$180$	− 4.46410i	− 0.332734i
$181$	−0.392305	−0.0291598	−0.0145799	−	0.999894i	$-0.504641\pi$
$181$	−0.392305	−0.0291598	−0.0145799	+	0.999894i	$0.504641\pi$
$182$	0	0
$183$	−22.9282	−1.69490
$184$	− 2.19615i	− 0.161903i
$185$	4.00000	0.294086
$186$	48.2487	3.53777
$187$	− 16.3923i	− 1.19872i
$188$	− 6.00000i	− 0.437595i
$189$	8.00000i	0.581914i
$190$	− 10.7321i	− 0.778585i
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	−2.73205	−0.197169
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	3.46410	0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.9282i	0.921096i	0.887635	+	0.460548i	$0.152347\pi$
$197$	12.9282i	0.921096i	−0.887635	+	0.460548i	$0.847653\pi$
$198$	36.5885	2.60023
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	− 1.73205i	− 0.122474i
$201$	− 17.4641i	− 1.23182i
$202$	1.60770i	0.113117i
$203$	− 5.07180i	− 0.355970i
$204$	−9.46410	−0.662620
$205$	3.46410	0.241943
$206$	0.339746i	0.0236712i
$207$	−5.66025	−0.393415
$208$	0	0
$209$	29.3205	2.02814
$210$	− 9.46410i	− 0.653085i
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−10.3923	−0.713746
$213$	− 12.9282i	− 0.885826i
$214$	30.5885i	2.09098i
$215$	0.196152i	0.0133775i
$216$	6.92820i	0.471405i
$217$	20.3923	1.38432
$218$	3.46410	0.234619
$219$	− 10.9282i	− 0.738460i
$220$	−4.73205	−0.319035
$221$	0	0
$222$	18.9282	1.27038
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	10.3923	0.694365
$225$	−4.46410	−0.297607
$226$	14.7846i	0.983458i
$227$	3.46410i	0.229920i	0.993370	+	0.114960i	$0.0366741\pi$
$227$	3.46410i	0.229920i	−0.993370	+	0.114960i	$0.963326\pi$
$228$	− 16.9282i	− 1.12110i
$229$	6.39230i	0.422415i	0.977441	+	0.211208i	$0.0677396\pi$
$229$	6.39230i	0.422415i	−0.977441	+	0.211208i	$0.932260\pi$
$230$	2.19615	0.144810
$231$	25.8564	1.70123
$232$	− 4.39230i	− 0.288369i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	− 9.12436i	− 0.593945i
$237$	−22.9282	−1.48935
$238$	−12.0000	−0.777844
$239$	14.1962i	0.918273i	0.888366	+	0.459136i	$0.151841\pi$
$239$	14.1962i	0.918273i	−0.888366	+	0.459136i	$0.848159\pi$
$240$	− 13.6603i	− 0.881766i
$241$	− 2.39230i	− 0.154102i	−0.997027	−	0.0770510i	$-0.975450\pi$
$241$	− 2.39230i	− 0.154102i	0.997027	−	0.0770510i	$-0.0245504\pi$
$242$	− 19.7321i	− 1.26842i
$243$	−18.7321	−1.20166
$244$	8.39230	0.537262
$245$	3.00000i	0.191663i
$246$	16.3923	1.04514
$247$	0	0
$248$	17.6603	1.12143
$249$	16.3923i	1.03882i
$250$	1.73205	0.109545
$251$	21.4641	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$251$	21.4641	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$252$	− 8.92820i	− 0.562424i
$253$	6.00000i	0.377217i
$254$	− 28.0526i	− 1.76017i
$255$	9.46410i	0.592665i
$256$	19.0000	1.18750
$257$	−19.8564	−1.23861	−0.619304	−	0.785151i	$-0.712585\pi$
$257$	−19.8564	−1.23861	−0.619304	+	0.785151i	$0.712585\pi$
$258$	0.928203i	0.0577874i
$259$	8.00000	0.497096
$260$	0	0
$261$	−11.3205	−0.700722
$262$	0	0
$263$	1.26795	0.0781851	0.0390925	−	0.999236i	$-0.487553\pi$
$263$	1.26795	0.0781851	0.0390925	+	0.999236i	$0.487553\pi$
$264$	22.3923	1.37815
$265$	10.3923i	0.638394i
$266$	− 21.4641i	− 1.31605i
$267$	− 35.3205i	− 2.16158i
$268$	6.39230i	0.390472i
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	−6.92820	−0.421637
$271$	30.9808i	1.88195i	0.338480	+	0.940974i	$0.390087\pi$
$271$	30.9808i	1.88195i	−0.338480	+	0.940974i	$0.609913\pi$
$272$	−17.3205	−1.05021
$273$	0	0
$274$	−1.60770	−0.0971244
$275$	4.73205i	0.285353i
$276$	3.46410	0.208514
$277$	26.3923	1.58576	0.792880	−	0.609378i	$-0.208581\pi$
$277$	26.3923	1.58576	0.792880	+	0.609378i	$0.208581\pi$
$278$	21.4641i	1.28733i
$279$	− 45.5167i	− 2.72501i
$280$	− 3.46410i	− 0.207020i
$281$	22.3923i	1.33581i	0.744245	+	0.667906i	$0.232809\pi$
$281$	22.3923i	1.33581i	−0.744245	+	0.667906i	$0.767191\pi$
$282$	−28.3923	−1.69074
$283$	−32.5885	−1.93718	−0.968591	−	0.248658i	$-0.920010\pi$
$283$	−32.5885	−1.93718	−0.968591	+	0.248658i	$0.920010\pi$
$284$	4.73205i	0.280796i
$285$	−16.9282	−1.00274
$286$	0	0
$287$	6.92820	0.408959
$288$	− 23.1962i	− 1.36685i
$289$	−5.00000	−0.294118
$290$	4.39230	0.257925
$291$	− 5.46410i	− 0.320311i
$292$	4.00000i	0.234082i
$293$	− 5.07180i	− 0.296298i	−0.988965	−	0.148149i	$-0.952669\pi$
$293$	− 5.07180i	− 0.296298i	0.988965	−	0.148149i	$-0.0473314\pi$
$294$	14.1962i	0.827936i
$295$	−9.12436	−0.531241
$296$	6.92820	0.402694
$297$	− 18.9282i	− 1.09833i
$298$	−13.6077	−0.788273
$299$	0	0
$300$	2.73205	0.157735
$301$	0.392305i	0.0226121i
$302$	3.12436	0.179786
$303$	2.53590	0.145684
$304$	− 30.9808i	− 1.77687i
$305$	− 8.39230i	− 0.480542i
$306$	26.7846i	1.53117i
$307$	− 18.7846i	− 1.07209i	−0.844188	−	0.536047i	$-0.819917\pi$
$307$	− 18.7846i	− 1.07209i	0.844188	−	0.536047i	$-0.180083\pi$
$308$	−9.46410	−0.539267
$309$	0.535898	0.0304862
$310$	17.6603i	1.00304i
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	− 17.3205i	− 0.977453i
$315$	−8.92820	−0.503047
$316$	8.39230	0.472104
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	49.1769i	2.75770i
$319$	12.0000i	0.671871i
$320$	− 1.00000i	− 0.0559017i
$321$	48.2487	2.69298
$322$	4.39230	0.244774
$323$	21.4641i	1.19429i
$324$	2.46410	0.136895
$325$	0	0
$326$	24.9282	1.38065
$327$	− 5.46410i	− 0.302166i
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	22.3923i	1.23266i
$331$	− 2.58846i	− 0.142274i	−0.997467	−	0.0711372i	$-0.977337\pi$
$331$	− 2.58846i	− 0.142274i	0.997467	−	0.0711372i	$-0.0226628\pi$
$332$	− 6.00000i	− 0.329293i
$333$	− 17.8564i	− 0.978525i
$334$	1.60770	0.0879692
$335$	6.39230	0.349249
$336$	− 27.3205i	− 1.49046i
$337$	26.3923	1.43768	0.718840	−	0.695175i	$-0.244673\pi$
$337$	26.3923	1.43768	0.718840	+	0.695175i	$0.244673\pi$
$338$	0	0
$339$	23.3205	1.26660
$340$	− 3.46410i	− 0.187867i
$341$	−48.2487	−2.61281
$342$	−47.9090	−2.59062
$343$	20.0000i	1.07990i
$344$	0.339746i	0.0183179i
$345$	− 3.46410i	− 0.186501i
$346$	− 14.7846i	− 0.794826i
$347$	−5.66025	−0.303858	−0.151929	−	0.988391i	$-0.548549\pi$
$347$	−5.66025	−0.303858	−0.151929	+	0.988391i	$0.548549\pi$
$348$	6.92820	0.371391
$349$	− 14.3923i	− 0.770402i	−0.922833	−	0.385201i	$-0.874132\pi$
$349$	− 14.3923i	− 0.770402i	0.922833	−	0.385201i	$-0.125868\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	−24.5885	−1.31057
$353$	27.7128i	1.47500i	0.675345	+	0.737502i	$0.263995\pi$
$353$	27.7128i	1.47500i	−0.675345	+	0.737502i	$0.736005\pi$
$354$	−43.1769	−2.29483
$355$	4.73205	0.251151
$356$	12.9282i	0.685193i
$357$	18.9282i	1.00179i
$358$	32.7846i	1.73272i
$359$	− 2.19615i	− 0.115908i	−0.998319	−	0.0579542i	$-0.981542\pi$
$359$	− 2.19615i	− 0.115908i	0.998319	−	0.0579542i	$-0.0184578\pi$
$360$	−7.73205	−0.407515
$361$	−19.3923	−1.02065
$362$	− 0.679492i	− 0.0357133i
$363$	−31.1244	−1.63361
$364$	0	0
$365$	4.00000	0.209370
$366$	− 39.7128i	− 2.07582i
$367$	11.8038	0.616156	0.308078	−	0.951361i	$-0.400314\pi$
$367$	11.8038	0.616156	0.308078	+	0.951361i	$0.400314\pi$
$368$	6.33975	0.330482
$369$	− 15.4641i	− 0.805029i
$370$	6.92820i	0.360180i
$371$	20.7846i	1.07908i
$372$	27.8564i	1.44429i
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	28.3923	1.46813
$375$	− 2.73205i	− 0.141082i
$376$	−10.3923	−0.535942
$377$	0	0
$378$	−13.8564	−0.712697
$379$	− 18.9808i	− 0.974976i	−0.873130	−	0.487488i	$-0.837913\pi$
$379$	− 18.9808i	− 0.974976i	0.873130	−	0.487488i	$-0.162087\pi$
$380$	6.19615	0.317856
$381$	−44.2487	−2.26693
$382$	− 8.78461i	− 0.449460i
$383$	− 12.9282i	− 0.660600i	−0.943876	−	0.330300i	$-0.892850\pi$
$383$	− 12.9282i	− 0.660600i	0.943876	−	0.330300i	$-0.107150\pi$
$384$	− 33.1244i	− 1.69037i
$385$	9.46410i	0.482335i
$386$	17.3205	0.881591
$387$	0.875644	0.0445115
$388$	2.00000i	0.101535i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.39230	−0.222128
$392$	5.19615i	0.262445i
$393$	0	0
$394$	−22.3923	−1.12811
$395$	− 8.39230i	− 0.422263i
$396$	21.1244i	1.06154i
$397$	28.7846i	1.44466i	0.691549	+	0.722329i	$0.256928\pi$
$397$	28.7846i	1.44466i	−0.691549	+	0.722329i	$0.743072\pi$
$398$	− 34.6410i	− 1.73640i
$399$	−33.8564	−1.69494
$400$	5.00000	0.250000
$401$	− 36.9282i	− 1.84411i	−0.387063	−	0.922053i	$-0.626510\pi$
$401$	− 36.9282i	− 1.84411i	0.387063	−	0.922053i	$-0.373490\pi$
$402$	30.2487	1.50867
$403$	0	0
$404$	−0.928203	−0.0461798
$405$	− 2.46410i	− 0.122442i
$406$	8.78461	0.435973
$407$	−18.9282	−0.938236
$408$	16.3923i	0.811540i
$409$	17.6077i	0.870644i	0.900275	+	0.435322i	$0.143366\pi$
$409$	17.6077i	0.870644i	−0.900275	+	0.435322i	$0.856634\pi$
$410$	6.00000i	0.296319i
$411$	2.53590i	0.125087i
$412$	−0.196152	−0.00966374
$413$	−18.2487	−0.897960
$414$	− 9.80385i	− 0.481833i
$415$	−6.00000	−0.294528
$416$	0	0
$417$	33.8564	1.65796
$418$	50.7846i	2.48396i
$419$	−2.53590	−0.123887	−0.0619434	−	0.998080i	$-0.519730\pi$
$419$	−2.53590	−0.123887	−0.0619434	+	0.998080i	$0.519730\pi$
$420$	5.46410	0.266621
$421$	30.7846i	1.50035i	0.661239	+	0.750175i	$0.270031\pi$
$421$	30.7846i	1.50035i	−0.661239	+	0.750175i	$0.729969\pi$
$422$	13.8564i	0.674519i
$423$	26.7846i	1.30231i
$424$	18.0000i	0.874157i
$425$	−3.46410	−0.168034
$426$	22.3923	1.08491
$427$	− 16.7846i	− 0.812264i
$428$	−17.6603	−0.853641
$429$	0	0
$430$	−0.339746	−0.0163840
$431$	25.5167i	1.22909i	0.788880	+	0.614547i	$0.210661\pi$
$431$	25.5167i	1.22909i	−0.788880	+	0.614547i	$0.789339\pi$
$432$	−20.0000	−0.962250
$433$	−34.7846	−1.67164	−0.835821	−	0.549002i	$-0.815008\pi$
$433$	−34.7846	−1.67164	−0.835821	+	0.549002i	$0.815008\pi$
$434$	35.3205i	1.69544i
$435$	− 6.92820i	− 0.332182i
$436$	2.00000i	0.0957826i
$437$	− 7.85641i	− 0.375823i
$438$	18.9282	0.904425
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	8.19615i	0.390736i
$441$	13.3923	0.637729
$442$	0	0
$443$	−16.9808	−0.806780	−0.403390	−	0.915028i	$-0.632168\pi$
$443$	−16.9808	−0.806780	−0.403390	+	0.915028i	$0.632168\pi$
$444$	10.9282i	0.518630i
$445$	12.9282	0.612856
$446$	3.46410	0.164030
$447$	21.4641i	1.01522i
$448$	− 2.00000i	− 0.0944911i
$449$	20.5359i	0.969149i	0.874750	+	0.484574i	$0.161026\pi$
$449$	20.5359i	0.969149i	−0.874750	+	0.484574i	$0.838974\pi$
$450$	− 7.73205i	− 0.364492i
$451$	−16.3923	−0.771883
$452$	−8.53590	−0.401495
$453$	− 4.92820i	− 0.231547i
$454$	−6.00000	−0.281594
$455$	0	0
$456$	−29.3205	−1.37306
$457$	− 10.7846i	− 0.504483i	−0.967664	−	0.252241i	$-0.918832\pi$
$457$	− 10.7846i	− 0.504483i	0.967664	−	0.252241i	$-0.0811677\pi$
$458$	−11.0718	−0.517351
$459$	13.8564	0.646762
$460$	1.26795i	0.0591184i
$461$	3.46410i	0.161339i	0.996741	+	0.0806696i	$0.0257059\pi$
$461$	3.46410i	0.161339i	−0.996741	+	0.0806696i	$0.974294\pi$
$462$	44.7846i	2.08357i
$463$	− 2.39230i	− 0.111180i	−0.998454	−	0.0555899i	$-0.982296\pi$
$463$	− 2.39230i	− 0.111180i	0.998454	−	0.0555899i	$-0.0177039\pi$
$464$	12.6795	0.588631
$465$	27.8564	1.29181
$466$	10.3923i	0.481414i
$467$	−27.8038	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$467$	−27.8038	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$468$	0	0
$469$	12.7846	0.590338
$470$	− 10.3923i	− 0.479361i
$471$	−27.3205	−1.25886
$472$	−15.8038	−0.727431
$473$	− 0.928203i	− 0.0426788i
$474$	− 39.7128i	− 1.82407i
$475$	− 6.19615i	− 0.284299i
$476$	− 6.92820i	− 0.317554i
$477$	46.3923	2.12416
$478$	−24.5885	−1.12465
$479$	35.6603i	1.62936i	0.579912	+	0.814679i	$0.303087\pi$
$479$	35.6603i	1.62936i	−0.579912	+	0.814679i	$0.696913\pi$
$480$	14.1962	0.647963
$481$	0	0
$482$	4.14359	0.188736
$483$	− 6.92820i	− 0.315244i
$484$	11.3923	0.517832
$485$	2.00000	0.0908153
$486$	− 32.4449i	− 1.47173i
$487$	26.3923i	1.19595i	0.801515	+	0.597975i	$0.204028\pi$
$487$	26.3923i	1.19595i	−0.801515	+	0.597975i	$0.795972\pi$
$488$	− 14.5359i	− 0.658009i
$489$	− 39.3205i	− 1.77813i
$490$	−5.19615	−0.234738
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	9.46410i	0.426675i
$493$	−8.78461	−0.395639
$494$	0	0
$495$	21.1244	0.949469
$496$	50.9808i	2.28910i
$497$	9.46410	0.424523
$498$	−28.3923	−1.27229
$499$	38.9808i	1.74502i	0.488598	+	0.872509i	$0.337509\pi$
$499$	38.9808i	1.74502i	−0.488598	+	0.872509i	$0.662491\pi$
$500$	1.00000i	0.0447214i
$501$	− 2.53590i	− 0.113296i
$502$	37.1769i	1.65929i
$503$	19.5167	0.870205	0.435102	−	0.900381i	$-0.356712\pi$
$503$	19.5167	0.870205	0.435102	+	0.900381i	$0.356712\pi$
$504$	−15.4641	−0.688826
$505$	0.928203i	0.0413045i
$506$	−10.3923	−0.461994
$507$	0	0
$508$	16.1962	0.718588
$509$	− 39.4641i	− 1.74922i	−0.484831	−	0.874608i	$-0.661119\pi$
$509$	− 39.4641i	− 1.74922i	0.484831	−	0.874608i	$-0.338881\pi$
$510$	−16.3923	−0.725863
$511$	8.00000	0.353899
$512$	8.66025i	0.382733i
$513$	24.7846i	1.09427i
$514$	− 34.3923i	− 1.51698i
$515$	0.196152i	0.00864351i
$516$	−0.535898	−0.0235916
$517$	28.3923	1.24869
$518$	13.8564i	0.608816i
$519$	−23.3205	−1.02366
$520$	0	0
$521$	−28.3923	−1.24389	−0.621945	−	0.783061i	$-0.713657\pi$
$521$	−28.3923	−1.24389	−0.621945	+	0.783061i	$0.713657\pi$
$522$	− 19.6077i	− 0.858206i
$523$	−24.1962	−1.05802	−0.529012	−	0.848614i	$-0.677437\pi$
$523$	−24.1962	−1.05802	−0.529012	+	0.848614i	$0.677437\pi$
$524$	0	0
$525$	− 5.46410i	− 0.238473i
$526$	2.19615i	0.0957568i
$527$	− 35.3205i	− 1.53859i
$528$	64.6410i	2.81314i
$529$	−21.3923	−0.930100
$530$	−18.0000	−0.781870
$531$	40.7321i	1.76762i
$532$	12.3923	0.537275
$533$	0	0
$534$	61.1769	2.64738
$535$	17.6603i	0.763519i
$536$	11.0718	0.478229
$537$	51.7128	2.23157
$538$	− 34.3923i	− 1.48276i
$539$	− 14.1962i	− 0.611472i
$540$	− 4.00000i	− 0.172133i
$541$	− 26.3923i	− 1.13469i	−0.823479	−	0.567347i	$-0.807970\pi$
$541$	− 26.3923i	− 1.13469i	0.823479	−	0.567347i	$-0.192030\pi$
$542$	−53.6603	−2.30491
$543$	−1.07180	−0.0459952
$544$	− 18.0000i	− 0.771744i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−12.1962	−0.521470	−0.260735	−	0.965410i	$-0.583965\pi$
$547$	−12.1962	−0.521470	−0.260735	+	0.965410i	$0.583965\pi$
$548$	− 0.928203i	− 0.0396509i
$549$	−37.4641	−1.59893
$550$	−8.19615	−0.349485
$551$	− 15.7128i	− 0.669388i
$552$	− 6.00000i	− 0.255377i
$553$	− 16.7846i	− 0.713754i
$554$	45.7128i	1.94215i
$555$	10.9282	0.463876
$556$	−12.3923	−0.525551
$557$	1.85641i	0.0786585i	0.999226	+	0.0393292i	$0.0125221\pi$
$557$	1.85641i	0.0786585i	−0.999226	+	0.0393292i	$0.987478\pi$
$558$	78.8372	3.33744
$559$	0	0
$560$	10.0000	0.422577
$561$	− 44.7846i	− 1.89081i
$562$	−38.7846	−1.63603
$563$	22.0526	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$564$	− 16.3923i	− 0.690241i
$565$	8.53590i	0.359108i
$566$	− 56.4449i	− 2.37255i
$567$	− 4.92820i	− 0.206965i
$568$	8.19615	0.343903
$569$	2.53590	0.106310	0.0531552	−	0.998586i	$-0.483072\pi$
$569$	2.53590	0.106310	0.0531552	+	0.998586i	$0.483072\pi$
$570$	− 29.3205i	− 1.22810i
$571$	−36.3923	−1.52297	−0.761485	−	0.648182i	$-0.775530\pi$
$571$	−36.3923	−1.52297	−0.761485	+	0.648182i	$0.775530\pi$
$572$	0	0
$573$	−13.8564	−0.578860
$574$	12.0000i	0.500870i
$575$	1.26795	0.0528771
$576$	−4.46410	−0.186004
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	− 8.66025i	− 0.360219i
$579$	− 27.3205i	− 1.13540i
$580$	2.53590i	0.105297i
$581$	−12.0000	−0.497844
$582$	9.46410	0.392300
$583$	− 49.1769i	− 2.03670i
$584$	6.92820	0.286691
$585$	0	0
$586$	8.78461	0.362889
$587$	8.53590i	0.352314i	0.984362	+	0.176157i	$0.0563667\pi$
$587$	8.53590i	0.352314i	−0.984362	+	0.176157i	$0.943633\pi$
$588$	−8.19615	−0.338004
$589$	63.1769	2.60316
$590$	− 15.8038i	− 0.650634i
$591$	35.3205i	1.45289i
$592$	20.0000i	0.821995i
$593$	26.7846i	1.09991i	0.835194	+	0.549956i	$0.185356\pi$
$593$	26.7846i	1.09991i	−0.835194	+	0.549956i	$0.814644\pi$
$594$	32.7846	1.34517
$595$	−6.92820	−0.284029
$596$	− 7.85641i	− 0.321811i
$597$	−54.6410	−2.23631
$598$	0	0
$599$	−7.60770	−0.310842	−0.155421	−	0.987848i	$-0.549673\pi$
$599$	−7.60770	−0.310842	−0.155421	+	0.987848i	$0.549673\pi$
$600$	− 4.73205i	− 0.193185i
$601$	43.5692	1.77723	0.888613	−	0.458658i	$-0.151670\pi$
$601$	43.5692	1.77723	0.888613	+	0.458658i	$0.151670\pi$
$602$	−0.679492	−0.0276940
$603$	− 28.5359i	− 1.16207i
$604$	1.80385i	0.0733975i
$605$	− 11.3923i	− 0.463163i
$606$	4.39230i	0.178425i
$607$	24.9808	1.01394	0.506969	−	0.861964i	$-0.330766\pi$
$607$	24.9808	1.01394	0.506969	+	0.861964i	$0.330766\pi$
$608$	32.1962	1.30573
$609$	− 13.8564i	− 0.561490i
$610$	14.5359	0.588541
$611$	0	0
$612$	−15.4641	−0.625099
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	32.5359	1.31304
$615$	9.46410	0.381629
$616$	16.3923i	0.660465i
$617$	− 33.7128i	− 1.35723i	−0.734496	−	0.678613i	$-0.762581\pi$
$617$	− 33.7128i	− 1.35723i	0.734496	−	0.678613i	$-0.237419\pi$
$618$	0.928203i	0.0373378i
$619$	6.98076i	0.280581i	0.990110	+	0.140290i	$0.0448036\pi$
$619$	6.98076i	0.280581i	−0.990110	+	0.140290i	$0.955196\pi$
$620$	−10.1962	−0.409487
$621$	−5.07180	−0.203524
$622$	28.3923i	1.13843i
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 24.9282i	− 0.996331i
$627$	80.1051	3.19909
$628$	10.0000	0.399043
$629$	− 13.8564i	− 0.552491i
$630$	− 15.4641i	− 0.616105i
$631$	5.80385i	0.231048i	0.993305	+	0.115524i	$0.0368546\pi$
$631$	5.80385i	0.231048i	−0.993305	+	0.115524i	$0.963145\pi$
$632$	− 14.5359i	− 0.578207i
$633$	21.8564	0.868714
$634$	41.5692	1.65092
$635$	− 16.1962i	− 0.642725i
$636$	−28.3923	−1.12583
$637$	0	0
$638$	−20.7846	−0.822871
$639$	− 21.1244i	− 0.835667i
$640$	12.1244	0.479257
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	83.5692i	3.29821i
$643$	6.78461i	0.267559i	0.991011	+	0.133779i	$0.0427114\pi$
$643$	6.78461i	0.267559i	−0.991011	+	0.133779i	$0.957289\pi$
$644$	2.53590i	0.0999284i
$645$	0.535898i	0.0211010i
$646$	−37.1769	−1.46271
$647$	−22.0526	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$647$	−22.0526	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$648$	− 4.26795i	− 0.167661i
$649$	43.1769	1.69484
$650$	0	0
$651$	55.7128	2.18356
$652$	14.3923i	0.563646i
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	9.46410	0.370076
$655$	0	0
$656$	17.3205i	0.676252i
$657$	− 17.8564i	− 0.696645i
$658$	− 20.7846i	− 0.810268i
$659$	−21.4641	−0.836123	−0.418061	−	0.908419i	$-0.637290\pi$
$659$	−21.4641	−0.836123	−0.418061	+	0.908419i	$0.637290\pi$
$660$	−12.9282	−0.503230
$661$	10.7846i	0.419473i	0.977758	+	0.209736i	$0.0672606\pi$
$661$	10.7846i	0.419473i	−0.977758	+	0.209736i	$0.932739\pi$
$662$	4.48334	0.174250
$663$	0	0
$664$	−10.3923	−0.403300
$665$	− 12.3923i	− 0.480553i
$666$	30.9282	1.19844
$667$	3.21539	0.124500
$668$	0.928203i	0.0359133i
$669$	− 5.46410i	− 0.211254i
$670$	11.0718i	0.427741i
$671$	39.7128i	1.53310i
$672$	28.3923	1.09526
$673$	14.3923	0.554783	0.277391	−	0.960757i	$-0.410530\pi$
$673$	14.3923	0.554783	0.277391	+	0.960757i	$0.410530\pi$
$674$	45.7128i	1.76079i
$675$	−4.00000	−0.153960
$676$	0	0
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	40.3923i	1.55126i
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	9.46410i	0.362665i
$682$	− 83.5692i	− 3.20003i
$683$	32.5359i	1.24495i	0.782639	+	0.622476i	$0.213873\pi$
$683$	32.5359i	1.24495i	−0.782639	+	0.622476i	$0.786127\pi$
$684$	− 27.6603i	− 1.05762i
$685$	−0.928203	−0.0354648
$686$	−34.6410	−1.32260
$687$	17.4641i	0.666297i
$688$	−0.980762	−0.0373912
$689$	0	0
$690$	6.00000	0.228416
$691$	47.7654i	1.81708i	0.417797	+	0.908540i	$0.362802\pi$
$691$	47.7654i	1.81708i	−0.417797	+	0.908540i	$0.637198\pi$
$692$	8.53590	0.324486
$693$	42.2487	1.60490
$694$	− 9.80385i	− 0.372149i
$695$	12.3923i	0.470067i
$696$	− 12.0000i	− 0.454859i
$697$	− 12.0000i	− 0.454532i
$698$	24.9282	0.943546
$699$	16.3923	0.620014
$700$	2.00000i	0.0755929i
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	24.7846	0.934769
$704$	4.73205i	0.178346i
$705$	−16.3923	−0.617370
$706$	−48.0000	−1.80650
$707$	1.85641i	0.0698174i
$708$	− 24.9282i	− 0.936859i
$709$	30.3923i	1.14141i	0.821156	+	0.570703i	$0.193329\pi$
$709$	30.3923i	1.14141i	−0.821156	+	0.570703i	$0.806671\pi$
$710$	8.19615i	0.307596i
$711$	−37.4641	−1.40501
$712$	22.3923	0.839187
$713$	12.9282i	0.484165i
$714$	−32.7846	−1.22693
$715$	0	0
$716$	−18.9282	−0.707380
$717$	38.7846i	1.44844i
$718$	3.80385	0.141958
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	− 22.3205i	− 0.831836i
$721$	0.392305i	0.0146102i
$722$	− 33.5885i	− 1.25003i
$723$	− 6.53590i	− 0.243073i
$724$	0.392305	0.0145799
$725$	2.53590	0.0941809
$726$	− 53.9090i	− 2.00075i
$727$	−44.5885	−1.65369	−0.826847	−	0.562427i	$-0.809868\pi$
$727$	−44.5885	−1.65369	−0.826847	+	0.562427i	$0.809868\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	6.92820i	0.256424i
$731$	0.679492	0.0251319
$732$	22.9282	0.847451
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	20.4449i	0.754634i
$735$	8.19615i	0.302320i
$736$	6.58846i	0.242854i
$737$	−30.2487	−1.11423
$738$	26.7846	0.985955
$739$	− 18.1962i	− 0.669356i	−0.942332	−	0.334678i	$-0.891372\pi$
$739$	− 18.1962i	− 0.669356i	0.942332	−	0.334678i	$-0.108628\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	16.1436i	0.592251i	0.955149	+	0.296126i	$0.0956947\pi$
$743$	16.1436i	0.592251i	−0.955149	+	0.296126i	$0.904305\pi$
$744$	48.2487	1.76888
$745$	−7.85641	−0.287836
$746$	− 17.3205i	− 0.634149i
$747$	26.7846i	0.979998i
$748$	16.3923i	0.599362i
$749$	35.3205i	1.29058i
$750$	4.73205	0.172790
$751$	−36.3923	−1.32797	−0.663987	−	0.747744i	$-0.731137\pi$
$751$	−36.3923	−1.32797	−0.663987	+	0.747744i	$0.731137\pi$
$752$	− 30.0000i	− 1.09399i
$753$	58.6410	2.13700
$754$	0	0
$755$	1.80385	0.0656487
$756$	− 8.00000i	− 0.290957i
$757$	−2.39230	−0.0869498	−0.0434749	−	0.999055i	$-0.513843\pi$
$757$	−2.39230	−0.0869498	−0.0434749	+	0.999055i	$0.513843\pi$
$758$	32.8756	1.19410
$759$	16.3923i	0.595003i
$760$	− 10.7321i	− 0.389292i
$761$	− 19.8564i	− 0.719794i	−0.932992	−	0.359897i	$-0.882812\pi$
$761$	− 19.8564i	− 0.719794i	0.932992	−	0.359897i	$-0.117188\pi$
$762$	− 76.6410i	− 2.77641i
$763$	4.00000	0.144810
$764$	5.07180	0.183491
$765$	15.4641i	0.559106i
$766$	22.3923	0.809067
$767$	0	0
$768$	51.9090	1.87310
$769$	− 34.7846i	− 1.25437i	−0.778872	−	0.627183i	$-0.784208\pi$
$769$	− 34.7846i	− 1.25437i	0.778872	−	0.627183i	$-0.215792\pi$
$770$	−16.3923	−0.590738
$771$	−54.2487	−1.95372
$772$	10.0000i	0.359908i
$773$	− 6.92820i	− 0.249190i	−0.992208	−	0.124595i	$-0.960237\pi$
$773$	− 6.92820i	− 0.249190i	0.992208	−	0.124595i	$-0.0397632\pi$
$774$	1.51666i	0.0545152i
$775$	10.1962i	0.366257i
$776$	3.46410	0.124354
$777$	21.8564	0.784094
$778$	− 10.3923i	− 0.372582i
$779$	21.4641	0.769031
$780$	0	0
$781$	−22.3923	−0.801260
$782$	− 7.60770i	− 0.272051i
$783$	−10.1436	−0.362502
$784$	−15.0000	−0.535714
$785$	− 10.0000i	− 0.356915i
$786$	0	0
$787$	31.5692i	1.12532i	0.826688	+	0.562661i	$0.190222\pi$
$787$	31.5692i	1.12532i	−0.826688	+	0.562661i	$0.809778\pi$
$788$	− 12.9282i	− 0.460548i
$789$	3.46410	0.123325
$790$	14.5359	0.517164
$791$	17.0718i	0.607003i
$792$	36.5885	1.30011
$793$	0	0
$794$	−49.8564	−1.76934
$795$	28.3923i	1.00697i
$796$	20.0000	0.708881
$797$	40.6410	1.43958	0.719789	−	0.694193i	$-0.244238\pi$
$797$	40.6410	1.43958	0.719789	+	0.694193i	$0.244238\pi$
$798$	− 58.6410i	− 2.07587i
$799$	20.7846i	0.735307i
$800$	5.19615i	0.183712i
$801$	− 57.7128i	− 2.03918i
$802$	63.9615	2.25856
$803$	−18.9282	−0.667962
$804$	17.4641i	0.615911i
$805$	2.53590	0.0893787
$806$	0	0
$807$	−54.2487	−1.90965
$808$	1.60770i	0.0565585i
$809$	−2.53590	−0.0891574	−0.0445787	−	0.999006i	$-0.514195\pi$
$809$	−2.53590	−0.0891574	−0.0445787	+	0.999006i	$0.514195\pi$
$810$	4.26795	0.149960
$811$	− 17.8038i	− 0.625178i	−0.949889	−	0.312589i	$-0.898804\pi$
$811$	− 17.8038i	− 0.625178i	0.949889	−	0.312589i	$-0.101196\pi$
$812$	5.07180i	0.177985i
$813$	84.6410i	2.96849i
$814$	− 32.7846i	− 1.14910i
$815$	14.3923	0.504140
$816$	−47.3205	−1.65655
$817$	1.21539i	0.0425211i
$818$	−30.4974	−1.06632
$819$	0	0
$820$	−3.46410	−0.120972
$821$	− 28.6410i	− 0.999578i	−0.866147	−	0.499789i	$-0.833411\pi$
$821$	− 28.6410i	− 0.999578i	0.866147	−	0.499789i	$-0.166589\pi$
$822$	−4.39230	−0.153199
$823$	15.4115	0.537213	0.268606	−	0.963250i	$-0.413437\pi$
$823$	15.4115	0.537213	0.268606	+	0.963250i	$0.413437\pi$
$824$	0.339746i	0.0118356i
$825$	12.9282i	0.450102i
$826$	− 31.6077i	− 1.09977i
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	5.66025	0.196707
$829$	−0.392305	−0.0136253	−0.00681266	−	0.999977i	$-0.502169\pi$
$829$	−0.392305	−0.0136253	−0.00681266	+	0.999977i	$0.502169\pi$
$830$	− 10.3923i	− 0.360722i
$831$	72.1051	2.50130
$832$	0	0
$833$	10.3923	0.360072
$834$	58.6410i	2.03057i
$835$	0.928203	0.0321218
$836$	−29.3205	−1.01407
$837$	− 40.7846i	− 1.40972i
$838$	− 4.39230i	− 0.151730i
$839$	0.339746i	0.0117293i	0.999983	+	0.00586467i	$0.00186679\pi$
$839$	0.339746i	0.0117293i	−0.999983	+	0.00586467i	$0.998133\pi$
$840$	− 9.46410i	− 0.326543i
$841$	−22.5692	−0.778249
$842$	−53.3205	−1.83755
$843$	61.1769i	2.10704i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−46.3923	−1.59500
$847$	− 22.7846i	− 0.782888i
$848$	−51.9615	−1.78437
$849$	−89.0333	−3.05562
$850$	− 6.00000i	− 0.205798i
$851$	5.07180i	0.173859i
$852$	12.9282i	0.442913i
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	29.0718	0.994816
$855$	−27.6603	−0.945961
$856$	30.5885i	1.04549i
$857$	35.5692	1.21502	0.607511	−	0.794311i	$-0.292168\pi$
$857$	35.5692	1.21502	0.607511	+	0.794311i	$0.292168\pi$
$858$	0	0
$859$	−17.1769	−0.586069	−0.293034	−	0.956102i	$-0.594665\pi$
$859$	−17.1769	−0.586069	−0.293034	+	0.956102i	$0.594665\pi$
$860$	− 0.196152i	− 0.00668874i
$861$	18.9282	0.645071
$862$	−44.1962	−1.50533
$863$	38.7846i	1.32024i	0.751159	+	0.660122i	$0.229495\pi$
$863$	38.7846i	1.32024i	−0.751159	+	0.660122i	$0.770505\pi$
$864$	− 20.7846i	− 0.707107i
$865$	− 8.53590i	− 0.290229i
$866$	− 60.2487i	− 2.04733i
$867$	−13.6603	−0.463927
$868$	−20.3923	−0.692160
$869$	39.7128i	1.34716i
$870$	12.0000	0.406838
$871$	0	0
$872$	3.46410	0.117309
$873$	− 8.92820i	− 0.302174i
$874$	13.6077	0.460287
$875$	2.00000	0.0676123
$876$	10.9282i	0.369230i
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	− 55.4256i	− 1.87052i
$879$	− 13.8564i	− 0.467365i
$880$	−23.6603	−0.797587
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	23.1962i	0.781055i
$883$	−23.8038	−0.801063	−0.400532	−	0.916283i	$-0.631175\pi$
$883$	−23.8038	−0.801063	−0.400532	+	0.916283i	$0.631175\pi$
$884$	0	0
$885$	−24.9282	−0.837952
$886$	− 29.4115i	− 0.988100i
$887$	−47.9090	−1.60863	−0.804313	−	0.594206i	$-0.797466\pi$
$887$	−47.9090	−1.60863	−0.804313	+	0.594206i	$0.797466\pi$
$888$	18.9282	0.635189
$889$	− 32.3923i	− 1.08640i
$890$	22.3923i	0.750592i
$891$	11.6603i	0.390633i
$892$	2.00000i	0.0669650i
$893$	−37.1769	−1.24408
$894$	−37.1769	−1.24338
$895$	18.9282i	0.632700i
$896$	24.2487	0.810093
$897$	0	0
$898$	−35.5692	−1.18696
$899$	25.8564i	0.862359i
$900$	4.46410	0.148803
$901$	36.0000	1.19933
$902$	− 28.3923i	− 0.945360i
$903$	1.07180i	0.0356672i
$904$	14.7846i	0.491729i
$905$	− 0.392305i	− 0.0130407i
$906$	8.53590	0.283586
$907$	53.7654	1.78525	0.892625	−	0.450800i	$-0.148861\pi$
$907$	53.7654	1.78525	0.892625	+	0.450800i	$0.148861\pi$
$908$	− 3.46410i	− 0.114960i
$909$	4.14359	0.137434
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	− 84.6410i	− 2.80274i
$913$	28.3923	0.939648
$914$	18.6795	0.617863
$915$	− 22.9282i	− 0.757983i
$916$	− 6.39230i	− 0.211208i
$917$	0	0
$918$	24.0000i	0.792118i
$919$	9.17691	0.302718	0.151359	−	0.988479i	$-0.451635\pi$
$919$	9.17691	0.302718	0.151359	+	0.988479i	$0.451635\pi$
$920$	2.19615	0.0724050
$921$	− 51.3205i	− 1.69107i
$922$	−6.00000	−0.197599
$923$	0	0
$924$	−25.8564	−0.850613
$925$	4.00000i	0.131519i
$926$	4.14359	0.136167
$927$	0.875644	0.0287599
$928$	13.1769i	0.432553i
$929$	− 44.5359i	− 1.46118i	−0.682819	−	0.730588i	$-0.739246\pi$
$929$	− 44.5359i	− 1.46118i	0.682819	−	0.730588i	$-0.260754\pi$
$930$	48.2487i	1.58214i
$931$	18.5885i	0.609212i
$932$	−6.00000	−0.196537
$933$	44.7846	1.46618
$934$	− 48.1577i	− 1.57577i
$935$	16.3923	0.536086
$936$	0	0
$937$	34.7846	1.13636	0.568182	−	0.822903i	$-0.307647\pi$
$937$	34.7846	1.13636	0.568182	+	0.822903i	$0.307647\pi$
$938$	22.1436i	0.723014i
$939$	−39.3205	−1.28318
$940$	6.00000	0.195698
$941$	− 31.1769i	− 1.01634i	−0.861257	−	0.508169i	$-0.830322\pi$
$941$	− 31.1769i	− 1.01634i	0.861257	−	0.508169i	$-0.169678\pi$
$942$	− 47.3205i	− 1.54179i
$943$	4.39230i	0.143033i
$944$	− 45.6218i	− 1.48486i
$945$	−8.00000	−0.260240
$946$	1.60770	0.0522707
$947$	40.6410i	1.32066i	0.750978	+	0.660328i	$0.229583\pi$
$947$	40.6410i	1.32066i	−0.750978	+	0.660328i	$0.770417\pi$
$948$	22.9282	0.744673
$949$	0	0
$950$	10.7321	0.348194
$951$	− 65.5692i	− 2.12623i
$952$	−12.0000	−0.388922
$953$	−0.928203	−0.0300675	−0.0150337	−	0.999887i	$-0.504786\pi$
$953$	−0.928203	−0.0300675	−0.0150337	+	0.999887i	$0.504786\pi$
$954$	80.3538i	2.60155i
$955$	− 5.07180i	− 0.164119i
$956$	− 14.1962i	− 0.459136i
$957$	32.7846i	1.05978i
$958$	−61.7654	−1.99555
$959$	−1.85641	−0.0599465
$960$	− 2.73205i	− 0.0881766i
$961$	−72.9615	−2.35360
$962$	0	0
$963$	78.8372	2.54049
$964$	2.39230i	0.0770510i
$965$	10.0000	0.321911
$966$	12.0000	0.386094
$967$	50.3923i	1.62051i	0.586079	+	0.810254i	$0.300671\pi$
$967$	50.3923i	1.62051i	−0.586079	+	0.810254i	$0.699329\pi$
$968$	− 19.7321i	− 0.634212i
$969$	58.6410i	1.88382i
$970$	3.46410i	0.111226i
$971$	18.9282	0.607435	0.303717	−	0.952762i	$-0.401772\pi$
$971$	18.9282	0.607435	0.303717	+	0.952762i	$0.401772\pi$
$972$	18.7321	0.600831
$973$	24.7846i	0.794558i
$974$	−45.7128	−1.46473
$975$	0	0
$976$	41.9615	1.34316
$977$	15.7128i	0.502697i	0.967897	+	0.251349i	$0.0808741\pi$
$977$	15.7128i	0.502697i	−0.967897	+	0.251349i	$0.919126\pi$
$978$	68.1051	2.17776
$979$	−61.1769	−1.95522
$980$	− 3.00000i	− 0.0958315i
$981$	− 8.92820i	− 0.285056i
$982$	4.39230i	0.140164i
$983$	− 34.3923i	− 1.09694i	−0.836169	−	0.548472i	$-0.815210\pi$
$983$	− 34.3923i	− 1.09694i	0.836169	−	0.548472i	$-0.184790\pi$
$984$	16.3923	0.522568
$985$	−12.9282	−0.411927
$986$	− 15.2154i	− 0.484557i
$987$	−32.7846	−1.04355
$988$	0	0
$989$	−0.248711	−0.00790856
$990$	36.5885i	1.16286i
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−52.9808	−1.68214
$993$	− 7.07180i	− 0.224417i
$994$	16.3923i	0.519932i
$995$	− 20.0000i	− 0.634043i
$996$	− 16.3923i	− 0.519410i
$997$	33.6077	1.06437	0.532183	−	0.846629i	$-0.321372\pi$
$997$	33.6077	1.06437	0.532183	+	0.846629i	$0.321372\pi$
$998$	−67.5167	−2.13720
$999$	− 16.0000i	− 0.506218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.e.506.4		4
13.2	odd	12		845.2.e.f.191.1		4
13.3	even	3		845.2.m.a.316.1		4
13.4	even	6		845.2.m.a.361.1		4
13.5	odd	4		845.2.a.d.1.2		2
13.6	odd	12		845.2.e.f.146.1		4
13.7	odd	12		845.2.e.e.146.2		4
13.8	odd	4		65.2.a.c.1.1	✓	2
13.9	even	3		845.2.m.c.361.1		4
13.10	even	6		845.2.m.c.316.1		4
13.11	odd	12		845.2.e.e.191.2		4
13.12	even	2	inner	845.2.c.e.506.2		4
39.5	even	4		7605.2.a.be.1.1		2
39.8	even	4		585.2.a.k.1.2		2
52.47	even	4		1040.2.a.h.1.1		2
65.8	even	4		325.2.b.e.274.4		4
65.34	odd	4		325.2.a.g.1.2		2
65.44	odd	4		4225.2.a.w.1.1		2
65.47	even	4		325.2.b.e.274.1		4
91.34	even	4		3185.2.a.k.1.1		2
104.21	odd	4		4160.2.a.y.1.1		2
104.99	even	4		4160.2.a.bj.1.2		2
143.21	even	4		7865.2.a.h.1.2		2
156.47	odd	4		9360.2.a.cm.1.1		2
195.8	odd	4		2925.2.c.v.2224.1		4
195.47	odd	4		2925.2.c.v.2224.4		4
195.164	even	4		2925.2.a.z.1.1		2
260.99	even	4		5200.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.1	✓	2	13.8	odd	4
325.2.a.g.1.2		2	65.34	odd	4
325.2.b.e.274.1		4	65.47	even	4
325.2.b.e.274.4		4	65.8	even	4
585.2.a.k.1.2		2	39.8	even	4
845.2.a.d.1.2		2	13.5	odd	4
845.2.c.e.506.2		4	13.12	even	2	inner
845.2.c.e.506.4		4	1.1	even	1	trivial
845.2.e.e.146.2		4	13.7	odd	12
845.2.e.e.191.2		4	13.11	odd	12
845.2.e.f.146.1		4	13.6	odd	12
845.2.e.f.191.1		4	13.2	odd	12
845.2.m.a.316.1		4	13.3	even	3
845.2.m.a.361.1		4	13.4	even	6
845.2.m.c.316.1		4	13.10	even	6
845.2.m.c.361.1		4	13.9	even	3
1040.2.a.h.1.1		2	52.47	even	4
2925.2.a.z.1.1		2	195.164	even	4
2925.2.c.v.2224.1		4	195.8	odd	4
2925.2.c.v.2224.4		4	195.47	odd	4
3185.2.a.k.1.1		2	91.34	even	4
4160.2.a.y.1.1		2	104.21	odd	4
4160.2.a.bj.1.2		2	104.99	even	4
4225.2.a.w.1.1		2	65.44	odd	4
5200.2.a.ca.1.2		2	260.99	even	4
7605.2.a.be.1.1		2	39.5	even	4
7865.2.a.h.1.2		2	143.21	even	4
9360.2.a.cm.1.1		2	156.47	odd	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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +2.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.00000i q^{7} +1.73205i q^{8} +4.46410 q^{9} -1.73205 q^{10} -4.73205i q^{11} -2.73205 q^{12} -3.46410 q^{14} +2.73205i q^{15} -5.00000 q^{16} +3.46410 q^{17} +7.73205i q^{18} +6.19615i q^{19} -1.00000i q^{20} +5.46410i q^{21} +8.19615 q^{22} -1.26795 q^{23} +4.73205i q^{24} -1.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -2.53590 q^{29} -4.73205 q^{30} -10.1962i q^{31} -5.19615i q^{32} -12.9282i q^{33} +6.00000i q^{34} -2.00000 q^{35} -4.46410 q^{36} -4.00000i q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410i q^{41} -9.46410 q^{42} +0.196152 q^{43} +4.73205i q^{44} +4.46410i q^{45} -2.19615i q^{46} +6.00000i q^{47} -13.6603 q^{48} +3.00000 q^{49} -1.73205i q^{50} +9.46410 q^{51} +10.3923 q^{53} +6.92820i q^{54} +4.73205 q^{55} -3.46410 q^{56} +16.9282i q^{57} -4.39230i q^{58} +9.12436i q^{59} -2.73205i q^{60} -8.39230 q^{61} +17.6603 q^{62} +8.92820i q^{63} -1.00000 q^{64} +22.3923 q^{66} -6.39230i q^{67} -3.46410 q^{68} -3.46410 q^{69} -3.46410i q^{70} -4.73205i q^{71} +7.73205i q^{72} -4.00000i q^{73} +6.92820 q^{74} -2.73205 q^{75} -6.19615i q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000i q^{80} -2.46410 q^{81} +6.00000 q^{82} +6.00000i q^{83} -5.46410i q^{84} +3.46410i q^{85} +0.339746i q^{86} -6.92820 q^{87} +8.19615 q^{88} -12.9282i q^{89} -7.73205 q^{90} +1.26795 q^{92} -27.8564i q^{93} -10.3923 q^{94} -6.19615 q^{95} -14.1962i q^{96} -2.00000i q^{97} +5.19615i q^{98} -21.1244i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49}+ \cdots - 4 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^12 - 20 * q^16 + 12 * q^22 - 12 * q^23 - 4 * q^25 + 16 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^35 - 4 * q^36 - 36 * q^38 - 24 * q^42 - 20 * q^43 - 20 * q^48 + 12 * q^49 + 24 * q^51 + 12 * q^55 + 8 * q^61 + 36 * q^62 - 4 * q^64 + 48 * q^66 - 4 * q^75 + 24 * q^77 + 8 * q^79 + 4 * q^81 + 24 * q^82 + 12 * q^88 - 24 * q^90 + 12 * q^92 - 4 * q^95

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