LMFDB - Embedded newform 845.2.c.c.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:	$\Q(i, \sqrt{5})$
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			506.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.c.506.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803i q^{2} -2.23607 q^{3} -0.618034 q^{4} +1.00000i q^{5} -3.61803i q^{6} +0.236068i q^{7} +2.23607i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.61803i q^{2} -2.23607 q^{3} -0.618034 q^{4} +1.00000i q^{5} -3.61803i q^{6} +0.236068i q^{7} +2.23607i q^{8} +2.00000 q^{9} -1.61803 q^{10} -4.23607i q^{11} +1.38197 q^{12} -0.381966 q^{14} -2.23607i q^{15} -4.85410 q^{16} -5.47214 q^{17} +3.23607i q^{18} -0.236068i q^{19} -0.618034i q^{20} -0.527864i q^{21} +6.85410 q^{22} -8.23607 q^{23} -5.00000i q^{24} -1.00000 q^{25} +2.23607 q^{27} -0.145898i q^{28} +1.47214 q^{29} +3.61803 q^{30} -3.38197i q^{32} +9.47214i q^{33} -8.85410i q^{34} -0.236068 q^{35} -1.23607 q^{36} -3.00000i q^{37} +0.381966 q^{38} -2.23607 q^{40} -5.94427i q^{41} +0.854102 q^{42} +1.76393 q^{43} +2.61803i q^{44} +2.00000i q^{45} -13.3262i q^{46} -12.9443i q^{47} +10.8541 q^{48} +6.94427 q^{49} -1.61803i q^{50} +12.2361 q^{51} +6.00000 q^{53} +3.61803i q^{54} +4.23607 q^{55} -0.527864 q^{56} +0.527864i q^{57} +2.38197i q^{58} -12.7082i q^{59} +1.38197i q^{60} -12.4164 q^{61} +0.472136i q^{63} -4.23607 q^{64} -15.3262 q^{66} +10.7082i q^{67} +3.38197 q^{68} +18.4164 q^{69} -0.381966i q^{70} -1.76393i q^{71} +4.47214i q^{72} +6.00000i q^{73} +4.85410 q^{74} +2.23607 q^{75} +0.145898i q^{76} +1.00000 q^{77} -4.85410i q^{80} -11.0000 q^{81} +9.61803 q^{82} -8.94427i q^{83} +0.326238i q^{84} -5.47214i q^{85} +2.85410i q^{86} -3.29180 q^{87} +9.47214 q^{88} +9.00000i q^{89} -3.23607 q^{90} +5.09017 q^{92} +20.9443 q^{94} +0.236068 q^{95} +7.56231i q^{96} +5.47214i q^{97} +11.2361i q^{98} -8.47214i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.61803i	1.14412i	0.820211	+	0.572061i	$0.193856\pi$
$2$	1.61803i	1.14412i	−0.820211	+	0.572061i	$0.806144\pi$
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	−0.618034	−0.309017
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 3.61803i	− 1.47706i
$7$	0.236068i	0.0892253i	0.999004	+	0.0446127i	$0.0142054\pi$
$7$	0.236068i	0.0892253i	−0.999004	+	0.0446127i	$0.985795\pi$
$8$	2.23607i	0.790569i
$9$	2.00000	0.666667
$10$	−1.61803	−0.511667
$11$	− 4.23607i	− 1.27722i	−0.769529	−	0.638611i	$-0.779509\pi$
$11$	− 4.23607i	− 1.27722i	0.769529	−	0.638611i	$-0.220491\pi$
$12$	1.38197	0.398939
$13$	0	0
$13$	0	0
$14$	−0.381966	−0.102085
$15$	− 2.23607i	− 0.577350i
$16$	−4.85410	−1.21353
$17$	−5.47214	−1.32719	−0.663594	−	0.748093i	$-0.730970\pi$
$17$	−5.47214	−1.32719	−0.663594	+	0.748093i	$0.730970\pi$
$18$	3.23607i	0.762749i
$19$	− 0.236068i	− 0.0541577i	−0.999633	−	0.0270789i	$-0.991379\pi$
$19$	− 0.236068i	− 0.0541577i	0.999633	−	0.0270789i	$-0.00862052\pi$
$20$	− 0.618034i	− 0.138197i
$21$	− 0.527864i	− 0.115189i
$22$	6.85410	1.46130
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	− 5.00000i	− 1.02062i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	2.23607	0.430331
$28$	− 0.145898i	− 0.0275721i
$29$	1.47214	0.273369	0.136684	−	0.990615i	$-0.456355\pi$
$29$	1.47214	0.273369	0.136684	+	0.990615i	$0.456355\pi$
$30$	3.61803	0.660560
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 3.38197i	− 0.597853i
$33$	9.47214i	1.64889i
$34$	− 8.85410i	− 1.51847i
$35$	−0.236068	−0.0399028
$36$	−1.23607	−0.206011
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0.381966	0.0619631
$39$	0	0
$40$	−2.23607	−0.353553
$41$	− 5.94427i	− 0.928339i	−0.885746	−	0.464170i	$-0.846353\pi$
$41$	− 5.94427i	− 0.928339i	0.885746	−	0.464170i	$-0.153647\pi$
$42$	0.854102	0.131791
$43$	1.76393	0.268997	0.134499	−	0.990914i	$-0.457058\pi$
$43$	1.76393	0.268997	0.134499	+	0.990914i	$0.457058\pi$
$44$	2.61803i	0.394683i
$45$	2.00000i	0.298142i
$46$	− 13.3262i	− 1.96485i
$47$	− 12.9443i	− 1.88812i	−0.329779	−	0.944058i	$-0.606974\pi$
$47$	− 12.9443i	− 1.88812i	0.329779	−	0.944058i	$-0.393026\pi$
$48$	10.8541	1.56665
$49$	6.94427	0.992039
$50$	− 1.61803i	− 0.228825i
$51$	12.2361	1.71339
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	3.61803i	0.492352i
$55$	4.23607	0.571191
$56$	−0.527864	−0.0705388
$57$	0.527864i	0.0699173i
$58$	2.38197i	0.312767i
$59$	− 12.7082i	− 1.65447i	−0.561858	−	0.827234i	$-0.689913\pi$
$59$	− 12.7082i	− 1.65447i	0.561858	−	0.827234i	$-0.310087\pi$
$60$	1.38197i	0.178411i
$61$	−12.4164	−1.58976	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	−12.4164	−1.58976	−0.794879	+	0.606768i	$0.792466\pi$
$62$	0	0
$63$	0.472136i	0.0594835i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	−15.3262	−1.88653
$67$	10.7082i	1.30822i	0.756401	+	0.654108i	$0.226956\pi$
$67$	10.7082i	1.30822i	−0.756401	+	0.654108i	$0.773044\pi$
$68$	3.38197	0.410124
$69$	18.4164	2.21707
$70$	− 0.381966i	− 0.0456537i
$71$	− 1.76393i	− 0.209340i	−0.994507	−	0.104670i	$-0.966621\pi$
$71$	− 1.76393i	− 0.209340i	0.994507	−	0.104670i	$-0.0333787\pi$
$72$	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	$0.885800\pi$
$74$	4.85410	0.564278
$75$	2.23607	0.258199
$76$	0.145898i	0.0167357i
$77$	1.00000	0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	− 4.85410i	− 0.542705i
$81$	−11.0000	−1.22222
$82$	9.61803	1.06213
$83$	− 8.94427i	− 0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	− 8.94427i	− 0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$	0.326238i	0.0355955i
$85$	− 5.47214i	− 0.593536i
$86$	2.85410i	0.307766i
$87$	−3.29180	−0.352918
$88$	9.47214	1.00973
$89$	9.00000i	0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$	9.00000i	0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	−3.23607	−0.341112
$91$	0	0
$92$	5.09017	0.530687
$93$	0	0
$94$	20.9443	2.16024
$95$	0.236068	0.0242201
$96$	7.56231i	0.771825i
$97$	5.47214i	0.555611i	0.960637	+	0.277806i	$0.0896071\pi$
$97$	5.47214i	0.555611i	−0.960637	+	0.277806i	$0.910393\pi$
$98$	11.2361i	1.13501i
$99$	− 8.47214i	− 0.851482i
$100$	0.618034	0.0618034
$101$	−9.47214	−0.942513	−0.471256	−	0.881996i	$-0.656199\pi$
$101$	−9.47214	−0.942513	−0.471256	+	0.881996i	$0.656199\pi$
$102$	19.7984i	1.96033i
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	0.527864	0.0515143
$106$	9.70820i	0.942944i
$107$	−10.2361	−0.989558	−0.494779	−	0.869019i	$-0.664751\pi$
$107$	−10.2361	−0.989558	−0.494779	+	0.869019i	$0.664751\pi$
$108$	−1.38197	−0.132980
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	6.85410i	0.653513i
$111$	6.70820i	0.636715i
$112$	− 1.14590i	− 0.108277i
$113$	−7.47214	−0.702919	−0.351460	−	0.936203i	$-0.614315\pi$
$113$	−7.47214	−0.702919	−0.351460	+	0.936203i	$0.614315\pi$
$114$	−0.854102	−0.0799940
$115$	− 8.23607i	− 0.768017i
$116$	−0.909830	−0.0844756
$117$	0	0
$118$	20.5623	1.89291
$119$	− 1.29180i	− 0.118419i
$120$	5.00000	0.456435
$121$	−6.94427	−0.631297
$122$	− 20.0902i	− 1.81888i
$123$	13.2918i	1.19848i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	−0.763932	−0.0680565
$127$	−0.236068	−0.0209476	−0.0104738	−	0.999945i	$-0.503334\pi$
$127$	−0.236068	−0.0209476	−0.0104738	+	0.999945i	$0.503334\pi$
$128$	− 13.6180i	− 1.20368i
$129$	−3.94427	−0.347274
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 5.85410i	− 0.509534i
$133$	0.0557281	0.00483224
$134$	−17.3262	−1.49676
$135$	2.23607i	0.192450i
$136$	− 12.2361i	− 1.04923i
$137$	7.47214i	0.638388i	0.947689	+	0.319194i	$0.103412\pi$
$137$	7.47214i	0.638388i	−0.947689	+	0.319194i	$0.896588\pi$
$138$	29.7984i	2.53661i
$139$	−3.29180	−0.279206	−0.139603	−	0.990208i	$-0.544583\pi$
$139$	−3.29180	−0.279206	−0.139603	+	0.990208i	$0.544583\pi$
$140$	0.145898	0.0123306
$141$	28.9443i	2.43755i
$142$	2.85410	0.239511
$143$	0	0
$144$	−9.70820	−0.809017
$145$	1.47214i	0.122254i
$146$	−9.70820	−0.803457
$147$	−15.5279	−1.28072
$148$	1.85410i	0.152406i
$149$	13.4721i	1.10368i	0.833950	+	0.551840i	$0.186074\pi$
$149$	13.4721i	1.10368i	−0.833950	+	0.551840i	$0.813926\pi$
$150$	3.61803i	0.295411i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	0.527864	0.0428154
$153$	−10.9443	−0.884792
$154$	1.61803i	0.130385i
$155$	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
$160$	3.38197	0.267368
$161$	− 1.94427i	− 0.153230i
$162$	− 17.7984i	− 1.39837i
$163$	− 1.29180i	− 0.101181i	−0.998719	−	0.0505906i	$-0.983890\pi$
$163$	− 1.29180i	− 0.101181i	0.998719	−	0.0505906i	$-0.0161104\pi$
$164$	3.67376i	0.286873i
$165$	−9.47214	−0.737405
$166$	14.4721	1.12326
$167$	− 5.18034i	− 0.400867i	−0.979707	−	0.200433i	$-0.935765\pi$
$167$	− 5.18034i	− 0.400867i	0.979707	−	0.200433i	$-0.0642350\pi$
$168$	1.18034	0.0910652
$169$	0	0
$170$	8.85410	0.679079
$171$	− 0.472136i	− 0.0361051i
$172$	−1.09017	−0.0831247
$173$	−16.8885	−1.28401	−0.642006	−	0.766700i	$-0.721898\pi$
$173$	−16.8885	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$174$	− 5.32624i	− 0.403781i
$175$	− 0.236068i	− 0.0178451i
$176$	20.5623i	1.54994i
$177$	28.4164i	2.13591i
$178$	−14.5623	−1.09149
$179$	3.76393	0.281329	0.140665	−	0.990057i	$-0.455076\pi$
$179$	3.76393	0.281329	0.140665	+	0.990057i	$0.455076\pi$
$180$	− 1.23607i	− 0.0921311i
$181$	−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
$183$	27.7639	2.05237
$184$	− 18.4164i	− 1.35768i
$185$	3.00000	0.220564
$186$	0	0
$187$	23.1803i	1.69511i
$188$	8.00000i	0.583460i
$189$	0.527864i	0.0383965i
$190$	0.381966i	0.0277107i
$191$	−4.81966	−0.348738	−0.174369	−	0.984680i	$-0.555789\pi$
$191$	−4.81966	−0.348738	−0.174369	+	0.984680i	$0.555789\pi$
$192$	9.47214	0.683593
$193$	− 5.47214i	− 0.393893i	−0.980414	−	0.196946i	$-0.936897\pi$
$193$	− 5.47214i	− 0.393893i	0.980414	−	0.196946i	$-0.0631025\pi$
$194$	−8.85410	−0.635687
$195$	0	0
$196$	−4.29180	−0.306557
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	13.7082	0.974200
$199$	14.7082	1.04264	0.521318	−	0.853362i	$-0.325440\pi$
$199$	14.7082	1.04264	0.521318	+	0.853362i	$0.325440\pi$
$200$	− 2.23607i	− 0.158114i
$201$	− 23.9443i	− 1.68890i
$202$	− 15.3262i	− 1.07835i
$203$	0.347524i	0.0243914i
$204$	−7.56231	−0.529467
$205$	5.94427	0.415166
$206$	8.00000i	0.557386i
$207$	−16.4721	−1.14489
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0.854102i	0.0589386i
$211$	9.18034	0.632001	0.316000	−	0.948759i	$-0.397660\pi$
$211$	9.18034	0.632001	0.316000	+	0.948759i	$0.397660\pi$
$212$	−3.70820	−0.254680
$213$	3.94427i	0.270257i
$214$	− 16.5623i	− 1.13218i
$215$	1.76393i	0.120299i
$216$	5.00000i	0.340207i
$217$	0	0
$218$	3.23607	0.219174
$219$	− 13.4164i	− 0.906597i
$220$	−2.61803	−0.176508
$221$	0	0
$222$	−10.8541	−0.728480
$223$	− 12.7082i	− 0.851004i	−0.904957	−	0.425502i	$-0.860097\pi$
$223$	− 12.7082i	− 0.851004i	0.904957	−	0.425502i	$-0.139903\pi$
$224$	0.798374	0.0533436
$225$	−2.00000	−0.133333
$226$	− 12.0902i	− 0.804226i
$227$	− 1.76393i	− 0.117076i	−0.998285	−	0.0585381i	$-0.981356\pi$
$227$	− 1.76393i	− 0.117076i	0.998285	−	0.0585381i	$-0.0186439\pi$
$228$	− 0.326238i	− 0.0216056i
$229$	19.8885i	1.31427i	0.753772	+	0.657136i	$0.228232\pi$
$229$	19.8885i	1.31427i	−0.753772	+	0.657136i	$0.771768\pi$
$230$	13.3262	0.878706
$231$	−2.23607	−0.147122
$232$	3.29180i	0.216117i
$233$	−19.8885	−1.30294	−0.651471	−	0.758674i	$-0.725848\pi$
$233$	−19.8885	−1.30294	−0.651471	+	0.758674i	$0.725848\pi$
$234$	0	0
$235$	12.9443	0.844391
$236$	7.85410i	0.511258i
$237$	0	0
$238$	2.09017	0.135486
$239$	9.88854i	0.639637i	0.947479	+	0.319818i	$0.103622\pi$
$239$	9.88854i	0.639637i	−0.947479	+	0.319818i	$0.896378\pi$
$240$	10.8541i	0.700629i
$241$	− 15.0000i	− 0.966235i	−0.875556	−	0.483117i	$-0.839504\pi$
$241$	− 15.0000i	− 0.966235i	0.875556	−	0.483117i	$-0.160496\pi$
$242$	− 11.2361i	− 0.722282i
$243$	17.8885	1.14755
$244$	7.67376	0.491262
$245$	6.94427i	0.443653i
$246$	−21.5066	−1.37121
$247$	0	0
$248$	0	0
$249$	20.0000i	1.26745i
$250$	1.61803	0.102333
$251$	−15.7639	−0.995011	−0.497505	−	0.867461i	$-0.665750\pi$
$251$	−15.7639	−0.995011	−0.497505	+	0.867461i	$0.665750\pi$
$252$	− 0.291796i	− 0.0183814i
$253$	34.8885i	2.19342i
$254$	− 0.381966i	− 0.0239667i
$255$	12.2361i	0.766252i
$256$	13.5623	0.847644
$257$	−8.52786	−0.531954	−0.265977	−	0.963979i	$-0.585694\pi$
$257$	−8.52786	−0.531954	−0.265977	+	0.963979i	$0.585694\pi$
$258$	− 6.38197i	− 0.397324i
$259$	0.708204	0.0440057
$260$	0	0
$261$	2.94427	0.182246
$262$	− 19.4164i	− 1.19955i
$263$	8.23607	0.507858	0.253929	−	0.967223i	$-0.418277\pi$
$263$	8.23607	0.507858	0.253929	+	0.967223i	$0.418277\pi$
$264$	−21.1803	−1.30356
$265$	6.00000i	0.368577i
$266$	0.0901699i	0.00552867i
$267$	− 20.1246i	− 1.23161i
$268$	− 6.61803i	− 0.404261i
$269$	−15.4721	−0.943353	−0.471676	−	0.881772i	$-0.656351\pi$
$269$	−15.4721	−0.943353	−0.471676	+	0.881772i	$0.656351\pi$
$270$	−3.61803	−0.220187
$271$	− 14.7082i	− 0.893460i	−0.894669	−	0.446730i	$-0.852589\pi$
$271$	− 14.7082i	− 0.893460i	0.894669	−	0.446730i	$-0.147411\pi$
$272$	26.5623	1.61058
$273$	0	0
$274$	−12.0902	−0.730394
$275$	4.23607i	0.255445i
$276$	−11.3820	−0.685114
$277$	18.8885	1.13490	0.567451	−	0.823407i	$-0.307930\pi$
$277$	18.8885	1.13490	0.567451	+	0.823407i	$0.307930\pi$
$278$	− 5.32624i	− 0.319447i
$279$	0	0
$280$	− 0.527864i	− 0.0315459i
$281$	− 19.8885i	− 1.18645i	−0.805036	−	0.593226i	$-0.797854\pi$
$281$	− 19.8885i	− 1.18645i	0.805036	−	0.593226i	$-0.202146\pi$
$282$	−46.8328	−2.78885
$283$	−2.70820	−0.160986	−0.0804930	−	0.996755i	$-0.525649\pi$
$283$	−2.70820	−0.160986	−0.0804930	+	0.996755i	$0.525649\pi$
$284$	1.09017i	0.0646897i
$285$	−0.527864	−0.0312680
$286$	0	0
$287$	1.40325	0.0828314
$288$	− 6.76393i	− 0.398569i
$289$	12.9443	0.761428
$290$	−2.38197	−0.139874
$291$	− 12.2361i	− 0.717291i
$292$	− 3.70820i	− 0.217006i
$293$	12.0557i	0.704303i	0.935943	+	0.352152i	$0.114550\pi$
$293$	12.0557i	0.704303i	−0.935943	+	0.352152i	$0.885450\pi$
$294$	− 25.1246i	− 1.46530i
$295$	12.7082	0.739900
$296$	6.70820	0.389906
$297$	− 9.47214i	− 0.549629i
$298$	−21.7984	−1.26275
$299$	0	0
$300$	−1.38197	−0.0797878
$301$	0.416408i	0.0240014i
$302$	12.9443	0.744859
$303$	21.1803	1.21678
$304$	1.14590i	0.0657218i
$305$	− 12.4164i	− 0.710961i
$306$	− 17.7082i	− 1.01231i
$307$	− 7.05573i	− 0.402692i	−0.979520	−	0.201346i	$-0.935468\pi$
$307$	− 7.05573i	− 0.402692i	0.979520	−	0.201346i	$-0.0645315\pi$
$308$	−0.618034	−0.0352158
$309$	−11.0557	−0.628938
$310$	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
$313$	−31.8885	−1.80245	−0.901224	−	0.433355i	$-0.857330\pi$
$313$	−31.8885	−1.80245	−0.901224	+	0.433355i	$0.857330\pi$
$314$	− 29.1246i	− 1.64360i
$315$	−0.472136	−0.0266018
$316$	0	0
$317$	23.8885i	1.34171i	0.741587	+	0.670857i	$0.234074\pi$
$317$	23.8885i	1.34171i	−0.741587	+	0.670857i	$0.765926\pi$
$318$	− 21.7082i	− 1.21734i
$319$	− 6.23607i	− 0.349153i
$320$	− 4.23607i	− 0.236803i
$321$	22.8885	1.27751
$322$	3.14590	0.175314
$323$	1.29180i	0.0718775i
$324$	6.79837	0.377687
$325$	0	0
$326$	2.09017	0.115764
$327$	4.47214i	0.247310i
$328$	13.2918	0.733917
$329$	3.05573	0.168468
$330$	− 15.3262i	− 0.843682i
$331$	25.6525i	1.40999i	0.709213	+	0.704994i	$0.249050\pi$
$331$	25.6525i	1.40999i	−0.709213	+	0.704994i	$0.750950\pi$
$332$	5.52786i	0.303381i
$333$	− 6.00000i	− 0.328798i
$334$	8.38197	0.458641
$335$	−10.7082	−0.585052
$336$	2.56231i	0.139785i
$337$	−27.8885	−1.51919	−0.759593	−	0.650399i	$-0.774602\pi$
$337$	−27.8885	−1.51919	−0.759593	+	0.650399i	$0.774602\pi$
$338$	0	0
$339$	16.7082	0.907465
$340$	3.38197i	0.183413i
$341$	0	0
$342$	0.763932	0.0413087
$343$	3.29180i	0.177740i
$344$	3.94427i	0.212661i
$345$	18.4164i	0.991506i
$346$	− 27.3262i	− 1.46907i
$347$	17.2918	0.928272	0.464136	−	0.885764i	$-0.346365\pi$
$347$	17.2918	0.928272	0.464136	+	0.885764i	$0.346365\pi$
$348$	2.03444	0.109058
$349$	24.4164i	1.30698i	0.756935	+	0.653490i	$0.226696\pi$
$349$	24.4164i	1.30698i	−0.756935	+	0.653490i	$0.773304\pi$
$350$	0.381966	0.0204169
$351$	0	0
$352$	−14.3262	−0.763591
$353$	− 10.5279i	− 0.560342i	−0.959950	−	0.280171i	$-0.909609\pi$
$353$	− 10.5279i	− 0.560342i	0.959950	−	0.280171i	$-0.0903911\pi$
$354$	−45.9787	−2.44374
$355$	1.76393	0.0936198
$356$	− 5.56231i	− 0.294802i
$357$	2.88854i	0.152878i
$358$	6.09017i	0.321875i
$359$	17.8885i	0.944121i	0.881566	+	0.472061i	$0.156490\pi$
$359$	17.8885i	0.944121i	−0.881566	+	0.472061i	$0.843510\pi$
$360$	−4.47214	−0.235702
$361$	18.9443	0.997067
$362$	− 9.70820i	− 0.510252i
$363$	15.5279	0.815001
$364$	0	0
$365$	−6.00000	−0.314054
$366$	44.9230i	2.34816i
$367$	−25.6525	−1.33905	−0.669524	−	0.742790i	$-0.733502\pi$
$367$	−25.6525	−1.33905	−0.669524	+	0.742790i	$0.733502\pi$
$368$	39.9787	2.08403
$369$	− 11.8885i	− 0.618893i
$370$	4.85410i	0.252353i
$371$	1.41641i	0.0735362i
$372$	0	0
$373$	10.0557	0.520666	0.260333	−	0.965519i	$-0.416168\pi$
$373$	10.0557	0.520666	0.260333	+	0.965519i	$0.416168\pi$
$374$	−37.5066	−1.93942
$375$	2.23607i	0.115470i
$376$	28.9443	1.49269
$377$	0	0
$378$	−0.854102	−0.0439303
$379$	33.1803i	1.70436i	0.523249	+	0.852180i	$0.324720\pi$
$379$	33.1803i	1.70436i	−0.523249	+	0.852180i	$0.675280\pi$
$380$	−0.145898	−0.00748441
$381$	0.527864	0.0270433
$382$	− 7.79837i	− 0.399000i
$383$	− 0.236068i	− 0.0120625i	−0.999982	−	0.00603126i	$-0.998080\pi$
$383$	− 0.236068i	− 0.0120625i	0.999982	−	0.00603126i	$-0.00191982\pi$
$384$	30.4508i	1.55394i
$385$	1.00000i	0.0509647i
$386$	8.85410	0.450662
$387$	3.52786	0.179331
$388$	− 3.38197i	− 0.171693i
$389$	35.8885	1.81962	0.909811	−	0.415023i	$-0.136227\pi$
$389$	35.8885	1.81962	0.909811	+	0.415023i	$0.136227\pi$
$390$	0	0
$391$	45.0689	2.27923
$392$	15.5279i	0.784276i
$393$	26.8328	1.35354
$394$	−4.85410	−0.244546
$395$	0	0
$396$	5.23607i	0.263122i
$397$	− 10.0557i	− 0.504683i	−0.967638	−	0.252341i	$-0.918799\pi$
$397$	− 10.0557i	− 0.504683i	0.967638	−	0.252341i	$-0.0812006\pi$
$398$	23.7984i	1.19290i
$399$	−0.124612	−0.00623839
$400$	4.85410	0.242705
$401$	26.8885i	1.34275i	0.741118	+	0.671375i	$0.234296\pi$
$401$	26.8885i	1.34275i	−0.741118	+	0.671375i	$0.765704\pi$
$402$	38.7426	1.93231
$403$	0	0
$404$	5.85410	0.291252
$405$	− 11.0000i	− 0.546594i
$406$	−0.562306	−0.0279068
$407$	−12.7082	−0.629922
$408$	27.3607i	1.35456i
$409$	− 10.8885i	− 0.538404i	−0.963084	−	0.269202i	$-0.913240\pi$
$409$	− 10.8885i	− 0.538404i	0.963084	−	0.269202i	$-0.0867599\pi$
$410$	9.61803i	0.475001i
$411$	− 16.7082i	− 0.824155i
$412$	−3.05573	−0.150545
$413$	3.00000	0.147620
$414$	− 26.6525i	− 1.30990i
$415$	8.94427	0.439057
$416$	0	0
$417$	7.36068	0.360454
$418$	− 1.61803i	− 0.0791406i
$419$	−13.6525	−0.666967	−0.333484	−	0.942756i	$-0.608224\pi$
$419$	−13.6525	−0.666967	−0.333484	+	0.942756i	$0.608224\pi$
$420$	−0.326238	−0.0159188
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	14.8541i	0.723086i
$423$	− 25.8885i	− 1.25874i
$424$	13.4164i	0.651558i
$425$	5.47214	0.265438
$426$	−6.38197	−0.309207
$427$	− 2.93112i	− 0.141847i
$428$	6.32624	0.305790
$429$	0	0
$430$	−2.85410	−0.137637
$431$	4.81966i	0.232155i	0.993240	+	0.116077i	$0.0370321\pi$
$431$	4.81966i	0.232155i	−0.993240	+	0.116077i	$0.962968\pi$
$432$	−10.8541	−0.522218
$433$	23.4721	1.12800	0.563999	−	0.825775i	$-0.309262\pi$
$433$	23.4721	1.12800	0.563999	+	0.825775i	$0.309262\pi$
$434$	0	0
$435$	− 3.29180i	− 0.157830i
$436$	1.23607i	0.0591969i
$437$	1.94427i	0.0930071i
$438$	21.7082	1.03726
$439$	9.29180	0.443473	0.221737	−	0.975107i	$-0.428827\pi$
$439$	9.29180	0.443473	0.221737	+	0.975107i	$0.428827\pi$
$440$	9.47214i	0.451566i
$441$	13.8885	0.661359
$442$	0	0
$443$	−16.9443	−0.805047	−0.402523	−	0.915410i	$-0.631867\pi$
$443$	−16.9443	−0.805047	−0.402523	+	0.915410i	$0.631867\pi$
$444$	− 4.14590i	− 0.196756i
$445$	−9.00000	−0.426641
$446$	20.5623	0.973653
$447$	− 30.1246i	− 1.42485i
$448$	− 1.00000i	− 0.0472456i
$449$	13.9443i	0.658071i	0.944318	+	0.329035i	$0.106724\pi$
$449$	13.9443i	0.658071i	−0.944318	+	0.329035i	$0.893276\pi$
$450$	− 3.23607i	− 0.152550i
$451$	−25.1803	−1.18570
$452$	4.61803	0.217214
$453$	17.8885i	0.840477i
$454$	2.85410	0.133950
$455$	0	0
$456$	−1.18034	−0.0552745
$457$	− 28.4164i	− 1.32926i	−0.747171	−	0.664632i	$-0.768588\pi$
$457$	− 28.4164i	− 1.32926i	0.747171	−	0.664632i	$-0.231412\pi$
$458$	−32.1803	−1.50369
$459$	−12.2361	−0.571131
$460$	5.09017i	0.237330i
$461$	− 28.4164i	− 1.32348i	−0.749732	−	0.661742i	$-0.769817\pi$
$461$	− 28.4164i	− 1.32348i	0.749732	−	0.661742i	$-0.230183\pi$
$462$	− 3.61803i	− 0.168326i
$463$	− 28.9443i	− 1.34515i	−0.740027	−	0.672577i	$-0.765187\pi$
$463$	− 28.9443i	− 1.34515i	0.740027	−	0.672577i	$-0.234813\pi$
$464$	−7.14590	−0.331740
$465$	0	0
$466$	− 32.1803i	− 1.49073i
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	−2.52786	−0.116726
$470$	20.9443i	0.966087i
$471$	40.2492	1.85459
$472$	28.4164	1.30797
$473$	− 7.47214i	− 0.343569i
$474$	0	0
$475$	0.236068i	0.0108315i
$476$	0.798374i	0.0365934i
$477$	12.0000	0.549442
$478$	−16.0000	−0.731823
$479$	8.12461i	0.371223i	0.982623	+	0.185611i	$0.0594266\pi$
$479$	8.12461i	0.371223i	−0.982623	+	0.185611i	$0.940573\pi$
$480$	−7.56231	−0.345170
$481$	0	0
$482$	24.2705	1.10549
$483$	4.34752i	0.197819i
$484$	4.29180	0.195082
$485$	−5.47214	−0.248477
$486$	28.9443i	1.31294i
$487$	− 26.1246i	− 1.18382i	−0.806004	−	0.591910i	$-0.798374\pi$
$487$	− 26.1246i	− 1.18382i	0.806004	−	0.591910i	$-0.201626\pi$
$488$	− 27.7639i	− 1.25681i
$489$	2.88854i	0.130624i
$490$	−11.2361	−0.507594
$491$	−4.23607	−0.191171	−0.0955855	−	0.995421i	$-0.530472\pi$
$491$	−4.23607	−0.191171	−0.0955855	+	0.995421i	$0.530472\pi$
$492$	− 8.21478i	− 0.370351i
$493$	−8.05573	−0.362812
$494$	0	0
$495$	8.47214	0.380794
$496$	0	0
$497$	0.416408	0.0186784
$498$	−32.3607	−1.45012
$499$	13.8885i	0.621737i	0.950453	+	0.310868i	$0.100620\pi$
$499$	13.8885i	0.621737i	−0.950453	+	0.310868i	$0.899380\pi$
$500$	0.618034i	0.0276393i
$501$	11.5836i	0.517517i
$502$	− 25.5066i	− 1.13841i
$503$	−13.1803	−0.587682	−0.293841	−	0.955854i	$-0.594934\pi$
$503$	−13.1803	−0.587682	−0.293841	+	0.955854i	$0.594934\pi$
$504$	−1.05573	−0.0470259
$505$	− 9.47214i	− 0.421505i
$506$	−56.4508	−2.50955
$507$	0	0
$508$	0.145898	0.00647318
$509$	11.3607i	0.503553i	0.967785	+	0.251777i	$0.0810148\pi$
$509$	11.3607i	0.503553i	−0.967785	+	0.251777i	$0.918985\pi$
$510$	−19.7984	−0.876687
$511$	−1.41641	−0.0626582
$512$	− 5.29180i	− 0.233867i
$513$	− 0.527864i	− 0.0233058i
$514$	− 13.7984i	− 0.608620i
$515$	4.94427i	0.217871i
$516$	2.43769	0.107313
$517$	−54.8328	−2.41154
$518$	1.14590i	0.0503479i
$519$	37.7639	1.65765
$520$	0	0
$521$	−41.7771	−1.83029	−0.915144	−	0.403128i	$-0.867923\pi$
$521$	−41.7771	−1.83029	−0.915144	+	0.403128i	$0.867923\pi$
$522$	4.76393i	0.208512i
$523$	18.7082	0.818053	0.409026	−	0.912523i	$-0.365868\pi$
$523$	18.7082	0.818053	0.409026	+	0.912523i	$0.365868\pi$
$524$	7.41641	0.323987
$525$	0.527864i	0.0230379i
$526$	13.3262i	0.581052i
$527$	0	0
$528$	− 45.9787i	− 2.00097i
$529$	44.8328	1.94925
$530$	−9.70820	−0.421697
$531$	− 25.4164i	− 1.10298i
$532$	−0.0344419	−0.00149324
$533$	0	0
$534$	32.5623	1.40911
$535$	− 10.2361i	− 0.442544i
$536$	−23.9443	−1.03424
$537$	−8.41641	−0.363195
$538$	− 25.0344i	− 1.07931i
$539$	− 29.4164i	− 1.26705i
$540$	− 1.38197i	− 0.0594703i
$541$	− 7.88854i	− 0.339155i	−0.985517	−	0.169577i	$-0.945760\pi$
$541$	− 7.88854i	− 0.339155i	0.985517	−	0.169577i	$-0.0542403\pi$
$542$	23.7984	1.02223
$543$	13.4164	0.575753
$544$	18.5066i	0.793463i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−34.8328	−1.48934	−0.744672	−	0.667431i	$-0.767394\pi$
$547$	−34.8328	−1.48934	−0.744672	+	0.667431i	$0.767394\pi$
$548$	− 4.61803i	− 0.197273i
$549$	−24.8328	−1.05984
$550$	−6.85410	−0.292260
$551$	− 0.347524i	− 0.0148050i
$552$	41.1803i	1.75275i
$553$	0	0
$554$	30.5623i	1.29847i
$555$	−6.70820	−0.284747
$556$	2.03444	0.0862796
$557$	− 7.94427i	− 0.336610i	−0.985735	−	0.168305i	$-0.946171\pi$
$557$	− 7.94427i	− 0.336610i	0.985735	−	0.168305i	$-0.0538293\pi$
$558$	0	0
$559$	0	0
$560$	1.14590	0.0484230
$561$	− 51.8328i	− 2.18838i
$562$	32.1803	1.35745
$563$	−17.2918	−0.728762	−0.364381	−	0.931250i	$-0.618719\pi$
$563$	−17.2918	−0.728762	−0.364381	+	0.931250i	$0.618719\pi$
$564$	− 17.8885i	− 0.753244i
$565$	− 7.47214i	− 0.314355i
$566$	− 4.38197i	− 0.184188i
$567$	− 2.59675i	− 0.109053i
$568$	3.94427	0.165498
$569$	18.8885	0.791849	0.395924	−	0.918283i	$-0.370424\pi$
$569$	18.8885	0.791849	0.395924	+	0.918283i	$0.370424\pi$
$570$	− 0.854102i	− 0.0357744i
$571$	−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
$573$	10.7771	0.450219
$574$	2.27051i	0.0947692i
$575$	8.23607	0.343468
$576$	−8.47214	−0.353006
$577$	− 7.88854i	− 0.328404i	−0.986427	−	0.164202i	$-0.947495\pi$
$577$	− 7.88854i	− 0.328404i	0.986427	−	0.164202i	$-0.0525050\pi$
$578$	20.9443i	0.871167i
$579$	12.2361i	0.508514i
$580$	− 0.909830i	− 0.0377786i
$581$	2.11146	0.0875980
$582$	19.7984	0.820669
$583$	− 25.4164i	− 1.05264i
$584$	−13.4164	−0.555175
$585$	0	0
$586$	−19.5066	−0.805809
$587$	5.76393i	0.237903i	0.992900	+	0.118951i	$0.0379533\pi$
$587$	5.76393i	0.237903i	−0.992900	+	0.118951i	$0.962047\pi$
$588$	9.59675	0.395763
$589$	0	0
$590$	20.5623i	0.846537i
$591$	− 6.70820i	− 0.275939i
$592$	14.5623i	0.598507i
$593$	− 27.8885i	− 1.14525i	−0.819819	−	0.572623i	$-0.805926\pi$
$593$	− 27.8885i	− 1.14525i	0.819819	−	0.572623i	$-0.194074\pi$
$594$	15.3262	0.628843
$595$	1.29180	0.0529585
$596$	− 8.32624i	− 0.341056i
$597$	−32.8885	−1.34604
$598$	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
$601$	26.0557	1.06284	0.531418	−	0.847110i	$-0.321659\pi$
$601$	26.0557	1.06284	0.531418	+	0.847110i	$0.321659\pi$
$602$	−0.673762	−0.0274605
$603$	21.4164i	0.872144i
$604$	4.94427i	0.201180i
$605$	− 6.94427i	− 0.282325i
$606$	34.2705i	1.39214i
$607$	12.7082	0.515810	0.257905	−	0.966170i	$-0.416968\pi$
$607$	12.7082	0.515810	0.257905	+	0.966170i	$0.416968\pi$
$608$	−0.798374	−0.0323783
$609$	− 0.777088i	− 0.0314892i
$610$	20.0902	0.813427
$611$	0	0
$612$	6.76393	0.273416
$613$	22.0557i	0.890822i	0.895326	+	0.445411i	$0.146943\pi$
$613$	22.0557i	0.890822i	−0.895326	+	0.445411i	$0.853057\pi$
$614$	11.4164	0.460729
$615$	−13.2918	−0.535977
$616$	2.23607i	0.0900937i
$617$	− 13.5836i	− 0.546855i	−0.961893	−	0.273427i	$-0.911843\pi$
$617$	− 13.5836i	− 0.546855i	0.961893	−	0.273427i	$-0.0881573\pi$
$618$	− 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
$621$	−18.4164	−0.739025
$622$	38.8328i	1.55705i
$623$	−2.12461	−0.0851208
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 51.5967i	− 2.06222i
$627$	2.23607	0.0893000
$628$	11.1246	0.443920
$629$	16.4164i	0.654565i
$630$	− 0.763932i	− 0.0304358i
$631$	− 24.1246i	− 0.960386i	−0.877163	−	0.480193i	$-0.840567\pi$
$631$	− 24.1246i	− 0.960386i	0.877163	−	0.480193i	$-0.159433\pi$
$632$	0	0
$633$	−20.5279	−0.815909
$634$	−38.6525	−1.53509
$635$	− 0.236068i	− 0.00936807i
$636$	8.29180	0.328791
$637$	0	0
$638$	10.0902	0.399474
$639$	− 3.52786i	− 0.139560i
$640$	13.6180	0.538300
$641$	26.8885	1.06203	0.531017	−	0.847361i	$-0.321810\pi$
$641$	26.8885	1.06203	0.531017	+	0.847361i	$0.321810\pi$
$642$	37.0344i	1.46163i
$643$	− 30.7082i	− 1.21101i	−0.795840	−	0.605507i	$-0.792970\pi$
$643$	− 30.7082i	− 1.21101i	0.795840	−	0.605507i	$-0.207030\pi$
$644$	1.20163i	0.0473507i
$645$	− 3.94427i	− 0.155306i
$646$	−2.09017	−0.0822366
$647$	46.5967	1.83191	0.915954	−	0.401284i	$-0.131436\pi$
$647$	46.5967	1.83191	0.915954	+	0.401284i	$0.131436\pi$
$648$	− 24.5967i	− 0.966252i
$649$	−53.8328	−2.11312
$650$	0	0
$651$	0	0
$652$	0.798374i	0.0312667i
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−7.23607	−0.282953
$655$	− 12.0000i	− 0.468879i
$656$	28.8541i	1.12656i
$657$	12.0000i	0.468165i
$658$	4.94427i	0.192748i
$659$	−48.2361	−1.87901	−0.939505	−	0.342535i	$-0.888715\pi$
$659$	−48.2361	−1.87901	−0.939505	+	0.342535i	$0.888715\pi$
$660$	5.85410	0.227871
$661$	− 3.36068i	− 0.130715i	−0.997862	−	0.0653576i	$-0.979181\pi$
$661$	− 3.36068i	− 0.130715i	0.997862	−	0.0653576i	$-0.0208188\pi$
$662$	−41.5066	−1.61320
$663$	0	0
$664$	20.0000	0.776151
$665$	0.0557281i	0.00216104i
$666$	9.70820	0.376185
$667$	−12.1246	−0.469467
$668$	3.20163i	0.123875i
$669$	28.4164i	1.09864i
$670$	− 17.3262i	− 0.669371i
$671$	52.5967i	2.03047i
$672$	−1.78522	−0.0688663
$673$	36.4164	1.40375	0.701875	−	0.712300i	$-0.252347\pi$
$673$	36.4164	1.40375	0.701875	+	0.712300i	$0.252347\pi$
$674$	− 45.1246i	− 1.73814i
$675$	−2.23607	−0.0860663
$676$	0	0
$677$	47.8885	1.84051	0.920253	−	0.391324i	$-0.127983\pi$
$677$	47.8885	1.84051	0.920253	+	0.391324i	$0.127983\pi$
$678$	27.0344i	1.03825i
$679$	−1.29180	−0.0495746
$680$	12.2361	0.469232
$681$	3.94427i	0.151145i
$682$	0	0
$683$	− 30.2361i	− 1.15695i	−0.815700	−	0.578475i	$-0.803648\pi$
$683$	− 30.2361i	− 1.15695i	0.815700	−	0.578475i	$-0.196352\pi$
$684$	0.291796i	0.0111571i
$685$	−7.47214	−0.285496
$686$	−5.32624	−0.203357
$687$	− 44.4721i	− 1.69672i
$688$	−8.56231	−0.326435
$689$	0	0
$690$	−29.7984	−1.13440
$691$	− 10.5967i	− 0.403119i	−0.979476	−	0.201560i	$-0.935399\pi$
$691$	− 10.5967i	− 0.403119i	0.979476	−	0.201560i	$-0.0646010\pi$
$692$	10.4377	0.396782
$693$	2.00000	0.0759737
$694$	27.9787i	1.06206i
$695$	− 3.29180i	− 0.124865i
$696$	− 7.36068i	− 0.279006i
$697$	32.5279i	1.23208i
$698$	−39.5066	−1.49535
$699$	44.4721	1.68209
$700$	0.145898i	0.00551443i
$701$	27.8885	1.05334	0.526668	−	0.850071i	$-0.323441\pi$
$701$	27.8885	1.05334	0.526668	+	0.850071i	$0.323441\pi$
$702$	0	0
$703$	−0.708204	−0.0267104
$704$	17.9443i	0.676300i
$705$	−28.9443	−1.09010
$706$	17.0344	0.641100
$707$	− 2.23607i	− 0.0840960i
$708$	− 17.5623i	− 0.660032i
$709$	38.3050i	1.43857i	0.694714	+	0.719286i	$0.255531\pi$
$709$	38.3050i	1.43857i	−0.694714	+	0.719286i	$0.744469\pi$
$710$	2.85410i	0.107113i
$711$	0	0
$712$	−20.1246	−0.754202
$713$	0	0
$714$	−4.67376	−0.174911
$715$	0	0
$716$	−2.32624	−0.0869356
$717$	− 22.1115i	− 0.825767i
$718$	−28.9443	−1.08019
$719$	−8.12461	−0.302997	−0.151498	−	0.988457i	$-0.548410\pi$
$719$	−8.12461	−0.302997	−0.151498	+	0.988457i	$0.548410\pi$
$720$	− 9.70820i	− 0.361803i
$721$	1.16718i	0.0434682i
$722$	30.6525i	1.14077i
$723$	33.5410i	1.24740i
$724$	3.70820	0.137814
$725$	−1.47214	−0.0546738
$726$	25.1246i	0.932462i
$727$	−11.0557	−0.410034	−0.205017	−	0.978758i	$-0.565725\pi$
$727$	−11.0557	−0.410034	−0.205017	+	0.978758i	$0.565725\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	− 9.70820i	− 0.359317i
$731$	−9.65248	−0.357010
$732$	−17.1591	−0.634217
$733$	− 8.11146i	− 0.299603i	−0.988716	−	0.149802i	$-0.952136\pi$
$733$	− 8.11146i	− 0.299603i	0.988716	−	0.149802i	$-0.0478635\pi$
$734$	− 41.5066i	− 1.53204i
$735$	− 15.5279i	− 0.572754i
$736$	27.8541i	1.02672i
$737$	45.3607	1.67088
$738$	19.2361	0.708089
$739$	39.5410i	1.45454i	0.686352	+	0.727270i	$0.259211\pi$
$739$	39.5410i	1.45454i	−0.686352	+	0.727270i	$0.740789\pi$
$740$	−1.85410	−0.0681581
$741$	0	0
$742$	−2.29180	−0.0841345
$743$	2.12461i	0.0779444i	0.999240	+	0.0389722i	$0.0124084\pi$
$743$	2.12461i	0.0779444i	−0.999240	+	0.0389722i	$0.987592\pi$
$744$	0	0
$745$	−13.4721	−0.493581
$746$	16.2705i	0.595706i
$747$	− 17.8885i	− 0.654508i
$748$	− 14.3262i	− 0.523819i
$749$	− 2.41641i	− 0.0882936i
$750$	−3.61803	−0.132112
$751$	17.2918	0.630987	0.315493	−	0.948928i	$-0.397830\pi$
$751$	17.2918	0.630987	0.315493	+	0.948928i	$0.397830\pi$
$752$	62.8328i	2.29128i
$753$	35.2492	1.28455
$754$	0	0
$755$	8.00000	0.291150
$756$	− 0.326238i	− 0.0118652i
$757$	28.8885	1.04997	0.524986	−	0.851111i	$-0.324071\pi$
$757$	28.8885	1.04997	0.524986	+	0.851111i	$0.324071\pi$
$758$	−53.6869	−1.95000
$759$	− 78.0132i	− 2.83170i
$760$	0.527864i	0.0191476i
$761$	20.0557i	0.727020i	0.931590	+	0.363510i	$0.118422\pi$
$761$	20.0557i	0.727020i	−0.931590	+	0.363510i	$0.881578\pi$
$762$	0.854102i	0.0309408i
$763$	0.472136	0.0170925
$764$	2.97871	0.107766
$765$	− 10.9443i	− 0.395691i
$766$	0.381966	0.0138010
$767$	0	0
$768$	−30.3262	−1.09430
$769$	− 23.8328i	− 0.859433i	−0.902964	−	0.429717i	$-0.858613\pi$
$769$	− 23.8328i	− 0.859433i	0.902964	−	0.429717i	$-0.141387\pi$
$770$	−1.61803	−0.0583099
$771$	19.0689	0.686749
$772$	3.38197i	0.121720i
$773$	− 37.9443i	− 1.36476i	−0.730997	−	0.682380i	$-0.760945\pi$
$773$	− 37.9443i	− 1.36476i	0.730997	−	0.682380i	$-0.239055\pi$
$774$	5.70820i	0.205177i
$775$	0	0
$776$	−12.2361	−0.439249
$777$	−1.58359	−0.0568111
$778$	58.0689i	2.08187i
$779$	−1.40325	−0.0502767
$780$	0	0
$781$	−7.47214	−0.267374
$782$	72.9230i	2.60772i
$783$	3.29180	0.117639
$784$	−33.7082	−1.20386
$785$	− 18.0000i	− 0.642448i
$786$	43.4164i	1.54861i
$787$	37.5410i	1.33819i	0.743176	+	0.669096i	$0.233319\pi$
$787$	37.5410i	1.33819i	−0.743176	+	0.669096i	$0.766681\pi$
$788$	− 1.85410i	− 0.0660496i
$789$	−18.4164	−0.655641
$790$	0	0
$791$	− 1.76393i	− 0.0627182i
$792$	18.9443	0.673155
$793$	0	0
$794$	16.2705	0.577419
$795$	− 13.4164i	− 0.475831i
$796$	−9.09017	−0.322193
$797$	−0.167184	−0.00592197	−0.00296099	−	0.999996i	$-0.500943\pi$
$797$	−0.167184	−0.00592197	−0.00296099	+	0.999996i	$0.500943\pi$
$798$	− 0.201626i	− 0.00713749i
$799$	70.8328i	2.50588i
$800$	3.38197i	0.119571i
$801$	18.0000i	0.635999i
$802$	−43.5066	−1.53627
$803$	25.4164	0.896926
$804$	14.7984i	0.521898i
$805$	1.94427	0.0685266
$806$	0	0
$807$	34.5967	1.21786
$808$	− 21.1803i	− 0.745122i
$809$	32.8885	1.15630	0.578150	−	0.815931i	$-0.303775\pi$
$809$	32.8885	1.15630	0.578150	+	0.815931i	$0.303775\pi$
$810$	17.7984	0.625371
$811$	− 39.7771i	− 1.39676i	−0.715726	−	0.698381i	$-0.753904\pi$
$811$	− 39.7771i	− 1.39676i	0.715726	−	0.698381i	$-0.246096\pi$
$812$	− 0.214782i	− 0.00753736i
$813$	32.8885i	1.15345i
$814$	− 20.5623i	− 0.720708i
$815$	1.29180	0.0452496
$816$	−59.3951	−2.07925
$817$	− 0.416408i	− 0.0145683i
$818$	17.6180	0.616000
$819$	0	0
$820$	−3.67376	−0.128293
$821$	19.3607i	0.675692i	0.941201	+	0.337846i	$0.109698\pi$
$821$	19.3607i	0.675692i	−0.941201	+	0.337846i	$0.890302\pi$
$822$	27.0344	0.942934
$823$	−22.5967	−0.787673	−0.393837	−	0.919180i	$-0.628852\pi$
$823$	−22.5967	−0.787673	−0.393837	+	0.919180i	$0.628852\pi$
$824$	11.0557i	0.385145i
$825$	− 9.47214i	− 0.329777i
$826$	4.85410i	0.168896i
$827$	− 8.94427i	− 0.311023i	−0.987834	−	0.155511i	$-0.950297\pi$
$827$	− 8.94427i	− 0.311023i	0.987834	−	0.155511i	$-0.0497025\pi$
$828$	10.1803	0.353791
$829$	−37.2492	−1.29372	−0.646860	−	0.762609i	$-0.723918\pi$
$829$	−37.2492	−1.29372	−0.646860	+	0.762609i	$0.723918\pi$
$830$	14.4721i	0.502335i
$831$	−42.2361	−1.46515
$832$	0	0
$833$	−38.0000	−1.31662
$834$	11.9098i	0.412404i
$835$	5.18034	0.179273
$836$	0.618034	0.0213752
$837$	0	0
$838$	− 22.0902i	− 0.763092i
$839$	− 17.2918i	− 0.596979i	−0.954413	−	0.298490i	$-0.903517\pi$
$839$	− 17.2918i	− 0.596979i	0.954413	−	0.298490i	$-0.0964828\pi$
$840$	1.18034i	0.0407256i
$841$	−26.8328	−0.925270
$842$	−9.70820	−0.334567
$843$	44.4721i	1.53170i
$844$	−5.67376	−0.195299
$845$	0	0
$846$	41.8885	1.44016
$847$	− 1.63932i	− 0.0563277i
$848$	−29.1246	−1.00014
$849$	6.05573	0.207832
$850$	8.85410i	0.303693i
$851$	24.7082i	0.846986i
$852$	− 2.43769i	− 0.0835140i
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	4.74265	0.162290
$855$	0.472136	0.0161467
$856$	− 22.8885i	− 0.782314i
$857$	−0.111456	−0.00380727	−0.00190364	−	0.999998i	$-0.500606\pi$
$857$	−0.111456	−0.00380727	−0.00190364	+	0.999998i	$0.500606\pi$
$858$	0	0
$859$	−13.8885	−0.473871	−0.236935	−	0.971525i	$-0.576143\pi$
$859$	−13.8885	−0.473871	−0.236935	+	0.971525i	$0.576143\pi$
$860$	− 1.09017i	− 0.0371745i
$861$	−3.13777	−0.106935
$862$	−7.79837	−0.265614
$863$	− 38.8328i	− 1.32188i	−0.750437	−	0.660942i	$-0.770157\pi$
$863$	− 38.8328i	− 1.32188i	0.750437	−	0.660942i	$-0.229843\pi$
$864$	− 7.56231i	− 0.257275i
$865$	− 16.8885i	− 0.574228i
$866$	37.9787i	1.29057i
$867$	−28.9443	−0.982999
$868$	0	0
$869$	0	0
$870$	5.32624	0.180576
$871$	0	0
$872$	4.47214	0.151446
$873$	10.9443i	0.370407i
$874$	−3.14590	−0.106412
$875$	0.236068	0.00798055
$876$	8.29180i	0.280154i
$877$	55.7214i	1.88158i	0.338995	+	0.940788i	$0.389913\pi$
$877$	55.7214i	1.88158i	−0.338995	+	0.940788i	$0.610087\pi$
$878$	15.0344i	0.507388i
$879$	− 26.9574i	− 0.909251i
$880$	−20.5623	−0.693155
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	22.4721i	0.756676i
$883$	−10.8328	−0.364553	−0.182277	−	0.983247i	$-0.558347\pi$
$883$	−10.8328	−0.364553	−0.182277	+	0.983247i	$0.558347\pi$
$884$	0	0
$885$	−28.4164	−0.955207
$886$	− 27.4164i	− 0.921072i
$887$	−40.4853	−1.35936	−0.679682	−	0.733507i	$-0.737882\pi$
$887$	−40.4853	−1.35936	−0.679682	+	0.733507i	$0.737882\pi$
$888$	−15.0000	−0.503367
$889$	− 0.0557281i	− 0.00186906i
$890$	− 14.5623i	− 0.488130i
$891$	46.5967i	1.56105i
$892$	7.85410i	0.262975i
$893$	−3.05573	−0.102256
$894$	48.7426	1.63020
$895$	3.76393i	0.125814i
$896$	3.21478	0.107398
$897$	0	0
$898$	−22.5623	−0.752914
$899$	0	0
$900$	1.23607	0.0412023
$901$	−32.8328	−1.09382
$902$	− 40.7426i	− 1.35658i
$903$	− 0.931116i	− 0.0309856i
$904$	− 16.7082i	− 0.555707i
$905$	− 6.00000i	− 0.199447i
$906$	−28.9443	−0.961609
$907$	20.1246	0.668227	0.334113	−	0.942533i	$-0.391563\pi$
$907$	20.1246	0.668227	0.334113	+	0.942533i	$0.391563\pi$
$908$	1.09017i	0.0361786i
$909$	−18.9443	−0.628342
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	− 2.56231i	− 0.0848464i
$913$	−37.8885	−1.25393
$914$	45.9787	1.52084
$915$	27.7639i	0.917847i
$916$	− 12.2918i	− 0.406132i
$917$	− 2.83282i	− 0.0935478i
$918$	− 19.7984i	− 0.653444i
$919$	33.2918	1.09820	0.549098	−	0.835758i	$-0.314972\pi$
$919$	33.2918	1.09820	0.549098	+	0.835758i	$0.314972\pi$
$920$	18.4164	0.607171
$921$	15.7771i	0.519873i
$922$	45.9787	1.51423
$923$	0	0
$924$	1.38197	0.0454633
$925$	3.00000i	0.0986394i
$926$	46.8328	1.53902
$927$	9.88854	0.324782
$928$	− 4.97871i	− 0.163434i
$929$	2.05573i	0.0674463i	0.999431	+	0.0337231i	$0.0107364\pi$
$929$	2.05573i	0.0674463i	−0.999431	+	0.0337231i	$0.989264\pi$
$930$	0	0
$931$	− 1.63932i	− 0.0537266i
$932$	12.2918	0.402631
$933$	−53.6656	−1.75693
$934$	14.4721i	0.473543i
$935$	−23.1803	−0.758078
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	− 4.09017i	− 0.133549i
$939$	71.3050	2.32695
$940$	−8.00000	−0.260931
$941$	23.8885i	0.778744i	0.921081	+	0.389372i	$0.127308\pi$
$941$	23.8885i	0.778744i	−0.921081	+	0.389372i	$0.872692\pi$
$942$	65.1246i	2.12187i
$943$	48.9574i	1.59427i
$944$	61.6869i	2.00774i
$945$	−0.527864	−0.0171714
$946$	12.0902	0.393085
$947$	− 16.8197i	− 0.546566i	−0.961934	−	0.273283i	$-0.911891\pi$
$947$	− 16.8197i	− 0.546566i	0.961934	−	0.273283i	$-0.0881095\pi$
$948$	0	0
$949$	0	0
$950$	−0.381966	−0.0123926
$951$	− 53.4164i	− 1.73215i
$952$	2.88854	0.0936182
$953$	56.1935	1.82029	0.910143	−	0.414294i	$-0.135972\pi$
$953$	56.1935	1.82029	0.910143	+	0.414294i	$0.135972\pi$
$954$	19.4164i	0.628629i
$955$	− 4.81966i	− 0.155961i
$956$	− 6.11146i	− 0.197659i
$957$	13.9443i	0.450754i
$958$	−13.1459	−0.424725
$959$	−1.76393	−0.0569603
$960$	9.47214i	0.305712i
$961$	31.0000	1.00000
$962$	0	0
$963$	−20.4721	−0.659705
$964$	9.27051i	0.298583i
$965$	5.47214	0.176154
$966$	−7.03444	−0.226329
$967$	− 1.16718i	− 0.0375341i	−0.999824	−	0.0187671i	$-0.994026\pi$
$967$	− 1.16718i	− 0.0375341i	0.999824	−	0.0187671i	$-0.00597409\pi$
$968$	− 15.5279i	− 0.499084i
$969$	− 2.88854i	− 0.0927934i
$970$	− 8.85410i	− 0.284288i
$971$	−16.2361	−0.521040	−0.260520	−	0.965468i	$-0.583894\pi$
$971$	−16.2361	−0.521040	−0.260520	+	0.965468i	$0.583894\pi$
$972$	−11.0557	−0.354613
$973$	− 0.777088i	− 0.0249123i
$974$	42.2705	1.35443
$975$	0	0
$976$	60.2705	1.92921
$977$	34.4164i	1.10108i	0.834809	+	0.550539i	$0.185578\pi$
$977$	34.4164i	1.10108i	−0.834809	+	0.550539i	$0.814422\pi$
$978$	−4.67376	−0.149450
$979$	38.1246	1.21847
$980$	− 4.29180i	− 0.137096i
$981$	− 4.00000i	− 0.127710i
$982$	− 6.85410i	− 0.218723i
$983$	14.8328i	0.473093i	0.971620	+	0.236547i	$0.0760156\pi$
$983$	14.8328i	0.473093i	−0.971620	+	0.236547i	$0.923984\pi$
$984$	−29.7214	−0.947482
$985$	−3.00000	−0.0955879
$986$	− 13.0344i	− 0.415101i
$987$	−6.83282	−0.217491
$988$	0	0
$989$	−14.5279	−0.461959
$990$	13.7082i	0.435675i
$991$	−33.2918	−1.05755	−0.528774	−	0.848762i	$-0.677348\pi$
$991$	−33.2918	−1.05755	−0.528774	+	0.848762i	$0.677348\pi$
$992$	0	0
$993$	− 57.3607i	− 1.82029i
$994$	0.673762i	0.0213704i
$995$	14.7082i	0.466281i
$996$	− 12.3607i	− 0.391663i
$997$	16.8885	0.534866	0.267433	−	0.963577i	$-0.413825\pi$
$997$	16.8885	0.534866	0.267433	+	0.963577i	$0.413825\pi$
$998$	−22.4721	−0.711343
$999$	− 6.70820i	− 0.212238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.c.506.4		4
13.2	odd	12		65.2.e.a.61.1	yes	4
13.3	even	3		845.2.m.e.316.1		8
13.4	even	6		845.2.m.e.361.1		8
13.5	odd	4		845.2.a.e.1.2		2
13.6	odd	12		65.2.e.a.16.1	✓	4
13.7	odd	12		845.2.e.g.146.2		4
13.8	odd	4		845.2.a.b.1.1		2
13.9	even	3		845.2.m.e.361.4		8
13.10	even	6		845.2.m.e.316.4		8
13.11	odd	12		845.2.e.g.191.2		4
13.12	even	2	inner	845.2.c.c.506.1		4
39.2	even	12		585.2.j.e.451.2		4
39.5	even	4		7605.2.a.ba.1.1		2
39.8	even	4		7605.2.a.bf.1.2		2
39.32	even	12		585.2.j.e.406.2		4
52.15	even	12		1040.2.q.n.321.1		4
52.19	even	12		1040.2.q.n.81.1		4
65.2	even	12		325.2.o.a.74.1		8
65.19	odd	12		325.2.e.b.276.2		4
65.28	even	12		325.2.o.a.74.4		8
65.32	even	12		325.2.o.a.224.4		8
65.34	odd	4		4225.2.a.y.1.2		2
65.44	odd	4		4225.2.a.u.1.1		2
65.54	odd	12		325.2.e.b.126.2		4
65.58	even	12		325.2.o.a.224.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.a.16.1	✓	4	13.6	odd	12
65.2.e.a.61.1	yes	4	13.2	odd	12
325.2.e.b.126.2		4	65.54	odd	12
325.2.e.b.276.2		4	65.19	odd	12
325.2.o.a.74.1		8	65.2	even	12
325.2.o.a.74.4		8	65.28	even	12
325.2.o.a.224.1		8	65.58	even	12
325.2.o.a.224.4		8	65.32	even	12
585.2.j.e.406.2		4	39.32	even	12
585.2.j.e.451.2		4	39.2	even	12
845.2.a.b.1.1		2	13.8	odd	4
845.2.a.e.1.2		2	13.5	odd	4
845.2.c.c.506.1		4	13.12	even	2	inner
845.2.c.c.506.4		4	1.1	even	1	trivial
845.2.e.g.146.2		4	13.7	odd	12
845.2.e.g.191.2		4	13.11	odd	12
845.2.m.e.316.1		8	13.3	even	3
845.2.m.e.316.4		8	13.10	even	6
845.2.m.e.361.1		8	13.4	even	6
845.2.m.e.361.4		8	13.9	even	3
1040.2.q.n.81.1		4	52.19	even	12
1040.2.q.n.321.1		4	52.15	even	12
4225.2.a.u.1.1		2	65.44	odd	4
4225.2.a.y.1.2		2	65.34	odd	4
7605.2.a.ba.1.1		2	39.5	even	4
7605.2.a.bf.1.2		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)

\(f(q)\)	\(=\)	\(q+1.61803i q^{2} -2.23607 q^{3} -0.618034 q^{4} +1.00000i q^{5} -3.61803i q^{6} +0.236068i q^{7} +2.23607i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.61803i q^{2} -2.23607 q^{3} -0.618034 q^{4} +1.00000i q^{5} -3.61803i q^{6} +0.236068i q^{7} +2.23607i q^{8} +2.00000 q^{9} -1.61803 q^{10} -4.23607i q^{11} +1.38197 q^{12} -0.381966 q^{14} -2.23607i q^{15} -4.85410 q^{16} -5.47214 q^{17} +3.23607i q^{18} -0.236068i q^{19} -0.618034i q^{20} -0.527864i q^{21} +6.85410 q^{22} -8.23607 q^{23} -5.00000i q^{24} -1.00000 q^{25} +2.23607 q^{27} -0.145898i q^{28} +1.47214 q^{29} +3.61803 q^{30} -3.38197i q^{32} +9.47214i q^{33} -8.85410i q^{34} -0.236068 q^{35} -1.23607 q^{36} -3.00000i q^{37} +0.381966 q^{38} -2.23607 q^{40} -5.94427i q^{41} +0.854102 q^{42} +1.76393 q^{43} +2.61803i q^{44} +2.00000i q^{45} -13.3262i q^{46} -12.9443i q^{47} +10.8541 q^{48} +6.94427 q^{49} -1.61803i q^{50} +12.2361 q^{51} +6.00000 q^{53} +3.61803i q^{54} +4.23607 q^{55} -0.527864 q^{56} +0.527864i q^{57} +2.38197i q^{58} -12.7082i q^{59} +1.38197i q^{60} -12.4164 q^{61} +0.472136i q^{63} -4.23607 q^{64} -15.3262 q^{66} +10.7082i q^{67} +3.38197 q^{68} +18.4164 q^{69} -0.381966i q^{70} -1.76393i q^{71} +4.47214i q^{72} +6.00000i q^{73} +4.85410 q^{74} +2.23607 q^{75} +0.145898i q^{76} +1.00000 q^{77} -4.85410i q^{80} -11.0000 q^{81} +9.61803 q^{82} -8.94427i q^{83} +0.326238i q^{84} -5.47214i q^{85} +2.85410i q^{86} -3.29180 q^{87} +9.47214 q^{88} +9.00000i q^{89} -3.23607 q^{90} +5.09017 q^{92} +20.9443 q^{94} +0.236068 q^{95} +7.56231i q^{96} +5.47214i q^{97} +11.2361i q^{98} -8.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{9}+O(q^{10}) \) 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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