LMFDB - Embedded newform 845.2.c.b.506.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("845.506");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.b.506.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.414214i q^{2} +1.41421 q^{3} +1.82843 q^{4} -1.00000i q^{5} -0.585786i q^{6} -0.828427i q^{7} -1.58579i q^{8} -1.00000 q^{9} -0.414214 q^{10} +0.585786i q^{11} +2.58579 q^{12} -0.343146 q^{14} -1.41421i q^{15} +3.00000 q^{16} +4.82843 q^{17} +0.414214i q^{18} -3.41421i q^{19} -1.82843i q^{20} -1.17157i q^{21} +0.242641 q^{22} +1.41421 q^{23} -2.24264i q^{24} -1.00000 q^{25} -5.65685 q^{27} -1.51472i q^{28} +5.65685 q^{29} -0.585786 q^{30} -10.2426i q^{31} -4.41421i q^{32} +0.828427i q^{33} -2.00000i q^{34} -0.828427 q^{35} -1.82843 q^{36} +8.48528i q^{37} -1.41421 q^{38} -1.58579 q^{40} +8.82843i q^{41} -0.485281 q^{42} -3.07107 q^{43} +1.07107i q^{44} +1.00000i q^{45} -0.585786i q^{46} +0.828427i q^{47} +4.24264 q^{48} +6.31371 q^{49} +0.414214i q^{50} +6.82843 q^{51} -14.4853 q^{53} +2.34315i q^{54} +0.585786 q^{55} -1.31371 q^{56} -4.82843i q^{57} -2.34315i q^{58} +10.2426i q^{59} -2.58579i q^{60} -8.00000 q^{61} -4.24264 q^{62} +0.828427i q^{63} +4.17157 q^{64} +0.343146 q^{66} +2.00000i q^{67} +8.82843 q^{68} +2.00000 q^{69} +0.343146i q^{70} +7.89949i q^{71} +1.58579i q^{72} -8.48528i q^{73} +3.51472 q^{74} -1.41421 q^{75} -6.24264i q^{76} +0.485281 q^{77} +8.48528 q^{79} -3.00000i q^{80} -5.00000 q^{81} +3.65685 q^{82} +8.82843i q^{83} -2.14214i q^{84} -4.82843i q^{85} +1.27208i q^{86} +8.00000 q^{87} +0.928932 q^{88} +6.00000i q^{89} +0.414214 q^{90} +2.58579 q^{92} -14.4853i q^{93} +0.343146 q^{94} -3.41421 q^{95} -6.24264i q^{96} -3.65685i q^{97} -2.61522i q^{98} -0.585786i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9} + 4 q^{10} + 16 q^{12} - 24 q^{14} + 12 q^{16} + 8 q^{17} - 16 q^{22} - 4 q^{25} - 8 q^{30} + 8 q^{35} + 4 q^{36} - 12 q^{40} + 32 q^{42} + 16 q^{43} - 20 q^{49} + 16 q^{51} - 24 q^{53}+ \cdots - 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 + 4 * q^10 + 16 * q^12 - 24 * q^14 + 12 * q^16 + 8 * q^17 - 16 * q^22 - 4 * q^25 - 8 * q^30 + 8 * q^35 + 4 * q^36 - 12 * q^40 + 32 * q^42 + 16 * q^43 - 20 * q^49 + 16 * q^51 - 24 * q^53 + 8 * q^55 + 40 * q^56 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 24 * q^68 + 8 * q^69 + 48 * q^74 - 32 * q^77 - 20 * q^81 - 8 * q^82 + 32 * q^87 + 32 * q^88 - 4 * q^90 + 16 * q^92 + 24 * q^94 - 8 * 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 $

$2$	− 0.414214i	− 0.292893i	−0.989219	−	0.146447i	$-0.953216\pi$
$2$	− 0.414214i	− 0.292893i	0.989219	−	0.146447i	$-0.0467837\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.82843	0.914214
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 0.585786i	− 0.239146i
$7$	− 0.828427i	− 0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$	− 0.828427i	− 0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$	− 1.58579i	− 0.560660i
$9$	−1.00000	−0.333333
$10$	−0.414214	−0.130986
$11$	0.585786i	0.176621i	0.996093	+	0.0883106i	$0.0281468\pi$
$11$	0.585786i	0.176621i	−0.996093	+	0.0883106i	$0.971853\pi$
$12$	2.58579	0.746452
$13$	0	0
$13$	0	0
$14$	−0.343146	−0.0917096
$15$	− 1.41421i	− 0.365148i
$16$	3.00000	0.750000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0.414214i	0.0976311i
$19$	− 3.41421i	− 0.783274i	−0.920120	−	0.391637i	$-0.871909\pi$
$19$	− 3.41421i	− 0.783274i	0.920120	−	0.391637i	$-0.128091\pi$
$20$	− 1.82843i	− 0.408849i
$21$	− 1.17157i	− 0.255658i
$22$	0.242641	0.0517312
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	− 2.24264i	− 0.457777i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	− 1.51472i	− 0.286255i
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	−0.585786	−0.106949
$31$	− 10.2426i	− 1.83963i	−0.392349	−	0.919816i	$-0.628338\pi$
$31$	− 10.2426i	− 1.83963i	0.392349	−	0.919816i	$-0.371662\pi$
$32$	− 4.41421i	− 0.780330i
$33$	0.828427i	0.144211i
$34$	− 2.00000i	− 0.342997i
$35$	−0.828427	−0.140030
$36$	−1.82843	−0.304738
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	−1.41421	−0.229416
$39$	0	0
$40$	−1.58579	−0.250735
$41$	8.82843i	1.37877i	0.724396	+	0.689384i	$0.242119\pi$
$41$	8.82843i	1.37877i	−0.724396	+	0.689384i	$0.757881\pi$
$42$	−0.485281	−0.0748805
$43$	−3.07107	−0.468333	−0.234167	−	0.972196i	$-0.575236\pi$
$43$	−3.07107	−0.468333	−0.234167	+	0.972196i	$0.575236\pi$
$44$	1.07107i	0.161470i
$45$	1.00000i	0.149071i
$46$	− 0.585786i	− 0.0863695i
$47$	0.828427i	0.120839i	0.998173	+	0.0604193i	$0.0192438\pi$
$47$	0.828427i	0.120839i	−0.998173	+	0.0604193i	$0.980756\pi$
$48$	4.24264	0.612372
$49$	6.31371	0.901958
$50$	0.414214i	0.0585786i
$51$	6.82843	0.956171
$52$	0	0
$53$	−14.4853	−1.98971	−0.994853	−	0.101327i	$-0.967691\pi$
$53$	−14.4853	−1.98971	−0.994853	+	0.101327i	$0.967691\pi$
$54$	2.34315i	0.318862i
$55$	0.585786	0.0789874
$56$	−1.31371	−0.175552
$57$	− 4.82843i	− 0.639541i
$58$	− 2.34315i	− 0.307670i
$59$	10.2426i	1.33348i	0.745291	+	0.666739i	$0.232310\pi$
$59$	10.2426i	1.33348i	−0.745291	+	0.666739i	$0.767690\pi$
$60$	− 2.58579i	− 0.333824i
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.24264	−0.538816
$63$	0.828427i	0.104372i
$64$	4.17157	0.521447
$65$	0	0
$66$	0.343146	0.0422383
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	8.82843	1.07060
$69$	2.00000	0.240772
$70$	0.343146i	0.0410138i
$71$	7.89949i	0.937498i	0.883332	+	0.468749i	$0.155295\pi$
$71$	7.89949i	0.937498i	−0.883332	+	0.468749i	$0.844705\pi$
$72$	1.58579i	0.186887i
$73$	− 8.48528i	− 0.993127i	−0.868000	−	0.496564i	$-0.834595\pi$
$73$	− 8.48528i	− 0.993127i	0.868000	−	0.496564i	$-0.165405\pi$
$74$	3.51472	0.408578
$75$	−1.41421	−0.163299
$76$	− 6.24264i	− 0.716080i
$77$	0.485281	0.0553029
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	− 3.00000i	− 0.335410i
$81$	−5.00000	−0.555556
$82$	3.65685	0.403832
$83$	8.82843i	0.969046i	0.874779	+	0.484523i	$0.161007\pi$
$83$	8.82843i	0.969046i	−0.874779	+	0.484523i	$0.838993\pi$
$84$	− 2.14214i	− 0.233726i
$85$	− 4.82843i	− 0.523716i
$86$	1.27208i	0.137172i
$87$	8.00000	0.857690
$88$	0.928932	0.0990245
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	0.414214	0.0436619
$91$	0	0
$92$	2.58579	0.269587
$93$	− 14.4853i	− 1.50205i
$94$	0.343146	0.0353928
$95$	−3.41421	−0.350291
$96$	− 6.24264i	− 0.637137i
$97$	− 3.65685i	− 0.371297i	−0.982616	−	0.185649i	$-0.940561\pi$
$97$	− 3.65685i	− 0.371297i	0.982616	−	0.185649i	$-0.0594386\pi$
$98$	− 2.61522i	− 0.264177i
$99$	− 0.585786i	− 0.0588738i
$100$	−1.82843	−0.182843
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	− 2.82843i	− 0.280056i
$103$	−17.4142	−1.71587	−0.857937	−	0.513755i	$-0.828254\pi$
$103$	−17.4142	−1.71587	−0.857937	+	0.513755i	$0.828254\pi$
$104$	0	0
$105$	−1.17157	−0.114334
$106$	6.00000i	0.582772i
$107$	−6.58579	−0.636672	−0.318336	−	0.947978i	$-0.603124\pi$
$107$	−6.58579	−0.636672	−0.318336	+	0.947978i	$0.603124\pi$
$108$	−10.3431	−0.995270
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	− 0.242641i	− 0.0231349i
$111$	12.0000i	1.13899i
$112$	− 2.48528i	− 0.234837i
$113$	−3.17157	−0.298356	−0.149178	−	0.988810i	$-0.547663\pi$
$113$	−3.17157	−0.298356	−0.149178	+	0.988810i	$0.547663\pi$
$114$	−2.00000	−0.187317
$115$	− 1.41421i	− 0.131876i
$116$	10.3431	0.960337
$117$	0	0
$118$	4.24264	0.390567
$119$	− 4.00000i	− 0.366679i
$120$	−2.24264	−0.204724
$121$	10.6569	0.968805
$122$	3.31371i	0.300009i
$123$	12.4853i	1.12576i
$124$	− 18.7279i	− 1.68182i
$125$	1.00000i	0.0894427i
$126$	0.343146	0.0305699
$127$	9.41421	0.835376	0.417688	−	0.908590i	$-0.362840\pi$
$127$	9.41421	0.835376	0.417688	+	0.908590i	$0.362840\pi$
$128$	− 10.5563i	− 0.933058i
$129$	−4.34315	−0.382393
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	1.51472i	0.131839i
$133$	−2.82843	−0.245256
$134$	0.828427	0.0715652
$135$	5.65685i	0.486864i
$136$	− 7.65685i	− 0.656570i
$137$	5.31371i	0.453981i	0.973897	+	0.226990i	$0.0728886\pi$
$137$	5.31371i	0.453981i	−0.973897	+	0.226990i	$0.927111\pi$
$138$	− 0.828427i	− 0.0705204i
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	−1.51472	−0.128017
$141$	1.17157i	0.0986642i
$142$	3.27208	0.274587
$143$	0	0
$144$	−3.00000	−0.250000
$145$	− 5.65685i	− 0.469776i
$146$	−3.51472	−0.290880
$147$	8.92893	0.736446
$148$	15.5147i	1.27530i
$149$	0.343146i	0.0281116i	0.999901	+	0.0140558i	$0.00447425\pi$
$149$	0.343146i	0.0281116i	−0.999901	+	0.0140558i	$0.995526\pi$
$150$	0.585786i	0.0478293i
$151$	18.2426i	1.48457i	0.670087	+	0.742283i	$0.266257\pi$
$151$	18.2426i	1.48457i	−0.670087	+	0.742283i	$0.733743\pi$
$152$	−5.41421	−0.439151
$153$	−4.82843	−0.390355
$154$	− 0.201010i	− 0.0161979i
$155$	−10.2426	−0.822709
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	− 3.51472i	− 0.279616i
$159$	−20.4853	−1.62459
$160$	−4.41421	−0.348974
$161$	− 1.17157i	− 0.0923329i
$162$	2.07107i	0.162718i
$163$	− 14.9706i	− 1.17258i	−0.810099	−	0.586292i	$-0.800587\pi$
$163$	− 14.9706i	− 1.17258i	0.810099	−	0.586292i	$-0.199413\pi$
$164$	16.1421i	1.26049i
$165$	0.828427	0.0644930
$166$	3.65685	0.283827
$167$	− 8.82843i	− 0.683164i	−0.939852	−	0.341582i	$-0.889037\pi$
$167$	− 8.82843i	− 0.683164i	0.939852	−	0.341582i	$-0.110963\pi$
$168$	−1.85786	−0.143337
$169$	0	0
$170$	−2.00000	−0.153393
$171$	3.41421i	0.261091i
$172$	−5.61522	−0.428157
$173$	−11.1716	−0.849359	−0.424679	−	0.905344i	$-0.639613\pi$
$173$	−11.1716	−0.849359	−0.424679	+	0.905344i	$0.639613\pi$
$174$	− 3.31371i	− 0.251212i
$175$	0.828427i	0.0626232i
$176$	1.75736i	0.132466i
$177$	14.4853i	1.08878i
$178$	2.48528	0.186280
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	1.82843i	0.136283i
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−11.3137	−0.836333
$184$	− 2.24264i	− 0.165330i
$185$	8.48528	0.623850
$186$	−6.00000	−0.439941
$187$	2.82843i	0.206835i
$188$	1.51472i	0.110472i
$189$	4.68629i	0.340878i
$190$	1.41421i	0.102598i
$191$	13.6569	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$191$	13.6569	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$192$	5.89949	0.425759
$193$	15.6569i	1.12701i	0.826114	+	0.563503i	$0.190546\pi$
$193$	15.6569i	1.12701i	−0.826114	+	0.563503i	$0.809454\pi$
$194$	−1.51472	−0.108750
$195$	0	0
$196$	11.5442	0.824583
$197$	22.9706i	1.63658i	0.574802	+	0.818292i	$0.305079\pi$
$197$	22.9706i	1.63658i	−0.574802	+	0.818292i	$0.694921\pi$
$198$	−0.242641	−0.0172437
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.58579i	0.112132i
$201$	2.82843i	0.199502i
$202$	3.17157i	0.223151i
$203$	− 4.68629i	− 0.328913i
$204$	12.4853	0.874145
$205$	8.82843	0.616604
$206$	7.21320i	0.502568i
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	2.00000	0.138343
$210$	0.485281i	0.0334876i
$211$	−19.3137	−1.32961	−0.664805	−	0.747017i	$-0.731485\pi$
$211$	−19.3137	−1.32961	−0.664805	+	0.747017i	$0.731485\pi$
$212$	−26.4853	−1.81902
$213$	11.1716i	0.765464i
$214$	2.72792i	0.186477i
$215$	3.07107i	0.209445i
$216$	8.97056i	0.610369i
$217$	−8.48528	−0.576018
$218$	0.828427	0.0561082
$219$	− 12.0000i	− 0.810885i
$220$	1.07107	0.0722114
$221$	0	0
$222$	4.97056	0.333602
$223$	− 26.4853i	− 1.77359i	−0.462167	−	0.886793i	$-0.652928\pi$
$223$	− 26.4853i	− 1.77359i	0.462167	−	0.886793i	$-0.347072\pi$
$224$	−3.65685	−0.244334
$225$	1.00000	0.0666667
$226$	1.31371i	0.0873866i
$227$	27.6569i	1.83565i	0.396985	+	0.917825i	$0.370056\pi$
$227$	27.6569i	1.83565i	−0.396985	+	0.917825i	$0.629944\pi$
$228$	− 8.82843i	− 0.584677i
$229$	0.828427i	0.0547440i	0.999625	+	0.0273720i	$0.00871387\pi$
$229$	0.828427i	0.0547440i	−0.999625	+	0.0273720i	$0.991286\pi$
$230$	−0.585786	−0.0386256
$231$	0.686292	0.0451547
$232$	− 8.97056i	− 0.588946i
$233$	−24.6274	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$233$	−24.6274	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	18.7279i	1.21908i
$237$	12.0000	0.779484
$238$	−1.65685	−0.107398
$239$	0.585786i	0.0378914i	0.999821	+	0.0189457i	$0.00603096\pi$
$239$	0.585786i	0.0378914i	−0.999821	+	0.0189457i	$0.993969\pi$
$240$	− 4.24264i	− 0.273861i
$241$	2.48528i	0.160091i	0.996791	+	0.0800455i	$0.0255066\pi$
$241$	2.48528i	0.160091i	−0.996791	+	0.0800455i	$0.974493\pi$
$242$	− 4.41421i	− 0.283756i
$243$	9.89949	0.635053
$244$	−14.6274	−0.936424
$245$	− 6.31371i	− 0.403368i
$246$	5.17157	0.329727
$247$	0	0
$248$	−16.2426	−1.03141
$249$	12.4853i	0.791223i
$250$	0.414214	0.0261972
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	1.51472i	0.0954183i
$253$	0.828427i	0.0520828i
$254$	− 3.89949i	− 0.244676i
$255$	− 6.82843i	− 0.427613i
$256$	3.97056	0.248160
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	1.79899i	0.112000i
$259$	7.02944	0.436788
$260$	0	0
$261$	−5.65685	−0.350150
$262$	− 7.02944i	− 0.434280i
$263$	−13.4142	−0.827156	−0.413578	−	0.910469i	$-0.635721\pi$
$263$	−13.4142	−0.827156	−0.413578	+	0.910469i	$0.635721\pi$
$264$	1.31371	0.0808532
$265$	14.4853i	0.889824i
$266$	1.17157i	0.0718337i
$267$	8.48528i	0.519291i
$268$	3.65685i	0.223378i
$269$	−2.68629	−0.163786	−0.0818930	−	0.996641i	$-0.526097\pi$
$269$	−2.68629	−0.163786	−0.0818930	+	0.996641i	$0.526097\pi$
$270$	2.34315	0.142599
$271$	1.27208i	0.0772732i	0.999253	+	0.0386366i	$0.0123015\pi$
$271$	1.27208i	0.0772732i	−0.999253	+	0.0386366i	$0.987699\pi$
$272$	14.4853	0.878299
$273$	0	0
$274$	2.20101	0.132968
$275$	− 0.585786i	− 0.0353243i
$276$	3.65685	0.220117
$277$	7.17157	0.430898	0.215449	−	0.976515i	$-0.430878\pi$
$277$	7.17157	0.430898	0.215449	+	0.976515i	$0.430878\pi$
$278$	5.17157i	0.310170i
$279$	10.2426i	0.613211i
$280$	1.31371i	0.0785091i
$281$	− 17.7990i	− 1.06180i	−0.847435	−	0.530899i	$-0.821854\pi$
$281$	− 17.7990i	− 1.06180i	0.847435	−	0.530899i	$-0.178146\pi$
$282$	0.485281	0.0288981
$283$	−8.72792	−0.518821	−0.259411	−	0.965767i	$-0.583528\pi$
$283$	−8.72792	−0.518821	−0.259411	+	0.965767i	$0.583528\pi$
$284$	14.4437i	0.857073i
$285$	−4.82843	−0.286011
$286$	0	0
$287$	7.31371	0.431715
$288$	4.41421i	0.260110i
$289$	6.31371	0.371395
$290$	−2.34315	−0.137594
$291$	− 5.17157i	− 0.303163i
$292$	− 15.5147i	− 0.907930i
$293$	− 2.14214i	− 0.125145i	−0.998040	−	0.0625724i	$-0.980070\pi$
$293$	− 2.14214i	− 0.125145i	0.998040	−	0.0625724i	$-0.0199304\pi$
$294$	− 3.69848i	− 0.215700i
$295$	10.2426	0.596350
$296$	13.4558	0.782105
$297$	− 3.31371i	− 0.192281i
$298$	0.142136	0.00823370
$299$	0	0
$300$	−2.58579	−0.149290
$301$	2.54416i	0.146643i
$302$	7.55635	0.434819
$303$	−10.8284	−0.622077
$304$	− 10.2426i	− 0.587456i
$305$	8.00000i	0.458079i
$306$	2.00000i	0.114332i
$307$	19.1716i	1.09418i	0.837074	+	0.547090i	$0.184264\pi$
$307$	19.1716i	1.09418i	−0.837074	+	0.547090i	$0.815736\pi$
$308$	0.887302	0.0505587
$309$	−24.6274	−1.40100
$310$	4.24264i	0.240966i
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	0.828427	0.0468255	0.0234127	−	0.999726i	$-0.492547\pi$
$313$	0.828427	0.0468255	0.0234127	+	0.999726i	$0.492547\pi$
$314$	− 7.45584i	− 0.420758i
$315$	0.828427	0.0466766
$316$	15.5147	0.872771
$317$	− 26.1421i	− 1.46829i	−0.678993	−	0.734144i	$-0.737584\pi$
$317$	− 26.1421i	− 1.46829i	0.678993	−	0.734144i	$-0.262416\pi$
$318$	8.48528i	0.475831i
$319$	3.31371i	0.185532i
$320$	− 4.17157i	− 0.233198i
$321$	−9.31371	−0.519841
$322$	−0.485281	−0.0270437
$323$	− 16.4853i	− 0.917266i
$324$	−9.14214	−0.507896
$325$	0	0
$326$	−6.20101	−0.343442
$327$	2.82843i	0.156412i
$328$	14.0000	0.773021
$329$	0.686292	0.0378365
$330$	− 0.343146i	− 0.0188896i
$331$	− 22.0416i	− 1.21152i	−0.795648	−	0.605759i	$-0.792870\pi$
$331$	− 22.0416i	− 1.21152i	0.795648	−	0.605759i	$-0.207130\pi$
$332$	16.1421i	0.885915i
$333$	− 8.48528i	− 0.464991i
$334$	−3.65685	−0.200094
$335$	2.00000	0.109272
$336$	− 3.51472i	− 0.191744i
$337$	−7.17157	−0.390660	−0.195330	−	0.980738i	$-0.562578\pi$
$337$	−7.17157	−0.390660	−0.195330	+	0.980738i	$0.562578\pi$
$338$	0	0
$339$	−4.48528	−0.243607
$340$	− 8.82843i	− 0.478789i
$341$	6.00000	0.324918
$342$	1.41421	0.0764719
$343$	− 11.0294i	− 0.595534i
$344$	4.87006i	0.262576i
$345$	− 2.00000i	− 0.107676i
$346$	4.62742i	0.248771i
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	14.6274	0.784112
$349$	1.51472i	0.0810810i	0.999178	+	0.0405405i	$0.0129080\pi$
$349$	1.51472i	0.0810810i	−0.999178	+	0.0405405i	$0.987092\pi$
$350$	0.343146	0.0183419
$351$	0	0
$352$	2.58579	0.137823
$353$	− 9.17157i	− 0.488154i	−0.969756	−	0.244077i	$-0.921515\pi$
$353$	− 9.17157i	− 0.488154i	0.969756	−	0.244077i	$-0.0784849\pi$
$354$	6.00000	0.318896
$355$	7.89949	0.419262
$356$	10.9706i	0.581439i
$357$	− 5.65685i	− 0.299392i
$358$	− 2.34315i	− 0.123839i
$359$	− 27.8995i	− 1.47248i	−0.676721	−	0.736240i	$-0.736600\pi$
$359$	− 27.8995i	− 1.47248i	0.676721	−	0.736240i	$-0.263400\pi$
$360$	1.58579	0.0835783
$361$	7.34315	0.386481
$362$	0	0
$363$	15.0711	0.791026
$364$	0	0
$365$	−8.48528	−0.444140
$366$	4.68629i	0.244956i
$367$	4.44365	0.231957	0.115978	−	0.993252i	$-0.463000\pi$
$367$	4.44365	0.231957	0.115978	+	0.993252i	$0.463000\pi$
$368$	4.24264	0.221163
$369$	− 8.82843i	− 0.459590i
$370$	− 3.51472i	− 0.182722i
$371$	12.0000i	0.623009i
$372$	− 26.4853i	− 1.37320i
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	1.17157	0.0605806
$375$	1.41421i	0.0730297i
$376$	1.31371	0.0677493
$377$	0	0
$378$	1.94113	0.0998407
$379$	− 14.9289i	− 0.766848i	−0.923572	−	0.383424i	$-0.874745\pi$
$379$	− 14.9289i	− 0.766848i	0.923572	−	0.383424i	$-0.125255\pi$
$380$	−6.24264	−0.320241
$381$	13.3137	0.682082
$382$	− 5.65685i	− 0.289430i
$383$	− 33.1127i	− 1.69198i	−0.533199	−	0.845990i	$-0.679010\pi$
$383$	− 33.1127i	− 1.69198i	0.533199	−	0.845990i	$-0.320990\pi$
$384$	− 14.9289i	− 0.761839i
$385$	− 0.485281i	− 0.0247322i
$386$	6.48528	0.330092
$387$	3.07107	0.156111
$388$	− 6.68629i	− 0.339445i
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	6.82843	0.345328
$392$	− 10.0122i	− 0.505692i
$393$	24.0000	1.21064
$394$	9.51472	0.479345
$395$	− 8.48528i	− 0.426941i
$396$	− 1.07107i	− 0.0538232i
$397$	− 27.7990i	− 1.39519i	−0.716492	−	0.697596i	$-0.754253\pi$
$397$	− 27.7990i	− 1.39519i	0.716492	−	0.697596i	$-0.245747\pi$
$398$	1.65685i	0.0830506i
$399$	−4.00000	−0.200250
$400$	−3.00000	−0.150000
$401$	17.3137i	0.864605i	0.901729	+	0.432303i	$0.142299\pi$
$401$	17.3137i	0.864605i	−0.901729	+	0.432303i	$0.857701\pi$
$402$	1.17157	0.0584327
$403$	0	0
$404$	−14.0000	−0.696526
$405$	5.00000i	0.248452i
$406$	−1.94113	−0.0963364
$407$	−4.97056	−0.246382
$408$	− 10.8284i	− 0.536087i
$409$	− 12.8284i	− 0.634325i	−0.948371	−	0.317162i	$-0.897270\pi$
$409$	− 12.8284i	− 0.634325i	0.948371	−	0.317162i	$-0.102730\pi$
$410$	− 3.65685i	− 0.180599i
$411$	7.51472i	0.370674i
$412$	−31.8406	−1.56867
$413$	8.48528	0.417533
$414$	0.585786i	0.0287898i
$415$	8.82843	0.433370
$416$	0	0
$417$	−17.6569	−0.864660
$418$	− 0.828427i	− 0.0405197i
$419$	5.17157	0.252648	0.126324	−	0.991989i	$-0.459682\pi$
$419$	5.17157	0.252648	0.126324	+	0.991989i	$0.459682\pi$
$420$	−2.14214	−0.104526
$421$	1.02944i	0.0501717i	0.999685	+	0.0250859i	$0.00798591\pi$
$421$	1.02944i	0.0501717i	−0.999685	+	0.0250859i	$0.992014\pi$
$422$	8.00000i	0.389434i
$423$	− 0.828427i	− 0.0402795i
$424$	22.9706i	1.11555i
$425$	−4.82843	−0.234213
$426$	4.62742	0.224199
$427$	6.62742i	0.320723i
$428$	−12.0416	−0.582054
$429$	0	0
$430$	1.27208	0.0613450
$431$	− 3.61522i	− 0.174139i	−0.996202	−	0.0870696i	$-0.972250\pi$
$431$	− 3.61522i	− 0.174139i	0.996202	−	0.0870696i	$-0.0277503\pi$
$432$	−16.9706	−0.816497
$433$	3.65685	0.175737	0.0878686	−	0.996132i	$-0.471994\pi$
$433$	3.65685	0.175737	0.0878686	+	0.996132i	$0.471994\pi$
$434$	3.51472i	0.168712i
$435$	− 8.00000i	− 0.383571i
$436$	3.65685i	0.175132i
$437$	− 4.82843i	− 0.230975i
$438$	−4.97056	−0.237503
$439$	32.9706	1.57360	0.786800	−	0.617209i	$-0.211737\pi$
$439$	32.9706	1.57360	0.786800	+	0.617209i	$0.211737\pi$
$440$	− 0.928932i	− 0.0442851i
$441$	−6.31371	−0.300653
$442$	0	0
$443$	−6.58579	−0.312900	−0.156450	−	0.987686i	$-0.550005\pi$
$443$	−6.58579	−0.312900	−0.156450	+	0.987686i	$0.550005\pi$
$444$	21.9411i	1.04128i
$445$	6.00000	0.284427
$446$	−10.9706	−0.519471
$447$	0.485281i	0.0229530i
$448$	− 3.45584i	− 0.163273i
$449$	29.1127i	1.37391i	0.726698	+	0.686957i	$0.241054\pi$
$449$	29.1127i	1.37391i	−0.726698	+	0.686957i	$0.758946\pi$
$450$	− 0.414214i	− 0.0195262i
$451$	−5.17157	−0.243520
$452$	−5.79899	−0.272762
$453$	25.7990i	1.21214i
$454$	11.4558	0.537649
$455$	0	0
$456$	−7.65685	−0.358565
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	0.343146	0.0160341
$459$	−27.3137	−1.27489
$460$	− 2.58579i	− 0.120563i
$461$	− 26.4853i	− 1.23354i	−0.787142	−	0.616771i	$-0.788440\pi$
$461$	− 26.4853i	− 1.23354i	0.787142	−	0.616771i	$-0.211560\pi$
$462$	− 0.284271i	− 0.0132255i
$463$	− 15.6569i	− 0.727636i	−0.931470	−	0.363818i	$-0.881473\pi$
$463$	− 15.6569i	− 0.727636i	0.931470	−	0.363818i	$-0.118527\pi$
$464$	16.9706	0.787839
$465$	−14.4853	−0.671739
$466$	10.2010i	0.472553i
$467$	10.5858	0.489852	0.244926	−	0.969542i	$-0.421236\pi$
$467$	10.5858	0.489852	0.244926	+	0.969542i	$0.421236\pi$
$468$	0	0
$469$	1.65685	0.0765064
$470$	− 0.343146i	− 0.0158281i
$471$	25.4558	1.17294
$472$	16.2426	0.747628
$473$	− 1.79899i	− 0.0827176i
$474$	− 4.97056i	− 0.228306i
$475$	3.41421i	0.156655i
$476$	− 7.31371i	− 0.335223i
$477$	14.4853	0.663235
$478$	0.242641	0.0110981
$479$	− 5.27208i	− 0.240887i	−0.992720	−	0.120444i	$-0.961568\pi$
$479$	− 5.27208i	− 0.240887i	0.992720	−	0.120444i	$-0.0384317\pi$
$480$	−6.24264	−0.284936
$481$	0	0
$482$	1.02944	0.0468896
$483$	− 1.65685i	− 0.0753895i
$484$	19.4853	0.885695
$485$	−3.65685	−0.166049
$486$	− 4.10051i	− 0.186003i
$487$	− 22.9706i	− 1.04090i	−0.853894	−	0.520448i	$-0.825765\pi$
$487$	− 22.9706i	− 1.04090i	0.853894	−	0.520448i	$-0.174235\pi$
$488$	12.6863i	0.574281i
$489$	− 21.1716i	− 0.957412i
$490$	−2.61522	−0.118144
$491$	−10.8284	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$491$	−10.8284	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$492$	22.8284i	1.02918i
$493$	27.3137	1.23015
$494$	0	0
$495$	−0.585786	−0.0263291
$496$	− 30.7279i	− 1.37972i
$497$	6.54416	0.293546
$498$	5.17157	0.231744
$499$	− 10.4437i	− 0.467522i	−0.972294	−	0.233761i	$-0.924897\pi$
$499$	− 10.4437i	− 0.467522i	0.972294	−	0.233761i	$-0.0751033\pi$
$500$	1.82843i	0.0817697i
$501$	− 12.4853i	− 0.557801i
$502$	− 8.20101i	− 0.366029i
$503$	18.1005	0.807062	0.403531	−	0.914966i	$-0.367783\pi$
$503$	18.1005	0.807062	0.403531	+	0.914966i	$0.367783\pi$
$504$	1.31371	0.0585172
$505$	7.65685i	0.340726i
$506$	0.343146	0.0152547
$507$	0	0
$508$	17.2132	0.763712
$509$	21.1127i	0.935804i	0.883780	+	0.467902i	$0.154990\pi$
$509$	21.1127i	0.935804i	−0.883780	+	0.467902i	$0.845010\pi$
$510$	−2.82843	−0.125245
$511$	−7.02944	−0.310964
$512$	− 22.7574i	− 1.00574i
$513$	19.3137i	0.852721i
$514$	6.76955i	0.298592i
$515$	17.4142i	0.767362i
$516$	−7.94113	−0.349589
$517$	−0.485281	−0.0213427
$518$	− 2.91169i	− 0.127932i
$519$	−15.7990	−0.693499
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	2.34315i	0.102557i
$523$	−28.2426	−1.23496	−0.617482	−	0.786585i	$-0.711847\pi$
$523$	−28.2426	−1.23496	−0.617482	+	0.786585i	$0.711847\pi$
$524$	31.0294	1.35553
$525$	1.17157i	0.0511316i
$526$	5.55635i	0.242268i
$527$	− 49.4558i	− 2.15433i
$528$	2.48528i	0.108158i
$529$	−21.0000	−0.913043
$530$	6.00000	0.260623
$531$	− 10.2426i	− 0.444493i
$532$	−5.17157	−0.224216
$533$	0	0
$534$	3.51472	0.152097
$535$	6.58579i	0.284728i
$536$	3.17157	0.136991
$537$	8.00000	0.345225
$538$	1.11270i	0.0479718i
$539$	3.69848i	0.159305i
$540$	10.3431i	0.445098i
$541$	− 12.8284i	− 0.551537i	−0.961224	−	0.275769i	$-0.911068\pi$
$541$	− 12.8284i	− 0.551537i	0.961224	−	0.275769i	$-0.0889323\pi$
$542$	0.526912	0.0226328
$543$	0	0
$544$	− 21.3137i	− 0.913818i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−29.2132	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$547$	−29.2132	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$548$	9.71573i	0.415035i
$549$	8.00000	0.341432
$550$	−0.242641	−0.0103462
$551$	− 19.3137i	− 0.822792i
$552$	− 3.17157i	− 0.134991i
$553$	− 7.02944i	− 0.298922i
$554$	− 2.97056i	− 0.126207i
$555$	12.0000	0.509372
$556$	−22.8284	−0.968141
$557$	3.79899i	0.160968i	0.996756	+	0.0804842i	$0.0256467\pi$
$557$	3.79899i	0.160968i	−0.996756	+	0.0804842i	$0.974353\pi$
$558$	4.24264	0.179605
$559$	0	0
$560$	−2.48528	−0.105022
$561$	4.00000i	0.168880i
$562$	−7.37258	−0.310994
$563$	16.2426	0.684546	0.342273	−	0.939601i	$-0.388803\pi$
$563$	16.2426	0.684546	0.342273	+	0.939601i	$0.388803\pi$
$564$	2.14214i	0.0902002i
$565$	3.17157i	0.133429i
$566$	3.61522i	0.151959i
$567$	4.14214i	0.173953i
$568$	12.5269	0.525618
$569$	21.6569	0.907903	0.453951	−	0.891027i	$-0.350014\pi$
$569$	21.6569	0.907903	0.453951	+	0.891027i	$0.350014\pi$
$570$	2.00000i	0.0837708i
$571$	28.4853	1.19207	0.596036	−	0.802958i	$-0.296742\pi$
$571$	28.4853	1.19207	0.596036	+	0.802958i	$0.296742\pi$
$572$	0	0
$573$	19.3137	0.806842
$574$	− 3.02944i	− 0.126446i
$575$	−1.41421	−0.0589768
$576$	−4.17157	−0.173816
$577$	29.1716i	1.21443i	0.794538	+	0.607214i	$0.207713\pi$
$577$	29.1716i	1.21443i	−0.794538	+	0.607214i	$0.792287\pi$
$578$	− 2.61522i	− 0.108779i
$579$	22.1421i	0.920196i
$580$	− 10.3431i	− 0.429476i
$581$	7.31371	0.303424
$582$	−2.14214	−0.0887944
$583$	− 8.48528i	− 0.351424i
$584$	−13.4558	−0.556807
$585$	0	0
$586$	−0.887302	−0.0366541
$587$	− 31.6569i	− 1.30662i	−0.757091	−	0.653309i	$-0.773380\pi$
$587$	− 31.6569i	− 1.30662i	0.757091	−	0.653309i	$-0.226620\pi$
$588$	16.3259	0.673269
$589$	−34.9706	−1.44094
$590$	− 4.24264i	− 0.174667i
$591$	32.4853i	1.33627i
$592$	25.4558i	1.04623i
$593$	20.6274i	0.847066i	0.905881	+	0.423533i	$0.139210\pi$
$593$	20.6274i	0.847066i	−0.905881	+	0.423533i	$0.860790\pi$
$594$	−1.37258	−0.0563178
$595$	−4.00000	−0.163984
$596$	0.627417i	0.0257000i
$597$	−5.65685	−0.231520
$598$	0	0
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	2.24264i	0.0915554i
$601$	0.627417	0.0255929	0.0127964	−	0.999918i	$-0.495927\pi$
$601$	0.627417	0.0255929	0.0127964	+	0.999918i	$0.495927\pi$
$602$	1.05382	0.0429507
$603$	− 2.00000i	− 0.0814463i
$604$	33.3553i	1.35721i
$605$	− 10.6569i	− 0.433263i
$606$	4.48528i	0.182202i
$607$	40.2426	1.63340	0.816699	−	0.577064i	$-0.195802\pi$
$607$	40.2426	1.63340	0.816699	+	0.577064i	$0.195802\pi$
$608$	−15.0711	−0.611213
$609$	− 6.62742i	− 0.268556i
$610$	3.31371	0.134168
$611$	0	0
$612$	−8.82843	−0.356868
$613$	37.3137i	1.50709i	0.657398	+	0.753543i	$0.271657\pi$
$613$	37.3137i	1.50709i	−0.657398	+	0.753543i	$0.728343\pi$
$614$	7.94113	0.320478
$615$	12.4853	0.503455
$616$	− 0.769553i	− 0.0310062i
$617$	22.9706i	0.924760i	0.886682	+	0.462380i	$0.153004\pi$
$617$	22.9706i	0.924760i	−0.886682	+	0.462380i	$0.846996\pi$
$618$	10.2010i	0.410345i
$619$	10.2426i	0.411686i	0.978585	+	0.205843i	$0.0659937\pi$
$619$	10.2426i	0.411686i	−0.978585	+	0.205843i	$0.934006\pi$
$620$	−18.7279	−0.752131
$621$	−8.00000	−0.321029
$622$	− 3.51472i	− 0.140927i
$623$	4.97056	0.199141
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 0.343146i	− 0.0137149i
$627$	2.82843	0.112956
$628$	32.9117	1.31332
$629$	40.9706i	1.63360i
$630$	− 0.343146i	− 0.0136713i
$631$	− 18.2426i	− 0.726228i	−0.931745	−	0.363114i	$-0.881714\pi$
$631$	− 18.2426i	− 0.726228i	0.931745	−	0.363114i	$-0.118286\pi$
$632$	− 13.4558i	− 0.535245i
$633$	−27.3137	−1.08562
$634$	−10.8284	−0.430052
$635$	− 9.41421i	− 0.373592i
$636$	−37.4558	−1.48522
$637$	0	0
$638$	1.37258	0.0543411
$639$	− 7.89949i	− 0.312499i
$640$	−10.5563	−0.417276
$641$	−36.3431	−1.43547	−0.717734	−	0.696317i	$-0.754821\pi$
$641$	−36.3431	−1.43547	−0.717734	+	0.696317i	$0.754821\pi$
$642$	3.85786i	0.152258i
$643$	− 26.4853i	− 1.04448i	−0.852799	−	0.522239i	$-0.825097\pi$
$643$	− 26.4853i	− 1.04448i	0.852799	−	0.522239i	$-0.174903\pi$
$644$	− 2.14214i	− 0.0844120i
$645$	4.34315i	0.171011i
$646$	−6.82843	−0.268661
$647$	−6.58579	−0.258914	−0.129457	−	0.991585i	$-0.541323\pi$
$647$	−6.58579	−0.258914	−0.129457	+	0.991585i	$0.541323\pi$
$648$	7.92893i	0.311478i
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−12.0000	−0.470317
$652$	− 27.3726i	− 1.07199i
$653$	13.0294	0.509881	0.254941	−	0.966957i	$-0.417944\pi$
$653$	13.0294	0.509881	0.254941	+	0.966957i	$0.417944\pi$
$654$	1.17157	0.0458121
$655$	− 16.9706i	− 0.663095i
$656$	26.4853i	1.03408i
$657$	8.48528i	0.331042i
$658$	− 0.284271i	− 0.0110820i
$659$	46.1421	1.79744	0.898721	−	0.438520i	$-0.144497\pi$
$659$	46.1421	1.79744	0.898721	+	0.438520i	$0.144497\pi$
$660$	1.51472	0.0589603
$661$	− 49.5980i	− 1.92914i	−0.263831	−	0.964569i	$-0.584986\pi$
$661$	− 49.5980i	− 1.92914i	0.263831	−	0.964569i	$-0.415014\pi$
$662$	−9.12994	−0.354845
$663$	0	0
$664$	14.0000	0.543305
$665$	2.82843i	0.109682i
$666$	−3.51472	−0.136193
$667$	8.00000	0.309761
$668$	− 16.1421i	− 0.624558i
$669$	− 37.4558i	− 1.44813i
$670$	− 0.828427i	− 0.0320049i
$671$	− 4.68629i	− 0.180912i
$672$	−5.17157	−0.199498
$673$	10.4853	0.404178	0.202089	−	0.979367i	$-0.435227\pi$
$673$	10.4853	0.404178	0.202089	+	0.979367i	$0.435227\pi$
$674$	2.97056i	0.114422i
$675$	5.65685	0.217732
$676$	0	0
$677$	8.14214	0.312928	0.156464	−	0.987684i	$-0.449991\pi$
$677$	8.14214	0.312928	0.156464	+	0.987684i	$0.449991\pi$
$678$	1.85786i	0.0713509i
$679$	−3.02944	−0.116259
$680$	−7.65685	−0.293627
$681$	39.1127i	1.49880i
$682$	− 2.48528i	− 0.0951663i
$683$	33.3137i	1.27471i	0.770569	+	0.637357i	$0.219972\pi$
$683$	33.3137i	1.27471i	−0.770569	+	0.637357i	$0.780028\pi$
$684$	6.24264i	0.238693i
$685$	5.31371	0.203026
$686$	−4.56854	−0.174428
$687$	1.17157i	0.0446983i
$688$	−9.21320	−0.351250
$689$	0	0
$690$	−0.828427	−0.0315377
$691$	− 21.0711i	− 0.801581i	−0.916170	−	0.400791i	$-0.868735\pi$
$691$	− 21.0711i	− 0.801581i	0.916170	−	0.400791i	$-0.131265\pi$
$692$	−20.4264	−0.776495
$693$	−0.485281	−0.0184343
$694$	− 1.75736i	− 0.0667084i
$695$	12.4853i	0.473594i
$696$	− 12.6863i	− 0.480873i
$697$	42.6274i	1.61463i
$698$	0.627417	0.0237481
$699$	−34.8284	−1.31733
$700$	1.51472i	0.0572510i
$701$	−37.3137	−1.40932	−0.704660	−	0.709545i	$-0.748900\pi$
$701$	−37.3137	−1.40932	−0.704660	+	0.709545i	$0.748900\pi$
$702$	0	0
$703$	28.9706	1.09265
$704$	2.44365i	0.0920986i
$705$	1.17157	0.0441240
$706$	−3.79899	−0.142977
$707$	6.34315i	0.238559i
$708$	26.4853i	0.995378i
$709$	− 17.1127i	− 0.642681i	−0.946964	−	0.321340i	$-0.895867\pi$
$709$	− 17.1127i	− 0.642681i	0.946964	−	0.321340i	$-0.104133\pi$
$710$	− 3.27208i	− 0.122799i
$711$	−8.48528	−0.318223
$712$	9.51472	0.356579
$713$	− 14.4853i	− 0.542478i
$714$	−2.34315	−0.0876900
$715$	0	0
$716$	10.3431	0.386542
$717$	0.828427i	0.0309382i
$718$	−11.5563	−0.431279
$719$	4.97056	0.185371	0.0926854	−	0.995695i	$-0.470455\pi$
$719$	4.97056	0.185371	0.0926854	+	0.995695i	$0.470455\pi$
$720$	3.00000i	0.111803i
$721$	14.4264i	0.537267i
$722$	− 3.04163i	− 0.113198i
$723$	3.51472i	0.130714i
$724$	0	0
$725$	−5.65685	−0.210090
$726$	− 6.24264i	− 0.231686i
$727$	−19.3553	−0.717850	−0.358925	−	0.933366i	$-0.616857\pi$
$727$	−19.3553	−0.717850	−0.358925	+	0.933366i	$0.616857\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	3.51472i	0.130086i
$731$	−14.8284	−0.548449
$732$	−20.6863	−0.764587
$733$	1.31371i	0.0485229i	0.999706	+	0.0242615i	$0.00772342\pi$
$733$	1.31371i	0.0485229i	−0.999706	+	0.0242615i	$0.992277\pi$
$734$	− 1.84062i	− 0.0679385i
$735$	− 8.92893i	− 0.329349i
$736$	− 6.24264i	− 0.230107i
$737$	−1.17157	−0.0431554
$738$	−3.65685	−0.134611
$739$	30.7279i	1.13034i	0.824973	+	0.565172i	$0.191190\pi$
$739$	30.7279i	1.13034i	−0.824973	+	0.565172i	$0.808810\pi$
$740$	15.5147	0.570332
$741$	0	0
$742$	4.97056	0.182475
$743$	38.4853i	1.41189i	0.708268	+	0.705944i	$0.249477\pi$
$743$	38.4853i	1.41189i	−0.708268	+	0.705944i	$0.750523\pi$
$744$	−22.9706	−0.842142
$745$	0.343146	0.0125719
$746$	10.4853i	0.383893i
$747$	− 8.82843i	− 0.323015i
$748$	5.17157i	0.189091i
$749$	5.45584i	0.199352i
$750$	0.585786	0.0213899
$751$	44.4853	1.62329	0.811645	−	0.584150i	$-0.198572\pi$
$751$	44.4853	1.62329	0.811645	+	0.584150i	$0.198572\pi$
$752$	2.48528i	0.0906289i
$753$	28.0000	1.02038
$754$	0	0
$755$	18.2426	0.663918
$756$	8.56854i	0.311635i
$757$	4.14214	0.150548	0.0752742	−	0.997163i	$-0.476017\pi$
$757$	4.14214	0.150548	0.0752742	+	0.997163i	$0.476017\pi$
$758$	−6.18377	−0.224605
$759$	1.17157i	0.0425254i
$760$	5.41421i	0.196394i
$761$	− 36.6274i	− 1.32774i	−0.747847	−	0.663871i	$-0.768912\pi$
$761$	− 36.6274i	− 1.32774i	0.747847	−	0.663871i	$-0.231088\pi$
$762$	− 5.51472i	− 0.199777i
$763$	1.65685	0.0599822
$764$	24.9706	0.903403
$765$	4.82843i	0.174572i
$766$	−13.7157	−0.495569
$767$	0	0
$768$	5.61522	0.202622
$769$	− 10.9706i	− 0.395609i	−0.980242	−	0.197804i	$-0.936619\pi$
$769$	− 10.9706i	− 0.395609i	0.980242	−	0.197804i	$-0.0633811\pi$
$770$	−0.201010	−0.00724390
$771$	−23.1127	−0.832383
$772$	28.6274i	1.03032i
$773$	− 6.14214i	− 0.220917i	−0.993881	−	0.110459i	$-0.964768\pi$
$773$	− 6.14214i	− 0.220917i	0.993881	−	0.110459i	$-0.0352320\pi$
$774$	− 1.27208i	− 0.0457239i
$775$	10.2426i	0.367927i
$776$	−5.79899	−0.208172
$777$	9.94113	0.356636
$778$	− 6.88730i	− 0.246922i
$779$	30.1421	1.07995
$780$	0	0
$781$	−4.62742	−0.165582
$782$	− 2.82843i	− 0.101144i
$783$	−32.0000	−1.14359
$784$	18.9411	0.676469
$785$	− 18.0000i	− 0.642448i
$786$	− 9.94113i	− 0.354588i
$787$	5.51472i	0.196578i	0.995158	+	0.0982892i	$0.0313370\pi$
$787$	5.51472i	0.196578i	−0.995158	+	0.0982892i	$0.968663\pi$
$788$	42.0000i	1.49619i
$789$	−18.9706	−0.675370
$790$	−3.51472	−0.125048
$791$	2.62742i	0.0934202i
$792$	−0.928932	−0.0330082
$793$	0	0
$794$	−11.5147	−0.408642
$795$	20.4853i	0.726538i
$796$	−7.31371	−0.259228
$797$	−10.9706	−0.388597	−0.194299	−	0.980942i	$-0.562243\pi$
$797$	−10.9706	−0.388597	−0.194299	+	0.980942i	$0.562243\pi$
$798$	1.65685i	0.0586520i
$799$	4.00000i	0.141510i
$800$	4.41421i	0.156066i
$801$	− 6.00000i	− 0.212000i
$802$	7.17157	0.253237
$803$	4.97056	0.175407
$804$	5.17157i	0.182387i
$805$	−1.17157	−0.0412925
$806$	0	0
$807$	−3.79899	−0.133731
$808$	12.1421i	0.427159i
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	2.07107	0.0727699
$811$	− 8.38478i	− 0.294429i	−0.989105	−	0.147215i	$-0.952969\pi$
$811$	− 8.38478i	− 0.294429i	0.989105	−	0.147215i	$-0.0470308\pi$
$812$	− 8.56854i	− 0.300697i
$813$	1.79899i	0.0630933i
$814$	2.05887i	0.0721635i
$815$	−14.9706	−0.524396
$816$	20.4853	0.717128
$817$	10.4853i	0.366834i
$818$	−5.31371	−0.185789
$819$	0	0
$820$	16.1421	0.563708
$821$	− 39.2548i	− 1.37000i	−0.728542	−	0.685002i	$-0.759801\pi$
$821$	− 39.2548i	− 1.37000i	0.728542	−	0.685002i	$-0.240199\pi$
$822$	3.11270	0.108568
$823$	−34.3848	−1.19858	−0.599289	−	0.800533i	$-0.704550\pi$
$823$	−34.3848	−1.19858	−0.599289	+	0.800533i	$0.704550\pi$
$824$	27.6152i	0.962022i
$825$	− 0.828427i	− 0.0288421i
$826$	− 3.51472i	− 0.122293i
$827$	27.8579i	0.968713i	0.874871	+	0.484356i	$0.160946\pi$
$827$	27.8579i	0.968713i	−0.874871	+	0.484356i	$0.839054\pi$
$828$	−2.58579	−0.0898623
$829$	−7.02944	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$829$	−7.02944	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$830$	− 3.65685i	− 0.126931i
$831$	10.1421	0.351827
$832$	0	0
$833$	30.4853	1.05625
$834$	7.31371i	0.253253i
$835$	−8.82843	−0.305520
$836$	3.65685	0.126475
$837$	57.9411i	2.00274i
$838$	− 2.14214i	− 0.0739988i
$839$	− 18.7279i	− 0.646560i	−0.946303	−	0.323280i	$-0.895214\pi$
$839$	− 18.7279i	− 0.646560i	0.946303	−	0.323280i	$-0.104786\pi$
$840$	1.85786i	0.0641024i
$841$	3.00000	0.103448
$842$	0.426407	0.0146950
$843$	− 25.1716i	− 0.866955i
$844$	−35.3137	−1.21555
$845$	0	0
$846$	−0.343146	−0.0117976
$847$	− 8.82843i	− 0.303348i
$848$	−43.4558	−1.49228
$849$	−12.3431	−0.423616
$850$	2.00000i	0.0685994i
$851$	12.0000i	0.411355i
$852$	20.4264i	0.699797i
$853$	37.4558i	1.28246i	0.767347	+	0.641232i	$0.221576\pi$
$853$	37.4558i	1.28246i	−0.767347	+	0.641232i	$0.778424\pi$
$854$	2.74517	0.0939376
$855$	3.41421	0.116764
$856$	10.4437i	0.356957i
$857$	−0.343146	−0.0117216	−0.00586082	−	0.999983i	$-0.501866\pi$
$857$	−0.343146	−0.0117216	−0.00586082	+	0.999983i	$0.501866\pi$
$858$	0	0
$859$	11.7990	0.402576	0.201288	−	0.979532i	$-0.435487\pi$
$859$	11.7990	0.402576	0.201288	+	0.979532i	$0.435487\pi$
$860$	5.61522i	0.191478i
$861$	10.3431	0.352493
$862$	−1.49747	−0.0510042
$863$	− 19.4558i	− 0.662285i	−0.943581	−	0.331142i	$-0.892566\pi$
$863$	− 19.4558i	− 0.662285i	0.943581	−	0.331142i	$-0.107434\pi$
$864$	24.9706i	0.849516i
$865$	11.1716i	0.379845i
$866$	− 1.51472i	− 0.0514722i
$867$	8.92893	0.303242
$868$	−15.5147	−0.526604
$869$	4.97056i	0.168615i
$870$	−3.31371	−0.112345
$871$	0	0
$872$	3.17157	0.107403
$873$	3.65685i	0.123766i
$874$	−2.00000	−0.0676510
$875$	0.828427	0.0280059
$876$	− 21.9411i	− 0.741322i
$877$	− 2.68629i	− 0.0907096i	−0.998971	−	0.0453548i	$-0.985558\pi$
$877$	− 2.68629i	− 0.0907096i	0.998971	−	0.0453548i	$-0.0144418\pi$
$878$	− 13.6569i	− 0.460897i
$879$	− 3.02944i	− 0.102180i
$880$	1.75736	0.0592406
$881$	−52.9706	−1.78462	−0.892312	−	0.451420i	$-0.850918\pi$
$881$	−52.9706	−1.78462	−0.892312	+	0.451420i	$0.850918\pi$
$882$	2.61522i	0.0880592i
$883$	32.2426	1.08505	0.542526	−	0.840039i	$-0.317468\pi$
$883$	32.2426	1.08505	0.542526	+	0.840039i	$0.317468\pi$
$884$	0	0
$885$	14.4853	0.486917
$886$	2.72792i	0.0916463i
$887$	14.3848	0.482994	0.241497	−	0.970402i	$-0.422362\pi$
$887$	14.3848	0.482994	0.241497	+	0.970402i	$0.422362\pi$
$888$	19.0294	0.638586
$889$	− 7.79899i	− 0.261570i
$890$	− 2.48528i	− 0.0833068i
$891$	− 2.92893i	− 0.0981229i
$892$	− 48.4264i	− 1.62144i
$893$	2.82843	0.0946497
$894$	0.201010	0.00672278
$895$	− 5.65685i	− 0.189088i
$896$	−8.74517	−0.292155
$897$	0	0
$898$	12.0589	0.402410
$899$	− 57.9411i	− 1.93244i
$900$	1.82843	0.0609476
$901$	−69.9411	−2.33008
$902$	2.14214i	0.0713253i
$903$	3.59798i	0.119733i
$904$	5.02944i	0.167277i
$905$	0	0
$906$	10.6863	0.355028
$907$	33.2132	1.10283	0.551413	−	0.834232i	$-0.314089\pi$
$907$	33.2132	1.10283	0.551413	+	0.834232i	$0.314089\pi$
$908$	50.5685i	1.67818i
$909$	7.65685	0.253962
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 14.4853i	− 0.479656i
$913$	−5.17157	−0.171154
$914$	7.45584	0.246617
$915$	11.3137i	0.374020i
$916$	1.51472i	0.0500477i
$917$	− 14.0589i	− 0.464265i
$918$	11.3137i	0.373408i
$919$	−16.4853	−0.543799	−0.271900	−	0.962326i	$-0.587652\pi$
$919$	−16.4853	−0.543799	−0.271900	+	0.962326i	$0.587652\pi$
$920$	−2.24264	−0.0739377
$921$	27.1127i	0.893394i
$922$	−10.9706	−0.361296
$923$	0	0
$924$	1.25483	0.0412810
$925$	− 8.48528i	− 0.278994i
$926$	−6.48528	−0.213120
$927$	17.4142	0.571958
$928$	− 24.9706i	− 0.819699i
$929$	− 11.1716i	− 0.366527i	−0.983064	−	0.183264i	$-0.941334\pi$
$929$	− 11.1716i	− 0.366527i	0.983064	−	0.183264i	$-0.0586662\pi$
$930$	6.00000i	0.196748i
$931$	− 21.5563i	− 0.706481i
$932$	−45.0294	−1.47499
$933$	12.0000	0.392862
$934$	− 4.38478i	− 0.143474i
$935$	2.82843	0.0924995
$936$	0	0
$937$	10.9706	0.358393	0.179196	−	0.983813i	$-0.442650\pi$
$937$	10.9706	0.358393	0.179196	+	0.983813i	$0.442650\pi$
$938$	− 0.686292i	− 0.0224082i
$939$	1.17157	0.0382328
$940$	1.51472	0.0494047
$941$	54.7696i	1.78544i	0.450615	+	0.892718i	$0.351205\pi$
$941$	54.7696i	1.78544i	−0.450615	+	0.892718i	$0.648795\pi$
$942$	− 10.5442i	− 0.343547i
$943$	12.4853i	0.406577i
$944$	30.7279i	1.00011i
$945$	4.68629	0.152445
$946$	−0.745166	−0.0242274
$947$	− 45.1127i	− 1.46597i	−0.680247	−	0.732983i	$-0.738128\pi$
$947$	− 45.1127i	− 1.46597i	0.680247	−	0.732983i	$-0.261872\pi$
$948$	21.9411	0.712615
$949$	0	0
$950$	1.41421	0.0458831
$951$	− 36.9706i	− 1.19885i
$952$	−6.34315	−0.205583
$953$	55.2548	1.78988	0.894940	−	0.446187i	$-0.147218\pi$
$953$	55.2548	1.78988	0.894940	+	0.446187i	$0.147218\pi$
$954$	− 6.00000i	− 0.194257i
$955$	− 13.6569i	− 0.441925i
$956$	1.07107i	0.0346408i
$957$	4.68629i	0.151486i
$958$	−2.18377	−0.0705543
$959$	4.40202	0.142149
$960$	− 5.89949i	− 0.190405i
$961$	−73.9117	−2.38425
$962$	0	0
$963$	6.58579	0.212224
$964$	4.54416i	0.146357i
$965$	15.6569	0.504012
$966$	−0.686292	−0.0220811
$967$	19.9411i	0.641263i	0.947204	+	0.320632i	$0.103895\pi$
$967$	19.9411i	0.641263i	−0.947204	+	0.320632i	$0.896105\pi$
$968$	− 16.8995i	− 0.543170i
$969$	− 23.3137i	− 0.748944i
$970$	1.51472i	0.0486347i
$971$	−12.2843	−0.394221	−0.197111	−	0.980381i	$-0.563156\pi$
$971$	−12.2843	−0.394221	−0.197111	+	0.980381i	$0.563156\pi$
$972$	18.1005	0.580574
$973$	10.3431i	0.331586i
$974$	−9.51472	−0.304871
$975$	0	0
$976$	−24.0000	−0.768221
$977$	56.4853i	1.80712i	0.428457	+	0.903562i	$0.359057\pi$
$977$	56.4853i	1.80712i	−0.428457	+	0.903562i	$0.640943\pi$
$978$	−8.76955	−0.280419
$979$	−3.51472	−0.112331
$980$	− 11.5442i	− 0.368765i
$981$	− 2.00000i	− 0.0638551i
$982$	4.48528i	0.143131i
$983$	− 34.9706i	− 1.11539i	−0.830047	−	0.557694i	$-0.811686\pi$
$983$	− 34.9706i	− 1.11539i	0.830047	−	0.557694i	$-0.188314\pi$
$984$	19.7990	0.631169
$985$	22.9706	0.731903
$986$	− 11.3137i	− 0.360302i
$987$	0.970563	0.0308934
$988$	0	0
$989$	−4.34315	−0.138104
$990$	0.242641i	0.00771163i
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	−45.2132	−1.43552
$993$	− 31.1716i	− 0.989200i
$994$	− 2.71068i	− 0.0859775i
$995$	4.00000i	0.126809i
$996$	22.8284i	0.723346i
$997$	23.1716	0.733851	0.366926	−	0.930250i	$-0.380410\pi$
$997$	23.1716	0.733851	0.366926	+	0.930250i	$0.380410\pi$
$998$	−4.32590	−0.136934
$999$	− 48.0000i	− 1.51865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.b.506.2		4
13.2	odd	12		845.2.e.c.191.2		4
13.3	even	3		845.2.m.f.316.3		8
13.4	even	6		845.2.m.f.361.3		8
13.5	odd	4		845.2.a.g.1.1		2
13.6	odd	12		845.2.e.c.146.2		4
13.7	odd	12		845.2.e.h.146.1		4
13.8	odd	4		65.2.a.b.1.2	✓	2
13.9	even	3		845.2.m.f.361.2		8
13.10	even	6		845.2.m.f.316.2		8
13.11	odd	12		845.2.e.h.191.1		4
13.12	even	2	inner	845.2.c.b.506.3		4
39.5	even	4		7605.2.a.x.1.2		2
39.8	even	4		585.2.a.m.1.1		2
52.47	even	4		1040.2.a.j.1.1		2
65.8	even	4		325.2.b.f.274.2		4
65.34	odd	4		325.2.a.i.1.1		2
65.44	odd	4		4225.2.a.r.1.2		2
65.47	even	4		325.2.b.f.274.3		4
91.34	even	4		3185.2.a.j.1.2		2
104.21	odd	4		4160.2.a.bf.1.1		2
104.99	even	4		4160.2.a.z.1.2		2
143.21	even	4		7865.2.a.j.1.1		2
156.47	odd	4		9360.2.a.cd.1.2		2
195.8	odd	4		2925.2.c.r.2224.3		4
195.47	odd	4		2925.2.c.r.2224.2		4
195.164	even	4		2925.2.a.u.1.2		2
260.99	even	4		5200.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	13.8	odd	4
325.2.a.i.1.1		2	65.34	odd	4
325.2.b.f.274.2		4	65.8	even	4
325.2.b.f.274.3		4	65.47	even	4
585.2.a.m.1.1		2	39.8	even	4
845.2.a.g.1.1		2	13.5	odd	4
845.2.c.b.506.2		4	1.1	even	1	trivial
845.2.c.b.506.3		4	13.12	even	2	inner
845.2.e.c.146.2		4	13.6	odd	12
845.2.e.c.191.2		4	13.2	odd	12
845.2.e.h.146.1		4	13.7	odd	12
845.2.e.h.191.1		4	13.11	odd	12
845.2.m.f.316.2		8	13.10	even	6
845.2.m.f.316.3		8	13.3	even	3
845.2.m.f.361.2		8	13.9	even	3
845.2.m.f.361.3		8	13.4	even	6
1040.2.a.j.1.1		2	52.47	even	4
2925.2.a.u.1.2		2	195.164	even	4
2925.2.c.r.2224.2		4	195.47	odd	4
2925.2.c.r.2224.3		4	195.8	odd	4
3185.2.a.j.1.2		2	91.34	even	4
4160.2.a.z.1.2		2	104.99	even	4
4160.2.a.bf.1.1		2	104.21	odd	4
4225.2.a.r.1.2		2	65.44	odd	4
5200.2.a.bu.1.2		2	260.99	even	4
7605.2.a.x.1.2		2	39.5	even	4
7865.2.a.j.1.1		2	143.21	even	4
9360.2.a.cd.1.2		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)

\(f(q)\)	\(=\)	\(q-0.414214i q^{2} +1.41421 q^{3} +1.82843 q^{4} -1.00000i q^{5} -0.585786i q^{6} -0.828427i q^{7} -1.58579i q^{8} -1.00000 q^{9} -0.414214 q^{10} +0.585786i q^{11} +2.58579 q^{12} -0.343146 q^{14} -1.41421i q^{15} +3.00000 q^{16} +4.82843 q^{17} +0.414214i q^{18} -3.41421i q^{19} -1.82843i q^{20} -1.17157i q^{21} +0.242641 q^{22} +1.41421 q^{23} -2.24264i q^{24} -1.00000 q^{25} -5.65685 q^{27} -1.51472i q^{28} +5.65685 q^{29} -0.585786 q^{30} -10.2426i q^{31} -4.41421i q^{32} +0.828427i q^{33} -2.00000i q^{34} -0.828427 q^{35} -1.82843 q^{36} +8.48528i q^{37} -1.41421 q^{38} -1.58579 q^{40} +8.82843i q^{41} -0.485281 q^{42} -3.07107 q^{43} +1.07107i q^{44} +1.00000i q^{45} -0.585786i q^{46} +0.828427i q^{47} +4.24264 q^{48} +6.31371 q^{49} +0.414214i q^{50} +6.82843 q^{51} -14.4853 q^{53} +2.34315i q^{54} +0.585786 q^{55} -1.31371 q^{56} -4.82843i q^{57} -2.34315i q^{58} +10.2426i q^{59} -2.58579i q^{60} -8.00000 q^{61} -4.24264 q^{62} +0.828427i q^{63} +4.17157 q^{64} +0.343146 q^{66} +2.00000i q^{67} +8.82843 q^{68} +2.00000 q^{69} +0.343146i q^{70} +7.89949i q^{71} +1.58579i q^{72} -8.48528i q^{73} +3.51472 q^{74} -1.41421 q^{75} -6.24264i q^{76} +0.485281 q^{77} +8.48528 q^{79} -3.00000i q^{80} -5.00000 q^{81} +3.65685 q^{82} +8.82843i q^{83} -2.14214i q^{84} -4.82843i q^{85} +1.27208i q^{86} +8.00000 q^{87} +0.928932 q^{88} +6.00000i q^{89} +0.414214 q^{90} +2.58579 q^{92} -14.4853i q^{93} +0.343146 q^{94} -3.41421 q^{95} -6.24264i q^{96} -3.65685i q^{97} -2.61522i q^{98} -0.585786i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9} + 4 q^{10} + 16 q^{12} - 24 q^{14} + 12 q^{16} + 8 q^{17} - 16 q^{22} - 4 q^{25} - 8 q^{30} + 8 q^{35} + 4 q^{36} - 12 q^{40} + 32 q^{42} + 16 q^{43} - 20 q^{49} + 16 q^{51} - 24 q^{53}+ \cdots - 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 + 4 * q^10 + 16 * q^12 - 24 * q^14 + 12 * q^16 + 8 * q^17 - 16 * q^22 - 4 * q^25 - 8 * q^30 + 8 * q^35 + 4 * q^36 - 12 * q^40 + 32 * q^42 + 16 * q^43 - 20 * q^49 + 16 * q^51 - 24 * q^53 + 8 * q^55 + 40 * q^56 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 24 * q^68 + 8 * q^69 + 48 * q^74 - 32 * q^77 - 20 * q^81 - 8 * q^82 + 32 * q^87 + 32 * q^88 - 4 * q^90 + 16 * q^92 + 24 * q^94 - 8 * q^95

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