LMFDB - Embedded newform 8034.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.a (trivial)

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:	yes
Analytic conductor:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +4.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +4.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -3.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} -1.00000 q^{39} +4.00000 q^{40} -1.00000 q^{42} -8.00000 q^{43} -3.00000 q^{44} +4.00000 q^{45} -2.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +11.0000 q^{50} -3.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -12.0000 q^{55} -1.00000 q^{56} +8.00000 q^{57} +8.00000 q^{58} -6.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -3.00000 q^{66} -3.00000 q^{67} -3.00000 q^{68} -2.00000 q^{69} -4.00000 q^{70} +2.00000 q^{71} +1.00000 q^{72} +11.0000 q^{73} -8.00000 q^{74} +11.0000 q^{75} +8.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{83} -1.00000 q^{84} -12.0000 q^{85} -8.00000 q^{86} +8.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} +1.00000 q^{91} -2.00000 q^{92} +8.00000 q^{93} +12.0000 q^{94} +32.0000 q^{95} +1.00000 q^{96} -12.0000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	4.00000	0.894427
$21$	−1.00000	−0.218218
$22$	−3.00000	−0.639602
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	4.00000	0.730297
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−3.00000	−0.514496
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	8.00000	1.29777
$39$	−1.00000	−0.160128
$40$	4.00000	0.632456
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−3.00000	−0.452267
$45$	4.00000	0.596285
$46$	−2.00000	−0.294884
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	11.0000	1.55563
$51$	−3.00000	−0.420084
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	1.00000	0.136083
$55$	−12.0000	−1.61808
$56$	−1.00000	−0.133631
$57$	8.00000	1.05963
$58$	8.00000	1.05045
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	4.00000	0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−3.00000	−0.369274
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−3.00000	−0.363803
$69$	−2.00000	−0.240772
$70$	−4.00000	−0.478091
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000	0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−8.00000	−0.929981
$75$	11.0000	1.27017
$76$	8.00000	0.917663
$77$	3.00000	0.341882
$78$	−1.00000	−0.113228
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	−0.109109
$85$	−12.0000	−1.30158
$86$	−8.00000	−0.862662
$87$	8.00000	0.857690
$88$	−3.00000	−0.319801
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	4.00000	0.421637
$91$	1.00000	0.104828
$92$	−2.00000	−0.208514
$93$	8.00000	0.829561
$94$	12.0000	1.23771
$95$	32.0000	3.28313
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−6.00000	−0.606092
$99$	−3.00000	−0.301511
$100$	11.0000	1.10000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	−3.00000	−0.297044
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	−4.00000	−0.390360
$106$	9.00000	0.874157
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−12.0000	−1.14416
$111$	−8.00000	−0.759326
$112$	−1.00000	−0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	8.00000	0.749269
$115$	−8.00000	−0.746004
$116$	8.00000	0.742781
$117$	−1.00000	−0.0924500
$118$	−6.00000	−0.552345
$119$	3.00000	0.275010
$120$	4.00000	0.365148
$121$	−2.00000	−0.181818
$122$	6.00000	0.543214
$123$	0	0
$124$	8.00000	0.718421
$125$	24.0000	2.14663
$126$	−1.00000	−0.0890871
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	−4.00000	−0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−3.00000	−0.261116
$133$	−8.00000	−0.693688
$134$	−3.00000	−0.259161
$135$	4.00000	0.344265
$136$	−3.00000	−0.257248
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−2.00000	−0.170251
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	−4.00000	−0.338062
$141$	12.0000	1.01058
$142$	2.00000	0.167836
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	32.0000	2.65746
$146$	11.0000	0.910366
$147$	−6.00000	−0.494872
$148$	−8.00000	−0.657596
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	11.0000	0.898146
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	8.00000	0.648886
$153$	−3.00000	−0.242536
$154$	3.00000	0.241747
$155$	32.0000	2.57030
$156$	−1.00000	−0.0800641
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	10.0000	0.795557
$159$	9.00000	0.713746
$160$	4.00000	0.316228
$161$	2.00000	0.157622
$162$	1.00000	0.0785674
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	−6.00000	−0.465690
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	−1.00000	−0.0771517
$169$	1.00000	0.0769231
$170$	−12.0000	−0.920358
$171$	8.00000	0.611775
$172$	−8.00000	−0.609994
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	8.00000	0.606478
$175$	−11.0000	−0.831522
$176$	−3.00000	−0.226134
$177$	−6.00000	−0.450988
$178$	−10.0000	−0.749532
$179$	7.00000	0.523205	0.261602	−	0.965176i	$-0.415749\pi$
$179$	7.00000	0.523205	0.261602	+	0.965176i	$0.415749\pi$
$180$	4.00000	0.298142
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	1.00000	0.0741249
$183$	6.00000	0.443533
$184$	−2.00000	−0.147442
$185$	−32.0000	−2.35269
$186$	8.00000	0.586588
$187$	9.00000	0.658145
$188$	12.0000	0.875190
$189$	−1.00000	−0.0727393
$190$	32.0000	2.32152
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−12.0000	−0.861550
$195$	−4.00000	−0.286446
$196$	−6.00000	−0.428571
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	11.0000	0.777817
$201$	−3.00000	−0.211604
$202$	5.00000	0.351799
$203$	−8.00000	−0.561490
$204$	−3.00000	−0.210042
$205$	0	0
$206$	1.00000	0.0696733
$207$	−2.00000	−0.139010
$208$	−1.00000	−0.0693375
$209$	−24.0000	−1.66011
$210$	−4.00000	−0.276026
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	9.00000	0.618123
$213$	2.00000	0.137038
$214$	−11.0000	−0.751945
$215$	−32.0000	−2.18238
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	2.00000	0.135457
$219$	11.0000	0.743311
$220$	−12.0000	−0.809040
$221$	3.00000	0.201802
$222$	−8.00000	−0.536925
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	−1.00000	−0.0668153
$225$	11.0000	0.733333
$226$	0	0
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	8.00000	0.529813
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−8.00000	−0.527504
$231$	3.00000	0.197386
$232$	8.00000	0.525226
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−1.00000	−0.0653720
$235$	48.0000	3.13117
$236$	−6.00000	−0.390567
$237$	10.0000	0.649570
$238$	3.00000	0.194461
$239$	−3.00000	−0.194054	−0.0970269	−	0.995282i	$-0.530933\pi$
$239$	−3.00000	−0.194054	−0.0970269	+	0.995282i	$0.530933\pi$
$240$	4.00000	0.258199
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	−24.0000	−1.53330
$246$	0	0
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	−6.00000	−0.380235
$250$	24.0000	1.51789
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	−1.00000	−0.0629941
$253$	6.00000	0.377217
$254$	−17.0000	−1.06667
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	−8.00000	−0.498058
$259$	8.00000	0.497096
$260$	−4.00000	−0.248069
$261$	8.00000	0.495188
$262$	−4.00000	−0.247121
$263$	−17.0000	−1.04826	−0.524132	−	0.851637i	$-0.675610\pi$
$263$	−17.0000	−1.04826	−0.524132	+	0.851637i	$0.675610\pi$
$264$	−3.00000	−0.184637
$265$	36.0000	2.21146
$266$	−8.00000	−0.490511
$267$	−10.0000	−0.611990
$268$	−3.00000	−0.183254
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	4.00000	0.243432
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	−3.00000	−0.181902
$273$	1.00000	0.0605228
$274$	14.0000	0.845771
$275$	−33.0000	−1.98997
$276$	−2.00000	−0.120386
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	1.00000	0.0599760
$279$	8.00000	0.478947
$280$	−4.00000	−0.239046
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	12.0000	0.714590
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	2.00000	0.118678
$285$	32.0000	1.89552
$286$	3.00000	0.177394
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	32.0000	1.87910
$291$	−12.0000	−0.703452
$292$	11.0000	0.643726
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	−6.00000	−0.349927
$295$	−24.0000	−1.39733
$296$	−8.00000	−0.464991
$297$	−3.00000	−0.174078
$298$	15.0000	0.868927
$299$	2.00000	0.115663
$300$	11.0000	0.635085
$301$	8.00000	0.461112
$302$	−12.0000	−0.690522
$303$	5.00000	0.287242
$304$	8.00000	0.458831
$305$	24.0000	1.37424
$306$	−3.00000	−0.171499
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	3.00000	0.170941
$309$	1.00000	0.0568880
$310$	32.0000	1.81748
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−1.00000	−0.0566139
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	13.0000	0.733632
$315$	−4.00000	−0.225374
$316$	10.0000	0.562544
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	9.00000	0.504695
$319$	−24.0000	−1.34374
$320$	4.00000	0.223607
$321$	−11.0000	−0.613960
$322$	2.00000	0.111456
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−11.0000	−0.610170
$326$	−14.0000	−0.775388
$327$	2.00000	0.110600
$328$	0	0
$329$	−12.0000	−0.661581
$330$	−12.0000	−0.660578
$331$	−35.0000	−1.92377	−0.961887	−	0.273447i	$-0.911836\pi$
$331$	−35.0000	−1.92377	−0.961887	+	0.273447i	$0.911836\pi$
$332$	−6.00000	−0.329293
$333$	−8.00000	−0.438397
$334$	19.0000	1.03963
$335$	−12.0000	−0.655630
$336$	−1.00000	−0.0545545
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−24.0000	−1.29967
$342$	8.00000	0.432590
$343$	13.0000	0.701934
$344$	−8.00000	−0.431331
$345$	−8.00000	−0.430706
$346$	−3.00000	−0.161281
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	8.00000	0.428845
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−11.0000	−0.587975
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	−6.00000	−0.318896
$355$	8.00000	0.424596
$356$	−10.0000	−0.529999
$357$	3.00000	0.158777
$358$	7.00000	0.369961
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	4.00000	0.210819
$361$	45.0000	2.36842
$362$	25.0000	1.31397
$363$	−2.00000	−0.104973
$364$	1.00000	0.0524142
$365$	44.0000	2.30307
$366$	6.00000	0.313625
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−2.00000	−0.104257
$369$	0	0
$370$	−32.0000	−1.66360
$371$	−9.00000	−0.467257
$372$	8.00000	0.414781
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	9.00000	0.465379
$375$	24.0000	1.23935
$376$	12.0000	0.618853
$377$	−8.00000	−0.412021
$378$	−1.00000	−0.0514344
$379$	−7.00000	−0.359566	−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000	−0.359566	−0.179783	+	0.983706i	$0.557540\pi$
$380$	32.0000	1.64157
$381$	−17.0000	−0.870936
$382$	−17.0000	−0.869796
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	1.00000	0.0510310
$385$	12.0000	0.611577
$386$	−2.00000	−0.101797
$387$	−8.00000	−0.406663
$388$	−12.0000	−0.609208
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	−4.00000	−0.202548
$391$	6.00000	0.303433
$392$	−6.00000	−0.303046
$393$	−4.00000	−0.201773
$394$	6.00000	0.302276
$395$	40.0000	2.01262
$396$	−3.00000	−0.150756
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	19.0000	0.952384
$399$	−8.00000	−0.400501
$400$	11.0000	0.550000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−3.00000	−0.149626
$403$	−8.00000	−0.398508
$404$	5.00000	0.248759
$405$	4.00000	0.198762
$406$	−8.00000	−0.397033
$407$	24.0000	1.18964
$408$	−3.00000	−0.148522
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	1.00000	0.0492665
$413$	6.00000	0.295241
$414$	−2.00000	−0.0982946
$415$	−24.0000	−1.17811
$416$	−1.00000	−0.0490290
$417$	1.00000	0.0489702
$418$	−24.0000	−1.17388
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	−4.00000	−0.195180
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	8.00000	0.389434
$423$	12.0000	0.583460
$424$	9.00000	0.437079
$425$	−33.0000	−1.60074
$426$	2.00000	0.0969003
$427$	−6.00000	−0.290360
$428$	−11.0000	−0.531705
$429$	3.00000	0.144841
$430$	−32.0000	−1.54318
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	−8.00000	−0.384012
$435$	32.0000	1.53428
$436$	2.00000	0.0957826
$437$	−16.0000	−0.765384
$438$	11.0000	0.525600
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	−12.0000	−0.572078
$441$	−6.00000	−0.285714
$442$	3.00000	0.142695
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−8.00000	−0.379663
$445$	−40.0000	−1.89618
$446$	21.0000	0.994379
$447$	15.0000	0.709476
$448$	−1.00000	−0.0472456
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	11.0000	0.518545
$451$	0	0
$452$	0	0
$453$	−12.0000	−0.563809
$454$	−7.00000	−0.328526
$455$	4.00000	0.187523
$456$	8.00000	0.374634
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	2.00000	0.0934539
$459$	−3.00000	−0.140028
$460$	−8.00000	−0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	3.00000	0.139573
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	8.00000	0.371391
$465$	32.0000	1.48396
$466$	6.00000	0.277945
$467$	−39.0000	−1.80470	−0.902352	−	0.430999i	$-0.858161\pi$
$467$	−39.0000	−1.80470	−0.902352	+	0.430999i	$0.858161\pi$
$468$	−1.00000	−0.0462250
$469$	3.00000	0.138527
$470$	48.0000	2.21407
$471$	13.0000	0.599008
$472$	−6.00000	−0.276172
$473$	24.0000	1.10352
$474$	10.0000	0.459315
$475$	88.0000	4.03772
$476$	3.00000	0.137505
$477$	9.00000	0.412082
$478$	−3.00000	−0.137217
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	4.00000	0.182574
$481$	8.00000	0.364769
$482$	−5.00000	−0.227744
$483$	2.00000	0.0910032
$484$	−2.00000	−0.0909091
$485$	−48.0000	−2.17957
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	6.00000	0.271607
$489$	−14.0000	−0.633102
$490$	−24.0000	−1.08421
$491$	−37.0000	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$491$	−37.0000	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	−8.00000	−0.359937
$495$	−12.0000	−0.539360
$496$	8.00000	0.359211
$497$	−2.00000	−0.0897123
$498$	−6.00000	−0.268866
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	24.0000	1.07331
$501$	19.0000	0.848857
$502$	4.00000	0.178529
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	−1.00000	−0.0445435
$505$	20.0000	0.889988
$506$	6.00000	0.266733
$507$	1.00000	0.0444116
$508$	−17.0000	−0.754253
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	−12.0000	−0.531369
$511$	−11.0000	−0.486611
$512$	1.00000	0.0441942
$513$	8.00000	0.353209
$514$	−12.0000	−0.529297
$515$	4.00000	0.176261
$516$	−8.00000	−0.352180
$517$	−36.0000	−1.58328
$518$	8.00000	0.351500
$519$	−3.00000	−0.131685
$520$	−4.00000	−0.175412
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	8.00000	0.350150
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	−4.00000	−0.174741
$525$	−11.0000	−0.480079
$526$	−17.0000	−0.741235
$527$	−24.0000	−1.04546
$528$	−3.00000	−0.130558
$529$	−19.0000	−0.826087
$530$	36.0000	1.56374
$531$	−6.00000	−0.260378
$532$	−8.00000	−0.346844
$533$	0	0
$534$	−10.0000	−0.432742
$535$	−44.0000	−1.90229
$536$	−3.00000	−0.129580
$537$	7.00000	0.302072
$538$	4.00000	0.172452
$539$	18.0000	0.775315
$540$	4.00000	0.172133
$541$	13.0000	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$542$	−14.0000	−0.601351
$543$	25.0000	1.07285
$544$	−3.00000	−0.128624
$545$	8.00000	0.342682
$546$	1.00000	0.0427960
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	14.0000	0.598050
$549$	6.00000	0.256074
$550$	−33.0000	−1.40712
$551$	64.0000	2.72649
$552$	−2.00000	−0.0851257
$553$	−10.0000	−0.425243
$554$	−2.00000	−0.0849719
$555$	−32.0000	−1.35832
$556$	1.00000	0.0424094
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	8.00000	0.338667
$559$	8.00000	0.338364
$560$	−4.00000	−0.169031
$561$	9.00000	0.379980
$562$	−22.0000	−0.928014
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−12.0000	−0.504398
$567$	−1.00000	−0.0419961
$568$	2.00000	0.0839181
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	32.0000	1.34033
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	3.00000	0.125436
$573$	−17.0000	−0.710185
$574$	0	0
$575$	−22.0000	−0.917463
$576$	1.00000	0.0416667
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	−8.00000	−0.332756
$579$	−2.00000	−0.0831172
$580$	32.0000	1.32873
$581$	6.00000	0.248922
$582$	−12.0000	−0.497416
$583$	−27.0000	−1.11823
$584$	11.0000	0.455183
$585$	−4.00000	−0.165380
$586$	−16.0000	−0.660954
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	−6.00000	−0.247436
$589$	64.0000	2.63707
$590$	−24.0000	−0.988064
$591$	6.00000	0.246807
$592$	−8.00000	−0.328798
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	−3.00000	−0.123091
$595$	12.0000	0.491952
$596$	15.0000	0.614424
$597$	19.0000	0.777618
$598$	2.00000	0.0817861
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	11.0000	0.449073
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	8.00000	0.326056
$603$	−3.00000	−0.122169
$604$	−12.0000	−0.488273
$605$	−8.00000	−0.325246
$606$	5.00000	0.203111
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	8.00000	0.324443
$609$	−8.00000	−0.324176
$610$	24.0000	0.971732
$611$	−12.0000	−0.485468
$612$	−3.00000	−0.121268
$613$	−35.0000	−1.41364	−0.706818	−	0.707395i	$-0.749870\pi$
$613$	−35.0000	−1.41364	−0.706818	+	0.707395i	$0.749870\pi$
$614$	−23.0000	−0.928204
$615$	0	0
$616$	3.00000	0.120873
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	1.00000	0.0402259
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	32.0000	1.28515
$621$	−2.00000	−0.0802572
$622$	−20.0000	−0.801927
$623$	10.0000	0.400642
$624$	−1.00000	−0.0400320
$625$	41.0000	1.64000
$626$	−22.0000	−0.879297
$627$	−24.0000	−0.958468
$628$	13.0000	0.518756
$629$	24.0000	0.956943
$630$	−4.00000	−0.159364
$631$	19.0000	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$631$	19.0000	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$632$	10.0000	0.397779
$633$	8.00000	0.317971
$634$	−27.0000	−1.07231
$635$	−68.0000	−2.69850
$636$	9.00000	0.356873
$637$	6.00000	0.237729
$638$	−24.0000	−0.950169
$639$	2.00000	0.0791188
$640$	4.00000	0.158114
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	−11.0000	−0.434135
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	2.00000	0.0788110
$645$	−32.0000	−1.26000
$646$	−24.0000	−0.944267
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	1.00000	0.0392837
$649$	18.0000	0.706562
$650$	−11.0000	−0.431455
$651$	−8.00000	−0.313545
$652$	−14.0000	−0.548282
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	2.00000	0.0782062
$655$	−16.0000	−0.625172
$656$	0	0
$657$	11.0000	0.429151
$658$	−12.0000	−0.467809
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	−12.0000	−0.467099
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−35.0000	−1.36031
$663$	3.00000	0.116510
$664$	−6.00000	−0.232845
$665$	−32.0000	−1.24091
$666$	−8.00000	−0.309994
$667$	−16.0000	−0.619522
$668$	19.0000	0.735132
$669$	21.0000	0.811907
$670$	−12.0000	−0.463600
$671$	−18.0000	−0.694882
$672$	−1.00000	−0.0385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−17.0000	−0.654816
$675$	11.0000	0.423390
$676$	1.00000	0.0384615
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	−12.0000	−0.460179
$681$	−7.00000	−0.268241
$682$	−24.0000	−0.919007
$683$	33.0000	1.26271	0.631355	−	0.775494i	$-0.282499\pi$
$683$	33.0000	1.26271	0.631355	+	0.775494i	$0.282499\pi$
$684$	8.00000	0.305888
$685$	56.0000	2.13965
$686$	13.0000	0.496342
$687$	2.00000	0.0763048
$688$	−8.00000	−0.304997
$689$	−9.00000	−0.342873
$690$	−8.00000	−0.304555
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	−3.00000	−0.114043
$693$	3.00000	0.113961
$694$	36.0000	1.36654
$695$	4.00000	0.151729
$696$	8.00000	0.303239
$697$	0	0
$698$	2.00000	0.0757011
$699$	6.00000	0.226941
$700$	−11.0000	−0.415761
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	−1.00000	−0.0377426
$703$	−64.0000	−2.41381
$704$	−3.00000	−0.113067
$705$	48.0000	1.80778
$706$	−31.0000	−1.16670
$707$	−5.00000	−0.188044
$708$	−6.00000	−0.225494
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	8.00000	0.300235
$711$	10.0000	0.375029
$712$	−10.0000	−0.374766
$713$	−16.0000	−0.599205
$714$	3.00000	0.112272
$715$	12.0000	0.448775
$716$	7.00000	0.261602
$717$	−3.00000	−0.112037
$718$	−25.0000	−0.932992
$719$	−47.0000	−1.75280	−0.876402	−	0.481580i	$-0.840063\pi$
$719$	−47.0000	−1.75280	−0.876402	+	0.481580i	$0.840063\pi$
$720$	4.00000	0.149071
$721$	−1.00000	−0.0372419
$722$	45.0000	1.67473
$723$	−5.00000	−0.185952
$724$	25.0000	0.929118
$725$	88.0000	3.26824
$726$	−2.00000	−0.0742270
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	44.0000	1.62851
$731$	24.0000	0.887672
$732$	6.00000	0.221766
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−22.0000	−0.812035
$735$	−24.0000	−0.885253
$736$	−2.00000	−0.0737210
$737$	9.00000	0.331519
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−32.0000	−1.17634
$741$	−8.00000	−0.293887
$742$	−9.00000	−0.330400
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	8.00000	0.293294
$745$	60.0000	2.19823
$746$	22.0000	0.805477
$747$	−6.00000	−0.219529
$748$	9.00000	0.329073
$749$	11.0000	0.401931
$750$	24.0000	0.876356
$751$	−52.0000	−1.89751	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000	−1.89751	−0.948753	+	0.316017i	$0.897654\pi$
$752$	12.0000	0.437595
$753$	4.00000	0.145768
$754$	−8.00000	−0.291343
$755$	−48.0000	−1.74690
$756$	−1.00000	−0.0363696
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−7.00000	−0.254251
$759$	6.00000	0.217786
$760$	32.0000	1.16076
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−17.0000	−0.615845
$763$	−2.00000	−0.0724049
$764$	−17.0000	−0.615038
$765$	−12.0000	−0.433861
$766$	34.0000	1.22847
$767$	6.00000	0.216647
$768$	1.00000	0.0360844
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	12.0000	0.432450
$771$	−12.0000	−0.432169
$772$	−2.00000	−0.0719816
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	−8.00000	−0.287554
$775$	88.0000	3.16105
$776$	−12.0000	−0.430775
$777$	8.00000	0.286998
$778$	9.00000	0.322666
$779$	0	0
$780$	−4.00000	−0.143223
$781$	−6.00000	−0.214697
$782$	6.00000	0.214560
$783$	8.00000	0.285897
$784$	−6.00000	−0.214286
$785$	52.0000	1.85596
$786$	−4.00000	−0.142675
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	6.00000	0.213741
$789$	−17.0000	−0.605216
$790$	40.0000	1.42314
$791$	0	0
$792$	−3.00000	−0.106600
$793$	−6.00000	−0.213066
$794$	−14.0000	−0.496841
$795$	36.0000	1.27679
$796$	19.0000	0.673437
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	−8.00000	−0.283197
$799$	−36.0000	−1.27359
$800$	11.0000	0.388909
$801$	−10.0000	−0.353333
$802$	−22.0000	−0.776847
$803$	−33.0000	−1.16454
$804$	−3.00000	−0.105802
$805$	8.00000	0.281963
$806$	−8.00000	−0.281788
$807$	4.00000	0.140807
$808$	5.00000	0.175899
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	4.00000	0.140546
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−8.00000	−0.280745
$813$	−14.0000	−0.491001
$814$	24.0000	0.841200
$815$	−56.0000	−1.96159
$816$	−3.00000	−0.105021
$817$	−64.0000	−2.23908
$818$	12.0000	0.419570
$819$	1.00000	0.0349428
$820$	0	0
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	14.0000	0.488306
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	1.00000	0.0348367
$825$	−33.0000	−1.14891
$826$	6.00000	0.208767
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−2.00000	−0.0695048
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−24.0000	−0.833052
$831$	−2.00000	−0.0693792
$832$	−1.00000	−0.0346688
$833$	18.0000	0.623663
$834$	1.00000	0.0346272
$835$	76.0000	2.63009
$836$	−24.0000	−0.830057
$837$	8.00000	0.276520
$838$	9.00000	0.310900
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	−4.00000	−0.138013
$841$	35.0000	1.20690
$842$	−17.0000	−0.585859
$843$	−22.0000	−0.757720
$844$	8.00000	0.275371
$845$	4.00000	0.137604
$846$	12.0000	0.412568
$847$	2.00000	0.0687208
$848$	9.00000	0.309061
$849$	−12.0000	−0.411839
$850$	−33.0000	−1.13189
$851$	16.0000	0.548473
$852$	2.00000	0.0685189
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	−6.00000	−0.205316
$855$	32.0000	1.09438
$856$	−11.0000	−0.375972
$857$	−19.0000	−0.649028	−0.324514	−	0.945881i	$-0.605201\pi$
$857$	−19.0000	−0.649028	−0.324514	+	0.945881i	$0.605201\pi$
$858$	3.00000	0.102418
$859$	54.0000	1.84246	0.921228	−	0.389023i	$-0.127187\pi$
$859$	54.0000	1.84246	0.921228	+	0.389023i	$0.127187\pi$
$860$	−32.0000	−1.09119
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	1.00000	0.0340207
$865$	−12.0000	−0.408012
$866$	32.0000	1.08740
$867$	−8.00000	−0.271694
$868$	−8.00000	−0.271538
$869$	−30.0000	−1.01768
$870$	32.0000	1.08490
$871$	3.00000	0.101651
$872$	2.00000	0.0677285
$873$	−12.0000	−0.406138
$874$	−16.0000	−0.541208
$875$	−24.0000	−0.811348
$876$	11.0000	0.371656
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	25.0000	0.843709
$879$	−16.0000	−0.539667
$880$	−12.0000	−0.404520
$881$	8.00000	0.269527	0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000	0.269527	0.134763	+	0.990878i	$0.456973\pi$
$882$	−6.00000	−0.202031
$883$	7.00000	0.235569	0.117784	−	0.993039i	$-0.462421\pi$
$883$	7.00000	0.235569	0.117784	+	0.993039i	$0.462421\pi$
$884$	3.00000	0.100901
$885$	−24.0000	−0.806751
$886$	−28.0000	−0.940678
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−8.00000	−0.268462
$889$	17.0000	0.570162
$890$	−40.0000	−1.34080
$891$	−3.00000	−0.100504
$892$	21.0000	0.703132
$893$	96.0000	3.21252
$894$	15.0000	0.501675
$895$	28.0000	0.935937
$896$	−1.00000	−0.0334077
$897$	2.00000	0.0667781
$898$	−15.0000	−0.500556
$899$	64.0000	2.13452
$900$	11.0000	0.366667
$901$	−27.0000	−0.899500
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	100.000	3.32411
$906$	−12.0000	−0.398673
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	−7.00000	−0.232303
$909$	5.00000	0.165840
$910$	4.00000	0.132599
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	8.00000	0.264906
$913$	18.0000	0.595713
$914$	18.0000	0.595387
$915$	24.0000	0.793416
$916$	2.00000	0.0660819
$917$	4.00000	0.132092
$918$	−3.00000	−0.0990148
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	−8.00000	−0.263752
$921$	−23.0000	−0.757876
$922$	30.0000	0.987997
$923$	−2.00000	−0.0658308
$924$	3.00000	0.0986928
$925$	−88.0000	−2.89342
$926$	−14.0000	−0.460069
$927$	1.00000	0.0328443
$928$	8.00000	0.262613
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	32.0000	1.04932
$931$	−48.0000	−1.57314
$932$	6.00000	0.196537
$933$	−20.0000	−0.654771
$934$	−39.0000	−1.27612
$935$	36.0000	1.17733
$936$	−1.00000	−0.0326860
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	3.00000	0.0979535
$939$	−22.0000	−0.717943
$940$	48.0000	1.56559
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	13.0000	0.423563
$943$	0	0
$944$	−6.00000	−0.195283
$945$	−4.00000	−0.130120
$946$	24.0000	0.780307
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	10.0000	0.324785
$949$	−11.0000	−0.357075
$950$	88.0000	2.85510
$951$	−27.0000	−0.875535
$952$	3.00000	0.0972306
$953$	−31.0000	−1.00419	−0.502094	−	0.864813i	$-0.667437\pi$
$953$	−31.0000	−1.00419	−0.502094	+	0.864813i	$0.667437\pi$
$954$	9.00000	0.291386
$955$	−68.0000	−2.20043
$956$	−3.00000	−0.0970269
$957$	−24.0000	−0.775810
$958$	14.0000	0.452319
$959$	−14.0000	−0.452084
$960$	4.00000	0.129099
$961$	33.0000	1.06452
$962$	8.00000	0.257930
$963$	−11.0000	−0.354470
$964$	−5.00000	−0.161039
$965$	−8.00000	−0.257529
$966$	2.00000	0.0643489
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−2.00000	−0.0642824
$969$	−24.0000	−0.770991
$970$	−48.0000	−1.54119
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	1.00000	0.0320750
$973$	−1.00000	−0.0320585
$974$	−32.0000	−1.02535
$975$	−11.0000	−0.352282
$976$	6.00000	0.192055
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	−14.0000	−0.447671
$979$	30.0000	0.958804
$980$	−24.0000	−0.766652
$981$	2.00000	0.0638551
$982$	−37.0000	−1.18072
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	−24.0000	−0.764316
$987$	−12.0000	−0.381964
$988$	−8.00000	−0.254514
$989$	16.0000	0.508770
$990$	−12.0000	−0.381385
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	8.00000	0.254000
$993$	−35.0000	−1.11069
$994$	−2.00000	−0.0634361
$995$	76.0000	2.40936
$996$	−6.00000	−0.190117
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	4.00000	0.126618
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.k.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.k.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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