LMFDB - Embedded newform 966.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, 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("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.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:	$7.71354883526$
Analytic rank:	$1$
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	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} +2.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−6.00000	−1.02899
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.00000	0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	−4.00000	−0.455842
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−1.00000	−0.109109
$85$	12.0000	1.30158
$86$	−4.00000	−0.431331
$87$	2.00000	0.214423
$88$	−4.00000	−0.426401
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	−1.00000	−0.104257
$93$	8.00000	0.829561
$94$	−8.00000	−0.825137
$95$	−8.00000	−0.820783
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	2.00000	0.195180
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	8.00000	0.762770
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−4.00000	−0.374634
$115$	2.00000	0.186501
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	4.00000	0.368230
$119$	−6.00000	−0.550019
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	1.00000	0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	1.00000	0.0851257
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−6.00000	−0.496564
$147$	−1.00000	−0.0824786
$148$	6.00000	0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	−4.00000	−0.322329
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	−2.00000	−0.158114
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−8.00000	−0.622799
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	2.00000	0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−2.00000	−0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	−12.0000	−0.882258
$186$	8.00000	0.586588
$187$	24.0000	1.75505
$188$	−8.00000	−0.583460
$189$	−1.00000	−0.0727393
$190$	−8.00000	−0.580381
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	10.0000	0.717958
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	8.00000	0.557386
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	−16.0000	−1.10674
$210$	2.00000	0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	−8.00000	−0.543075
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	8.00000	0.539360
$221$	12.0000	0.807207
$222$	−6.00000	−0.402694
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	2.00000	0.131876
$231$	4.00000	0.263181
$232$	−2.00000	−0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−2.00000	−0.130744
$235$	16.0000	1.04372
$236$	4.00000	0.260378
$237$	0	0
$238$	−6.00000	−0.388922
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	2.00000	0.129099
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−2.00000	−0.127775
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	−8.00000	−0.508001
$249$	12.0000	0.760469
$250$	12.0000	0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000	0.0629941
$253$	4.00000	0.251478
$254$	−16.0000	−1.00393
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	4.00000	0.249029
$259$	6.00000	0.372822
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	4.00000	0.246183
$265$	−12.0000	−0.737154
$266$	4.00000	0.245256
$267$	−2.00000	−0.122398
$268$	4.00000	0.244339
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−6.00000	−0.363803
$273$	2.00000	0.121046
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−12.0000	−0.719712
$279$	−8.00000	−0.478947
$280$	−2.00000	−0.119523
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	8.00000	0.476393
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−8.00000	−0.474713
$285$	8.00000	0.473879
$286$	8.00000	0.473050
$287$	−6.00000	−0.354169
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	4.00000	0.234888
$291$	−10.0000	−0.586210
$292$	−6.00000	−0.351123
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−1.00000	−0.0583212
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	6.00000	0.347571
$299$	2.00000	0.115663
$300$	1.00000	0.0577350
$301$	−4.00000	−0.230556
$302$	8.00000	0.460348
$303$	−6.00000	−0.344691
$304$	4.00000	0.229416
$305$	20.0000	1.14520
$306$	−6.00000	−0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−4.00000	−0.227921
$309$	−8.00000	−0.455104
$310$	16.0000	0.908739
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	2.00000	0.113228
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	22.0000	1.24153
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−6.00000	−0.336463
$319$	8.00000	0.447914
$320$	−2.00000	−0.111803
$321$	−12.0000	−0.669775
$322$	−1.00000	−0.0557278
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	4.00000	0.221540
$327$	−14.0000	−0.774202
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	−8.00000	−0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−12.0000	−0.658586
$333$	6.00000	0.328798
$334$	0	0
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	−9.00000	−0.489535
$339$	−18.0000	−0.977626
$340$	12.0000	0.650791
$341$	32.0000	1.73290
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	−2.00000	−0.107676
$346$	−2.00000	−0.107521
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	2.00000	0.107211
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−1.00000	−0.0534522
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−4.00000	−0.212598
$355$	16.0000	0.849192
$356$	2.00000	0.106000
$357$	6.00000	0.317554
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	−5.00000	−0.262432
$364$	−2.00000	−0.104828
$365$	12.0000	0.628109
$366$	10.0000	0.522708
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−1.00000	−0.0521286
$369$	−6.00000	−0.312348
$370$	−12.0000	−0.623850
$371$	6.00000	0.311504
$372$	8.00000	0.414781
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	24.0000	1.24101
$375$	−12.0000	−0.619677
$376$	−8.00000	−0.412568
$377$	4.00000	0.206010
$378$	−1.00000	−0.0514344
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	−8.00000	−0.410391
$381$	16.0000	0.819705
$382$	−16.0000	−0.818631
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	8.00000	0.407718
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.00000	−0.202548
$391$	6.00000	0.303433
$392$	1.00000	0.0505076
$393$	−12.0000	−0.605320
$394$	−26.0000	−1.30986
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−24.0000	−1.20301
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	−4.00000	−0.199502
$403$	16.0000	0.797017
$404$	6.00000	0.298511
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	−24.0000	−1.18964
$408$	6.00000	0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	12.0000	0.592638
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	4.00000	0.196827
$414$	−1.00000	−0.0491473
$415$	24.0000	1.17811
$416$	−2.00000	−0.0980581
$417$	12.0000	0.587643
$418$	−16.0000	−0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	2.00000	0.0975900
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	−8.00000	−0.388973
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	8.00000	0.387601
$427$	−10.0000	−0.483934
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	8.00000	0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−8.00000	−0.384012
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	−4.00000	−0.191346
$438$	6.00000	0.286691
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	8.00000	0.381385
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−6.00000	−0.284747
$445$	−4.00000	−0.189618
$446$	24.0000	1.13643
$447$	−6.00000	−0.283790
$448$	1.00000	0.0472456
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	18.0000	0.846649
$453$	−8.00000	−0.375873
$454$	−12.0000	−0.563188
$455$	4.00000	0.187523
$456$	−4.00000	−0.187317
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	14.0000	0.654177
$459$	6.00000	0.280056
$460$	2.00000	0.0932505
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	4.00000	0.186097
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−2.00000	−0.0928477
$465$	−16.0000	−0.741982
$466$	10.0000	0.463241
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	−2.00000	−0.0924500
$469$	4.00000	0.184703
$470$	16.0000	0.738025
$471$	−22.0000	−1.01371
$472$	4.00000	0.184115
$473$	16.0000	0.735681
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−6.00000	−0.275010
$477$	6.00000	0.274721
$478$	−16.0000	−0.731823
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	2.00000	0.0912871
$481$	−12.0000	−0.547153
$482$	−22.0000	−1.00207
$483$	1.00000	0.0455016
$484$	5.00000	0.227273
$485$	−20.0000	−0.908153
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−10.0000	−0.452679
$489$	−4.00000	−0.180886
$490$	−2.00000	−0.0903508
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	6.00000	0.270501
$493$	12.0000	0.540453
$494$	−8.00000	−0.359937
$495$	8.00000	0.359573
$496$	−8.00000	−0.359211
$497$	−8.00000	−0.358849
$498$	12.0000	0.537733
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	12.0000	0.535586
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	1.00000	0.0445435
$505$	−12.0000	−0.533993
$506$	4.00000	0.177822
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	−12.0000	−0.531369
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	−16.0000	−0.705044
$516$	4.00000	0.176090
$517$	32.0000	1.40736
$518$	6.00000	0.263625
$519$	2.00000	0.0877903
$520$	4.00000	0.175412
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	12.0000	0.524222
$525$	1.00000	0.0436436
$526$	−24.0000	−1.04645
$527$	48.0000	2.09091
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	4.00000	0.173585
$532$	4.00000	0.173422
$533$	12.0000	0.519778
$534$	−2.00000	−0.0865485
$535$	−24.0000	−1.03761
$536$	4.00000	0.172774
$537$	12.0000	0.517838
$538$	−2.00000	−0.0862261
$539$	−4.00000	−0.172292
$540$	2.00000	0.0860663
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	24.0000	1.03089
$543$	2.00000	0.0858282
$544$	−6.00000	−0.257248
$545$	−28.0000	−1.19939
$546$	2.00000	0.0855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	1.00000	0.0425628
$553$	0	0
$554$	22.0000	0.934690
$555$	12.0000	0.509372
$556$	−12.0000	−0.508913
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−8.00000	−0.338667
$559$	8.00000	0.338364
$560$	−2.00000	−0.0845154
$561$	−24.0000	−1.01328
$562$	10.0000	0.421825
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	8.00000	0.336861
$565$	−36.0000	−1.51453
$566$	−20.0000	−0.840663
$567$	1.00000	0.0419961
$568$	−8.00000	−0.335673
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	8.00000	0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	16.0000	0.668410
$574$	−6.00000	−0.250435
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	19.0000	0.790296
$579$	−2.00000	−0.0831172
$580$	4.00000	0.166091
$581$	−12.0000	−0.497844
$582$	−10.0000	−0.414513
$583$	−24.0000	−0.993978
$584$	−6.00000	−0.248282
$585$	4.00000	0.165380
$586$	−2.00000	−0.0826192
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−1.00000	−0.0412393
$589$	−32.0000	−1.31854
$590$	−8.00000	−0.329355
$591$	26.0000	1.06950
$592$	6.00000	0.246598
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	4.00000	0.164122
$595$	12.0000	0.491952
$596$	6.00000	0.245770
$597$	24.0000	0.982255
$598$	2.00000	0.0817861
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−4.00000	−0.163028
$603$	4.00000	0.162893
$604$	8.00000	0.325515
$605$	−10.0000	−0.406558
$606$	−6.00000	−0.243733
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	4.00000	0.162221
$609$	2.00000	0.0810441
$610$	20.0000	0.809776
$611$	16.0000	0.647291
$612$	−6.00000	−0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	12.0000	0.484281
$615$	−12.0000	−0.483887
$616$	−4.00000	−0.161165
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−8.00000	−0.321807
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	16.0000	0.642575
$621$	1.00000	0.0401286
$622$	−16.0000	−0.641542
$623$	2.00000	0.0801283
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−14.0000	−0.559553
$627$	16.0000	0.638978
$628$	22.0000	0.877896
$629$	−36.0000	−1.43541
$630$	−2.00000	−0.0796819
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−2.00000	−0.0794301
$635$	32.0000	1.26988
$636$	−6.00000	−0.237915
$637$	−2.00000	−0.0792429
$638$	8.00000	0.316723
$639$	−8.00000	−0.316475
$640$	−2.00000	−0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	−1.00000	−0.0394055
$645$	−8.00000	−0.315000
$646$	−24.0000	−0.944267
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	2.00000	0.0784465
$651$	8.00000	0.313545
$652$	4.00000	0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−14.0000	−0.547443
$655$	−24.0000	−0.937758
$656$	−6.00000	−0.234261
$657$	−6.00000	−0.234082
$658$	−8.00000	−0.311872
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	−8.00000	−0.311400
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−4.00000	−0.155464
$663$	−12.0000	−0.466041
$664$	−12.0000	−0.465690
$665$	−8.00000	−0.310227
$666$	6.00000	0.232495
$667$	2.00000	0.0774403
$668$	0	0
$669$	−24.0000	−0.927894
$670$	−8.00000	−0.309067
$671$	40.0000	1.54418
$672$	−1.00000	−0.0385758
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−30.0000	−1.15556
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	−18.0000	−0.691286
$679$	10.0000	0.383765
$680$	12.0000	0.460179
$681$	12.0000	0.459841
$682$	32.0000	1.22534
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	4.00000	0.152944
$685$	12.0000	0.458496
$686$	1.00000	0.0381802
$687$	−14.0000	−0.534133
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	−2.00000	−0.0761387
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	−2.00000	−0.0760286
$693$	−4.00000	−0.151947
$694$	−36.0000	−1.36654
$695$	24.0000	0.910372
$696$	2.00000	0.0758098
$697$	36.0000	1.36360
$698$	−2.00000	−0.0757011
$699$	−10.0000	−0.378235
$700$	−1.00000	−0.0377964
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	2.00000	0.0754851
$703$	24.0000	0.905177
$704$	−4.00000	−0.150756
$705$	−16.0000	−0.602595
$706$	−30.0000	−1.12906
$707$	6.00000	0.225653
$708$	−4.00000	−0.150329
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	2.00000	0.0749532
$713$	8.00000	0.299602
$714$	6.00000	0.224544
$715$	−16.0000	−0.598366
$716$	−12.0000	−0.448461
$717$	16.0000	0.597531
$718$	−24.0000	−0.895672
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−2.00000	−0.0745356
$721$	8.00000	0.297936
$722$	−3.00000	−0.111648
$723$	22.0000	0.818189
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	24.0000	0.887672
$732$	10.0000	0.369611
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	−1.00000	−0.0368605
$737$	−16.0000	−0.589368
$738$	−6.00000	−0.220863
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−12.0000	−0.441129
$741$	8.00000	0.293887
$742$	6.00000	0.220267
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	8.00000	0.293294
$745$	−12.0000	−0.439646
$746$	6.00000	0.219676
$747$	−12.0000	−0.439057
$748$	24.0000	0.877527
$749$	12.0000	0.438470
$750$	−12.0000	−0.438178
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−8.00000	−0.291730
$753$	−12.0000	−0.437304
$754$	4.00000	0.145671
$755$	−16.0000	−0.582300
$756$	−1.00000	−0.0363696
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−36.0000	−1.30758
$759$	−4.00000	−0.145191
$760$	−8.00000	−0.290191
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	16.0000	0.579619
$763$	14.0000	0.506834
$764$	−16.0000	−0.578860
$765$	12.0000	0.433861
$766$	16.0000	0.578103
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	8.00000	0.288300
$771$	−18.0000	−0.648254
$772$	2.00000	0.0719816
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	−4.00000	−0.143777
$775$	8.00000	0.287368
$776$	10.0000	0.358979
$777$	−6.00000	−0.215249
$778$	6.00000	0.215110
$779$	−24.0000	−0.859889
$780$	−4.00000	−0.143223
$781$	32.0000	1.14505
$782$	6.00000	0.214560
$783$	2.00000	0.0714742
$784$	1.00000	0.0357143
$785$	−44.0000	−1.57043
$786$	−12.0000	−0.428026
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	−26.0000	−0.926212
$789$	24.0000	0.854423
$790$	0	0
$791$	18.0000	0.640006
$792$	−4.00000	−0.142134
$793$	20.0000	0.710221
$794$	−2.00000	−0.0709773
$795$	12.0000	0.425596
$796$	−24.0000	−0.850657
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	−4.00000	−0.141598
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	2.00000	0.0706665
$802$	34.0000	1.20058
$803$	24.0000	0.846942
$804$	−4.00000	−0.141069
$805$	2.00000	0.0704907
$806$	16.0000	0.563576
$807$	2.00000	0.0704033
$808$	6.00000	0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	−2.00000	−0.0702728
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	−2.00000	−0.0701862
$813$	−24.0000	−0.841717
$814$	−24.0000	−0.841200
$815$	−8.00000	−0.280228
$816$	6.00000	0.210042
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	−2.00000	−0.0698857
$820$	12.0000	0.419058
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	6.00000	0.209274
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	8.00000	0.278693
$825$	−4.00000	−0.139262
$826$	4.00000	0.139178
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−1.00000	−0.0347524
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	24.0000	0.833052
$831$	−22.0000	−0.763172
$832$	−2.00000	−0.0693375
$833$	−6.00000	−0.207888
$834$	12.0000	0.415526
$835$	0	0
$836$	−16.0000	−0.553372
$837$	8.00000	0.276520
$838$	−12.0000	−0.414533
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	2.00000	0.0690066
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	−10.0000	−0.344418
$844$	4.00000	0.137686
$845$	18.0000	0.619219
$846$	−8.00000	−0.275046
$847$	5.00000	0.171802
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	6.00000	0.205798
$851$	−6.00000	−0.205677
$852$	8.00000	0.274075
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−10.0000	−0.342193
$855$	−8.00000	−0.273594
$856$	12.0000	0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−8.00000	−0.273115
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	8.00000	0.272798
$861$	6.00000	0.204479
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−1.00000	−0.0340207
$865$	4.00000	0.136004
$866$	26.0000	0.883516
$867$	−19.0000	−0.645274
$868$	−8.00000	−0.271538
$869$	0	0
$870$	−4.00000	−0.135613
$871$	−8.00000	−0.271070
$872$	14.0000	0.474100
$873$	10.0000	0.338449
$874$	−4.00000	−0.135302
$875$	12.0000	0.405674
$876$	6.00000	0.202721
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−16.0000	−0.539974
$879$	2.00000	0.0674583
$880$	8.00000	0.269680
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	1.00000	0.0336718
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	12.0000	0.403604
$885$	8.00000	0.268917
$886$	−20.0000	−0.671913
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−6.00000	−0.201347
$889$	−16.0000	−0.536623
$890$	−4.00000	−0.134080
$891$	−4.00000	−0.134005
$892$	24.0000	0.803579
$893$	−32.0000	−1.07084
$894$	−6.00000	−0.200670
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	−2.00000	−0.0667781
$898$	2.00000	0.0667409
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−36.0000	−1.19933
$902$	24.0000	0.799113
$903$	4.00000	0.133112
$904$	18.0000	0.598671
$905$	4.00000	0.132964
$906$	−8.00000	−0.265782
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	4.00000	0.132599
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−4.00000	−0.132453
$913$	48.0000	1.58857
$914$	−22.0000	−0.727695
$915$	−20.0000	−0.661180
$916$	14.0000	0.462573
$917$	12.0000	0.396275
$918$	6.00000	0.198030
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	2.00000	0.0659380
$921$	−12.0000	−0.395413
$922$	30.0000	0.987997
$923$	16.0000	0.526646
$924$	4.00000	0.131590
$925$	−6.00000	−0.197279
$926$	0	0
$927$	8.00000	0.262754
$928$	−2.00000	−0.0656532
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−16.0000	−0.524661
$931$	4.00000	0.131095
$932$	10.0000	0.327561
$933$	16.0000	0.523816
$934$	4.00000	0.130884
$935$	−48.0000	−1.56977
$936$	−2.00000	−0.0653720
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	4.00000	0.130605
$939$	14.0000	0.456873
$940$	16.0000	0.521862
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−22.0000	−0.716799
$943$	6.00000	0.195387
$944$	4.00000	0.130189
$945$	2.00000	0.0650600
$946$	16.0000	0.520205
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	−4.00000	−0.129777
$951$	2.00000	0.0648544
$952$	−6.00000	−0.194461
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	32.0000	1.03550
$956$	−16.0000	−0.517477
$957$	−8.00000	−0.258603
$958$	−32.0000	−1.03387
$959$	−6.00000	−0.193750
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	−22.0000	−0.708572
$965$	−4.00000	−0.128765
$966$	1.00000	0.0321745
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	5.00000	0.160706
$969$	24.0000	0.770991
$970$	−20.0000	−0.642161
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	−1.00000	−0.0320750
$973$	−12.0000	−0.384702
$974$	8.00000	0.256337
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	−62.0000	−1.98356	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000	−1.98356	−0.991778	+	0.127971i	$0.959153\pi$
$978$	−4.00000	−0.127906
$979$	−8.00000	−0.255681
$980$	−2.00000	−0.0638877
$981$	14.0000	0.446986
$982$	−4.00000	−0.127645
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	6.00000	0.191273
$985$	52.0000	1.65686
$986$	12.0000	0.382158
$987$	8.00000	0.254643
$988$	−8.00000	−0.254514
$989$	4.00000	0.127193
$990$	8.00000	0.254257
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−8.00000	−0.254000
$993$	4.00000	0.126936
$994$	−8.00000	−0.253745
$995$	48.0000	1.52170
$996$	12.0000	0.380235
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−28.0000	−0.886325
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.a.g.1.1	✓	1
3.2	odd	2		2898.2.a.i.1.1		1
4.3	odd	2		7728.2.a.o.1.1		1
7.6	odd	2		6762.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.g.1.1	✓	1	1.1	even	1	trivial
2898.2.a.i.1.1		1	3.2	odd	2
6762.2.a.bl.1.1		1	7.6	odd	2
7728.2.a.o.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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