LMFDB - Embedded newform 966.2.a.j.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:	$0$
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	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +2.00000 q^{70} -16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{77} +2.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -16.0000 q^{83} -1.00000 q^{84} -12.0000 q^{85} +12.0000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} +4.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	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	$0.335828\pi$
$38$	0	0
$39$	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
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	4.00000	0.603023
$45$	−2.00000	−0.298142
$46$	1.00000	0.147442
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	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
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−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	$0.721142\pi$
$62$	4.00000	0.508001
$63$	−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
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	2.00000	0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−1.00000	−0.109109
$85$	−12.0000	−1.30158
$86$	12.0000	1.29399
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	1.00000	0.104257
$93$	4.00000	0.414781
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	6.00000	0.594089
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	2.00000	0.195180
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−8.00000	−0.762770
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−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
$124$	4.00000	0.359211
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	−4.00000	−0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	−2.00000	−0.172133
$136$	6.00000	0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	1.00000	0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	2.00000	0.169031
$141$	−12.0000	−1.01058
$142$	−16.0000	−1.34269
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	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
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	8.00000	0.636446
$159$	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	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.00000	−0.468521
$165$	−8.00000	−0.622799
$166$	−16.0000	−1.24184
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	0	0
$172$	12.0000	0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−2.00000	−0.151620
$175$	1.00000	0.0755929
$176$	4.00000	0.301511
$177$	−4.00000	−0.300658
$178$	6.00000	0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−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
$186$	4.00000	0.293294
$187$	24.0000	1.75505
$188$	−12.0000	−0.875190
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	−2.00000	−0.143592
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−14.0000	−0.985037
$203$	2.00000	0.140372
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	0	0
$207$	1.00000	0.0695048
$208$	2.00000	0.138675
$209$	0	0
$210$	2.00000	0.138013
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	6.00000	0.412082
$213$	−16.0000	−1.09630
$214$	−12.0000	−0.820303
$215$	−24.0000	−1.63679
$216$	1.00000	0.0680414
$217$	−4.00000	−0.271538
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	−8.00000	−0.539360
$221$	12.0000	0.807207
$222$	6.00000	0.402694
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−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
$235$	24.0000	1.56559
$236$	−4.00000	−0.260378
$237$	8.00000	0.519656
$238$	−6.00000	−0.388922
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−2.00000	−0.129099
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	−2.00000	−0.127775
$246$	−6.00000	−0.382546
$247$	0	0
$248$	4.00000	0.254000
$249$	−16.0000	−1.01396
$250$	12.0000	0.758947
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	−8.00000	−0.501965
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	12.0000	0.747087
$259$	−6.00000	−0.372822
$260$	−4.00000	−0.248069
$261$	−2.00000	−0.123797
$262$	4.00000	0.247121
$263$	−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
$266$	0	0
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	−2.00000	−0.121046
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	1.00000	0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	2.00000	0.119523
$281$	−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
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	8.00000	0.473050
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	4.00000	0.234888
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	−34.0000	−1.98630	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−34.0000	−1.98630	−0.993151	+	0.116841i	$0.962723\pi$
$294$	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
$301$	−12.0000	−0.691669
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	20.0000	1.14520
$306$	6.00000	0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	−8.00000	−0.454369
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−18.0000	−1.01580
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	6.00000	0.336463
$319$	−8.00000	−0.447914
$320$	−2.00000	−0.111803
$321$	−12.0000	−0.669775
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	−6.00000	−0.331295
$329$	12.0000	0.661581
$330$	−8.00000	−0.440386
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−16.0000	−0.878114
$333$	6.00000	0.328798
$334$	20.0000	1.09435
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	−12.0000	−0.650791
$341$	16.0000	0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	12.0000	0.646997
$345$	−2.00000	−0.107676
$346$	−6.00000	−0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−2.00000	−0.107211
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−4.00000	−0.212598
$355$	32.0000	1.69838
$356$	6.00000	0.317999
$357$	−6.00000	−0.317554
$358$	4.00000	0.211407
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	−2.00000	−0.105409
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	5.00000	0.262432
$364$	−2.00000	−0.104828
$365$	−4.00000	−0.209370
$366$	−10.0000	−0.522708
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	1.00000	0.0521286
$369$	−6.00000	−0.312348
$370$	−12.0000	−0.623850
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	24.0000	1.24101
$375$	12.0000	0.619677
$376$	−12.0000	−0.618853
$377$	−4.00000	−0.206010
$378$	−1.00000	−0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	8.00000	0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	8.00000	0.407718
$386$	18.0000	0.916176
$387$	12.0000	0.609994
$388$	−2.00000	−0.101535
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	−4.00000	−0.202548
$391$	6.00000	0.303433
$392$	1.00000	0.0505076
$393$	4.00000	0.201773
$394$	−18.0000	−0.906827
$395$	−16.0000	−0.805047
$396$	4.00000	0.201008
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000	0.199502
$403$	8.00000	0.398508
$404$	−14.0000	−0.696526
$405$	−2.00000	−0.0993808
$406$	2.00000	0.0992583
$407$	24.0000	1.18964
$408$	6.00000	0.297044
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	12.0000	0.592638
$411$	10.0000	0.493264
$412$	0	0
$413$	4.00000	0.196827
$414$	1.00000	0.0491473
$415$	32.0000	1.57082
$416$	2.00000	0.0980581
$417$	4.00000	0.195881
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	2.00000	0.0975900
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	28.0000	1.36302
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	−16.0000	−0.775203
$427$	10.0000	0.483934
$428$	−12.0000	−0.580042
$429$	8.00000	0.386244
$430$	−24.0000	−1.15738
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−4.00000	−0.192006
$435$	4.00000	0.191785
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	2.00000	0.0955637
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−8.00000	−0.381385
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	6.00000	0.284747
$445$	−12.0000	−0.568855
$446$	−4.00000	−0.189405
$447$	6.00000	0.283790
$448$	−1.00000	−0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−24.0000	−1.12638
$455$	4.00000	0.187523
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−18.0000	−0.841085
$459$	6.00000	0.280056
$460$	−2.00000	−0.0932505
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	−4.00000	−0.186097
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−2.00000	−0.0928477
$465$	−8.00000	−0.370991
$466$	10.0000	0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	2.00000	0.0924500
$469$	−4.00000	−0.184703
$470$	24.0000	1.10704
$471$	−18.0000	−0.829396
$472$	−4.00000	−0.184115
$473$	48.0000	2.20704
$474$	8.00000	0.367452
$475$	0	0
$476$	−6.00000	−0.275010
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	−2.00000	−0.0912871
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	−1.00000	−0.0455016
$484$	5.00000	0.227273
$485$	4.00000	0.181631
$486$	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
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	16.0000	0.717698
$498$	−16.0000	−0.716977
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	12.0000	0.536656
$501$	20.0000	0.893534
$502$	24.0000	1.07117
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−1.00000	−0.0445435
$505$	28.0000	1.24598
$506$	4.00000	0.177822
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−12.0000	−0.531369
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	12.0000	0.528271
$517$	−48.0000	−2.11104
$518$	−6.00000	−0.263625
$519$	−6.00000	−0.263371
$520$	−4.00000	−0.175412
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−2.00000	−0.0875376
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	4.00000	0.174741
$525$	1.00000	0.0436436
$526$	−24.0000	−1.04645
$527$	24.0000	1.04546
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−12.0000	−0.519778
$534$	6.00000	0.259645
$535$	24.0000	1.03761
$536$	4.00000	0.172774
$537$	4.00000	0.172613
$538$	−6.00000	−0.258678
$539$	4.00000	0.172292
$540$	−2.00000	−0.0860663
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	20.0000	0.859074
$543$	−2.00000	−0.0858282
$544$	6.00000	0.257248
$545$	4.00000	0.171341
$546$	−2.00000	−0.0855921
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	10.0000	0.427179
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	0	0
$552$	1.00000	0.0425628
$553$	−8.00000	−0.340195
$554$	−10.0000	−0.424859
$555$	−12.0000	−0.509372
$556$	4.00000	0.169638
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	4.00000	0.169334
$559$	24.0000	1.01509
$560$	2.00000	0.0845154
$561$	24.0000	1.01328
$562$	−22.0000	−0.928014
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−12.0000	−0.505291
$565$	12.0000	0.504844
$566$	24.0000	1.00880
$567$	−1.00000	−0.0419961
$568$	−16.0000	−0.671345
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−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
$573$	8.00000	0.334205
$574$	6.00000	0.250435
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	19.0000	0.790296
$579$	18.0000	0.748054
$580$	4.00000	0.166091
$581$	16.0000	0.663792
$582$	−2.00000	−0.0829027
$583$	24.0000	0.993978
$584$	2.00000	0.0827606
$585$	−4.00000	−0.165380
$586$	−34.0000	−1.40453
$587$	−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
$589$	0	0
$590$	8.00000	0.329355
$591$	−18.0000	−0.740421
$592$	6.00000	0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	4.00000	0.164122
$595$	12.0000	0.491952
$596$	6.00000	0.245770
$597$	0	0
$598$	2.00000	0.0817861
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	−1.00000	−0.0408248
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	4.00000	0.162893
$604$	0	0
$605$	−10.0000	−0.406558
$606$	−14.0000	−0.568711
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	20.0000	0.809776
$611$	−24.0000	−0.970936
$612$	6.00000	0.242536
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−20.0000	−0.807134
$615$	12.0000	0.483887
$616$	−4.00000	−0.161165
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−8.00000	−0.321288
$621$	1.00000	0.0401286
$622$	12.0000	0.481156
$623$	−6.00000	−0.240385
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	0	0
$628$	−18.0000	−0.718278
$629$	36.0000	1.43541
$630$	2.00000	0.0796819
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	8.00000	0.318223
$633$	28.0000	1.11290
$634$	−10.0000	−0.397151
$635$	16.0000	0.634941
$636$	6.00000	0.237915
$637$	2.00000	0.0792429
$638$	−8.00000	−0.316723
$639$	−16.0000	−0.632950
$640$	−2.00000	−0.0790569
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	−1.00000	−0.0394055
$645$	−24.0000	−0.944999
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	−2.00000	−0.0784465
$651$	−4.00000	−0.156772
$652$	−4.00000	−0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−2.00000	−0.0782062
$655$	−8.00000	−0.312586
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	12.0000	0.467809
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−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	$0.613839\pi$
$662$	28.0000	1.08825
$663$	12.0000	0.466041
$664$	−16.0000	−0.620920
$665$	0	0
$666$	6.00000	0.232495
$667$	−2.00000	−0.0774403
$668$	20.0000	0.773823
$669$	−4.00000	−0.154649
$670$	−8.00000	−0.309067
$671$	−40.0000	−1.54418
$672$	−1.00000	−0.0385758
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−22.0000	−0.847408
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−6.00000	−0.230429
$679$	2.00000	0.0767530
$680$	−12.0000	−0.460179
$681$	−24.0000	−0.919682
$682$	16.0000	0.612672
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−20.0000	−0.764161
$686$	−1.00000	−0.0381802
$687$	−18.0000	−0.686743
$688$	12.0000	0.457496
$689$	12.0000	0.457164
$690$	−2.00000	−0.0761387
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	−4.00000	−0.151947
$694$	−12.0000	−0.455514
$695$	−8.00000	−0.303457
$696$	−2.00000	−0.0758098
$697$	−36.0000	−1.36360
$698$	−6.00000	−0.227103
$699$	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
$703$	0	0
$704$	4.00000	0.150756
$705$	24.0000	0.903892
$706$	−14.0000	−0.526897
$707$	14.0000	0.526524
$708$	−4.00000	−0.150329
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	32.0000	1.20094
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	4.00000	0.149801
$714$	−6.00000	−0.224544
$715$	−16.0000	−0.598366
$716$	4.00000	0.149487
$717$	8.00000	0.298765
$718$	−16.0000	−0.597115
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−10.0000	−0.371904
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	5.00000	0.185567
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	72.0000	2.66302
$732$	−10.0000	−0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−24.0000	−0.885856
$735$	−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	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−6.00000	−0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	4.00000	0.146647
$745$	−12.0000	−0.439646
$746$	22.0000	0.805477
$747$	−16.0000	−0.585409
$748$	24.0000	0.877527
$749$	12.0000	0.438470
$750$	12.0000	0.438178
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−12.0000	−0.437595
$753$	24.0000	0.874609
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	20.0000	0.726433
$759$	4.00000	0.145191
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−8.00000	−0.289809
$763$	2.00000	0.0724049
$764$	8.00000	0.289430
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−8.00000	−0.288863
$768$	1.00000	0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	8.00000	0.288300
$771$	−6.00000	−0.216085
$772$	18.0000	0.647834
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	12.0000	0.431331
$775$	−4.00000	−0.143684
$776$	−2.00000	−0.0717958
$777$	−6.00000	−0.215249
$778$	38.0000	1.36237
$779$	0	0
$780$	−4.00000	−0.143223
$781$	−64.0000	−2.29010
$782$	6.00000	0.214560
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	36.0000	1.28490
$786$	4.00000	0.142675
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−18.0000	−0.641223
$789$	−24.0000	−0.854423
$790$	−16.0000	−0.569254
$791$	6.00000	0.213335
$792$	4.00000	0.142134
$793$	−20.0000	−0.710221
$794$	34.0000	1.20661
$795$	−12.0000	−0.425596
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	8.00000	0.282314
$804$	4.00000	0.141069
$805$	2.00000	0.0704907
$806$	8.00000	0.281788
$807$	−6.00000	−0.211210
$808$	−14.0000	−0.492518
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−2.00000	−0.0702728
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	2.00000	0.0701862
$813$	20.0000	0.701431
$814$	24.0000	0.841200
$815$	8.00000	0.280228
$816$	6.00000	0.210042
$817$	0	0
$818$	26.0000	0.909069
$819$	−2.00000	−0.0698857
$820$	12.0000	0.419058
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	10.0000	0.348790
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	4.00000	0.139178
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	1.00000	0.0347524
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	32.0000	1.11074
$831$	−10.0000	−0.346896
$832$	2.00000	0.0693375
$833$	6.00000	0.207888
$834$	4.00000	0.138509
$835$	−40.0000	−1.38426
$836$	0	0
$837$	4.00000	0.138260
$838$	16.0000	0.552711
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	2.00000	0.0690066
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	−22.0000	−0.757720
$844$	28.0000	0.963800
$845$	18.0000	0.619219
$846$	−12.0000	−0.412568
$847$	−5.00000	−0.171802
$848$	6.00000	0.206041
$849$	24.0000	0.823678
$850$	−6.00000	−0.205798
$851$	6.00000	0.205677
$852$	−16.0000	−0.548151
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	8.00000	0.273115
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	−24.0000	−0.818393
$861$	6.00000	0.204479
$862$	16.0000	0.544962
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	1.00000	0.0340207
$865$	12.0000	0.408012
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	−4.00000	−0.135769
$869$	32.0000	1.08553
$870$	4.00000	0.135613
$871$	8.00000	0.271070
$872$	−2.00000	−0.0677285
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−12.0000	−0.405674
$876$	2.00000	0.0675737
$877$	−26.0000	−0.877958	−0.438979	−	0.898497i	$-0.644660\pi$
$877$	−26.0000	−0.877958	−0.438979	+	0.898497i	$0.644660\pi$
$878$	20.0000	0.674967
$879$	−34.0000	−1.14679
$880$	−8.00000	−0.269680
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	1.00000	0.0336718
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	12.0000	0.403604
$885$	8.00000	0.268917
$886$	20.0000	0.671913
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	6.00000	0.201347
$889$	8.00000	0.268311
$890$	−12.0000	−0.402241
$891$	4.00000	0.134005
$892$	−4.00000	−0.133930
$893$	0	0
$894$	6.00000	0.200670
$895$	−8.00000	−0.267411
$896$	−1.00000	−0.0334077
$897$	2.00000	0.0667781
$898$	−30.0000	−1.00111
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	36.0000	1.19933
$902$	−24.0000	−0.799113
$903$	−12.0000	−0.399335
$904$	−6.00000	−0.199557
$905$	4.00000	0.132964
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−24.0000	−0.796468
$909$	−14.0000	−0.464351
$910$	4.00000	0.132599
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−64.0000	−2.11809
$914$	−6.00000	−0.198462
$915$	20.0000	0.661180
$916$	−18.0000	−0.594737
$917$	−4.00000	−0.132092
$918$	6.00000	0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−2.00000	−0.0659380
$921$	−20.0000	−0.659022
$922$	−22.0000	−0.724531
$923$	−32.0000	−1.05329
$924$	−4.00000	−0.131590
$925$	−6.00000	−0.197279
$926$	−32.0000	−1.05159
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	10.0000	0.327561
$933$	12.0000	0.392862
$934$	8.00000	0.261768
$935$	−48.0000	−1.56977
$936$	2.00000	0.0653720
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	−4.00000	−0.130605
$939$	14.0000	0.456873
$940$	24.0000	0.782794
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−18.0000	−0.586472
$943$	−6.00000	−0.195387
$944$	−4.00000	−0.130189
$945$	2.00000	0.0650600
$946$	48.0000	1.56061
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	4.00000	0.129845
$950$	0	0
$951$	−10.0000	−0.324272
$952$	−6.00000	−0.194461
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	6.00000	0.194257
$955$	−16.0000	−0.517748
$956$	8.00000	0.258738
$957$	−8.00000	−0.258603
$958$	40.0000	1.29234
$959$	−10.0000	−0.322917
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	−36.0000	−1.15888
$966$	−1.00000	−0.0321745
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	4.00000	0.128432
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	8.00000	0.256337
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−4.00000	−0.127906
$979$	24.0000	0.767043
$980$	−2.00000	−0.0638877
$981$	−2.00000	−0.0638551
$982$	−12.0000	−0.382935
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−6.00000	−0.191273
$985$	36.0000	1.14706
$986$	−12.0000	−0.382158
$987$	12.0000	0.381964
$988$	0	0
$989$	12.0000	0.381578
$990$	−8.00000	−0.254257
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	4.00000	0.127000
$993$	28.0000	0.888553
$994$	16.0000	0.507489
$995$	0	0
$996$	−16.0000	−0.506979
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	−36.0000	−1.13956
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.a.j.1.1	✓	1
3.2	odd	2		2898.2.a.f.1.1		1
4.3	odd	2		7728.2.a.b.1.1		1
7.6	odd	2		6762.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.j.1.1	✓	1	1.1	even	1	trivial
2898.2.a.f.1.1		1	3.2	odd	2
6762.2.a.bc.1.1		1	7.6	odd	2
7728.2.a.b.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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