LMFDB - Embedded newform 966.2.a.i.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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +8.00000 q^{19} -3.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +8.00000 q^{38} +5.00000 q^{39} -3.00000 q^{40} +9.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} -3.00000 q^{45} -1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} +3.00000 q^{58} -6.00000 q^{59} -3.00000 q^{60} +14.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -15.0000 q^{65} -4.00000 q^{67} -1.00000 q^{69} -3.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -7.00000 q^{74} +4.00000 q^{75} +8.00000 q^{76} +5.00000 q^{78} -16.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} -1.00000 q^{86} +3.00000 q^{87} +6.00000 q^{89} -3.00000 q^{90} +5.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} -3.00000 q^{94} -24.0000 q^{95} +1.00000 q^{96} -1.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	1.00000	0.267261
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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	$0.130073\pi$
$20$	−3.00000	−0.670820
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−3.00000	−0.547723
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	8.00000	1.29777
$39$	5.00000	0.800641
$40$	−3.00000	−0.474342
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	1.00000	0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	−1.00000	−0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	8.00000	1.05963
$58$	3.00000	0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−3.00000	−0.387298
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	2.00000	0.254000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	−3.00000	−0.358569
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−7.00000	−0.813733
$75$	4.00000	0.461880
$76$	8.00000	0.917663
$77$	0	0
$78$	5.00000	0.566139
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−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
$85$	0	0
$86$	−1.00000	−0.107833
$87$	3.00000	0.321634
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−3.00000	−0.316228
$91$	5.00000	0.524142
$92$	−1.00000	−0.104257
$93$	2.00000	0.207390
$94$	−3.00000	−0.309426
$95$	−24.0000	−2.46235
$96$	1.00000	0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	5.00000	0.490290
$105$	−3.00000	−0.292770
$106$	−12.0000	−1.16554
$107$	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
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	1.00000	0.0944911
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	8.00000	0.749269
$115$	3.00000	0.279751
$116$	3.00000	0.278543
$117$	5.00000	0.462250
$118$	−6.00000	−0.552345
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−11.0000	−1.00000
$122$	14.0000	1.26750
$123$	9.00000	0.811503
$124$	2.00000	0.179605
$125$	3.00000	0.268328
$126$	1.00000	0.0890871
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	−15.0000	−1.31559
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	−4.00000	−0.345547
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−1.00000	−0.0851257
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−3.00000	−0.253546
$141$	−3.00000	−0.252646
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−9.00000	−0.747409
$146$	−4.00000	−0.331042
$147$	1.00000	0.0824786
$148$	−7.00000	−0.575396
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	4.00000	0.326599
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	8.00000	0.648886
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	5.00000	0.400320
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−16.0000	−1.27289
$159$	−12.0000	−0.951662
$160$	−3.00000	−0.237171
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	1.00000	0.0771517
$169$	12.0000	0.923077
$170$	0	0
$171$	8.00000	0.611775
$172$	−1.00000	−0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	3.00000	0.227429
$175$	4.00000	0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	6.00000	0.449719
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	−3.00000	−0.223607
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	5.00000	0.370625
$183$	14.0000	1.03491
$184$	−1.00000	−0.0737210
$185$	21.0000	1.54395
$186$	2.00000	0.146647
$187$	0	0
$188$	−3.00000	−0.218797
$189$	1.00000	0.0727393
$190$	−24.0000	−1.74114
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	−1.00000	−0.0717958
$195$	−15.0000	−1.07417
$196$	1.00000	0.0714286
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	4.00000	0.282843
$201$	−4.00000	−0.282138
$202$	−6.00000	−0.422159
$203$	3.00000	0.210559
$204$	0	0
$205$	−27.0000	−1.88576
$206$	−1.00000	−0.0696733
$207$	−1.00000	−0.0695048
$208$	5.00000	0.346688
$209$	0	0
$210$	−3.00000	−0.207020
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−12.0000	−0.824163
$213$	6.00000	0.411113
$214$	12.0000	0.820303
$215$	3.00000	0.204598
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	−19.0000	−1.28684
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	−7.00000	−0.469809
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	1.00000	0.0668153
$225$	4.00000	0.266667
$226$	15.0000	0.997785
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	8.00000	0.529813
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	5.00000	0.326860
$235$	9.00000	0.587095
$236$	−6.00000	−0.390567
$237$	−16.0000	−1.03931
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−3.00000	−0.193649
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	−3.00000	−0.191663
$246$	9.00000	0.573819
$247$	40.0000	2.54514
$248$	2.00000	0.127000
$249$	−12.0000	−0.760469
$250$	3.00000	0.189737
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−13.0000	−0.815693
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−1.00000	−0.0622573
$259$	−7.00000	−0.434959
$260$	−15.0000	−0.930261
$261$	3.00000	0.185695
$262$	−18.0000	−1.11204
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	8.00000	0.490511
$267$	6.00000	0.367194
$268$	−4.00000	−0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−3.00000	−0.182574
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	5.00000	0.302614
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	5.00000	0.299880
$279$	2.00000	0.119737
$280$	−3.00000	−0.179284
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	−3.00000	−0.178647
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	−24.0000	−1.42164
$286$	0	0
$287$	9.00000	0.531253
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	−9.00000	−0.528498
$291$	−1.00000	−0.0586210
$292$	−4.00000	−0.234082
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	1.00000	0.0583212
$295$	18.0000	1.04800
$296$	−7.00000	−0.406867
$297$	0	0
$298$	0	0
$299$	−5.00000	−0.289157
$300$	4.00000	0.230940
$301$	−1.00000	−0.0576390
$302$	11.0000	0.632979
$303$	−6.00000	−0.344691
$304$	8.00000	0.458831
$305$	−42.0000	−2.40491
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	−6.00000	−0.340777
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	5.00000	0.283069
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−4.00000	−0.225733
$315$	−3.00000	−0.169031
$316$	−16.0000	−0.900070
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	−3.00000	−0.167705
$321$	12.0000	0.669775
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	20.0000	1.10770
$327$	−19.0000	−1.05070
$328$	9.00000	0.496942
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	−12.0000	−0.658586
$333$	−7.00000	−0.383598
$334$	−12.0000	−0.656611
$335$	12.0000	0.655630
$336$	1.00000	0.0545545
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	12.0000	0.652714
$339$	15.0000	0.814688
$340$	0	0
$341$	0	0
$342$	8.00000	0.432590
$343$	1.00000	0.0539949
$344$	−1.00000	−0.0539164
$345$	3.00000	0.161515
$346$	6.00000	0.322562
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	3.00000	0.160817
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	4.00000	0.213809
$351$	5.00000	0.266880
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	−6.00000	−0.318896
$355$	−18.0000	−0.955341
$356$	6.00000	0.317999
$357$	0	0
$358$	15.0000	0.792775
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	−3.00000	−0.158114
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	−11.0000	−0.577350
$364$	5.00000	0.262071
$365$	12.0000	0.628109
$366$	14.0000	0.731792
$367$	29.0000	1.51379	0.756894	−	0.653538i	$-0.226716\pi$
$367$	29.0000	1.51379	0.756894	+	0.653538i	$0.226716\pi$
$368$	−1.00000	−0.0521286
$369$	9.00000	0.468521
$370$	21.0000	1.09174
$371$	−12.0000	−0.623009
$372$	2.00000	0.103695
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	−3.00000	−0.154713
$377$	15.0000	0.772539
$378$	1.00000	0.0514344
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	−24.0000	−1.23117
$381$	−13.0000	−0.666010
$382$	−12.0000	−0.613973
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−1.00000	−0.0508987
$387$	−1.00000	−0.0508329
$388$	−1.00000	−0.0507673
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−15.0000	−0.759555
$391$	0	0
$392$	1.00000	0.0505076
$393$	−18.0000	−0.907980
$394$	9.00000	0.453413
$395$	48.0000	2.41514
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	17.0000	0.852133
$399$	8.00000	0.400501
$400$	4.00000	0.200000
$401$	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
$403$	10.0000	0.498135
$404$	−6.00000	−0.298511
$405$	−3.00000	−0.149071
$406$	3.00000	0.148888
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−27.0000	−1.33343
$411$	−9.00000	−0.443937
$412$	−1.00000	−0.0492665
$413$	−6.00000	−0.295241
$414$	−1.00000	−0.0491473
$415$	36.0000	1.76717
$416$	5.00000	0.245145
$417$	5.00000	0.244851
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	−3.00000	−0.146385
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	8.00000	0.389434
$423$	−3.00000	−0.145865
$424$	−12.0000	−0.582772
$425$	0	0
$426$	6.00000	0.290701
$427$	14.0000	0.677507
$428$	12.0000	0.580042
$429$	0	0
$430$	3.00000	0.144673
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	1.00000	0.0481125
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	2.00000	0.0960031
$435$	−9.00000	−0.431517
$436$	−19.0000	−0.909935
$437$	−8.00000	−0.382692
$438$	−4.00000	−0.191127
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−27.0000	−1.28281	−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000	−1.28281	−0.641404	+	0.767203i	$0.721648\pi$
$444$	−7.00000	−0.332205
$445$	−18.0000	−0.853282
$446$	14.0000	0.662919
$447$	0	0
$448$	1.00000	0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	15.0000	0.705541
$453$	11.0000	0.516825
$454$	−15.0000	−0.703985
$455$	−15.0000	−0.703211
$456$	8.00000	0.374634
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	3.00000	0.139876
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	3.00000	0.139272
$465$	−6.00000	−0.278243
$466$	−6.00000	−0.277945
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	5.00000	0.231125
$469$	−4.00000	−0.184703
$470$	9.00000	0.415139
$471$	−4.00000	−0.184310
$472$	−6.00000	−0.276172
$473$	0	0
$474$	−16.0000	−0.734904
$475$	32.0000	1.46826
$476$	0	0
$477$	−12.0000	−0.549442
$478$	12.0000	0.548867
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	−3.00000	−0.136931
$481$	−35.0000	−1.59586
$482$	−19.0000	−0.865426
$483$	−1.00000	−0.0455016
$484$	−11.0000	−0.500000
$485$	3.00000	0.136223
$486$	1.00000	0.0453609
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	14.0000	0.633750
$489$	20.0000	0.904431
$490$	−3.00000	−0.135526
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	9.00000	0.405751
$493$	0	0
$494$	40.0000	1.79969
$495$	0	0
$496$	2.00000	0.0898027
$497$	6.00000	0.269137
$498$	−12.0000	−0.537733
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	3.00000	0.134164
$501$	−12.0000	−0.536120
$502$	−15.0000	−0.669483
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	1.00000	0.0445435
$505$	18.0000	0.800989
$506$	0	0
$507$	12.0000	0.532939
$508$	−13.0000	−0.576782
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	8.00000	0.353209
$514$	6.00000	0.264649
$515$	3.00000	0.132196
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	−7.00000	−0.307562
$519$	6.00000	0.263371
$520$	−15.0000	−0.657794
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	3.00000	0.131306
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	−18.0000	−0.786334
$525$	4.00000	0.174574
$526$	−9.00000	−0.392419
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	36.0000	1.56374
$531$	−6.00000	−0.260378
$532$	8.00000	0.346844
$533$	45.0000	1.94917
$534$	6.00000	0.259645
$535$	−36.0000	−1.55642
$536$	−4.00000	−0.172774
$537$	15.0000	0.647298
$538$	−18.0000	−0.776035
$539$	0	0
$540$	−3.00000	−0.129099
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−28.0000	−1.20270
$543$	2.00000	0.0858282
$544$	0	0
$545$	57.0000	2.44161
$546$	5.00000	0.213980
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	−9.00000	−0.384461
$549$	14.0000	0.597505
$550$	0	0
$551$	24.0000	1.02243
$552$	−1.00000	−0.0425628
$553$	−16.0000	−0.680389
$554$	−4.00000	−0.169944
$555$	21.0000	0.891400
$556$	5.00000	0.212047
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	2.00000	0.0846668
$559$	−5.00000	−0.211477
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−15.0000	−0.632737
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	−3.00000	−0.126323
$565$	−45.0000	−1.89316
$566$	14.0000	0.588464
$567$	1.00000	0.0419961
$568$	6.00000	0.251754
$569$	45.0000	1.88650	0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000	1.88650	0.943249	+	0.332086i	$0.107752\pi$
$570$	−24.0000	−1.00525
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	9.00000	0.375653
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−17.0000	−0.707107
$579$	−1.00000	−0.0415586
$580$	−9.00000	−0.373705
$581$	−12.0000	−0.497844
$582$	−1.00000	−0.0414513
$583$	0	0
$584$	−4.00000	−0.165521
$585$	−15.0000	−0.620174
$586$	−30.0000	−1.23929
$587$	−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
$589$	16.0000	0.659269
$590$	18.0000	0.741048
$591$	9.00000	0.370211
$592$	−7.00000	−0.287698
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.0000	0.695764
$598$	−5.00000	−0.204465
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	4.00000	0.163299
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	−1.00000	−0.0407570
$603$	−4.00000	−0.162893
$604$	11.0000	0.447584
$605$	33.0000	1.34164
$606$	−6.00000	−0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	8.00000	0.324443
$609$	3.00000	0.121566
$610$	−42.0000	−1.70053
$611$	−15.0000	−0.606835
$612$	0	0
$613$	−1.00000	−0.0403896	−0.0201948	−	0.999796i	$-0.506429\pi$
$613$	−1.00000	−0.0403896	−0.0201948	+	0.999796i	$0.506429\pi$
$614$	11.0000	0.443924
$615$	−27.0000	−1.08875
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−1.00000	−0.0402259
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−6.00000	−0.240966
$621$	−1.00000	−0.0401286
$622$	12.0000	0.481156
$623$	6.00000	0.240385
$624$	5.00000	0.200160
$625$	−29.0000	−1.16000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−4.00000	−0.159617
$629$	0	0
$630$	−3.00000	−0.119523
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−16.0000	−0.636446
$633$	8.00000	0.317971
$634$	−27.0000	−1.07231
$635$	39.0000	1.54767
$636$	−12.0000	−0.475831
$637$	5.00000	0.198107
$638$	0	0
$639$	6.00000	0.237356
$640$	−3.00000	−0.118585
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	12.0000	0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−1.00000	−0.0394055
$645$	3.00000	0.118125
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	20.0000	0.784465
$651$	2.00000	0.0783862
$652$	20.0000	0.783260
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	−19.0000	−0.742959
$655$	54.0000	2.10995
$656$	9.00000	0.351391
$657$	−4.00000	−0.156055
$658$	−3.00000	−0.116952
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−16.0000	−0.621858
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−24.0000	−0.930680
$666$	−7.00000	−0.271244
$667$	−3.00000	−0.116160
$668$	−12.0000	−0.464294
$669$	14.0000	0.541271
$670$	12.0000	0.463600
$671$	0	0
$672$	1.00000	0.0385758
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	−4.00000	−0.154074
$675$	4.00000	0.153960
$676$	12.0000	0.461538
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	15.0000	0.576072
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	8.00000	0.305888
$685$	27.0000	1.03162
$686$	1.00000	0.0381802
$687$	−4.00000	−0.152610
$688$	−1.00000	−0.0381246
$689$	−60.0000	−2.28582
$690$	3.00000	0.114208
$691$	−49.0000	−1.86405	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−49.0000	−1.86405	−0.932024	+	0.362397i	$0.881959\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	3.00000	0.113878
$695$	−15.0000	−0.568982
$696$	3.00000	0.113715
$697$	0	0
$698$	−10.0000	−0.378506
$699$	−6.00000	−0.226941
$700$	4.00000	0.151186
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	5.00000	0.188713
$703$	−56.0000	−2.11208
$704$	0	0
$705$	9.00000	0.338960
$706$	27.0000	1.01616
$707$	−6.00000	−0.225653
$708$	−6.00000	−0.225494
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−18.0000	−0.675528
$711$	−16.0000	−0.600047
$712$	6.00000	0.224860
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	15.0000	0.560576
$717$	12.0000	0.448148
$718$	21.0000	0.783713
$719$	45.0000	1.67822	0.839108	−	0.543964i	$-0.183077\pi$
$719$	45.0000	1.67822	0.839108	+	0.543964i	$0.183077\pi$
$720$	−3.00000	−0.111803
$721$	−1.00000	−0.0372419
$722$	45.0000	1.67473
$723$	−19.0000	−0.706618
$724$	2.00000	0.0743294
$725$	12.0000	0.445669
$726$	−11.0000	−0.408248
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	5.00000	0.185312
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	0	0
$732$	14.0000	0.517455
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	29.0000	1.07041
$735$	−3.00000	−0.110657
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	9.00000	0.331295
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	21.0000	0.771975
$741$	40.0000	1.46944
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−22.0000	−0.805477
$747$	−12.0000	−0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	3.00000	0.109545
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−3.00000	−0.109399
$753$	−15.0000	−0.546630
$754$	15.0000	0.546268
$755$	−33.0000	−1.20099
$756$	1.00000	0.0363696
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	29.0000	1.05333
$759$	0	0
$760$	−24.0000	−0.870572
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−13.0000	−0.470940
$763$	−19.0000	−0.687846
$764$	−12.0000	−0.434145
$765$	0	0
$766$	36.0000	1.30073
$767$	−30.0000	−1.08324
$768$	1.00000	0.0360844
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−1.00000	−0.0359908
$773$	−51.0000	−1.83434	−0.917171	−	0.398493i	$-0.869533\pi$
$773$	−51.0000	−1.83434	−0.917171	+	0.398493i	$0.869533\pi$
$774$	−1.00000	−0.0359443
$775$	8.00000	0.287368
$776$	−1.00000	−0.0358979
$777$	−7.00000	−0.251124
$778$	−30.0000	−1.07555
$779$	72.0000	2.57967
$780$	−15.0000	−0.537086
$781$	0	0
$782$	0	0
$783$	3.00000	0.107211
$784$	1.00000	0.0357143
$785$	12.0000	0.428298
$786$	−18.0000	−0.642039
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	9.00000	0.320612
$789$	−9.00000	−0.320408
$790$	48.0000	1.70776
$791$	15.0000	0.533339
$792$	0	0
$793$	70.0000	2.48577
$794$	−22.0000	−0.780751
$795$	36.0000	1.27679
$796$	17.0000	0.602549
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	8.00000	0.283197
$799$	0	0
$800$	4.00000	0.141421
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	−4.00000	−0.141069
$805$	3.00000	0.105736
$806$	10.0000	0.352235
$807$	−18.0000	−0.633630
$808$	−6.00000	−0.211079
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	−3.00000	−0.105409
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	3.00000	0.105279
$813$	−28.0000	−0.982003
$814$	0	0
$815$	−60.0000	−2.10171
$816$	0	0
$817$	−8.00000	−0.279885
$818$	32.0000	1.11885
$819$	5.00000	0.174714
$820$	−27.0000	−0.942881
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	−9.00000	−0.313911
$823$	−13.0000	−0.453152	−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000	−0.453152	−0.226576	+	0.973994i	$0.572753\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	−6.00000	−0.208767
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	−1.00000	−0.0347524
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	36.0000	1.24958
$831$	−4.00000	−0.138758
$832$	5.00000	0.173344
$833$	0	0
$834$	5.00000	0.173136
$835$	36.0000	1.24583
$836$	0	0
$837$	2.00000	0.0691301
$838$	36.0000	1.24360
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	−3.00000	−0.103510
$841$	−20.0000	−0.689655
$842$	−19.0000	−0.654783
$843$	−15.0000	−0.516627
$844$	8.00000	0.275371
$845$	−36.0000	−1.23844
$846$	−3.00000	−0.103142
$847$	−11.0000	−0.377964
$848$	−12.0000	−0.412082
$849$	14.0000	0.480479
$850$	0	0
$851$	7.00000	0.239957
$852$	6.00000	0.205557
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	14.0000	0.479070
$855$	−24.0000	−0.820783
$856$	12.0000	0.410152
$857$	−15.0000	−0.512390	−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000	−0.512390	−0.256195	+	0.966625i	$0.582469\pi$
$858$	0	0
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	3.00000	0.102299
$861$	9.00000	0.306719
$862$	15.0000	0.510902
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	17.0000	0.577684
$867$	−17.0000	−0.577350
$868$	2.00000	0.0678844
$869$	0	0
$870$	−9.00000	−0.305129
$871$	−20.0000	−0.677674
$872$	−19.0000	−0.643421
$873$	−1.00000	−0.0338449
$874$	−8.00000	−0.270604
$875$	3.00000	0.101419
$876$	−4.00000	−0.135147
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	26.0000	0.877457
$879$	−30.0000	−1.01187
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	1.00000	0.0336718
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	−27.0000	−0.907083
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−7.00000	−0.234905
$889$	−13.0000	−0.436006
$890$	−18.0000	−0.603361
$891$	0	0
$892$	14.0000	0.468755
$893$	−24.0000	−0.803129
$894$	0	0
$895$	−45.0000	−1.50418
$896$	1.00000	0.0334077
$897$	−5.00000	−0.166945
$898$	18.0000	0.600668
$899$	6.00000	0.200111
$900$	4.00000	0.133333
$901$	0	0
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	15.0000	0.498893
$905$	−6.00000	−0.199447
$906$	11.0000	0.365451
$907$	41.0000	1.36138	0.680691	−	0.732570i	$-0.261680\pi$
$907$	41.0000	1.36138	0.680691	+	0.732570i	$0.261680\pi$
$908$	−15.0000	−0.497792
$909$	−6.00000	−0.199007
$910$	−15.0000	−0.497245
$911$	21.0000	0.695761	0.347881	−	0.937539i	$-0.386901\pi$
$911$	21.0000	0.695761	0.347881	+	0.937539i	$0.386901\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	−28.0000	−0.926158
$915$	−42.0000	−1.38848
$916$	−4.00000	−0.132164
$917$	−18.0000	−0.594412
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	3.00000	0.0989071
$921$	11.0000	0.362462
$922$	−30.0000	−0.987997
$923$	30.0000	0.987462
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−31.0000	−1.01872
$927$	−1.00000	−0.0328443
$928$	3.00000	0.0984798
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	−6.00000	−0.196748
$931$	8.00000	0.262189
$932$	−6.00000	−0.196537
$933$	12.0000	0.392862
$934$	−33.0000	−1.07979
$935$	0	0
$936$	5.00000	0.163430
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	−4.00000	−0.130605
$939$	−10.0000	−0.326338
$940$	9.00000	0.293548
$941$	−51.0000	−1.66255	−0.831276	−	0.555860i	$-0.812389\pi$
$941$	−51.0000	−1.66255	−0.831276	+	0.555860i	$0.812389\pi$
$942$	−4.00000	−0.130327
$943$	−9.00000	−0.293080
$944$	−6.00000	−0.195283
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	−16.0000	−0.519656
$949$	−20.0000	−0.649227
$950$	32.0000	1.03822
$951$	−27.0000	−0.875535
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−12.0000	−0.388514
$955$	36.0000	1.16493
$956$	12.0000	0.388108
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−9.00000	−0.290625
$960$	−3.00000	−0.0968246
$961$	−27.0000	−0.870968
$962$	−35.0000	−1.12845
$963$	12.0000	0.386695
$964$	−19.0000	−0.611949
$965$	3.00000	0.0965734
$966$	−1.00000	−0.0321745
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	3.00000	0.0963242
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	1.00000	0.0320750
$973$	5.00000	0.160293
$974$	−31.0000	−0.993304
$975$	20.0000	0.640513
$976$	14.0000	0.448129
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	−19.0000	−0.606623
$982$	12.0000	0.382935
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	9.00000	0.286910
$985$	−27.0000	−0.860292
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	40.0000	1.27257
$989$	1.00000	0.0317982
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	2.00000	0.0635001
$993$	−16.0000	−0.507745
$994$	6.00000	0.190308
$995$	−51.0000	−1.61681
$996$	−12.0000	−0.380235
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	14.0000	0.443162
$999$	−7.00000	−0.221470

Twists

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

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

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