LMFDB - Embedded newform 966.2.a.h.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +1.00000 q^{40} +7.00000 q^{41} -1.00000 q^{42} -3.00000 q^{43} +1.00000 q^{45} +1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -4.00000 q^{51} -1.00000 q^{52} -1.00000 q^{54} +1.00000 q^{56} -4.00000 q^{57} -1.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} -6.00000 q^{61} +10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -2.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -5.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -3.00000 q^{86} +1.00000 q^{87} -6.00000 q^{89} +1.00000 q^{90} -1.00000 q^{91} +1.00000 q^{92} -10.0000 q^{93} +3.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -1.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	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	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−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
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−1.00000	−0.182574
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	1.00000	0.158114
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	−1.00000	−0.154303
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	−4.00000	−0.560112
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	−1.00000	−0.131306
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	10.0000	1.27000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	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	$0.575209\pi$
$74$	−5.00000	−0.581238
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000	0.113228
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−1.00000	−0.109109
$85$	4.00000	0.433861
$86$	−3.00000	−0.323498
$87$	1.00000	0.107211
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	−1.00000	−0.104828
$92$	1.00000	0.104257
$93$	−10.0000	−1.03695
$94$	3.00000	0.309426
$95$	4.00000	0.410391
$96$	−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
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−4.00000	−0.396059
$103$	−17.0000	−1.67506	−0.837530	−	0.546392i	$-0.816001\pi$
$103$	−17.0000	−1.67506	−0.837530	+	0.546392i	$0.816001\pi$
$104$	−1.00000	−0.0980581
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	1.00000	0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−4.00000	−0.374634
$115$	1.00000	0.0932505
$116$	−1.00000	−0.0928477
$117$	−1.00000	−0.0924500
$118$	6.00000	0.552345
$119$	4.00000	0.366679
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	−7.00000	−0.631169
$124$	10.0000	0.898027
$125$	−9.00000	−0.804984
$126$	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
$129$	3.00000	0.264135
$130$	−1.00000	−0.0877058
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−1.00000	−0.0851257
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	1.00000	0.0845154
$141$	−3.00000	−0.252646
$142$	−2.00000	−0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	−4.00000	−0.331042
$147$	−1.00000	−0.0824786
$148$	−5.00000	−0.410997
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	4.00000	0.326599
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	4.00000	0.324443
$153$	4.00000	0.323381
$154$	0	0
$155$	10.0000	0.803219
$156$	1.00000	0.0800641
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	4.00000	0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−1.00000	−0.0771517
$169$	−12.0000	−0.923077
$170$	4.00000	0.306786
$171$	4.00000	0.305888
$172$	−3.00000	−0.228748
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	1.00000	0.0758098
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−6.00000	−0.449719
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	1.00000	0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−1.00000	−0.0741249
$183$	6.00000	0.443533
$184$	1.00000	0.0737210
$185$	−5.00000	−0.367607
$186$	−10.0000	−0.733236
$187$	0	0
$188$	3.00000	0.218797
$189$	−1.00000	−0.0727393
$190$	4.00000	0.290191
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−1.00000	−0.0721688
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	−1.00000	−0.0717958
$195$	1.00000	0.0716115
$196$	1.00000	0.0714286
$197$	5.00000	0.356235	0.178118	−	0.984009i	$-0.442999\pi$
$197$	5.00000	0.356235	0.178118	+	0.984009i	$0.442999\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	−4.00000	−0.282843
$201$	−4.00000	−0.282138
$202$	2.00000	0.140720
$203$	−1.00000	−0.0701862
$204$	−4.00000	−0.280056
$205$	7.00000	0.488901
$206$	−17.0000	−1.18445
$207$	1.00000	0.0695048
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	−3.00000	−0.204598
$216$	−1.00000	−0.0680414
$217$	10.0000	0.678844
$218$	−17.0000	−1.15139
$219$	4.00000	0.270295
$220$	0	0
$221$	−4.00000	−0.269069
$222$	5.00000	0.335578
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	1.00000	0.0665190
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	−4.00000	−0.264906
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−1.00000	−0.0653720
$235$	3.00000	0.195698
$236$	6.00000	0.390567
$237$	−12.0000	−0.779484
$238$	4.00000	0.259281
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−1.00000	−0.0645497
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	−7.00000	−0.446304
$247$	−4.00000	−0.254514
$248$	10.0000	0.635001
$249$	−4.00000	−0.253490
$250$	−9.00000	−0.569210
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−13.0000	−0.815693
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	3.00000	0.186772
$259$	−5.00000	−0.310685
$260$	−1.00000	−0.0620174
$261$	−1.00000	−0.0618984
$262$	−14.0000	−0.864923
$263$	25.0000	1.54157	0.770783	−	0.637098i	$-0.219865\pi$
$263$	25.0000	1.54157	0.770783	+	0.637098i	$0.219865\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	4.00000	0.242536
$273$	1.00000	0.0605228
$274$	9.00000	0.543710
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−32.0000	−1.92269	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−32.0000	−1.92269	−0.961347	+	0.275340i	$0.911209\pi$
$278$	−1.00000	−0.0599760
$279$	10.0000	0.598684
$280$	1.00000	0.0597614
$281$	23.0000	1.37206	0.686032	−	0.727571i	$-0.259351\pi$
$281$	23.0000	1.37206	0.686032	+	0.727571i	$0.259351\pi$
$282$	−3.00000	−0.178647
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−2.00000	−0.118678
$285$	−4.00000	−0.236940
$286$	0	0
$287$	7.00000	0.413197
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	−1.00000	−0.0587220
$291$	1.00000	0.0586210
$292$	−4.00000	−0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−1.00000	−0.0583212
$295$	6.00000	0.349334
$296$	−5.00000	−0.290619
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−1.00000	−0.0578315
$300$	4.00000	0.230940
$301$	−3.00000	−0.172917
$302$	−5.00000	−0.287718
$303$	−2.00000	−0.114897
$304$	4.00000	0.229416
$305$	−6.00000	−0.343559
$306$	4.00000	0.228665
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	17.0000	0.967096
$310$	10.0000	0.567962
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	1.00000	0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	12.0000	0.677199
$315$	1.00000	0.0563436
$316$	12.0000	0.675053
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	1.00000	0.0557278
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−16.0000	−0.886158
$327$	17.0000	0.940102
$328$	7.00000	0.386510
$329$	3.00000	0.165395
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	4.00000	0.219529
$333$	−5.00000	−0.273998
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	−1.00000	−0.0545545
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	−12.0000	−0.652714
$339$	−1.00000	−0.0543125
$340$	4.00000	0.216930
$341$	0	0
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	−3.00000	−0.161749
$345$	−1.00000	−0.0538382
$346$	−18.0000	−0.967686
$347$	−33.0000	−1.77153	−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000	−1.77153	−0.885766	+	0.464131i	$0.846367\pi$
$348$	1.00000	0.0536056
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−4.00000	−0.213809
$351$	1.00000	0.0533761
$352$	0	0
$353$	13.0000	0.691920	0.345960	−	0.938249i	$-0.387553\pi$
$353$	13.0000	0.691920	0.345960	+	0.938249i	$0.387553\pi$
$354$	−6.00000	−0.318896
$355$	−2.00000	−0.106149
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	3.00000	0.158555
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	11.0000	0.577350
$364$	−1.00000	−0.0524142
$365$	−4.00000	−0.209370
$366$	6.00000	0.313625
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	1.00000	0.0521286
$369$	7.00000	0.364405
$370$	−5.00000	−0.259938
$371$	0	0
$372$	−10.0000	−0.518476
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	3.00000	0.154713
$377$	1.00000	0.0515026
$378$	−1.00000	−0.0514344
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	4.00000	0.205196
$381$	13.0000	0.666010
$382$	−20.0000	−1.02329
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−17.0000	−0.865277
$387$	−3.00000	−0.152499
$388$	−1.00000	−0.0507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	1.00000	0.0506370
$391$	4.00000	0.202289
$392$	1.00000	0.0505076
$393$	14.0000	0.706207
$394$	5.00000	0.251896
$395$	12.0000	0.603786
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−7.00000	−0.350878
$399$	−4.00000	−0.200250
$400$	−4.00000	−0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−4.00000	−0.199502
$403$	−10.0000	−0.498135
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	−1.00000	−0.0496292
$407$	0	0
$408$	−4.00000	−0.198030
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	7.00000	0.345705
$411$	−9.00000	−0.443937
$412$	−17.0000	−0.837530
$413$	6.00000	0.295241
$414$	1.00000	0.0491473
$415$	4.00000	0.196352
$416$	−1.00000	−0.0490290
$417$	1.00000	0.0489702
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−1.00000	−0.0487950
$421$	−9.00000	−0.438633	−0.219317	−	0.975654i	$-0.570383\pi$
$421$	−9.00000	−0.438633	−0.219317	+	0.975654i	$0.570383\pi$
$422$	0	0
$423$	3.00000	0.145865
$424$	0	0
$425$	−16.0000	−0.776114
$426$	2.00000	0.0969003
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	−3.00000	−0.144673
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	−1.00000	−0.0481125
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	10.0000	0.480015
$435$	1.00000	0.0479463
$436$	−17.0000	−0.814152
$437$	4.00000	0.191346
$438$	4.00000	0.191127
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−4.00000	−0.190261
$443$	1.00000	0.0475114	0.0237557	−	0.999718i	$-0.492438\pi$
$443$	1.00000	0.0475114	0.0237557	+	0.999718i	$0.492438\pi$
$444$	5.00000	0.237289
$445$	−6.00000	−0.284427
$446$	−2.00000	−0.0947027
$447$	4.00000	0.189194
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	1.00000	0.0470360
$453$	5.00000	0.234920
$454$	21.0000	0.985579
$455$	−1.00000	−0.0468807
$456$	−4.00000	−0.187317
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	4.00000	0.186908
$459$	−4.00000	−0.186704
$460$	1.00000	0.0466252
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	25.0000	1.16185	0.580924	−	0.813958i	$-0.302691\pi$
$463$	25.0000	1.16185	0.580924	+	0.813958i	$0.302691\pi$
$464$	−1.00000	−0.0464238
$465$	−10.0000	−0.463739
$466$	−6.00000	−0.277945
$467$	−29.0000	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$467$	−29.0000	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$468$	−1.00000	−0.0462250
$469$	4.00000	0.184703
$470$	3.00000	0.138380
$471$	−12.0000	−0.552931
$472$	6.00000	0.276172
$473$	0	0
$474$	−12.0000	−0.551178
$475$	−16.0000	−0.734130
$476$	4.00000	0.183340
$477$	0	0
$478$	12.0000	0.548867
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	−1.00000	−0.0456435
$481$	5.00000	0.227980
$482$	−3.00000	−0.136646
$483$	−1.00000	−0.0455016
$484$	−11.0000	−0.500000
$485$	−1.00000	−0.0454077
$486$	−1.00000	−0.0453609
$487$	−15.0000	−0.679715	−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000	−0.679715	−0.339857	+	0.940477i	$0.610379\pi$
$488$	−6.00000	−0.271607
$489$	16.0000	0.723545
$490$	1.00000	0.0451754
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−7.00000	−0.315584
$493$	−4.00000	−0.180151
$494$	−4.00000	−0.179969
$495$	0	0
$496$	10.0000	0.449013
$497$	−2.00000	−0.0897123
$498$	−4.00000	−0.179244
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	−9.00000	−0.402492
$501$	−12.0000	−0.536120
$502$	21.0000	0.937276
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	1.00000	0.0445435
$505$	2.00000	0.0889988
$506$	0	0
$507$	12.0000	0.532939
$508$	−13.0000	−0.576782
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	−4.00000	−0.177123
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−22.0000	−0.970378
$515$	−17.0000	−0.749110
$516$	3.00000	0.132068
$517$	0	0
$518$	−5.00000	−0.219687
$519$	18.0000	0.790112
$520$	−1.00000	−0.0438529
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−1.00000	−0.0437688
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−14.0000	−0.611593
$525$	4.00000	0.174574
$526$	25.0000	1.09005
$527$	40.0000	1.74243
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.00000	0.260378
$532$	4.00000	0.173422
$533$	−7.00000	−0.303204
$534$	6.00000	0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	−3.00000	−0.129460
$538$	18.0000	0.776035
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	24.0000	1.03089
$543$	14.0000	0.600798
$544$	4.00000	0.171499
$545$	−17.0000	−0.728200
$546$	1.00000	0.0427960
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	9.00000	0.384461
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−4.00000	−0.170406
$552$	−1.00000	−0.0425628
$553$	12.0000	0.510292
$554$	−32.0000	−1.35955
$555$	5.00000	0.212238
$556$	−1.00000	−0.0424094
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	10.0000	0.423334
$559$	3.00000	0.126886
$560$	1.00000	0.0422577
$561$	0	0
$562$	23.0000	0.970196
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	−3.00000	−0.126323
$565$	1.00000	0.0420703
$566$	−6.00000	−0.252199
$567$	1.00000	0.0419961
$568$	−2.00000	−0.0839181
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	−4.00000	−0.167542
$571$	−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
$573$	20.0000	0.835512
$574$	7.00000	0.292174
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	−1.00000	−0.0415945
$579$	17.0000	0.706496
$580$	−1.00000	−0.0415227
$581$	4.00000	0.165948
$582$	1.00000	0.0414513
$583$	0	0
$584$	−4.00000	−0.165521
$585$	−1.00000	−0.0413449
$586$	−14.0000	−0.578335
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−1.00000	−0.0412393
$589$	40.0000	1.64817
$590$	6.00000	0.247016
$591$	−5.00000	−0.205673
$592$	−5.00000	−0.205499
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−4.00000	−0.163846
$597$	7.00000	0.286491
$598$	−1.00000	−0.0408930
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	4.00000	0.163299
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	−3.00000	−0.122271
$603$	4.00000	0.162893
$604$	−5.00000	−0.203447
$605$	−11.0000	−0.447214
$606$	−2.00000	−0.0812444
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	4.00000	0.162221
$609$	1.00000	0.0405220
$610$	−6.00000	−0.242933
$611$	−3.00000	−0.121367
$612$	4.00000	0.161690
$613$	−27.0000	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$613$	−27.0000	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$614$	−23.0000	−0.928204
$615$	−7.00000	−0.282267
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	17.0000	0.683840
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	10.0000	0.401610
$621$	−1.00000	−0.0401286
$622$	4.00000	0.160385
$623$	−6.00000	−0.240385
$624$	1.00000	0.0400320
$625$	11.0000	0.440000
$626$	14.0000	0.559553
$627$	0	0
$628$	12.0000	0.478852
$629$	−20.0000	−0.797452
$630$	1.00000	0.0398410
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	17.0000	0.675156
$635$	−13.0000	−0.515889
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	1.00000	0.0395285
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	1.00000	0.0394055
$645$	3.00000	0.118125
$646$	16.0000	0.629512
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.00000	0.156893
$651$	−10.0000	−0.391931
$652$	−16.0000	−0.626608
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	17.0000	0.664753
$655$	−14.0000	−0.547025
$656$	7.00000	0.273304
$657$	−4.00000	−0.156055
$658$	3.00000	0.116952
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−8.00000	−0.310929
$663$	4.00000	0.155347
$664$	4.00000	0.155230
$665$	4.00000	0.155113
$666$	−5.00000	−0.193746
$667$	−1.00000	−0.0387202
$668$	12.0000	0.464294
$669$	2.00000	0.0773245
$670$	4.00000	0.154533
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	43.0000	1.65753	0.828764	−	0.559598i	$-0.189045\pi$
$673$	43.0000	1.65753	0.828764	+	0.559598i	$0.189045\pi$
$674$	24.0000	0.924445
$675$	4.00000	0.153960
$676$	−12.0000	−0.461538
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−1.00000	−0.0384048
$679$	−1.00000	−0.0383765
$680$	4.00000	0.153393
$681$	−21.0000	−0.804722
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	4.00000	0.152944
$685$	9.00000	0.343872
$686$	1.00000	0.0381802
$687$	−4.00000	−0.152610
$688$	−3.00000	−0.114374
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	29.0000	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$691$	29.0000	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−33.0000	−1.25266
$695$	−1.00000	−0.0379322
$696$	1.00000	0.0379049
$697$	28.0000	1.06058
$698$	10.0000	0.378506
$699$	6.00000	0.226941
$700$	−4.00000	−0.151186
$701$	−52.0000	−1.96401	−0.982006	−	0.188847i	$-0.939525\pi$
$701$	−52.0000	−1.96401	−0.982006	+	0.188847i	$0.939525\pi$
$702$	1.00000	0.0377426
$703$	−20.0000	−0.754314
$704$	0	0
$705$	−3.00000	−0.112987
$706$	13.0000	0.489261
$707$	2.00000	0.0752177
$708$	−6.00000	−0.225494
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−2.00000	−0.0750587
$711$	12.0000	0.450035
$712$	−6.00000	−0.224860
$713$	10.0000	0.374503
$714$	−4.00000	−0.149696
$715$	0	0
$716$	3.00000	0.112115
$717$	−12.0000	−0.448148
$718$	3.00000	0.111959
$719$	51.0000	1.90198	0.950990	−	0.309223i	$-0.100069\pi$
$719$	51.0000	1.90198	0.950990	+	0.309223i	$0.100069\pi$
$720$	1.00000	0.0372678
$721$	−17.0000	−0.633113
$722$	−3.00000	−0.111648
$723$	3.00000	0.111571
$724$	−14.0000	−0.520306
$725$	4.00000	0.148556
$726$	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	$0.673782\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	−12.0000	−0.443836
$732$	6.00000	0.221766
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	13.0000	0.479839
$735$	−1.00000	−0.0368856
$736$	1.00000	0.0368605
$737$	0	0
$738$	7.00000	0.257674
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−5.00000	−0.183804
$741$	4.00000	0.146944
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−10.0000	−0.366618
$745$	−4.00000	−0.146549
$746$	−10.0000	−0.366126
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	9.00000	0.328634
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	3.00000	0.109399
$753$	−21.0000	−0.765283
$754$	1.00000	0.0364179
$755$	−5.00000	−0.181969
$756$	−1.00000	−0.0363696
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	23.0000	0.835398
$759$	0	0
$760$	4.00000	0.145095
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	13.0000	0.470940
$763$	−17.0000	−0.615441
$764$	−20.0000	−0.723575
$765$	4.00000	0.144620
$766$	−4.00000	−0.144526
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	51.0000	1.83911	0.919554	−	0.392965i	$-0.128551\pi$
$769$	51.0000	1.83911	0.919554	+	0.392965i	$0.128551\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	−17.0000	−0.611843
$773$	−23.0000	−0.827253	−0.413626	−	0.910447i	$-0.635738\pi$
$773$	−23.0000	−0.827253	−0.413626	+	0.910447i	$0.635738\pi$
$774$	−3.00000	−0.107833
$775$	−40.0000	−1.43684
$776$	−1.00000	−0.0358979
$777$	5.00000	0.179374
$778$	2.00000	0.0717035
$779$	28.0000	1.00320
$780$	1.00000	0.0358057
$781$	0	0
$782$	4.00000	0.143040
$783$	1.00000	0.0357371
$784$	1.00000	0.0357143
$785$	12.0000	0.428298
$786$	14.0000	0.499363
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	5.00000	0.178118
$789$	−25.0000	−0.890024
$790$	12.0000	0.426941
$791$	1.00000	0.0355559
$792$	0	0
$793$	6.00000	0.213066
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	−4.00000	−0.141598
$799$	12.0000	0.424529
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	−4.00000	−0.141069
$805$	1.00000	0.0352454
$806$	−10.0000	−0.352235
$807$	−18.0000	−0.633630
$808$	2.00000	0.0703598
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	1.00000	0.0351364
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	−1.00000	−0.0350931
$813$	−24.0000	−0.841717
$814$	0	0
$815$	−16.0000	−0.560456
$816$	−4.00000	−0.140028
$817$	−12.0000	−0.419827
$818$	28.0000	0.978997
$819$	−1.00000	−0.0349428
$820$	7.00000	0.244451
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	−9.00000	−0.313911
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	6.00000	0.208767
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	1.00000	0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	4.00000	0.138842
$831$	32.0000	1.11007
$832$	−1.00000	−0.0346688
$833$	4.00000	0.138592
$834$	1.00000	0.0346272
$835$	12.0000	0.415277
$836$	0	0
$837$	−10.0000	−0.345651
$838$	12.0000	0.414533
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	−1.00000	−0.0345033
$841$	−28.0000	−0.965517
$842$	−9.00000	−0.310160
$843$	−23.0000	−0.792162
$844$	0	0
$845$	−12.0000	−0.412813
$846$	3.00000	0.103142
$847$	−11.0000	−0.377964
$848$	0	0
$849$	6.00000	0.205919
$850$	−16.0000	−0.548795
$851$	−5.00000	−0.171398
$852$	2.00000	0.0685189
$853$	−27.0000	−0.924462	−0.462231	−	0.886759i	$-0.652951\pi$
$853$	−27.0000	−0.924462	−0.462231	+	0.886759i	$0.652951\pi$
$854$	−6.00000	−0.205316
$855$	4.00000	0.136797
$856$	0	0
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	43.0000	1.46714	0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000	1.46714	0.733571	+	0.679613i	$0.237852\pi$
$860$	−3.00000	−0.102299
$861$	−7.00000	−0.238559
$862$	33.0000	1.12398
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	−7.00000	−0.237870
$867$	1.00000	0.0339618
$868$	10.0000	0.339422
$869$	0	0
$870$	1.00000	0.0339032
$871$	−4.00000	−0.135535
$872$	−17.0000	−0.575693
$873$	−1.00000	−0.0338449
$874$	4.00000	0.135302
$875$	−9.00000	−0.304256
$876$	4.00000	0.135147
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−34.0000	−1.14744
$879$	14.0000	0.472208
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	1.00000	0.0336718
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	−4.00000	−0.134535
$885$	−6.00000	−0.201688
$886$	1.00000	0.0335957
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	5.00000	0.167789
$889$	−13.0000	−0.436006
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	12.0000	0.401565
$894$	4.00000	0.133780
$895$	3.00000	0.100279
$896$	1.00000	0.0334077
$897$	1.00000	0.0333890
$898$	6.00000	0.200223
$899$	−10.0000	−0.333519
$900$	−4.00000	−0.133333
$901$	0	0
$902$	0	0
$903$	3.00000	0.0998337
$904$	1.00000	0.0332595
$905$	−14.0000	−0.465376
$906$	5.00000	0.166114
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	21.0000	0.696909
$909$	2.00000	0.0663358
$910$	−1.00000	−0.0331497
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−8.00000	−0.264616
$915$	6.00000	0.198354
$916$	4.00000	0.132164
$917$	−14.0000	−0.462321
$918$	−4.00000	−0.132020
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	1.00000	0.0329690
$921$	23.0000	0.757876
$922$	2.00000	0.0658665
$923$	2.00000	0.0658308
$924$	0	0
$925$	20.0000	0.657596
$926$	25.0000	0.821551
$927$	−17.0000	−0.558353
$928$	−1.00000	−0.0328266
$929$	−25.0000	−0.820223	−0.410112	−	0.912035i	$-0.634510\pi$
$929$	−25.0000	−0.820223	−0.410112	+	0.912035i	$0.634510\pi$
$930$	−10.0000	−0.327913
$931$	4.00000	0.131095
$932$	−6.00000	−0.196537
$933$	−4.00000	−0.130954
$934$	−29.0000	−0.948909
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	4.00000	0.130605
$939$	−14.0000	−0.456873
$940$	3.00000	0.0978492
$941$	57.0000	1.85815	0.929073	−	0.369895i	$-0.120606\pi$
$941$	57.0000	1.85815	0.929073	+	0.369895i	$0.120606\pi$
$942$	−12.0000	−0.390981
$943$	7.00000	0.227951
$944$	6.00000	0.195283
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	33.0000	1.07236	0.536178	−	0.844105i	$-0.319868\pi$
$947$	33.0000	1.07236	0.536178	+	0.844105i	$0.319868\pi$
$948$	−12.0000	−0.389742
$949$	4.00000	0.129845
$950$	−16.0000	−0.519109
$951$	−17.0000	−0.551263
$952$	4.00000	0.129641
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	12.0000	0.388108
$957$	0	0
$958$	10.0000	0.323085
$959$	9.00000	0.290625
$960$	−1.00000	−0.0322749
$961$	69.0000	2.22581
$962$	5.00000	0.161206
$963$	0	0
$964$	−3.00000	−0.0966235
$965$	−17.0000	−0.547249
$966$	−1.00000	−0.0321745
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	−11.0000	−0.353553
$969$	−16.0000	−0.513994
$970$	−1.00000	−0.0321081
$971$	−56.0000	−1.79713	−0.898563	−	0.438845i	$-0.855388\pi$
$971$	−56.0000	−1.79713	−0.898563	+	0.438845i	$0.855388\pi$
$972$	−1.00000	−0.0320750
$973$	−1.00000	−0.0320585
$974$	−15.0000	−0.480631
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	1.00000	0.0319438
$981$	−17.0000	−0.542768
$982$	−20.0000	−0.638226
$983$	58.0000	1.84991	0.924956	−	0.380073i	$-0.124101\pi$
$983$	58.0000	1.84991	0.924956	+	0.380073i	$0.124101\pi$
$984$	−7.00000	−0.223152
$985$	5.00000	0.159313
$986$	−4.00000	−0.127386
$987$	−3.00000	−0.0954911
$988$	−4.00000	−0.127257
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	10.0000	0.317500
$993$	8.00000	0.253872
$994$	−2.00000	−0.0634361
$995$	−7.00000	−0.221915
$996$	−4.00000	−0.126745
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	18.0000	0.569780
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.a.h.1.1	✓	1
3.2	odd	2		2898.2.a.d.1.1		1
4.3	odd	2		7728.2.a.s.1.1		1
7.6	odd	2		6762.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.h.1.1	✓	1	1.1	even	1	trivial
2898.2.a.d.1.1		1	3.2	odd	2
6762.2.a.bi.1.1		1	7.6	odd	2
7728.2.a.s.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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