LMFDB - Embedded newform 966.2.a.k.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} +3.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +3.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} +3.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -9.00000 q^{37} -3.00000 q^{39} +3.00000 q^{40} +9.00000 q^{41} -1.00000 q^{42} -3.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -4.00000 q^{51} -3.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} +3.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +10.0000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -9.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -4.00000 q^{68} +1.00000 q^{69} -3.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} -9.00000 q^{74} +4.00000 q^{75} -4.00000 q^{77} -3.00000 q^{78} +8.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} -12.0000 q^{85} -3.00000 q^{86} +3.00000 q^{87} +4.00000 q^{88} -14.0000 q^{89} +3.00000 q^{90} +3.00000 q^{91} +1.00000 q^{92} -6.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} -7.00000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})

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	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	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
$11$	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
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−1.00000	−0.267261
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.00000	0.670820
$21$	−1.00000	−0.218218
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	−3.00000	−0.588348
$27$	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
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	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
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	4.00000	0.603023
$45$	3.00000	0.447214
$46$	1.00000	0.147442
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	−4.00000	−0.560112
$52$	−3.00000	−0.416025
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	1.00000	0.136083
$55$	12.0000	1.61808
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	3.00000	0.387298
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.00000	−0.762001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−9.00000	−1.11631
$66$	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	$0.421429\pi$
$68$	−4.00000	−0.485071
$69$	1.00000	0.120386
$70$	−3.00000	−0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−9.00000	−1.04623
$75$	4.00000	0.461880
$76$	0	0
$77$	−4.00000	−0.455842
$78$	−3.00000	−0.339683
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	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
$85$	−12.0000	−1.30158
$86$	−3.00000	−0.323498
$87$	3.00000	0.321634
$88$	4.00000	0.426401
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	3.00000	0.316228
$91$	3.00000	0.314485
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	1.00000	0.101015
$99$	4.00000	0.402015
$100$	4.00000	0.400000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−4.00000	−0.396059
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	−3.00000	−0.294174
$105$	−3.00000	−0.292770
$106$	−4.00000	−0.388514
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	12.0000	1.14416
$111$	−9.00000	−0.854242
$112$	−1.00000	−0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	3.00000	0.278543
$117$	−3.00000	−0.277350
$118$	6.00000	0.552345
$119$	4.00000	0.366679
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	9.00000	0.811503
$124$	−6.00000	−0.538816
$125$	−3.00000	−0.268328
$126$	−1.00000	−0.0890871
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	−3.00000	−0.264135
$130$	−9.00000	−0.789352
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	3.00000	0.258199
$136$	−4.00000	−0.342997
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	1.00000	0.0851257
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	−3.00000	−0.253546
$141$	−7.00000	−0.589506
$142$	−6.00000	−0.503509
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	−8.00000	−0.662085
$147$	1.00000	0.0824786
$148$	−9.00000	−0.739795
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	4.00000	0.326599
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	−4.00000	−0.322329
$155$	−18.0000	−1.44579
$156$	−3.00000	−0.240192
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	8.00000	0.636446
$159$	−4.00000	−0.317221
$160$	3.00000	0.237171
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	9.00000	0.702782
$165$	12.0000	0.934199
$166$	4.00000	0.310460
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	−1.00000	−0.0771517
$169$	−4.00000	−0.307692
$170$	−12.0000	−0.920358
$171$	0	0
$172$	−3.00000	−0.228748
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	3.00000	0.227429
$175$	−4.00000	−0.302372
$176$	4.00000	0.301511
$177$	6.00000	0.450988
$178$	−14.0000	−1.04934
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	3.00000	0.223607
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	3.00000	0.222375
$183$	10.0000	0.739221
$184$	1.00000	0.0737210
$185$	−27.0000	−1.98508
$186$	−6.00000	−0.439941
$187$	−16.0000	−1.17004
$188$	−7.00000	−0.510527
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−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
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	−7.00000	−0.502571
$195$	−9.00000	−0.644503
$196$	1.00000	0.0714286
$197$	−23.0000	−1.63868	−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000	−1.63868	−0.819341	+	0.573306i	$0.805660\pi$
$198$	4.00000	0.284268
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	4.00000	0.282843
$201$	4.00000	0.282138
$202$	−14.0000	−0.985037
$203$	−3.00000	−0.210559
$204$	−4.00000	−0.280056
$205$	27.0000	1.88576
$206$	5.00000	0.348367
$207$	1.00000	0.0695048
$208$	−3.00000	−0.208013
$209$	0	0
$210$	−3.00000	−0.207020
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−4.00000	−0.274721
$213$	−6.00000	−0.411113
$214$	8.00000	0.546869
$215$	−9.00000	−0.613795
$216$	1.00000	0.0680414
$217$	6.00000	0.407307
$218$	3.00000	0.203186
$219$	−8.00000	−0.540590
$220$	12.0000	0.809040
$221$	12.0000	0.807207
$222$	−9.00000	−0.604040
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−1.00000	−0.0668153
$225$	4.00000	0.266667
$226$	9.00000	0.598671
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	3.00000	0.197814
$231$	−4.00000	−0.263181
$232$	3.00000	0.196960
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−3.00000	−0.196116
$235$	−21.0000	−1.36989
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	4.00000	0.259281
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	3.00000	0.193649
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	3.00000	0.191663
$246$	9.00000	0.573819
$247$	0	0
$248$	−6.00000	−0.381000
$249$	4.00000	0.253490
$250$	−3.00000	−0.189737
$251$	19.0000	1.19927	0.599635	−	0.800274i	$-0.295313\pi$
$251$	19.0000	1.19927	0.599635	+	0.800274i	$0.295313\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	7.00000	0.439219
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	−3.00000	−0.186772
$259$	9.00000	0.559233
$260$	−9.00000	−0.558156
$261$	3.00000	0.185695
$262$	−6.00000	−0.370681
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	4.00000	0.246183
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−14.0000	−0.856786
$268$	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	$0.558553\pi$
$270$	3.00000	0.182574
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−4.00000	−0.242536
$273$	3.00000	0.181568
$274$	−15.0000	−0.906183
$275$	16.0000	0.964836
$276$	1.00000	0.0601929
$277$	20.0000	1.20168	0.600842	−	0.799368i	$-0.294832\pi$
$277$	20.0000	1.20168	0.600842	+	0.799368i	$0.294832\pi$
$278$	9.00000	0.539784
$279$	−6.00000	−0.359211
$280$	−3.00000	−0.179284
$281$	23.0000	1.37206	0.686032	−	0.727571i	$-0.259351\pi$
$281$	23.0000	1.37206	0.686032	+	0.727571i	$0.259351\pi$
$282$	−7.00000	−0.416844
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−9.00000	−0.531253
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	9.00000	0.528498
$291$	−7.00000	−0.410347
$292$	−8.00000	−0.468165
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	1.00000	0.0583212
$295$	18.0000	1.04800
$296$	−9.00000	−0.523114
$297$	4.00000	0.232104
$298$	16.0000	0.926855
$299$	−3.00000	−0.173494
$300$	4.00000	0.230940
$301$	3.00000	0.172917
$302$	15.0000	0.863153
$303$	−14.0000	−0.804279
$304$	0	0
$305$	30.0000	1.71780
$306$	−4.00000	−0.228665
$307$	15.0000	0.856095	0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000	0.856095	0.428048	+	0.903756i	$0.359202\pi$
$308$	−4.00000	−0.227921
$309$	5.00000	0.284440
$310$	−18.0000	−1.02233
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	−3.00000	−0.169842
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	−8.00000	−0.451466
$315$	−3.00000	−0.169031
$316$	8.00000	0.450035
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	−4.00000	−0.224309
$319$	12.0000	0.671871
$320$	3.00000	0.167705
$321$	8.00000	0.446516
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	−12.0000	−0.665640
$326$	16.0000	0.886158
$327$	3.00000	0.165900
$328$	9.00000	0.496942
$329$	7.00000	0.385922
$330$	12.0000	0.660578
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	4.00000	0.219529
$333$	−9.00000	−0.493197
$334$	−20.0000	−1.09435
$335$	12.0000	0.655630
$336$	−1.00000	−0.0545545
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	−4.00000	−0.217571
$339$	9.00000	0.488813
$340$	−12.0000	−0.650791
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.00000	−0.161749
$345$	3.00000	0.161515
$346$	−6.00000	−0.322562
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	3.00000	0.160817
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−4.00000	−0.213809
$351$	−3.00000	−0.160128
$352$	4.00000	0.213201
$353$	−29.0000	−1.54351	−0.771757	−	0.635917i	$-0.780622\pi$
$353$	−29.0000	−1.54351	−0.771757	+	0.635917i	$0.780622\pi$
$354$	6.00000	0.318896
$355$	−18.0000	−0.955341
$356$	−14.0000	−0.741999
$357$	4.00000	0.211702
$358$	19.0000	1.00418
$359$	−1.00000	−0.0527780	−0.0263890	−	0.999652i	$-0.508401\pi$
$359$	−1.00000	−0.0527780	−0.0263890	+	0.999652i	$0.508401\pi$
$360$	3.00000	0.158114
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	5.00000	0.262432
$364$	3.00000	0.157243
$365$	−24.0000	−1.25622
$366$	10.0000	0.522708
$367$	31.0000	1.61819	0.809093	−	0.587680i	$-0.199959\pi$
$367$	31.0000	1.61819	0.809093	+	0.587680i	$0.199959\pi$
$368$	1.00000	0.0521286
$369$	9.00000	0.468521
$370$	−27.0000	−1.40366
$371$	4.00000	0.207670
$372$	−6.00000	−0.311086
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−16.0000	−0.827340
$375$	−3.00000	−0.154919
$376$	−7.00000	−0.360997
$377$	−9.00000	−0.463524
$378$	−1.00000	−0.0514344
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	−12.0000	−0.611577
$386$	−17.0000	−0.865277
$387$	−3.00000	−0.152499
$388$	−7.00000	−0.355371
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−9.00000	−0.455733
$391$	−4.00000	−0.202289
$392$	1.00000	0.0505076
$393$	−6.00000	−0.302660
$394$	−23.0000	−1.15872
$395$	24.0000	1.20757
$396$	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	$0.174655\pi$
$398$	−5.00000	−0.250627
$399$	0	0
$400$	4.00000	0.200000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	4.00000	0.199502
$403$	18.0000	0.896644
$404$	−14.0000	−0.696526
$405$	3.00000	0.149071
$406$	−3.00000	−0.148888
$407$	−36.0000	−1.78445
$408$	−4.00000	−0.198030
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	27.0000	1.33343
$411$	−15.0000	−0.739895
$412$	5.00000	0.246332
$413$	−6.00000	−0.295241
$414$	1.00000	0.0491473
$415$	12.0000	0.589057
$416$	−3.00000	−0.147087
$417$	9.00000	0.440732
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−3.00000	−0.146385
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	−12.0000	−0.584151
$423$	−7.00000	−0.340352
$424$	−4.00000	−0.194257
$425$	−16.0000	−0.776114
$426$	−6.00000	−0.290701
$427$	−10.0000	−0.483934
$428$	8.00000	0.386695
$429$	−12.0000	−0.579365
$430$	−9.00000	−0.434019
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	1.00000	0.0481125
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	6.00000	0.288009
$435$	9.00000	0.431517
$436$	3.00000	0.143674
$437$	0	0
$438$	−8.00000	−0.382255
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	12.0000	0.572078
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	25.0000	1.18779	0.593893	−	0.804544i	$-0.297590\pi$
$443$	25.0000	1.18779	0.593893	+	0.804544i	$0.297590\pi$
$444$	−9.00000	−0.427121
$445$	−42.0000	−1.99099
$446$	−14.0000	−0.662919
$447$	16.0000	0.756774
$448$	−1.00000	−0.0472456
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	4.00000	0.188562
$451$	36.0000	1.69517
$452$	9.00000	0.423324
$453$	15.0000	0.704761
$454$	11.0000	0.516256
$455$	9.00000	0.421927
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	12.0000	0.560723
$459$	−4.00000	−0.186704
$460$	3.00000	0.139876
$461$	−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
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	3.00000	0.139272
$465$	−18.0000	−0.834730
$466$	−10.0000	−0.463241
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	−3.00000	−0.138675
$469$	−4.00000	−0.184703
$470$	−21.0000	−0.968658
$471$	−8.00000	−0.368621
$472$	6.00000	0.276172
$473$	−12.0000	−0.551761
$474$	8.00000	0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	−4.00000	−0.183147
$478$	−12.0000	−0.548867
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	3.00000	0.136931
$481$	27.0000	1.23109
$482$	−5.00000	−0.227744
$483$	−1.00000	−0.0455016
$484$	5.00000	0.227273
$485$	−21.0000	−0.953561
$486$	1.00000	0.0453609
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	10.0000	0.452679
$489$	16.0000	0.723545
$490$	3.00000	0.135526
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	9.00000	0.405751
$493$	−12.0000	−0.540453
$494$	0	0
$495$	12.0000	0.539360
$496$	−6.00000	−0.269408
$497$	6.00000	0.269137
$498$	4.00000	0.179244
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	−3.00000	−0.134164
$501$	−20.0000	−0.893534
$502$	19.0000	0.848012
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	−1.00000	−0.0445435
$505$	−42.0000	−1.86898
$506$	4.00000	0.177822
$507$	−4.00000	−0.177646
$508$	7.00000	0.310575
$509$	44.0000	1.95027	0.975133	−	0.221621i	$-0.0711348\pi$
$509$	44.0000	1.95027	0.975133	+	0.221621i	$0.0711348\pi$
$510$	−12.0000	−0.531369
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	0	0
$514$	−26.0000	−1.14681
$515$	15.0000	0.660979
$516$	−3.00000	−0.132068
$517$	−28.0000	−1.23144
$518$	9.00000	0.395437
$519$	−6.00000	−0.263371
$520$	−9.00000	−0.394676
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	3.00000	0.131306
$523$	42.0000	1.83653	0.918266	−	0.395964i	$-0.129590\pi$
$523$	42.0000	1.83653	0.918266	+	0.395964i	$0.129590\pi$
$524$	−6.00000	−0.262111
$525$	−4.00000	−0.174574
$526$	21.0000	0.915644
$527$	24.0000	1.04546
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	6.00000	0.260378
$532$	0	0
$533$	−27.0000	−1.16950
$534$	−14.0000	−0.605839
$535$	24.0000	1.03761
$536$	4.00000	0.172774
$537$	19.0000	0.819911
$538$	−6.00000	−0.258678
$539$	4.00000	0.172292
$540$	3.00000	0.129099
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	20.0000	0.859074
$543$	−2.00000	−0.0858282
$544$	−4.00000	−0.171499
$545$	9.00000	0.385518
$546$	3.00000	0.128388
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−15.0000	−0.640768
$549$	10.0000	0.426790
$550$	16.0000	0.682242
$551$	0	0
$552$	1.00000	0.0425628
$553$	−8.00000	−0.340195
$554$	20.0000	0.849719
$555$	−27.0000	−1.14609
$556$	9.00000	0.381685
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	−6.00000	−0.254000
$559$	9.00000	0.380659
$560$	−3.00000	−0.126773
$561$	−16.0000	−0.675521
$562$	23.0000	0.970196
$563$	15.0000	0.632175	0.316087	−	0.948730i	$-0.397631\pi$
$563$	15.0000	0.632175	0.316087	+	0.948730i	$0.397631\pi$
$564$	−7.00000	−0.294753
$565$	27.0000	1.13590
$566$	−6.00000	−0.252199
$567$	−1.00000	−0.0419961
$568$	−6.00000	−0.251754
$569$	−13.0000	−0.544988	−0.272494	−	0.962157i	$-0.587849\pi$
$569$	−13.0000	−0.544988	−0.272494	+	0.962157i	$0.587849\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−12.0000	−0.501745
$573$	−12.0000	−0.501307
$574$	−9.00000	−0.375653
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−1.00000	−0.0415945
$579$	−17.0000	−0.706496
$580$	9.00000	0.373705
$581$	−4.00000	−0.165948
$582$	−7.00000	−0.290159
$583$	−16.0000	−0.662652
$584$	−8.00000	−0.331042
$585$	−9.00000	−0.372104
$586$	6.00000	0.247858
$587$	−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
$590$	18.0000	0.741048
$591$	−23.0000	−0.946094
$592$	−9.00000	−0.369898
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	4.00000	0.164122
$595$	12.0000	0.491952
$596$	16.0000	0.655386
$597$	−5.00000	−0.204636
$598$	−3.00000	−0.122679
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	4.00000	0.163299
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	3.00000	0.122271
$603$	4.00000	0.162893
$604$	15.0000	0.610341
$605$	15.0000	0.609837
$606$	−14.0000	−0.568711
$607$	30.0000	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$607$	30.0000	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$608$	0	0
$609$	−3.00000	−0.121566
$610$	30.0000	1.21466
$611$	21.0000	0.849569
$612$	−4.00000	−0.161690
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	15.0000	0.605351
$615$	27.0000	1.08875
$616$	−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	$0.538538\pi$
$618$	5.00000	0.201129
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−18.0000	−0.722897
$621$	1.00000	0.0401286
$622$	−28.0000	−1.12270
$623$	14.0000	0.560898
$624$	−3.00000	−0.120096
$625$	−29.0000	−1.16000
$626$	34.0000	1.35891
$627$	0	0
$628$	−8.00000	−0.319235
$629$	36.0000	1.43541
$630$	−3.00000	−0.119523
$631$	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
$633$	−12.0000	−0.476957
$634$	5.00000	0.198575
$635$	21.0000	0.833360
$636$	−4.00000	−0.158610
$637$	−3.00000	−0.118864
$638$	12.0000	0.475085
$639$	−6.00000	−0.237356
$640$	3.00000	0.118585
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	8.00000	0.315735
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	−1.00000	−0.0394055
$645$	−9.00000	−0.354375
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000	0.0392837
$649$	24.0000	0.942082
$650$	−12.0000	−0.470679
$651$	6.00000	0.235159
$652$	16.0000	0.626608
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	3.00000	0.117309
$655$	−18.0000	−0.703318
$656$	9.00000	0.351391
$657$	−8.00000	−0.312110
$658$	7.00000	0.272888
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	12.0000	0.467099
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	8.00000	0.310929
$663$	12.0000	0.466041
$664$	4.00000	0.155230
$665$	0	0
$666$	−9.00000	−0.348743
$667$	3.00000	0.116160
$668$	−20.0000	−0.773823
$669$	−14.0000	−0.541271
$670$	12.0000	0.463600
$671$	40.0000	1.54418
$672$	−1.00000	−0.0385758
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−12.0000	−0.462223
$675$	4.00000	0.153960
$676$	−4.00000	−0.153846
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	9.00000	0.345643
$679$	7.00000	0.268635
$680$	−12.0000	−0.460179
$681$	11.0000	0.421521
$682$	−24.0000	−0.919007
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−45.0000	−1.71936
$686$	−1.00000	−0.0381802
$687$	12.0000	0.457829
$688$	−3.00000	−0.114374
$689$	12.0000	0.457164
$690$	3.00000	0.114208
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	−6.00000	−0.228086
$693$	−4.00000	−0.151947
$694$	23.0000	0.873068
$695$	27.0000	1.02417
$696$	3.00000	0.113715
$697$	−36.0000	−1.36360
$698$	−26.0000	−0.984115
$699$	−10.0000	−0.378235
$700$	−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	$0.262045\pi$
$702$	−3.00000	−0.113228
$703$	0	0
$704$	4.00000	0.150756
$705$	−21.0000	−0.790906
$706$	−29.0000	−1.09143
$707$	14.0000	0.526524
$708$	6.00000	0.225494
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	−18.0000	−0.675528
$711$	8.00000	0.300023
$712$	−14.0000	−0.524672
$713$	−6.00000	−0.224702
$714$	4.00000	0.149696
$715$	−36.0000	−1.34632
$716$	19.0000	0.710063
$717$	−12.0000	−0.448148
$718$	−1.00000	−0.0373197
$719$	−7.00000	−0.261056	−0.130528	−	0.991445i	$-0.541667\pi$
$719$	−7.00000	−0.261056	−0.130528	+	0.991445i	$0.541667\pi$
$720$	3.00000	0.111803
$721$	−5.00000	−0.186210
$722$	−19.0000	−0.707107
$723$	−5.00000	−0.185952
$724$	−2.00000	−0.0743294
$725$	12.0000	0.445669
$726$	5.00000	0.185567
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	−24.0000	−0.888280
$731$	12.0000	0.443836
$732$	10.0000	0.369611
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	31.0000	1.14423
$735$	3.00000	0.110657
$736$	1.00000	0.0368605
$737$	16.0000	0.589368
$738$	9.00000	0.331295
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	−27.0000	−0.992540
$741$	0	0
$742$	4.00000	0.146845
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	−6.00000	−0.219971
$745$	48.0000	1.75858
$746$	22.0000	0.805477
$747$	4.00000	0.146352
$748$	−16.0000	−0.585018
$749$	−8.00000	−0.292314
$750$	−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	$0.760392\pi$
$752$	−7.00000	−0.255264
$753$	19.0000	0.692398
$754$	−9.00000	−0.327761
$755$	45.0000	1.63772
$756$	−1.00000	−0.0363696
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−25.0000	−0.908041
$759$	4.00000	0.145191
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	7.00000	0.253583
$763$	−3.00000	−0.108607
$764$	−12.0000	−0.434145
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−18.0000	−0.649942
$768$	1.00000	0.0360844
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	−12.0000	−0.432450
$771$	−26.0000	−0.936367
$772$	−17.0000	−0.611843
$773$	19.0000	0.683383	0.341691	−	0.939812i	$-0.389000\pi$
$773$	19.0000	0.683383	0.341691	+	0.939812i	$0.389000\pi$
$774$	−3.00000	−0.107833
$775$	−24.0000	−0.862105
$776$	−7.00000	−0.251285
$777$	9.00000	0.322873
$778$	18.0000	0.645331
$779$	0	0
$780$	−9.00000	−0.322252
$781$	−24.0000	−0.858788
$782$	−4.00000	−0.143040
$783$	3.00000	0.107211
$784$	1.00000	0.0357143
$785$	−24.0000	−0.856597
$786$	−6.00000	−0.214013
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−23.0000	−0.819341
$789$	21.0000	0.747620
$790$	24.0000	0.853882
$791$	−9.00000	−0.320003
$792$	4.00000	0.142134
$793$	−30.0000	−1.06533
$794$	34.0000	1.20661
$795$	−12.0000	−0.425596
$796$	−5.00000	−0.177220
$797$	−49.0000	−1.73567	−0.867835	−	0.496853i	$-0.834489\pi$
$797$	−49.0000	−1.73567	−0.867835	+	0.496853i	$0.834489\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	4.00000	0.141421
$801$	−14.0000	−0.494666
$802$	38.0000	1.34183
$803$	−32.0000	−1.12926
$804$	4.00000	0.141069
$805$	−3.00000	−0.105736
$806$	18.0000	0.634023
$807$	−6.00000	−0.211210
$808$	−14.0000	−0.492518
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	3.00000	0.105409
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	−3.00000	−0.105279
$813$	20.0000	0.701431
$814$	−36.0000	−1.26180
$815$	48.0000	1.68137
$816$	−4.00000	−0.140028
$817$	0	0
$818$	16.0000	0.559427
$819$	3.00000	0.104828
$820$	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	$0.675375\pi$
$822$	−15.0000	−0.523185
$823$	7.00000	0.244005	0.122002	−	0.992530i	$-0.461068\pi$
$823$	7.00000	0.244005	0.122002	+	0.992530i	$0.461068\pi$
$824$	5.00000	0.174183
$825$	16.0000	0.557048
$826$	−6.00000	−0.208767
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	1.00000	0.0347524
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	12.0000	0.416526
$831$	20.0000	0.693792
$832$	−3.00000	−0.104006
$833$	−4.00000	−0.138592
$834$	9.00000	0.311645
$835$	−60.0000	−2.07639
$836$	0	0
$837$	−6.00000	−0.207390
$838$	−4.00000	−0.138178
$839$	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
$842$	−21.0000	−0.723708
$843$	23.0000	0.792162
$844$	−12.0000	−0.413057
$845$	−12.0000	−0.412813
$846$	−7.00000	−0.240665
$847$	−5.00000	−0.171802
$848$	−4.00000	−0.137361
$849$	−6.00000	−0.205919
$850$	−16.0000	−0.548795
$851$	−9.00000	−0.308516
$852$	−6.00000	−0.205557
$853$	7.00000	0.239675	0.119838	−	0.992793i	$-0.461763\pi$
$853$	7.00000	0.239675	0.119838	+	0.992793i	$0.461763\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	8.00000	0.273434
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	−12.0000	−0.409673
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	−9.00000	−0.306897
$861$	−9.00000	−0.306719
$862$	21.0000	0.715263
$863$	−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
$865$	−18.0000	−0.612018
$866$	23.0000	0.781572
$867$	−1.00000	−0.0339618
$868$	6.00000	0.203653
$869$	32.0000	1.08553
$870$	9.00000	0.305129
$871$	−12.0000	−0.406604
$872$	3.00000	0.101593
$873$	−7.00000	−0.236914
$874$	0	0
$875$	3.00000	0.101419
$876$	−8.00000	−0.270295
$877$	−36.0000	−1.21563	−0.607817	−	0.794077i	$-0.707955\pi$
$877$	−36.0000	−1.21563	−0.607817	+	0.794077i	$0.707955\pi$
$878$	−10.0000	−0.337484
$879$	6.00000	0.202375
$880$	12.0000	0.404520
$881$	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
$883$	54.0000	1.81724	0.908622	−	0.417619i	$-0.137135\pi$
$883$	54.0000	1.81724	0.908622	+	0.417619i	$0.137135\pi$
$884$	12.0000	0.403604
$885$	18.0000	0.605063
$886$	25.0000	0.839891
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−9.00000	−0.302020
$889$	−7.00000	−0.234772
$890$	−42.0000	−1.40784
$891$	4.00000	0.134005
$892$	−14.0000	−0.468755
$893$	0	0
$894$	16.0000	0.535120
$895$	57.0000	1.90530
$896$	−1.00000	−0.0334077
$897$	−3.00000	−0.100167
$898$	30.0000	1.00111
$899$	−18.0000	−0.600334
$900$	4.00000	0.133333
$901$	16.0000	0.533037
$902$	36.0000	1.19867
$903$	3.00000	0.0998337
$904$	9.00000	0.299336
$905$	−6.00000	−0.199447
$906$	15.0000	0.498342
$907$	−53.0000	−1.75984	−0.879918	−	0.475125i	$-0.842403\pi$
$907$	−53.0000	−1.75984	−0.879918	+	0.475125i	$0.842403\pi$
$908$	11.0000	0.365048
$909$	−14.0000	−0.464351
$910$	9.00000	0.298347
$911$	−49.0000	−1.62344	−0.811721	−	0.584045i	$-0.801469\pi$
$911$	−49.0000	−1.62344	−0.811721	+	0.584045i	$0.801469\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	24.0000	0.793849
$915$	30.0000	0.991769
$916$	12.0000	0.396491
$917$	6.00000	0.198137
$918$	−4.00000	−0.132020
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	3.00000	0.0989071
$921$	15.0000	0.494267
$922$	−22.0000	−0.724531
$923$	18.0000	0.592477
$924$	−4.00000	−0.131590
$925$	−36.0000	−1.18367
$926$	13.0000	0.427207
$927$	5.00000	0.164222
$928$	3.00000	0.0984798
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	−18.0000	−0.590243
$931$	0	0
$932$	−10.0000	−0.327561
$933$	−28.0000	−0.916679
$934$	13.0000	0.425373
$935$	−48.0000	−1.56977
$936$	−3.00000	−0.0980581
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	−4.00000	−0.130605
$939$	34.0000	1.10955
$940$	−21.0000	−0.684944
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	−8.00000	−0.260654
$943$	9.00000	0.293080
$944$	6.00000	0.195283
$945$	−3.00000	−0.0975900
$946$	−12.0000	−0.390154
$947$	−31.0000	−1.00736	−0.503682	−	0.863889i	$-0.668022\pi$
$947$	−31.0000	−1.00736	−0.503682	+	0.863889i	$0.668022\pi$
$948$	8.00000	0.259828
$949$	24.0000	0.779073
$950$	0	0
$951$	5.00000	0.162136
$952$	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	$0.314367\pi$
$954$	−4.00000	−0.129505
$955$	−36.0000	−1.16493
$956$	−12.0000	−0.388108
$957$	12.0000	0.387905
$958$	30.0000	0.969256
$959$	15.0000	0.484375
$960$	3.00000	0.0968246
$961$	5.00000	0.161290
$962$	27.0000	0.870515
$963$	8.00000	0.257796
$964$	−5.00000	−0.161039
$965$	−51.0000	−1.64175
$966$	−1.00000	−0.0321745
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−21.0000	−0.674269
$971$	−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
$973$	−9.00000	−0.288527
$974$	13.0000	0.416547
$975$	−12.0000	−0.384308
$976$	10.0000	0.320092
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	16.0000	0.511624
$979$	−56.0000	−1.78977
$980$	3.00000	0.0958315
$981$	3.00000	0.0957826
$982$	−12.0000	−0.382935
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	9.00000	0.286910
$985$	−69.0000	−2.19852
$986$	−12.0000	−0.382158
$987$	7.00000	0.222812
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	12.0000	0.381385
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	−6.00000	−0.190500
$993$	8.00000	0.253872
$994$	6.00000	0.190308
$995$	−15.0000	−0.475532
$996$	4.00000	0.126745
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	34.0000	1.07625
$999$	−9.00000	−0.284747

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.k.1.1	✓	1	1.1	even	1	trivial
2898.2.a.a.1.1		1	3.2	odd	2
6762.2.a.y.1.1		1	7.6	odd	2
7728.2.a.j.1.1		1	4.3	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	966.2.a.k.1.1

Level	$966$
Weight	$2$
Character	966.1
Self dual	yes
Analytic conductor	$7.714$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$