LMFDB - Embedded newform 1110.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} -1.00000 q^{30} +7.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +7.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -11.0000 q^{41} +1.00000 q^{42} -11.0000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} +2.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{57} -9.00000 q^{58} -10.0000 q^{59} +1.00000 q^{60} -1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -7.00000 q^{68} +1.00000 q^{70} -1.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +11.0000 q^{82} -6.00000 q^{83} -1.00000 q^{84} +7.00000 q^{85} +11.0000 q^{86} -9.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -7.00000 q^{93} -8.00000 q^{94} +2.00000 q^{95} +1.00000 q^{96} -19.0000 q^{97} +6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−1.00000	−0.182574
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	7.00000	1.20049
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.00000	0.324443
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	1.00000	0.154303
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	7.00000	0.980196
$52$	2.00000	0.277350
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	2.00000	0.264906
$58$	−9.00000	−1.18176
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	1.00000	0.129099
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−7.00000	−0.889001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−1.00000	−0.123091
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	1.00000	0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	1.00000	0.116248
$75$	−1.00000	−0.115470
$76$	−2.00000	−0.229416
$77$	−1.00000	−0.113961
$78$	2.00000	0.226455
$79$	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
$82$	11.0000	1.21475
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	−0.109109
$85$	7.00000	0.759257
$86$	11.0000	1.18616
$87$	−9.00000	−0.964901
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	0	0
$93$	−7.00000	−0.725866
$94$	−8.00000	−0.825137
$95$	2.00000	0.205196
$96$	1.00000	0.102062
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	6.00000	0.606092
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−20.0000	−1.99007	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−20.0000	−1.99007	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	−7.00000	−0.693103
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	11.0000	1.06841
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−1.00000	−0.0953463
$111$	1.00000	0.0949158
$112$	1.00000	0.0944911
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	9.00000	0.835629
$117$	2.00000	0.184900
$118$	10.0000	0.920575
$119$	−7.00000	−0.641689
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	1.00000	0.0905357
$123$	11.0000	0.991837
$124$	7.00000	0.628619
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	11.0000	0.968496
$130$	2.00000	0.175412
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	1.00000	0.0870388
$133$	−2.00000	−0.173422
$134$	8.00000	0.691095
$135$	1.00000	0.0860663
$136$	7.00000	0.600245
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	21.0000	1.78120	0.890598	−	0.454791i	$-0.150286\pi$
$139$	21.0000	1.78120	0.890598	+	0.454791i	$0.150286\pi$
$140$	−1.00000	−0.0845154
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	−9.00000	−0.747409
$146$	−4.00000	−0.331042
$147$	6.00000	0.494872
$148$	−1.00000	−0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	2.00000	0.162221
$153$	−7.00000	−0.565916
$154$	1.00000	0.0805823
$155$	−7.00000	−0.562254
$156$	−2.00000	−0.160128
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	−12.0000	−0.954669
$159$	11.0000	0.872357
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	−11.0000	−0.858956
$165$	−1.00000	−0.0778499
$166$	6.00000	0.465690
$167$	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
$169$	−9.00000	−0.692308
$170$	−7.00000	−0.536875
$171$	−2.00000	−0.152944
$172$	−11.0000	−0.838742
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	9.00000	0.682288
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	10.0000	0.751646
$178$	−6.00000	−0.449719
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−1.00000	−0.0745356
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	−2.00000	−0.148250
$183$	1.00000	0.0739221
$184$	0	0
$185$	1.00000	0.0735215
$186$	7.00000	0.513265
$187$	7.00000	0.511891
$188$	8.00000	0.583460
$189$	−1.00000	−0.0727393
$190$	−2.00000	−0.145095
$191$	19.0000	1.37479	0.687396	−	0.726283i	$-0.258754\pi$
$191$	19.0000	1.37479	0.687396	+	0.726283i	$0.258754\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	19.0000	1.36412
$195$	2.00000	0.143223
$196$	−6.00000	−0.428571
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	1.00000	0.0710669
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	20.0000	1.40720
$203$	9.00000	0.631676
$204$	7.00000	0.490098
$205$	11.0000	0.768273
$206$	12.0000	0.836080
$207$	0	0
$208$	2.00000	0.138675
$209$	2.00000	0.138343
$210$	−1.00000	−0.0690066
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−11.0000	−0.755483
$213$	0	0
$214$	6.00000	0.410152
$215$	11.0000	0.750194
$216$	1.00000	0.0680414
$217$	7.00000	0.475191
$218$	−7.00000	−0.474100
$219$	−4.00000	−0.270295
$220$	1.00000	0.0674200
$221$	−14.0000	−0.941742
$222$	−1.00000	−0.0671156
$223$	3.00000	0.200895	0.100447	−	0.994942i	$-0.467973\pi$
$223$	3.00000	0.200895	0.100447	+	0.994942i	$0.467973\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	13.0000	0.864747
$227$	−23.0000	−1.52656	−0.763282	−	0.646066i	$-0.776413\pi$
$227$	−23.0000	−1.52656	−0.763282	+	0.646066i	$0.776413\pi$
$228$	2.00000	0.132453
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−9.00000	−0.590879
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	−2.00000	−0.130744
$235$	−8.00000	−0.521862
$236$	−10.0000	−0.650945
$237$	−12.0000	−0.779484
$238$	7.00000	0.453743
$239$	27.0000	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$239$	27.0000	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$240$	1.00000	0.0645497
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	6.00000	0.383326
$246$	−11.0000	−0.701334
$247$	−4.00000	−0.254514
$248$	−7.00000	−0.444500
$249$	6.00000	0.380235
$250$	1.00000	0.0632456
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	16.0000	1.00393
$255$	−7.00000	−0.438357
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−11.0000	−0.684830
$259$	−1.00000	−0.0621370
$260$	−2.00000	−0.124035
$261$	9.00000	0.557086
$262$	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	$0.275832\pi$
$264$	−1.00000	−0.0615457
$265$	11.0000	0.675725
$266$	2.00000	0.122628
$267$	−6.00000	−0.367194
$268$	−8.00000	−0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−7.00000	−0.424437
$273$	−2.00000	−0.121046
$274$	6.00000	0.362473
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−21.0000	−1.25950
$279$	7.00000	0.419079
$280$	1.00000	0.0597614
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	8.00000	0.476393
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	2.00000	0.118262
$287$	−11.0000	−0.649309
$288$	−1.00000	−0.0589256
$289$	32.0000	1.88235
$290$	9.00000	0.528498
$291$	19.0000	1.11380
$292$	4.00000	0.234082
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−6.00000	−0.349927
$295$	10.0000	0.582223
$296$	1.00000	0.0581238
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−11.0000	−0.634029
$302$	6.00000	0.345261
$303$	20.0000	1.14897
$304$	−2.00000	−0.114708
$305$	1.00000	0.0572598
$306$	7.00000	0.400163
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	−1.00000	−0.0569803
$309$	12.0000	0.682656
$310$	7.00000	0.397573
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	2.00000	0.113228
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	−23.0000	−1.29797
$315$	−1.00000	−0.0563436
$316$	12.0000	0.675053
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	−11.0000	−0.616849
$319$	−9.00000	−0.503903
$320$	−1.00000	−0.0559017
$321$	6.00000	0.334887
$322$	0	0
$323$	14.0000	0.778981
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−13.0000	−0.720003
$327$	−7.00000	−0.387101
$328$	11.0000	0.607373
$329$	8.00000	0.441054
$330$	1.00000	0.0550482
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−6.00000	−0.329293
$333$	−1.00000	−0.0547997
$334$	−12.0000	−0.656611
$335$	8.00000	0.437087
$336$	−1.00000	−0.0545545
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	9.00000	0.489535
$339$	13.0000	0.706063
$340$	7.00000	0.379628
$341$	−7.00000	−0.379071
$342$	2.00000	0.108148
$343$	−13.0000	−0.701934
$344$	11.0000	0.593080
$345$	0	0
$346$	3.00000	0.161281
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−9.00000	−0.482451
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	−1.00000	−0.0534522
$351$	−2.00000	−0.106752
$352$	1.00000	0.0533002
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	6.00000	0.317999
$357$	7.00000	0.370479
$358$	−24.0000	−1.26844
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	26.0000	1.36653
$363$	10.0000	0.524864
$364$	2.00000	0.104828
$365$	−4.00000	−0.209370
$366$	−1.00000	−0.0522708
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	−11.0000	−0.572637
$370$	−1.00000	−0.0519875
$371$	−11.0000	−0.571092
$372$	−7.00000	−0.362933
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−7.00000	−0.361961
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	18.0000	0.927047
$378$	1.00000	0.0514344
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	2.00000	0.102598
$381$	16.0000	0.819705
$382$	−19.0000	−0.972125
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	1.00000	0.0509647
$386$	−14.0000	−0.712581
$387$	−11.0000	−0.559161
$388$	−19.0000	−0.964579
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	6.00000	0.303046
$393$	6.00000	0.302660
$394$	18.0000	0.906827
$395$	−12.0000	−0.603786
$396$	−1.00000	−0.0502519
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	4.00000	0.200502
$399$	2.00000	0.100125
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−8.00000	−0.399004
$403$	14.0000	0.697390
$404$	−20.0000	−0.995037
$405$	−1.00000	−0.0496904
$406$	−9.00000	−0.446663
$407$	1.00000	0.0495682
$408$	−7.00000	−0.346552
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	−11.0000	−0.543251
$411$	6.00000	0.295958
$412$	−12.0000	−0.591198
$413$	−10.0000	−0.492068
$414$	0	0
$415$	6.00000	0.294528
$416$	−2.00000	−0.0980581
$417$	−21.0000	−1.02837
$418$	−2.00000	−0.0978232
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	1.00000	0.0487950
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	5.00000	0.243396
$423$	8.00000	0.388973
$424$	11.0000	0.534207
$425$	−7.00000	−0.339550
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	−6.00000	−0.290021
$429$	2.00000	0.0965609
$430$	−11.0000	−0.530467
$431$	−1.00000	−0.0481683	−0.0240842	−	0.999710i	$-0.507667\pi$
$431$	−1.00000	−0.0481683	−0.0240842	+	0.999710i	$0.507667\pi$
$432$	−1.00000	−0.0481125
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	−7.00000	−0.336011
$435$	9.00000	0.431517
$436$	7.00000	0.335239
$437$	0	0
$438$	4.00000	0.191127
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	−1.00000	−0.0476731
$441$	−6.00000	−0.285714
$442$	14.0000	0.665912
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	1.00000	0.0474579
$445$	−6.00000	−0.284427
$446$	−3.00000	−0.142054
$447$	−6.00000	−0.283790
$448$	1.00000	0.0472456
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	−1.00000	−0.0471405
$451$	11.0000	0.517970
$452$	−13.0000	−0.611469
$453$	6.00000	0.281905
$454$	23.0000	1.07944
$455$	−2.00000	−0.0937614
$456$	−2.00000	−0.0936586
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	16.0000	0.747631
$459$	7.00000	0.326732
$460$	0	0
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	−1.00000	−0.0465242
$463$	2.00000	0.0929479	0.0464739	−	0.998920i	$-0.485202\pi$
$463$	2.00000	0.0929479	0.0464739	+	0.998920i	$0.485202\pi$
$464$	9.00000	0.417815
$465$	7.00000	0.324617
$466$	−24.0000	−1.11178
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	2.00000	0.0924500
$469$	−8.00000	−0.369406
$470$	8.00000	0.369012
$471$	−23.0000	−1.05978
$472$	10.0000	0.460287
$473$	11.0000	0.505781
$474$	12.0000	0.551178
$475$	−2.00000	−0.0917663
$476$	−7.00000	−0.320844
$477$	−11.0000	−0.503655
$478$	−27.0000	−1.23495
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−1.00000	−0.0456435
$481$	−2.00000	−0.0911922
$482$	−6.00000	−0.273293
$483$	0	0
$484$	−10.0000	−0.454545
$485$	19.0000	0.862746
$486$	1.00000	0.0453609
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	1.00000	0.0452679
$489$	−13.0000	−0.587880
$490$	−6.00000	−0.271052
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	11.0000	0.495918
$493$	−63.0000	−2.83738
$494$	4.00000	0.179969
$495$	1.00000	0.0449467
$496$	7.00000	0.314309
$497$	0	0
$498$	−6.00000	−0.268866
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−1.00000	−0.0447214
$501$	−12.0000	−0.536120
$502$	16.0000	0.714115
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−1.00000	−0.0445435
$505$	20.0000	0.889988
$506$	0	0
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	7.00000	0.309965
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	−14.0000	−0.617514
$515$	12.0000	0.528783
$516$	11.0000	0.484248
$517$	−8.00000	−0.351840
$518$	1.00000	0.0439375
$519$	3.00000	0.131685
$520$	2.00000	0.0877058
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	−9.00000	−0.393919
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	−6.00000	−0.262111
$525$	−1.00000	−0.0436436
$526$	−21.0000	−0.915644
$527$	−49.0000	−2.13447
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	−11.0000	−0.477809
$531$	−10.0000	−0.433963
$532$	−2.00000	−0.0867110
$533$	−22.0000	−0.952926
$534$	6.00000	0.259645
$535$	6.00000	0.259403
$536$	8.00000	0.345547
$537$	−24.0000	−1.03568
$538$	0	0
$539$	6.00000	0.258438
$540$	1.00000	0.0430331
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	26.0000	1.11577
$544$	7.00000	0.300123
$545$	−7.00000	−0.299847
$546$	2.00000	0.0855921
$547$	−11.0000	−0.470326	−0.235163	−	0.971956i	$-0.575562\pi$
$547$	−11.0000	−0.470326	−0.235163	+	0.971956i	$0.575562\pi$
$548$	−6.00000	−0.256307
$549$	−1.00000	−0.0426790
$550$	1.00000	0.0426401
$551$	−18.0000	−0.766826
$552$	0	0
$553$	12.0000	0.510292
$554$	−8.00000	−0.339887
$555$	−1.00000	−0.0424476
$556$	21.0000	0.890598
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−7.00000	−0.296334
$559$	−22.0000	−0.930501
$560$	−1.00000	−0.0422577
$561$	−7.00000	−0.295540
$562$	−6.00000	−0.253095
$563$	13.0000	0.547885	0.273942	−	0.961746i	$-0.411672\pi$
$563$	13.0000	0.547885	0.273942	+	0.961746i	$0.411672\pi$
$564$	−8.00000	−0.336861
$565$	13.0000	0.546914
$566$	−12.0000	−0.504398
$567$	1.00000	0.0419961
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	2.00000	0.0837708
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	−2.00000	−0.0836242
$573$	−19.0000	−0.793736
$574$	11.0000	0.459131
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−32.0000	−1.33102
$579$	−14.0000	−0.581820
$580$	−9.00000	−0.373705
$581$	−6.00000	−0.248922
$582$	−19.0000	−0.787575
$583$	11.0000	0.455573
$584$	−4.00000	−0.165521
$585$	−2.00000	−0.0826898
$586$	9.00000	0.371787
$587$	43.0000	1.77480	0.887400	−	0.461000i	$-0.152509\pi$
$587$	43.0000	1.77480	0.887400	+	0.461000i	$0.152509\pi$
$588$	6.00000	0.247436
$589$	−14.0000	−0.576860
$590$	−10.0000	−0.411693
$591$	18.0000	0.740421
$592$	−1.00000	−0.0410997
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−1.00000	−0.0410305
$595$	7.00000	0.286972
$596$	6.00000	0.245770
$597$	4.00000	0.163709
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	1.00000	0.0408248
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	11.0000	0.448327
$603$	−8.00000	−0.325785
$604$	−6.00000	−0.244137
$605$	10.0000	0.406558
$606$	−20.0000	−0.812444
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	2.00000	0.0811107
$609$	−9.00000	−0.364698
$610$	−1.00000	−0.0404888
$611$	16.0000	0.647291
$612$	−7.00000	−0.282958
$613$	33.0000	1.33286	0.666429	−	0.745569i	$-0.267822\pi$
$613$	33.0000	1.33286	0.666429	+	0.745569i	$0.267822\pi$
$614$	−10.0000	−0.403567
$615$	−11.0000	−0.443563
$616$	1.00000	0.0402911
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	−12.0000	−0.482711
$619$	−19.0000	−0.763674	−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000	−0.763674	−0.381837	+	0.924230i	$0.624709\pi$
$620$	−7.00000	−0.281127
$621$	0	0
$622$	9.00000	0.360867
$623$	6.00000	0.240385
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	18.0000	0.719425
$627$	−2.00000	−0.0798723
$628$	23.0000	0.917800
$629$	7.00000	0.279108
$630$	1.00000	0.0398410
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−12.0000	−0.477334
$633$	5.00000	0.198732
$634$	−5.00000	−0.198575
$635$	16.0000	0.634941
$636$	11.0000	0.436178
$637$	−12.0000	−0.475457
$638$	9.00000	0.356313
$639$	0	0
$640$	1.00000	0.0395285
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	−6.00000	−0.236801
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	−11.0000	−0.433125
$646$	−14.0000	−0.550823
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−1.00000	−0.0392837
$649$	10.0000	0.392534
$650$	−2.00000	−0.0784465
$651$	−7.00000	−0.274352
$652$	13.0000	0.509119
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	7.00000	0.273722
$655$	6.00000	0.234439
$656$	−11.0000	−0.429478
$657$	4.00000	0.156055
$658$	−8.00000	−0.311872
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−1.00000	−0.0389249
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	8.00000	0.310929
$663$	14.0000	0.543715
$664$	6.00000	0.232845
$665$	2.00000	0.0775567
$666$	1.00000	0.0387492
$667$	0	0
$668$	12.0000	0.464294
$669$	−3.00000	−0.115987
$670$	−8.00000	−0.309067
$671$	1.00000	0.0386046
$672$	1.00000	0.0385758
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	−22.0000	−0.847408
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−13.0000	−0.499262
$679$	−19.0000	−0.729153
$680$	−7.00000	−0.268438
$681$	23.0000	0.881362
$682$	7.00000	0.268044
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	−2.00000	−0.0764719
$685$	6.00000	0.229248
$686$	13.0000	0.496342
$687$	16.0000	0.610438
$688$	−11.0000	−0.419371
$689$	−22.0000	−0.838133
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	−3.00000	−0.114043
$693$	−1.00000	−0.0379869
$694$	−28.0000	−1.06287
$695$	−21.0000	−0.796575
$696$	9.00000	0.341144
$697$	77.0000	2.91658
$698$	32.0000	1.21122
$699$	−24.0000	−0.907763
$700$	1.00000	0.0377964
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	2.00000	0.0754851
$703$	2.00000	0.0754314
$704$	−1.00000	−0.0376889
$705$	8.00000	0.301297
$706$	31.0000	1.16670
$707$	−20.0000	−0.752177
$708$	10.0000	0.375823
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	−6.00000	−0.224860
$713$	0	0
$714$	−7.00000	−0.261968
$715$	2.00000	0.0747958
$716$	24.0000	0.896922
$717$	−27.0000	−1.00833
$718$	−14.0000	−0.522475
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	−1.00000	−0.0372678
$721$	−12.0000	−0.446903
$722$	15.0000	0.558242
$723$	−6.00000	−0.223142
$724$	−26.0000	−0.966282
$725$	9.00000	0.334252
$726$	−10.0000	−0.371135
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	77.0000	2.84795
$732$	1.00000	0.0369611
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	−17.0000	−0.627481
$735$	−6.00000	−0.221313
$736$	0	0
$737$	8.00000	0.294684
$738$	11.0000	0.404916
$739$	−31.0000	−1.14035	−0.570177	−	0.821522i	$-0.693125\pi$
$739$	−31.0000	−1.14035	−0.570177	+	0.821522i	$0.693125\pi$
$740$	1.00000	0.0367607
$741$	4.00000	0.146944
$742$	11.0000	0.403823
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	7.00000	0.256632
$745$	−6.00000	−0.219823
$746$	6.00000	0.219676
$747$	−6.00000	−0.219529
$748$	7.00000	0.255945
$749$	−6.00000	−0.219235
$750$	−1.00000	−0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	8.00000	0.291730
$753$	16.0000	0.583072
$754$	−18.0000	−0.655521
$755$	6.00000	0.218362
$756$	−1.00000	−0.0363696
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	−43.0000	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$761$	−43.0000	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$762$	−16.0000	−0.579619
$763$	7.00000	0.253417
$764$	19.0000	0.687396
$765$	7.00000	0.253086
$766$	−24.0000	−0.867155
$767$	−20.0000	−0.722158
$768$	−1.00000	−0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	−1.00000	−0.0360375
$771$	−14.0000	−0.504198
$772$	14.0000	0.503871
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	11.0000	0.395387
$775$	7.00000	0.251447
$776$	19.0000	0.682060
$777$	1.00000	0.0358748
$778$	21.0000	0.752886
$779$	22.0000	0.788232
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	−9.00000	−0.321634
$784$	−6.00000	−0.214286
$785$	−23.0000	−0.820905
$786$	−6.00000	−0.214013
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	−18.0000	−0.641223
$789$	−21.0000	−0.747620
$790$	12.0000	0.426941
$791$	−13.0000	−0.462227
$792$	1.00000	0.0355335
$793$	−2.00000	−0.0710221
$794$	14.0000	0.496841
$795$	−11.0000	−0.390130
$796$	−4.00000	−0.141776
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	−2.00000	−0.0707992
$799$	−56.0000	−1.98114
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	6.00000	0.211867
$803$	−4.00000	−0.141157
$804$	8.00000	0.282138
$805$	0	0
$806$	−14.0000	−0.493129
$807$	0	0
$808$	20.0000	0.703598
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	1.00000	0.0351364
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	9.00000	0.315838
$813$	−8.00000	−0.280572
$814$	−1.00000	−0.0350500
$815$	−13.0000	−0.455370
$816$	7.00000	0.245049
$817$	22.0000	0.769683
$818$	18.0000	0.629355
$819$	2.00000	0.0698857
$820$	11.0000	0.384137
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	−6.00000	−0.209274
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	12.0000	0.418040
$825$	1.00000	0.0348155
$826$	10.0000	0.347945
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	−6.00000	−0.208263
$831$	−8.00000	−0.277517
$832$	2.00000	0.0693375
$833$	42.0000	1.45521
$834$	21.0000	0.727171
$835$	−12.0000	−0.415277
$836$	2.00000	0.0691714
$837$	−7.00000	−0.241955
$838$	−28.0000	−0.967244
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	−1.00000	−0.0345033
$841$	52.0000	1.79310
$842$	2.00000	0.0689246
$843$	−6.00000	−0.206651
$844$	−5.00000	−0.172107
$845$	9.00000	0.309609
$846$	−8.00000	−0.275046
$847$	−10.0000	−0.343604
$848$	−11.0000	−0.377742
$849$	−12.0000	−0.411839
$850$	7.00000	0.240098
$851$	0	0
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	1.00000	0.0342193
$855$	2.00000	0.0683986
$856$	6.00000	0.205076
$857$	31.0000	1.05894	0.529470	−	0.848329i	$-0.322391\pi$
$857$	31.0000	1.05894	0.529470	+	0.848329i	$0.322391\pi$
$858$	−2.00000	−0.0682789
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	11.0000	0.375097
$861$	11.0000	0.374879
$862$	1.00000	0.0340601
$863$	27.0000	0.919091	0.459545	−	0.888154i	$-0.348012\pi$
$863$	27.0000	0.919091	0.459545	+	0.888154i	$0.348012\pi$
$864$	1.00000	0.0340207
$865$	3.00000	0.102003
$866$	−20.0000	−0.679628
$867$	−32.0000	−1.08678
$868$	7.00000	0.237595
$869$	−12.0000	−0.407072
$870$	−9.00000	−0.305129
$871$	−16.0000	−0.542139
$872$	−7.00000	−0.237050
$873$	−19.0000	−0.643053
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	−4.00000	−0.135147
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	−1.00000	−0.0337484
$879$	9.00000	0.303562
$880$	1.00000	0.0337100
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	6.00000	0.202031
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	−14.0000	−0.470871
$885$	−10.0000	−0.336146
$886$	30.0000	1.00787
$887$	−47.0000	−1.57811	−0.789053	−	0.614325i	$-0.789428\pi$
$887$	−47.0000	−1.57811	−0.789053	+	0.614325i	$0.789428\pi$
$888$	−1.00000	−0.0335578
$889$	−16.0000	−0.536623
$890$	6.00000	0.201120
$891$	−1.00000	−0.0335013
$892$	3.00000	0.100447
$893$	−16.0000	−0.535420
$894$	6.00000	0.200670
$895$	−24.0000	−0.802232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−8.00000	−0.266963
$899$	63.0000	2.10117
$900$	1.00000	0.0333333
$901$	77.0000	2.56524
$902$	−11.0000	−0.366260
$903$	11.0000	0.366057
$904$	13.0000	0.432374
$905$	26.0000	0.864269
$906$	−6.00000	−0.199337
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−23.0000	−0.763282
$909$	−20.0000	−0.663358
$910$	2.00000	0.0662994
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	2.00000	0.0662266
$913$	6.00000	0.198571
$914$	11.0000	0.363848
$915$	−1.00000	−0.0330590
$916$	−16.0000	−0.528655
$917$	−6.00000	−0.198137
$918$	−7.00000	−0.231034
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	13.0000	0.428132
$923$	0	0
$924$	1.00000	0.0328976
$925$	−1.00000	−0.0328798
$926$	−2.00000	−0.0657241
$927$	−12.0000	−0.394132
$928$	−9.00000	−0.295439
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	−7.00000	−0.229539
$931$	12.0000	0.393284
$932$	24.0000	0.786146
$933$	9.00000	0.294647
$934$	−3.00000	−0.0981630
$935$	−7.00000	−0.228924
$936$	−2.00000	−0.0653720
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	8.00000	0.261209
$939$	18.0000	0.587408
$940$	−8.00000	−0.260931
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	23.0000	0.749380
$943$	0	0
$944$	−10.0000	−0.325472
$945$	1.00000	0.0325300
$946$	−11.0000	−0.357641
$947$	−47.0000	−1.52729	−0.763647	−	0.645634i	$-0.776593\pi$
$947$	−47.0000	−1.52729	−0.763647	+	0.645634i	$0.776593\pi$
$948$	−12.0000	−0.389742
$949$	8.00000	0.259691
$950$	2.00000	0.0648886
$951$	−5.00000	−0.162136
$952$	7.00000	0.226871
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	11.0000	0.356138
$955$	−19.0000	−0.614826
$956$	27.0000	0.873242
$957$	9.00000	0.290929
$958$	−24.0000	−0.775405
$959$	−6.00000	−0.193750
$960$	1.00000	0.0322749
$961$	18.0000	0.580645
$962$	2.00000	0.0644826
$963$	−6.00000	−0.193347
$964$	6.00000	0.193247
$965$	−14.0000	−0.450676
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	10.0000	0.321412
$969$	−14.0000	−0.449745
$970$	−19.0000	−0.610053
$971$	49.0000	1.57248	0.786242	−	0.617918i	$-0.212024\pi$
$971$	49.0000	1.57248	0.786242	+	0.617918i	$0.212024\pi$
$972$	−1.00000	−0.0320750
$973$	21.0000	0.673229
$974$	−18.0000	−0.576757
$975$	−2.00000	−0.0640513
$976$	−1.00000	−0.0320092
$977$	5.00000	0.159964	0.0799821	−	0.996796i	$-0.474514\pi$
$977$	5.00000	0.159964	0.0799821	+	0.996796i	$0.474514\pi$
$978$	13.0000	0.415694
$979$	−6.00000	−0.191761
$980$	6.00000	0.191663
$981$	7.00000	0.223493
$982$	12.0000	0.382935
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	−11.0000	−0.350667
$985$	18.0000	0.573528
$986$	63.0000	2.00633
$987$	−8.00000	−0.254643
$988$	−4.00000	−0.127257
$989$	0	0
$990$	−1.00000	−0.0317821
$991$	37.0000	1.17534	0.587672	−	0.809099i	$-0.300045\pi$
$991$	37.0000	1.17534	0.587672	+	0.809099i	$0.300045\pi$
$992$	−7.00000	−0.222250
$993$	8.00000	0.253872
$994$	0	0
$995$	4.00000	0.126809
$996$	6.00000	0.190117
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	28.0000	0.886325
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.b.1.1	✓	1
3.2	odd	2		3330.2.a.x.1.1		1
4.3	odd	2		8880.2.a.s.1.1		1
5.4	even	2		5550.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.b.1.1	✓	1	1.1	even	1	trivial
3330.2.a.x.1.1		1	3.2	odd	2
5550.2.a.bk.1.1		1	5.4	even	2
8880.2.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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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