LMFDB - Embedded newform 462.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	462.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:	$3.68908857338$
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	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -14.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +2.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +12.0000 q^{85} -8.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} -2.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -4.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} +18.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000	0.447214
$21$	1.00000	0.218218
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−6.00000	−1.02899
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	8.00000	1.29777
$39$	−2.00000	−0.320256
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−1.00000	−0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	−4.00000	−0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	2.00000	0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	1.00000	0.133631
$57$	8.00000	1.05963
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−2.00000	−0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−8.00000	−1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	−4.00000	−0.481543
$70$	2.00000	0.239046
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	−8.00000	−0.917663
$77$	−1.00000	−0.113961
$78$	2.00000	0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	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
$86$	−8.00000	−0.862662
$87$	−2.00000	−0.214423
$88$	−1.00000	−0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	4.00000	0.417029
$93$	−8.00000	−0.829561
$94$	−4.00000	−0.412568
$95$	−16.0000	−1.64157
$96$	1.00000	0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	−0.196116
$105$	2.00000	0.195180
$106$	−10.0000	−0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	−6.00000	−0.569495
$112$	−1.00000	−0.0944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−8.00000	−0.749269
$115$	8.00000	0.746004
$116$	2.00000	0.185695
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	−6.00000	−0.550019
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−4.00000	−0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−1.00000	−0.0870388
$133$	8.00000	0.693688
$134$	4.00000	0.345547
$135$	−2.00000	−0.172133
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	4.00000	0.340503
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−2.00000	−0.169031
$141$	−4.00000	−0.336861
$142$	4.00000	0.335673
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	−1.00000	−0.0824786
$148$	6.00000	0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	8.00000	0.648886
$153$	6.00000	0.485071
$154$	1.00000	0.0805823
$155$	16.0000	1.28515
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	8.00000	0.636446
$159$	−10.0000	−0.793052
$160$	−2.00000	−0.158114
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	−8.00000	−0.611775
$172$	8.00000	0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	2.00000	0.151620
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	−4.00000	−0.300658
$178$	14.0000	1.04934
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	14.0000	1.03491
$184$	−4.00000	−0.294884
$185$	12.0000	0.882258
$186$	8.00000	0.586588
$187$	6.00000	0.438763
$188$	4.00000	0.291730
$189$	1.00000	0.0727393
$190$	16.0000	1.16076
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−18.0000	−1.29232
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	−6.00000	−0.420084
$205$	12.0000	0.838116
$206$	16.0000	1.11477
$207$	4.00000	0.278019
$208$	2.00000	0.138675
$209$	−8.00000	−0.553372
$210$	−2.00000	−0.138013
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	10.0000	0.686803
$213$	4.00000	0.274075
$214$	12.0000	0.820303
$215$	16.0000	1.09119
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	−10.0000	−0.677285
$219$	14.0000	0.946032
$220$	2.00000	0.134840
$221$	12.0000	0.807207
$222$	6.00000	0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	−10.0000	−0.665190
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	8.00000	0.529813
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	1.00000	0.0657952
$232$	−2.00000	−0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	8.00000	0.521862
$236$	4.00000	0.260378
$237$	8.00000	0.519656
$238$	6.00000	0.388922
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	2.00000	0.127775
$246$	6.00000	0.382546
$247$	−16.0000	−1.01806
$248$	−8.00000	−0.508001
$249$	−4.00000	−0.253490
$250$	12.0000	0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	−8.00000	−0.501965
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	8.00000	0.498058
$259$	−6.00000	−0.372822
$260$	4.00000	0.248069
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.00000	0.0615457
$265$	20.0000	1.22859
$266$	−8.00000	−0.490511
$267$	14.0000	0.856786
$268$	−4.00000	−0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	2.00000	0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	2.00000	0.121046
$274$	6.00000	0.362473
$275$	−1.00000	−0.0603023
$276$	−4.00000	−0.240772
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	16.0000	0.959616
$279$	8.00000	0.478947
$280$	2.00000	0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	4.00000	0.238197
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−4.00000	−0.237356
$285$	16.0000	0.947758
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	−18.0000	−1.05518
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	1.00000	0.0583212
$295$	8.00000	0.465778
$296$	−6.00000	−0.348743
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	8.00000	0.462652
$300$	1.00000	0.0577350
$301$	−8.00000	−0.461112
$302$	−16.0000	−0.920697
$303$	6.00000	0.344691
$304$	−8.00000	−0.458831
$305$	−28.0000	−1.60328
$306$	−6.00000	−0.342997
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−1.00000	−0.0569803
$309$	16.0000	0.910208
$310$	−16.0000	−0.908739
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	2.00000	0.113228
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	2.00000	0.112867
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	10.0000	0.560772
$319$	2.00000	0.111979
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	4.00000	0.222911
$323$	−48.0000	−2.67079
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−12.0000	−0.664619
$327$	−10.0000	−0.553001
$328$	−6.00000	−0.331295
$329$	−4.00000	−0.220527
$330$	2.00000	0.110096
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	8.00000	0.437741
$335$	−8.00000	−0.437087
$336$	1.00000	0.0545545
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	9.00000	0.489535
$339$	−10.0000	−0.543125
$340$	12.0000	0.650791
$341$	8.00000	0.433224
$342$	8.00000	0.432590
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	−8.00000	−0.430706
$346$	14.0000	0.752645
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	−2.00000	−0.107211
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−1.00000	−0.0534522
$351$	−2.00000	−0.106752
$352$	−1.00000	−0.0533002
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	4.00000	0.212598
$355$	−8.00000	−0.424596
$356$	−14.0000	−0.741999
$357$	6.00000	0.317554
$358$	12.0000	0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	−1.00000	−0.0524864
$364$	−2.00000	−0.104828
$365$	−28.0000	−1.46559
$366$	−14.0000	−0.731792
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00000	0.208514
$369$	6.00000	0.312348
$370$	−12.0000	−0.623850
$371$	−10.0000	−0.519174
$372$	−8.00000	−0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−6.00000	−0.310253
$375$	12.0000	0.619677
$376$	−4.00000	−0.206284
$377$	4.00000	0.206010
$378$	−1.00000	−0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−16.0000	−0.820783
$381$	−8.00000	−0.409852
$382$	4.00000	0.204658
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	1.00000	0.0510310
$385$	−2.00000	−0.101929
$386$	6.00000	0.305392
$387$	8.00000	0.406663
$388$	18.0000	0.913812
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	4.00000	0.202548
$391$	24.0000	1.21373
$392$	−1.00000	−0.0505076
$393$	4.00000	0.201773
$394$	6.00000	0.302276
$395$	−16.0000	−0.805047
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	8.00000	0.401004
$399$	−8.00000	−0.400501
$400$	−1.00000	−0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	−4.00000	−0.199502
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	2.00000	0.0992583
$407$	6.00000	0.297409
$408$	6.00000	0.297044
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−12.0000	−0.592638
$411$	6.00000	0.295958
$412$	−16.0000	−0.788263
$413$	−4.00000	−0.196827
$414$	−4.00000	−0.196589
$415$	8.00000	0.392705
$416$	−2.00000	−0.0980581
$417$	16.0000	0.783523
$418$	8.00000	0.391293
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	2.00000	0.0975900
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−8.00000	−0.389434
$423$	4.00000	0.194487
$424$	−10.0000	−0.485643
$425$	−6.00000	−0.291043
$426$	−4.00000	−0.193801
$427$	14.0000	0.677507
$428$	−12.0000	−0.580042
$429$	−2.00000	−0.0965609
$430$	−16.0000	−0.771589
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	8.00000	0.384012
$435$	−4.00000	−0.191785
$436$	10.0000	0.478913
$437$	−32.0000	−1.53077
$438$	−14.0000	−0.668946
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	−2.00000	−0.0953463
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−6.00000	−0.284747
$445$	−28.0000	−1.32733
$446$	−16.0000	−0.757622
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	1.00000	0.0471405
$451$	6.00000	0.282529
$452$	10.0000	0.470360
$453$	−16.0000	−0.751746
$454$	4.00000	0.187729
$455$	−4.00000	−0.187523
$456$	−8.00000	−0.374634
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	10.0000	0.467269
$459$	−6.00000	−0.280056
$460$	8.00000	0.373002
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−1.00000	−0.0465242
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	2.00000	0.0928477
$465$	−16.0000	−0.741982
$466$	18.0000	0.833834
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	−8.00000	−0.369012
$471$	2.00000	0.0921551
$472$	−4.00000	−0.184115
$473$	8.00000	0.367840
$474$	−8.00000	−0.367452
$475$	8.00000	0.367065
$476$	−6.00000	−0.275010
$477$	10.0000	0.457869
$478$	8.00000	0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	2.00000	0.0912871
$481$	12.0000	0.547153
$482$	−2.00000	−0.0910975
$483$	4.00000	0.182006
$484$	1.00000	0.0454545
$485$	36.0000	1.63468
$486$	1.00000	0.0453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	14.0000	0.633750
$489$	−12.0000	−0.542659
$490$	−2.00000	−0.0903508
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−6.00000	−0.270501
$493$	12.0000	0.540453
$494$	16.0000	0.719874
$495$	2.00000	0.0898933
$496$	8.00000	0.359211
$497$	4.00000	0.179425
$498$	4.00000	0.179244
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−12.0000	−0.536656
$501$	8.00000	0.357414
$502$	20.0000	0.892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	1.00000	0.0445435
$505$	−12.0000	−0.533993
$506$	−4.00000	−0.177822
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	12.0000	0.531369
$511$	14.0000	0.619324
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	−26.0000	−1.14681
$515$	−32.0000	−1.41009
$516$	−8.00000	−0.352180
$517$	4.00000	0.175920
$518$	6.00000	0.263625
$519$	14.0000	0.614532
$520$	−4.00000	−0.175412
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−4.00000	−0.174741
$525$	−1.00000	−0.0436436
$526$	−24.0000	−1.04645
$527$	48.0000	2.09091
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	−20.0000	−0.868744
$531$	4.00000	0.173585
$532$	8.00000	0.346844
$533$	12.0000	0.519778
$534$	−14.0000	−0.605839
$535$	−24.0000	−1.03761
$536$	4.00000	0.172774
$537$	12.0000	0.517838
$538$	−2.00000	−0.0862261
$539$	1.00000	0.0430730
$540$	−2.00000	−0.0860663
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−8.00000	−0.343629
$543$	10.0000	0.429141
$544$	−6.00000	−0.257248
$545$	20.0000	0.856706
$546$	−2.00000	−0.0855921
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	−6.00000	−0.256307
$549$	−14.0000	−0.597505
$550$	1.00000	0.0426401
$551$	−16.0000	−0.681623
$552$	4.00000	0.170251
$553$	8.00000	0.340195
$554$	6.00000	0.254916
$555$	−12.0000	−0.509372
$556$	−16.0000	−0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−8.00000	−0.338667
$559$	16.0000	0.676728
$560$	−2.00000	−0.0845154
$561$	−6.00000	−0.253320
$562$	−30.0000	−1.26547
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−4.00000	−0.168430
$565$	20.0000	0.841406
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	4.00000	0.167836
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	−16.0000	−0.670166
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	2.00000	0.0836242
$573$	4.00000	0.167102
$574$	6.00000	0.250435
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−19.0000	−0.790296
$579$	6.00000	0.249351
$580$	4.00000	0.166091
$581$	−4.00000	−0.165948
$582$	18.0000	0.746124
$583$	10.0000	0.414158
$584$	14.0000	0.579324
$585$	4.00000	0.165380
$586$	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
$589$	−64.0000	−2.63707
$590$	−8.00000	−0.329355
$591$	6.00000	0.246807
$592$	6.00000	0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	1.00000	0.0410305
$595$	−12.0000	−0.491952
$596$	10.0000	0.409616
$597$	8.00000	0.327418
$598$	−8.00000	−0.327144
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−1.00000	−0.0408248
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	16.0000	0.651031
$605$	2.00000	0.0813116
$606$	−6.00000	−0.243733
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	8.00000	0.324443
$609$	2.00000	0.0810441
$610$	28.0000	1.13369
$611$	8.00000	0.323645
$612$	6.00000	0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	8.00000	0.322854
$615$	−12.0000	−0.483887
$616$	1.00000	0.0402911
$617$	−46.0000	−1.85189	−0.925945	−	0.377658i	$-0.876729\pi$
$617$	−46.0000	−1.85189	−0.925945	+	0.377658i	$0.876729\pi$
$618$	−16.0000	−0.643614
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	16.0000	0.642575
$621$	−4.00000	−0.160514
$622$	−12.0000	−0.481156
$623$	14.0000	0.560898
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	8.00000	0.319489
$628$	−2.00000	−0.0798087
$629$	36.0000	1.43541
$630$	2.00000	0.0796819
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	8.00000	0.318223
$633$	−8.00000	−0.317971
$634$	−18.0000	−0.714871
$635$	16.0000	0.634941
$636$	−10.0000	−0.396526
$637$	2.00000	0.0792429
$638$	−2.00000	−0.0791808
$639$	−4.00000	−0.158238
$640$	−2.00000	−0.0790569
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−12.0000	−0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−4.00000	−0.157622
$645$	−16.0000	−0.629999
$646$	48.0000	1.88853
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	4.00000	0.157014
$650$	2.00000	0.0784465
$651$	8.00000	0.313545
$652$	12.0000	0.469956
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	10.0000	0.391031
$655$	−8.00000	−0.312586
$656$	6.00000	0.234261
$657$	−14.0000	−0.546192
$658$	4.00000	0.155936
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	−2.00000	−0.0778499
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−4.00000	−0.155464
$663$	−12.0000	−0.466041
$664$	−4.00000	−0.155230
$665$	16.0000	0.620453
$666$	−6.00000	−0.232495
$667$	8.00000	0.309761
$668$	−8.00000	−0.309529
$669$	−16.0000	−0.618596
$670$	8.00000	0.309067
$671$	−14.0000	−0.540464
$672$	−1.00000	−0.0385758
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	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	$0.536783\pi$
$678$	10.0000	0.384048
$679$	−18.0000	−0.690777
$680$	−12.0000	−0.460179
$681$	4.00000	0.153280
$682$	−8.00000	−0.306336
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−8.00000	−0.305888
$685$	−12.0000	−0.458496
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	8.00000	0.304997
$689$	20.0000	0.761939
$690$	8.00000	0.304555
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−14.0000	−0.532200
$693$	−1.00000	−0.0379869
$694$	36.0000	1.36654
$695$	−32.0000	−1.21383
$696$	2.00000	0.0758098
$697$	36.0000	1.36360
$698$	22.0000	0.832712
$699$	18.0000	0.680823
$700$	1.00000	0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	2.00000	0.0754851
$703$	−48.0000	−1.81035
$704$	1.00000	0.0376889
$705$	−8.00000	−0.301297
$706$	−26.0000	−0.978523
$707$	6.00000	0.225653
$708$	−4.00000	−0.150329
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	8.00000	0.300235
$711$	−8.00000	−0.300023
$712$	14.0000	0.524672
$713$	32.0000	1.19841
$714$	−6.00000	−0.224544
$715$	4.00000	0.149592
$716$	−12.0000	−0.448461
$717$	8.00000	0.298765
$718$	−24.0000	−0.895672
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	2.00000	0.0745356
$721$	16.0000	0.595871
$722$	−45.0000	−1.67473
$723$	−2.00000	−0.0743808
$724$	−10.0000	−0.371647
$725$	−2.00000	−0.0742781
$726$	1.00000	0.0371135
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	48.0000	1.77534
$732$	14.0000	0.517455
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	−4.00000	−0.147442
$737$	−4.00000	−0.147342
$738$	−6.00000	−0.220863
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	12.0000	0.441129
$741$	16.0000	0.587775
$742$	10.0000	0.367112
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	8.00000	0.293294
$745$	20.0000	0.732743
$746$	22.0000	0.805477
$747$	4.00000	0.146352
$748$	6.00000	0.219382
$749$	12.0000	0.438470
$750$	−12.0000	−0.438178
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	4.00000	0.145865
$753$	20.0000	0.728841
$754$	−4.00000	−0.145671
$755$	32.0000	1.16460
$756$	1.00000	0.0363696
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	−20.0000	−0.726433
$759$	−4.00000	−0.145191
$760$	16.0000	0.580381
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	8.00000	0.289809
$763$	−10.0000	−0.362024
$764$	−4.00000	−0.144715
$765$	12.0000	0.433861
$766$	28.0000	1.01168
$767$	8.00000	0.288863
$768$	−1.00000	−0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	2.00000	0.0720750
$771$	−26.0000	−0.936367
$772$	−6.00000	−0.215945
$773$	−22.0000	−0.791285	−0.395643	−	0.918405i	$-0.629478\pi$
$773$	−22.0000	−0.791285	−0.395643	+	0.918405i	$0.629478\pi$
$774$	−8.00000	−0.287554
$775$	−8.00000	−0.287368
$776$	−18.0000	−0.646162
$777$	6.00000	0.215249
$778$	30.0000	1.07555
$779$	−48.0000	−1.71978
$780$	−4.00000	−0.143223
$781$	−4.00000	−0.143131
$782$	−24.0000	−0.858238
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	−4.00000	−0.142766
$786$	−4.00000	−0.142675
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−6.00000	−0.213741
$789$	−24.0000	−0.854423
$790$	16.0000	0.569254
$791$	−10.0000	−0.355559
$792$	−1.00000	−0.0355335
$793$	−28.0000	−0.994309
$794$	18.0000	0.638796
$795$	−20.0000	−0.709327
$796$	−8.00000	−0.283552
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	8.00000	0.283197
$799$	24.0000	0.849059
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−34.0000	−1.20058
$803$	−14.0000	−0.494049
$804$	4.00000	0.141069
$805$	−8.00000	−0.281963
$806$	−16.0000	−0.563576
$807$	−2.00000	−0.0704033
$808$	6.00000	0.211079
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−2.00000	−0.0702728
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	−2.00000	−0.0701862
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	24.0000	0.840683
$816$	−6.00000	−0.210042
$817$	−64.0000	−2.23908
$818$	22.0000	0.769212
$819$	−2.00000	−0.0698857
$820$	12.0000	0.419058
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−6.00000	−0.209274
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	16.0000	0.557386
$825$	1.00000	0.0348155
$826$	4.00000	0.139178
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	4.00000	0.139010
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	−8.00000	−0.277684
$831$	6.00000	0.208138
$832$	2.00000	0.0693375
$833$	6.00000	0.207888
$834$	−16.0000	−0.554035
$835$	−16.0000	−0.553703
$836$	−8.00000	−0.276686
$837$	−8.00000	−0.276520
$838$	12.0000	0.414533
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	−30.0000	−1.03325
$844$	8.00000	0.275371
$845$	−18.0000	−0.619219
$846$	−4.00000	−0.137523
$847$	−1.00000	−0.0343604
$848$	10.0000	0.343401
$849$	0	0
$850$	6.00000	0.205798
$851$	24.0000	0.822709
$852$	4.00000	0.137038
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−14.0000	−0.479070
$855$	−16.0000	−0.547188
$856$	12.0000	0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	2.00000	0.0682789
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	16.0000	0.545595
$861$	6.00000	0.204479
$862$	−32.0000	−1.08992
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	1.00000	0.0340207
$865$	−28.0000	−0.952029
$866$	−2.00000	−0.0679628
$867$	−19.0000	−0.645274
$868$	−8.00000	−0.271538
$869$	−8.00000	−0.271381
$870$	4.00000	0.135613
$871$	−8.00000	−0.271070
$872$	−10.0000	−0.338643
$873$	18.0000	0.609208
$874$	32.0000	1.08242
$875$	12.0000	0.405674
$876$	14.0000	0.473016
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	40.0000	1.34993
$879$	6.00000	0.202375
$880$	2.00000	0.0674200
$881$	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
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	12.0000	0.403604
$885$	−8.00000	−0.268917
$886$	−4.00000	−0.134383
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	6.00000	0.201347
$889$	−8.00000	−0.268311
$890$	28.0000	0.938562
$891$	1.00000	0.0335013
$892$	16.0000	0.535720
$893$	−32.0000	−1.07084
$894$	10.0000	0.334450
$895$	−24.0000	−0.802232
$896$	1.00000	0.0334077
$897$	−8.00000	−0.267112
$898$	6.00000	0.200223
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	60.0000	1.99889
$902$	−6.00000	−0.199778
$903$	8.00000	0.266223
$904$	−10.0000	−0.332595
$905$	−20.0000	−0.664822
$906$	16.0000	0.531564
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−4.00000	−0.132745
$909$	−6.00000	−0.199007
$910$	4.00000	0.132599
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	8.00000	0.264906
$913$	4.00000	0.132381
$914$	−10.0000	−0.330771
$915$	28.0000	0.925651
$916$	−10.0000	−0.330409
$917$	4.00000	0.132092
$918$	6.00000	0.198030
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−8.00000	−0.263752
$921$	8.00000	0.263609
$922$	−2.00000	−0.0658665
$923$	−8.00000	−0.263323
$924$	1.00000	0.0328976
$925$	−6.00000	−0.197279
$926$	24.0000	0.788689
$927$	−16.0000	−0.525509
$928$	−2.00000	−0.0656532
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	16.0000	0.524661
$931$	−8.00000	−0.262189
$932$	−18.0000	−0.589610
$933$	−12.0000	−0.392862
$934$	−20.0000	−0.654420
$935$	12.0000	0.392442
$936$	−2.00000	−0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−4.00000	−0.130605
$939$	6.00000	0.195803
$940$	8.00000	0.260931
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−2.00000	−0.0651635
$943$	24.0000	0.781548
$944$	4.00000	0.130189
$945$	2.00000	0.0650600
$946$	−8.00000	−0.260102
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	−28.0000	−0.908918
$950$	−8.00000	−0.259554
$951$	−18.0000	−0.583690
$952$	6.00000	0.194461
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−10.0000	−0.323762
$955$	−8.00000	−0.258874
$956$	−8.00000	−0.258738
$957$	−2.00000	−0.0646508
$958$	−24.0000	−0.775405
$959$	6.00000	0.193750
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	−12.0000	−0.386294
$966$	−4.00000	−0.128698
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−1.00000	−0.0321412
$969$	48.0000	1.54198
$970$	−36.0000	−1.15589
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	24.0000	0.769010
$975$	2.00000	0.0640513
$976$	−14.0000	−0.448129
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	12.0000	0.383718
$979$	−14.0000	−0.447442
$980$	2.00000	0.0638877
$981$	10.0000	0.319275
$982$	−20.0000	−0.638226
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	6.00000	0.191273
$985$	−12.0000	−0.382352
$986$	−12.0000	−0.382158
$987$	4.00000	0.127321
$988$	−16.0000	−0.509028
$989$	32.0000	1.01754
$990$	−2.00000	−0.0635642
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−8.00000	−0.254000
$993$	−4.00000	−0.126936
$994$	−4.00000	−0.126872
$995$	−16.0000	−0.507234
$996$	−4.00000	−0.126745
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	28.0000	0.886325
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.2.a.c.1.1	✓	1
3.2	odd	2		1386.2.a.g.1.1		1
4.3	odd	2		3696.2.a.bb.1.1		1
7.6	odd	2		3234.2.a.i.1.1		1
11.10	odd	2		5082.2.a.v.1.1		1
21.20	even	2		9702.2.a.by.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.c.1.1	✓	1	1.1	even	1	trivial
1386.2.a.g.1.1		1	3.2	odd	2
3234.2.a.i.1.1		1	7.6	odd	2
3696.2.a.bb.1.1		1	4.3	odd	2
5082.2.a.v.1.1		1	11.10	odd	2
9702.2.a.by.1.1		1	21.20	even	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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more