LMFDB - Embedded newform 234.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$234 = 2 \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	234.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:	$1.86849940730$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	234.1

$q$ -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^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -1.00000 q^{43} -6.00000 q^{44} -3.00000 q^{47} -6.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} -18.0000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{73} -7.00000 q^{74} +2.00000 q^{76} +6.00000 q^{77} +8.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -3.00000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -6.00000 q^{98} +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
$3$	0	0
$3$	0	0
$4$	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	$0.265942\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−6.00000	−1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	−3.00000	−0.358569
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	6.00000	0.683763
$78$	0	0
$79$	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
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	−3.00000	−0.309426
$95$	6.00000	0.615587
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	4.00000	0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−18.0000	−1.71623
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	−3.00000	−0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	8.00000	0.724286
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	14.0000	1.20942
$135$	0	0
$136$	3.00000	0.257248
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	3.00000	0.251754
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−18.0000	−1.49482
$146$	2.00000	0.165521
$147$	0	0
$148$	−7.00000	−0.575396
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	6.00000	0.483494
$155$	−12.0000	−0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	3.00000	0.237171
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	9.00000	0.690268
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	−6.00000	−0.452267
$177$	0	0
$178$	6.00000	0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	−18.0000	−1.31629
$188$	−3.00000	−0.218797
$189$	0	0
$190$	6.00000	0.435286
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	12.0000	0.844317
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000	0.0693375
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−3.00000	−0.204598
$216$	0	0
$217$	4.00000	0.271538
$218$	−7.00000	−0.474100
$219$	0	0
$220$	−18.0000	−1.21356
$221$	3.00000	0.201802
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	6.00000	0.390567
$237$	0	0
$238$	−3.00000	−0.194461
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	8.00000	0.512148
$245$	−18.0000	−1.14998
$246$	0	0
$247$	2.00000	0.127257
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	3.00000	0.186052
$261$	0	0
$262$	21.0000	1.29738
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	14.0000	0.855186
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	−3.00000	−0.179284
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−18.0000	−1.05700
$291$	0	0
$292$	2.00000	0.117041
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	−7.00000	−0.406867
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	17.0000	0.978240
$303$	0	0
$304$	2.00000	0.114708
$305$	24.0000	1.37424
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	−12.0000	−0.681554
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	4.00000	0.221880
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	42.0000	2.29471
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	9.00000	0.488094
$341$	24.0000	1.29967
$342$	0	0
$343$	13.0000	0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	6.00000	0.317999
$357$	0	0
$358$	−3.00000	−0.158555
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	20.0000	1.05118
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	6.00000	0.314054
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	0	0
$369$	0	0
$370$	−21.0000	−1.09174
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−6.00000	−0.309016
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	18.0000	0.920960
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	18.0000	0.917365
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−3.00000	−0.151138
$395$	24.0000	1.20757
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	2.00000	0.100251
$399$	0	0
$400$	4.00000	0.200000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	12.0000	0.597022
$405$	0	0
$406$	6.00000	0.297775
$407$	42.0000	2.08186
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−36.0000	−1.76717
$416$	1.00000	0.0490290
$417$	0	0
$418$	−12.0000	−0.586939
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−13.0000	−0.632830
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−3.00000	−0.144673
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−7.00000	−0.335239
$437$	0	0
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	3.00000	0.142695
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	−19.0000	−0.899676
$447$	0	0
$448$	−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	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−13.0000	−0.607450
$459$	0	0
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	27.0000	1.25075
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	−9.00000	−0.415139
$471$	0	0
$472$	6.00000	0.276172
$473$	6.00000	0.275880
$474$	0	0
$475$	8.00000	0.367065
$476$	−3.00000	−0.137505
$477$	0	0
$478$	−15.0000	−0.686084
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−10.0000	−0.455488
$483$	0	0
$484$	25.0000	1.13636
$485$	−30.0000	−1.36223
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−18.0000	−0.813157
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	2.00000	0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−3.00000	−0.134568
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−24.0000	−1.07117
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	0	0
$508$	20.0000	0.887357
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.00000	−0.396973
$515$	−12.0000	−0.528783
$516$	0	0
$517$	18.0000	0.791639
$518$	7.00000	0.307562
$519$	0	0
$520$	3.00000	0.131559
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	21.0000	0.917389
$525$	0	0
$526$	12.0000	0.523225
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	0	0
$535$	−36.0000	−1.55642
$536$	14.0000	0.604708
$537$	0	0
$538$	−24.0000	−1.03471
$539$	36.0000	1.55063
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	11.0000	0.472490
$543$	0	0
$544$	3.00000	0.128624
$545$	−21.0000	−0.899541
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	0	0
$550$	−24.0000	−1.02336
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−13.0000	−0.551323
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	−3.00000	−0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	−4.00000	−0.168133
$567$	0	0
$568$	3.00000	0.125877
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−18.0000	−0.747409
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−21.0000	−0.867502
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	18.0000	0.741048
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	17.0000	0.691720
$605$	75.0000	3.04918
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	24.0000	0.971732
$611$	−3.00000	−0.121367
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	6.00000	0.241747
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	30.0000	1.20289
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	14.0000	0.558661
$629$	−21.0000	−0.837325
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	6.00000	0.238290
$635$	60.0000	2.38103
$636$	0	0
$637$	−6.00000	−0.237729
$638$	36.0000	1.42525
$639$	0	0
$640$	3.00000	0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	4.00000	0.156893
$651$	0	0
$652$	−16.0000	−0.626608
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	63.0000	2.46161
$656$	0	0
$657$	0	0
$658$	3.00000	0.116952
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	42.0000	1.62260
$671$	−48.0000	−1.85302
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	23.0000	0.885927
$675$	0	0
$676$	1.00000	0.0384615
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	9.00000	0.345134
$681$	0	0
$682$	24.0000	0.919007
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	−3.00000	−0.113878
$695$	−39.0000	−1.47935
$696$	0	0
$697$	0	0
$698$	−19.0000	−0.719161
$699$	0	0
$700$	−4.00000	−0.151186
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−24.0000	−0.903252
$707$	−12.0000	−0.451306
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	9.00000	0.337764
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−18.0000	−0.673162
$716$	−3.00000	−0.112115
$717$	0	0
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−15.0000	−0.558242
$723$	0	0
$724$	20.0000	0.743294
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	6.00000	0.222070
$731$	−3.00000	−0.110959
$732$	0	0
$733$	23.0000	0.849524	0.424762	−	0.905305i	$-0.360358\pi$
$733$	23.0000	0.849524	0.424762	+	0.905305i	$0.360358\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	0	0
$737$	−84.0000	−3.09418
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−18.0000	−0.658145
$749$	12.0000	0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	−6.00000	−0.218507
$755$	51.0000	1.85608
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	6.00000	0.217643
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	18.0000	0.651217
$765$	0	0
$766$	−21.0000	−0.758761
$767$	6.00000	0.216647
$768$	0	0
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	18.0000	0.648675
$771$	0	0
$772$	−4.00000	−0.143963
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−10.0000	−0.358979
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	42.0000	1.49904
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	24.0000	0.853882
$791$	−6.00000	−0.213335
$792$	0	0
$793$	8.00000	0.284088
$794$	−34.0000	−1.20661
$795$	0	0
$796$	2.00000	0.0708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	4.00000	0.141421
$801$	0	0
$802$	−36.0000	−1.27120
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	12.0000	0.422159
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	42.0000	1.47210
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	32.0000	1.11885
$819$	0	0
$820$	0	0
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−36.0000	−1.24958
$831$	0	0
$832$	1.00000	0.0346688
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	−9.00000	−0.310900
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	17.0000	0.585859
$843$	0	0
$844$	−13.0000	−0.447478
$845$	3.00000	0.103203
$846$	0	0
$847$	−25.0000	−0.859010
$848$	0	0
$849$	0	0
$850$	12.0000	0.411597
$851$	0	0
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	33.0000	1.12398
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	0	0
$865$	0	0
$866$	−25.0000	−0.849535
$867$	0	0
$868$	4.00000	0.135769
$869$	−48.0000	−1.62829
$870$	0	0
$871$	14.0000	0.474372
$872$	−7.00000	−0.237050
$873$	0	0
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	26.0000	0.877457
$879$	0	0
$880$	−18.0000	−0.606780
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−21.0000	−0.705509
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	18.0000	0.603361
$891$	0	0
$892$	−19.0000	−0.636167
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−9.00000	−0.300837
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	60.0000	1.99447
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	0	0
$909$	0	0
$910$	−3.00000	−0.0994490
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−13.0000	−0.429532
$917$	−21.0000	−0.693481
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	−9.00000	−0.296399
$923$	3.00000	0.0987462
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−40.0000	−1.31448
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	27.0000	0.884414
$933$	0	0
$934$	−36.0000	−1.17796
$935$	−54.0000	−1.76599
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	−14.0000	−0.457116
$939$	0	0
$940$	−9.00000	−0.293548
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	6.00000	0.195077
$947$	−6.00000	−0.194974	−0.0974869	−	0.995237i	$-0.531080\pi$
$947$	−6.00000	−0.194974	−0.0974869	+	0.995237i	$0.531080\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	8.00000	0.259554
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	54.0000	1.74740
$956$	−15.0000	−0.485135
$957$	0	0
$958$	21.0000	0.678479
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−7.00000	−0.225689
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−30.0000	−0.963242
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	−16.0000	−0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	−18.0000	−0.574989
$981$	0	0
$982$	9.00000	0.287202
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	−18.0000	−0.573237
$987$	0	0
$988$	2.00000	0.0636285
$989$	0	0
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	−3.00000	−0.0951542
$995$	6.00000	0.190213
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−40.0000	−1.26618
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.2.a.e.1.1		1
3.2	odd	2		26.2.a.a.1.1	✓	1
4.3	odd	2		1872.2.a.q.1.1		1
5.2	odd	4		5850.2.e.a.5149.2		2
5.3	odd	4		5850.2.e.a.5149.1		2
5.4	even	2		5850.2.a.p.1.1		1
8.3	odd	2		7488.2.a.h.1.1		1
8.5	even	2		7488.2.a.g.1.1		1
9.2	odd	6		2106.2.e.ba.1405.1		2
9.4	even	3		2106.2.e.b.703.1		2
9.5	odd	6		2106.2.e.ba.703.1		2
9.7	even	3		2106.2.e.b.1405.1		2
12.11	even	2		208.2.a.a.1.1		1
13.5	odd	4		3042.2.b.a.1351.1		2
13.8	odd	4		3042.2.b.a.1351.2		2
13.12	even	2		3042.2.a.a.1.1		1
15.2	even	4		650.2.b.d.599.1		2
15.8	even	4		650.2.b.d.599.2		2
15.14	odd	2		650.2.a.j.1.1		1
21.2	odd	6		1274.2.f.p.1145.1		2
21.5	even	6		1274.2.f.r.1145.1		2
21.11	odd	6		1274.2.f.p.79.1		2
21.17	even	6		1274.2.f.r.79.1		2
21.20	even	2		1274.2.a.d.1.1		1
24.5	odd	2		832.2.a.d.1.1		1
24.11	even	2		832.2.a.i.1.1		1
33.32	even	2		3146.2.a.n.1.1		1
39.2	even	12		338.2.e.a.147.1		4
39.5	even	4		338.2.b.c.337.2		2
39.8	even	4		338.2.b.c.337.1		2
39.11	even	12		338.2.e.a.147.2		4
39.17	odd	6		338.2.c.a.315.1		2
39.20	even	12		338.2.e.a.23.1		4
39.23	odd	6		338.2.c.a.191.1		2
39.29	odd	6		338.2.c.d.191.1		2
39.32	even	12		338.2.e.a.23.2		4
39.35	odd	6		338.2.c.d.315.1		2
39.38	odd	2		338.2.a.f.1.1		1
48.5	odd	4		3328.2.b.m.1665.1		2
48.11	even	4		3328.2.b.j.1665.2		2
48.29	odd	4		3328.2.b.m.1665.2		2
48.35	even	4		3328.2.b.j.1665.1		2
51.50	odd	2		7514.2.a.c.1.1		1
57.56	even	2		9386.2.a.j.1.1		1
60.59	even	2		5200.2.a.x.1.1		1
156.47	odd	4		2704.2.f.d.337.1		2
156.83	odd	4		2704.2.f.d.337.2		2
156.155	even	2		2704.2.a.f.1.1		1
195.194	odd	2		8450.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	3.2	odd	2
208.2.a.a.1.1		1	12.11	even	2
234.2.a.e.1.1		1	1.1	even	1	trivial
338.2.a.f.1.1		1	39.38	odd	2
338.2.b.c.337.1		2	39.8	even	4
338.2.b.c.337.2		2	39.5	even	4
338.2.c.a.191.1		2	39.23	odd	6
338.2.c.a.315.1		2	39.17	odd	6
338.2.c.d.191.1		2	39.29	odd	6
338.2.c.d.315.1		2	39.35	odd	6
338.2.e.a.23.1		4	39.20	even	12
338.2.e.a.23.2		4	39.32	even	12
338.2.e.a.147.1		4	39.2	even	12
338.2.e.a.147.2		4	39.11	even	12
650.2.a.j.1.1		1	15.14	odd	2
650.2.b.d.599.1		2	15.2	even	4
650.2.b.d.599.2		2	15.8	even	4
832.2.a.d.1.1		1	24.5	odd	2
832.2.a.i.1.1		1	24.11	even	2
1274.2.a.d.1.1		1	21.20	even	2
1274.2.f.p.79.1		2	21.11	odd	6
1274.2.f.p.1145.1		2	21.2	odd	6
1274.2.f.r.79.1		2	21.17	even	6
1274.2.f.r.1145.1		2	21.5	even	6
1872.2.a.q.1.1		1	4.3	odd	2
2106.2.e.b.703.1		2	9.4	even	3
2106.2.e.b.1405.1		2	9.7	even	3
2106.2.e.ba.703.1		2	9.5	odd	6
2106.2.e.ba.1405.1		2	9.2	odd	6
2704.2.a.f.1.1		1	156.155	even	2
2704.2.f.d.337.1		2	156.47	odd	4
2704.2.f.d.337.2		2	156.83	odd	4
3042.2.a.a.1.1		1	13.12	even	2
3042.2.b.a.1351.1		2	13.5	odd	4
3042.2.b.a.1351.2		2	13.8	odd	4
3146.2.a.n.1.1		1	33.32	even	2
3328.2.b.j.1665.1		2	48.35	even	4
3328.2.b.j.1665.2		2	48.11	even	4
3328.2.b.m.1665.1		2	48.5	odd	4
3328.2.b.m.1665.2		2	48.29	odd	4
5200.2.a.x.1.1		1	60.59	even	2
5850.2.a.p.1.1		1	5.4	even	2
5850.2.e.a.5149.1		2	5.3	odd	4
5850.2.e.a.5149.2		2	5.2	odd	4
7488.2.a.g.1.1		1	8.5	even	2
7488.2.a.h.1.1		1	8.3	odd	2
7514.2.a.c.1.1		1	51.50	odd	2
8450.2.a.c.1.1		1	195.194	odd	2
9386.2.a.j.1.1		1	57.56	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	234.2.a.e.1.1

Level	$234$
Weight	$2$
Character	234.1
Self dual	yes
Analytic conductor	$1.868$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$