LMFDB - Embedded newform 3920.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3920 = 2^{4} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3920.1

$q$ -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^{3} -1.00000 q^{5} -2.00000 q^{9} +6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} -2.00000 q^{19} +3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -8.00000 q^{31} -6.00000 q^{33} -4.00000 q^{37} +4.00000 q^{39} +9.00000 q^{41} +7.00000 q^{43} +2.00000 q^{45} -6.00000 q^{53} -6.00000 q^{55} +2.00000 q^{57} +6.00000 q^{59} +5.00000 q^{61} +4.00000 q^{65} -5.00000 q^{67} -3.00000 q^{69} +6.00000 q^{71} -16.0000 q^{73} -1.00000 q^{75} -2.00000 q^{79} +1.00000 q^{81} -3.00000 q^{83} +3.00000 q^{87} -15.0000 q^{89} +8.00000 q^{93} +2.00000 q^{95} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	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
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	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
$115$	−3.00000	−0.279751
$116$	0	0
$117$	8.00000	0.739600
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−7.00000	−0.477396
$216$	0	0
$217$	0	0
$218$	0	0
$219$	16.0000	1.08118
$220$	0	0
$221$	0	0
$222$	0	0
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.00000	0.129914
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	3.00000	0.190117
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	15.0000	0.917985
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	30.0000	1.74078
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−15.0000	−0.861727
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	−5.00000	−0.285365	−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000	−0.285365	−0.142683	+	0.989769i	$0.545573\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	5.00000	0.273179
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−48.0000	−2.59935
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−25.0000	−1.31216
$364$	0	0
$365$	16.0000	0.837478
$366$	0	0
$367$	−35.0000	−1.82699	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−35.0000	−1.82699	−0.913493	+	0.406855i	$0.866625\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	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	$0.533022\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.0000	−0.711660
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.00000	0.147264
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−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
$445$	15.0000	0.711068
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	54.0000	2.54276
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	42.0000	1.93116
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	−15.0000	−0.648507
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	−11.0000	−0.472055
$544$	0	0
$545$	−11.0000	−0.471188
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−28.0000	−1.18427
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	0	0
$605$	−25.0000	−1.01639
$606$	0	0
$607$	−23.0000	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	7.00000	0.275625
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−48.0000	−1.87839	−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000	−1.87839	−0.939193	+	0.343391i	$0.888424\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.0000	1.24844
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	−28.0000	−1.08254
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.0000	−0.379322
$696$	0	0
$697$	0	0
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30.0000	−1.10506
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	0	0
$759$	−18.0000	−0.653359
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	22.0000	0.785214
$786$	0	0
$787$	43.0000	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$787$	43.0000	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$788$	0	0
$789$	−21.0000	−0.747620
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	−96.0000	−3.38777
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−14.0000	−0.489798
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	−40.0000	−1.38260
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.0000	−0.919091	−0.459545	−	0.888154i	$-0.651988\pi$
$863$	−27.0000	−0.919091	−0.459545	+	0.888154i	$0.651988\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	57.0000	1.92038	0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000	1.92038	0.960189	+	0.279350i	$0.0901189\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.0000	−0.365652
$906$	0	0
$907$	25.0000	0.830111	0.415056	−	0.909796i	$-0.363762\pi$
$907$	25.0000	0.830111	0.415056	+	0.909796i	$0.363762\pi$
$908$	0	0
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	5.00000	0.165295
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18.0000	−0.589294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	27.0000	0.879241
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	60.0000	1.94359	0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000	1.94359	0.971795	+	0.235826i	$0.0757795\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−30.0000	−0.966736
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−35.0000	−1.12552	−0.562762	−	0.826619i	$-0.690261\pi$
$967$	−35.0000	−1.12552	−0.562762	+	0.826619i	$0.690261\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	−90.0000	−2.87641
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	−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
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.0000	0.667761
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	−20.0000	−0.632772

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	3920.2.a.p.1.1		1
4.3	odd	2		490.2.a.c.1.1		1
7.2	even	3		560.2.q.g.81.1		2
7.4	even	3		560.2.q.g.401.1		2
7.6	odd	2		3920.2.a.bc.1.1		1
12.11	even	2		4410.2.a.bm.1.1		1
20.3	even	4		2450.2.c.g.99.2		2
20.7	even	4		2450.2.c.g.99.1		2
20.19	odd	2		2450.2.a.w.1.1		1
28.3	even	6		490.2.e.h.471.1		2
28.11	odd	6		70.2.e.c.51.1	yes	2
28.19	even	6		490.2.e.h.361.1		2
28.23	odd	6		70.2.e.c.11.1	✓	2
28.27	even	2		490.2.a.b.1.1		1
84.11	even	6		630.2.k.b.541.1		2
84.23	even	6		630.2.k.b.361.1		2
84.83	odd	2		4410.2.a.bd.1.1		1
140.23	even	12		350.2.j.b.249.2		4
140.27	odd	4		2450.2.c.l.99.1		2
140.39	odd	6		350.2.e.e.51.1		2
140.67	even	12		350.2.j.b.149.2		4
140.79	odd	6		350.2.e.e.151.1		2
140.83	odd	4		2450.2.c.l.99.2		2
140.107	even	12		350.2.j.b.249.1		4
140.123	even	12		350.2.j.b.149.1		4
140.139	even	2		2450.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.c.11.1	✓	2	28.23	odd	6
70.2.e.c.51.1	yes	2	28.11	odd	6
350.2.e.e.51.1		2	140.39	odd	6
350.2.e.e.151.1		2	140.79	odd	6
350.2.j.b.149.1		4	140.123	even	12
350.2.j.b.149.2		4	140.67	even	12
350.2.j.b.249.1		4	140.107	even	12
350.2.j.b.249.2		4	140.23	even	12
490.2.a.b.1.1		1	28.27	even	2
490.2.a.c.1.1		1	4.3	odd	2
490.2.e.h.361.1		2	28.19	even	6
490.2.e.h.471.1		2	28.3	even	6
560.2.q.g.81.1		2	7.2	even	3
560.2.q.g.401.1		2	7.4	even	3
630.2.k.b.361.1		2	84.23	even	6
630.2.k.b.541.1		2	84.11	even	6
2450.2.a.w.1.1		1	20.19	odd	2
2450.2.a.bc.1.1		1	140.139	even	2
2450.2.c.g.99.1		2	20.7	even	4
2450.2.c.g.99.2		2	20.3	even	4
2450.2.c.l.99.1		2	140.27	odd	4
2450.2.c.l.99.2		2	140.83	odd	4
3920.2.a.p.1.1		1	1.1	even	1	trivial
3920.2.a.bc.1.1		1	7.6	odd	2
4410.2.a.bd.1.1		1	84.83	odd	2
4410.2.a.bm.1.1		1	12.11	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

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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	3920.2.a.p.1.1

Level	$3920$
Weight	$2$
Character	3920.1
Self dual	yes
Analytic conductor	$31.301$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$