LMFDB - Embedded newform 416.4.f.b.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,4,Mod(129,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("416.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	416.f (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$24.5447945624$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 7x^{2} + 9$ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	416.129
Dual form			416.4.f.b.129.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.972244i q^{7} -27.0000 q^{9} +19.7611i q^{11} +46.8722 q^{13} -7.21110 q^{17} -146.772i q^{19} +125.000 q^{25} -252.389 q^{29} -276.572i q^{31} -225.638i q^{47} +342.055 q^{49} +310.000 q^{53} -791.549i q^{59} -882.000 q^{61} +26.2506i q^{63} -1081.97i q^{67} -1042.15i q^{71} +19.2126 q^{77} +729.000 q^{81} +1219.25i q^{83} -45.5712i q^{91} -533.551i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 108 q^{9} + 500 q^{25} - 1372 q^{49} + 1240 q^{53} - 3528 q^{61} + 5240 q^{77} + 2916 q^{81}+O(q^{100})$ 4 * q - 108 * q^9 + 500 * q^25 - 1372 * q^49 + 1240 * q^53 - 3528 * q^61 + 5240 * q^77 + 2916 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$ .

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 0.972244i	− 0.0524962i	−0.999655	−	0.0262481i	$-0.991644\pi$
$7$	− 0.972244i	− 0.0524962i	0.999655	−	0.0262481i	$-0.00835599\pi$
$8$	0	0
$9$	−27.0000	−1.00000
$10$	0	0
$11$	19.7611i	0.541655i	0.962628	+	0.270828i	$0.0872973\pi$
$11$	19.7611i	0.541655i	−0.962628	+	0.270828i	$0.912703\pi$
$12$	0	0
$13$	46.8722	1.00000
$13$	46.8722	1.00000
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.21110	−0.102879	−0.0514397	−	0.998676i	$-0.516381\pi$
$17$	−7.21110	−0.102879	−0.0514397	+	0.998676i	$0.516381\pi$
$18$	0	0
$19$	− 146.772i	− 1.77220i	−0.463493	−	0.886101i	$-0.653404\pi$
$19$	− 146.772i	− 1.77220i	0.463493	−	0.886101i	$-0.346596\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−252.389	−1.61612	−0.808058	−	0.589102i	$-0.799481\pi$
$29$	−252.389	−1.61612	−0.808058	+	0.589102i	$0.799481\pi$
$30$	0	0
$31$	− 276.572i	− 1.60238i	−0.598410	−	0.801190i	$-0.704201\pi$
$31$	− 276.572i	− 1.60238i	0.598410	−	0.801190i	$-0.295799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 225.638i	− 0.700271i	−0.936699	−	0.350136i	$-0.886136\pi$
$47$	− 225.638i	− 0.700271i	0.936699	−	0.350136i	$-0.113864\pi$
$48$	0	0
$49$	342.055	0.997244
$50$	0	0
$51$	0	0
$52$	0	0
$53$	310.000	0.803430	0.401715	−	0.915765i	$-0.368414\pi$
$53$	310.000	0.803430	0.401715	+	0.915765i	$0.368414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 791.549i	− 1.74663i	−0.487159	−	0.873313i	$-0.661967\pi$
$59$	− 791.549i	− 1.74663i	0.487159	−	0.873313i	$-0.338033\pi$
$60$	0	0
$61$	−882.000	−1.85129	−0.925644	−	0.378396i	$-0.876476\pi$
$61$	−882.000	−1.85129	−0.925644	+	0.378396i	$0.876476\pi$
$62$	0	0
$63$	26.2506i	0.0524962i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1081.97i	− 1.97289i	−0.164082	−	0.986447i	$-0.552466\pi$
$67$	− 1081.97i	− 1.97289i	0.164082	−	0.986447i	$-0.447534\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 1042.15i	− 1.74198i	−0.491303	−	0.870989i	$-0.663479\pi$
$71$	− 1042.15i	− 1.74198i	0.491303	−	0.870989i	$-0.336521\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.2126	0.0284349
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	1219.25i	1.61241i	0.591636	+	0.806205i	$0.298482\pi$
$83$	1219.25i	1.61241i	−0.591636	+	0.806205i	$0.701518\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 45.5712i	− 0.0524962i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 533.551i	− 0.541655i
$100$	0	0
$101$	−1498.00	−1.47581	−0.737904	−	0.674906i	$-0.764184\pi$
$101$	−1498.00	−1.47581	−0.737904	+	0.674906i	$0.764184\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2069.59	1.72292	0.861462	−	0.507823i	$-0.169550\pi$
$113$	2069.59	1.72292	0.861462	+	0.507823i	$0.169550\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1265.55	−1.00000
$118$	0	0
$119$	7.01095i	0.00540078i
$120$	0	0
$121$	940.497	0.706610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−142.698	−0.0930339
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	926.247i	0.541655i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	585.430i	0.315507i	0.987479	+	0.157754i	$0.0504252\pi$
$151$	585.430i	0.315507i	−0.987479	+	0.157754i	$0.949575\pi$
$152$	0	0
$153$	194.700	0.102879
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3021.45	−1.53591	−0.767956	−	0.640503i	$-0.778726\pi$
$157$	−3021.45	−1.53591	−0.767956	+	0.640503i	$0.778726\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2415.47i	1.16070i	0.814367	+	0.580350i	$0.197084\pi$
$163$	2415.47i	1.16070i	−0.814367	+	0.580350i	$0.802916\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4135.14i	1.91609i	0.286624	+	0.958043i	$0.407467\pi$
$167$	4135.14i	1.91609i	−0.286624	+	0.958043i	$0.592533\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	3962.85i	1.77220i
$172$	0	0
$173$	−770.000	−0.338393	−0.169197	−	0.985582i	$-0.554117\pi$
$173$	−770.000	−0.338393	−0.169197	+	0.985582i	$0.554117\pi$
$174$	0	0
$175$	− 121.530i	− 0.0524962i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1622.50	0.666295	0.333147	−	0.942875i	$-0.391889\pi$
$181$	1622.50	0.666295	0.333147	+	0.942875i	$0.391889\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 142.500i	− 0.0557252i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	245.383i	0.0848400i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2900.38	0.959922
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−268.895	−0.0841189
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−338.000	−0.102879
$222$	0	0
$223$	4354.47i	1.30761i	0.756663	+	0.653805i	$0.226828\pi$
$223$	4354.47i	1.30761i	−0.756663	+	0.653805i	$0.773172\pi$
$224$	0	0
$225$	−3375.00	−1.00000
$226$	0	0
$227$	1859.36i	0.543658i	0.962346	+	0.271829i	$0.0876285\pi$
$227$	1859.36i	0.543658i	−0.962346	+	0.271829i	$0.912372\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5990.00	−1.68420	−0.842099	−	0.539324i	$-0.818680\pi$
$233$	−5990.00	−1.68420	−0.842099	+	0.539324i	$0.818680\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5129.44i	1.38827i	0.719846	+	0.694134i	$0.244213\pi$
$239$	5129.44i	1.38827i	−0.719846	+	0.694134i	$0.755787\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6879.53i	− 1.77220i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8050.00	1.95387	0.976936	−	0.213531i	$-0.0684966\pi$
$257$	8050.00	1.95387	0.976936	+	0.213531i	$0.0684966\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6814.49	1.61612
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5444.38	−1.23401	−0.617007	−	0.786958i	$-0.711655\pi$
$269$	−5444.38	−1.23401	−0.617007	+	0.786958i	$0.711655\pi$
$270$	0	0
$271$	− 8472.57i	− 1.89916i	−0.313527	−	0.949579i	$-0.601511\pi$
$271$	− 8472.57i	− 1.89916i	0.313527	−	0.949579i	$-0.398489\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2470.14i	0.541655i
$276$	0	0
$277$	2070.00	0.449004	0.224502	−	0.974474i	$-0.427924\pi$
$277$	2070.00	0.449004	0.224502	+	0.974474i	$0.427924\pi$
$278$	0	0
$279$	7467.44i	1.60238i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4861.00	−0.989416
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10716.5i	− 1.99226i	−0.0878848	−	0.996131i	$-0.528011\pi$
$307$	− 10716.5i	− 1.99226i	0.0878848	−	0.996131i	$-0.471989\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10189.3	−1.84004	−0.920020	−	0.391871i	$-0.871828\pi$
$313$	−10189.3	−1.84004	−0.920020	+	0.391871i	$0.871828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	− 4987.49i	− 0.875378i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1058.39i	0.182323i
$324$	0	0
$325$	5859.02	1.00000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−219.375	−0.0367616
$330$	0	0
$331$	− 11807.1i	− 1.96066i	−0.197369	−	0.980329i	$-0.563240\pi$
$331$	− 11807.1i	− 1.96066i	0.197369	−	0.980329i	$-0.436760\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6915.45	1.11783	0.558915	−	0.829225i	$-0.311218\pi$
$337$	6915.45	1.11783	0.558915	+	0.829225i	$0.311218\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5465.38	0.867938
$342$	0	0
$343$	− 666.040i	− 0.104848i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 13207.1i	− 1.94162i	−0.239849	−	0.970810i	$-0.577098\pi$
$359$	− 13207.1i	− 1.94162i	0.239849	−	0.970810i	$-0.422902\pi$
$360$	0	0
$361$	−14683.0	−2.14070
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 301.396i	− 0.0421770i
$372$	0	0
$373$	14083.3	1.95497	0.977486	−	0.210999i	$-0.0676717\pi$
$373$	14083.3	1.95497	0.977486	+	0.210999i	$0.0676717\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11830.0	−1.61612
$378$	0	0
$379$	6387.28i	0.865679i	0.901471	+	0.432840i	$0.142488\pi$
$379$	6387.28i	0.865679i	−0.901471	+	0.432840i	$0.857512\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 14602.9i	− 1.94823i	−0.226060	−	0.974113i	$-0.572585\pi$
$383$	− 14602.9i	− 1.94823i	0.226060	−	0.974113i	$-0.427415\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4795.38	0.625027	0.312514	−	0.949913i	$-0.398829\pi$
$389$	4795.38	0.625027	0.312514	+	0.949913i	$0.398829\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 12963.5i	− 1.60238i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−769.579	−0.0916913
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	6092.24i	0.700271i
$424$	0	0
$425$	−901.388	−0.102879
$426$	0	0
$427$	857.519i	0.0971856i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13124.5i	1.46679i	0.679802	+	0.733395i	$0.262066\pi$
$431$	13124.5i	1.46679i	−0.679802	+	0.733395i	$0.737934\pi$
$432$	0	0
$433$	11970.0	1.32850	0.664251	−	0.747509i	$-0.268751\pi$
$433$	11970.0	1.32850	0.664251	+	0.747509i	$0.268751\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−9235.48	−0.997244
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 10884.5i	− 1.09254i	−0.837608	−	0.546271i	$-0.816047\pi$
$463$	− 10884.5i	− 1.09254i	0.837608	−	0.546271i	$-0.183953\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−1051.94	−0.103569
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 18346.5i	− 1.77220i
$476$	0	0
$477$	−8370.00	−0.803430
$478$	0	0
$479$	20117.4i	1.91898i	0.281751	+	0.959488i	$0.409085\pi$
$479$	20117.4i	1.91898i	−0.281751	+	0.959488i	$0.590915\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 19465.0i	− 1.81117i	−0.424160	−	0.905587i	$-0.639431\pi$
$487$	− 19465.0i	− 1.81117i	0.424160	−	0.905587i	$-0.360569\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	1820.00	0.166265
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1013.22	−0.0914472
$498$	0	0
$499$	20353.3i	1.82593i	0.408042	+	0.912963i	$0.366212\pi$
$499$	20353.3i	1.82593i	−0.408042	+	0.912963i	$0.633788\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4458.87	0.379306
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9482.60	0.797390	0.398695	−	0.917084i	$-0.369463\pi$
$521$	9482.60	0.797390	0.398695	+	0.917084i	$0.369463\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1994.39i	0.164852i
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	21371.8i	1.74663i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6759.39i	0.540163i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	23814.0	1.85129
$550$	0	0
$551$	37043.6i	2.86408i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 708.766i	− 0.0524962i
$568$	0	0
$569$	9994.00	0.736327	0.368164	−	0.929761i	$-0.379987\pi$
$569$	9994.00	0.736327	0.368164	+	0.929761i	$0.379987\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1185.41	0.0846454
$582$	0	0
$583$	6125.95i	0.435182i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27904.2i	1.96206i	0.193848	+	0.981032i	$0.437903\pi$
$587$	27904.2i	1.96206i	−0.193848	+	0.981032i	$0.562097\pi$
$588$	0	0
$589$	−40593.0	−2.83974
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−28880.5	−1.96016	−0.980082	−	0.198592i	$-0.936363\pi$
$601$	−28880.5	−1.96016	−0.980082	+	0.198592i	$0.936363\pi$
$602$	0	0
$603$	29213.2i	1.97289i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 10576.2i	− 0.700271i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 27356.8i	− 1.77635i	−0.459504	−	0.888175i	$-0.651973\pi$
$619$	− 27356.8i	− 1.77635i	0.459504	−	0.888175i	$-0.348027\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 452.146i	− 0.0285256i	−0.999898	−	0.0142628i	$-0.995460\pi$
$631$	− 452.146i	− 0.0285256i	0.999898	−	0.0142628i	$-0.00454015\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16032.8	0.997244
$638$	0	0
$639$	28138.0i	1.74198i
$640$	0	0
$641$	31548.6	1.94398	0.971992	−	0.235014i	$-0.0755136\pi$
$641$	31548.6	1.94398	0.971992	+	0.235014i	$0.0755136\pi$
$642$	0	0
$643$	− 22840.5i	− 1.40084i	−0.713730	−	0.700421i	$-0.752996\pi$
$643$	− 22840.5i	− 1.40084i	0.713730	−	0.700421i	$-0.247004\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	15641.9	0.946069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29731.4	−1.78174	−0.890872	−	0.454255i	$-0.849906\pi$
$653$	−29731.4	−1.78174	−0.890872	+	0.454255i	$0.849906\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 17429.3i	− 1.00276i
$672$	0	0
$673$	26702.7	1.52944	0.764721	−	0.644362i	$-0.222877\pi$
$673$	26702.7	1.52944	0.764721	+	0.644362i	$0.222877\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23450.0	−1.33125	−0.665625	−	0.746286i	$-0.731835\pi$
$677$	−23450.0	−1.33125	−0.665625	+	0.746286i	$0.731835\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35517.1i	1.98979i	0.100929	+	0.994894i	$0.467818\pi$
$683$	35517.1i	1.98979i	−0.100929	+	0.994894i	$0.532182\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14530.4	0.803430
$690$	0	0
$691$	17224.4i	0.948261i	0.880455	+	0.474131i	$0.157238\pi$
$691$	17224.4i	0.948261i	−0.880455	+	0.474131i	$0.842762\pi$
$692$	0	0
$693$	−518.741	−0.0284349
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22462.6	−1.21027	−0.605136	−	0.796122i	$-0.706881\pi$
$701$	−22462.6	−1.21027	−0.605136	+	0.796122i	$0.706881\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1456.42i	0.0774743i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−31548.6	−1.61612
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−19683.0	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21381.0	1.06863
$738$	0	0
$739$	8285.24i	0.412419i	0.978508	+	0.206209i	$0.0661128\pi$
$739$	8285.24i	0.412419i	−0.978508	+	0.206209i	$0.933887\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40412.2i	1.99540i	0.0677989	+	0.997699i	$0.478402\pi$
$743$	40412.2i	1.99540i	−0.0677989	+	0.997699i	$0.521598\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 32919.7i	− 1.61241i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	37910.0	1.82016	0.910081	−	0.414431i	$-0.136019\pi$
$757$	37910.0	1.82016	0.910081	+	0.414431i	$0.136019\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 37101.6i	− 1.74663i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 34571.5i	− 1.60238i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	20594.1	0.943551
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 33492.9i	− 1.51702i	−0.651662	−	0.758510i	$-0.725928\pi$
$787$	− 33492.9i	− 1.51702i	0.651662	−	0.758510i	$-0.274072\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2012.14i	− 0.0904470i
$792$	0	0
$793$	−41341.3	−1.85129
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16866.8	−0.749626	−0.374813	−	0.927101i	$-0.622293\pi$
$797$	−16866.8	−0.749626	−0.374813	+	0.927101i	$0.622293\pi$
$798$	0	0
$799$	1627.10i	0.0720435i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40634.6	1.76593	0.882964	−	0.469441i	$-0.155545\pi$
$809$	40634.6	1.76593	0.882964	+	0.469441i	$0.155545\pi$
$810$	0	0
$811$	− 11933.3i	− 0.516691i	−0.966053	−	0.258345i	$-0.916823\pi$
$811$	− 11933.3i	− 0.516691i	0.966053	−	0.258345i	$-0.0831773\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1230.42i	0.0524962i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 47245.6i	− 1.98657i	−0.115707	−	0.993283i	$-0.536913\pi$
$827$	− 47245.6i	− 1.98657i	0.115707	−	0.993283i	$-0.463087\pi$
$828$	0	0
$829$	−39634.0	−1.66049	−0.830245	−	0.557399i	$-0.811799\pi$
$829$	−39634.0	−1.66049	−0.830245	+	0.557399i	$0.811799\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2466.59	−0.102596
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 45011.9i	− 1.85219i	−0.377295	−	0.926093i	$-0.623146\pi$
$839$	− 45011.9i	− 1.85219i	0.377295	−	0.926093i	$-0.376854\pi$
$840$	0	0
$841$	39311.0	1.61183
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 914.393i	− 0.0370943i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24710.0	−0.984921	−0.492461	−	0.870335i	$-0.663902\pi$
$857$	−24710.0	−0.984921	−0.492461	+	0.870335i	$0.663902\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19626.1i	0.774137i	0.922051	+	0.387068i	$0.126512\pi$
$863$	19626.1i	0.774137i	−0.922051	+	0.387068i	$0.873488\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 50714.3i	− 1.97289i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34258.0	1.31008	0.655040	−	0.755594i	$-0.272652\pi$
$881$	34258.0	1.31008	0.655040	+	0.755594i	$0.272652\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	14405.9i	0.541655i
$892$	0	0
$893$	−33117.4	−1.24102
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	69803.6i	2.58963i
$900$	0	0
$901$	−2235.44	−0.0826563
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	40446.0	1.47581
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−24093.8	−0.873371
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 48847.8i	− 1.74198i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 50204.1i	− 1.76732i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5670.00	−0.197685	−0.0988425	−	0.995103i	$-0.531514\pi$
$937$	−5670.00	−0.197685	−0.0988425	+	0.995103i	$0.531514\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 57461.7i	− 1.97176i	−0.167459	−	0.985879i	$-0.553556\pi$
$947$	− 57461.7i	− 1.97176i	0.167459	−	0.985879i	$-0.446444\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52250.0	1.77602	0.888008	−	0.459828i	$-0.152089\pi$
$953$	52250.0	1.77602	0.888008	+	0.459828i	$0.152089\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−46701.0	−1.56762
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31684.0i	1.05366i	0.849970	+	0.526830i	$0.176620\pi$
$967$	31684.0i	1.05366i	−0.849970	+	0.526830i	$0.823380\pi$
$968$	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
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 2839.78i	− 0.0921415i	−0.998938	−	0.0460707i	$-0.985330\pi$
$983$	− 2839.78i	− 0.0921415i	0.998938	−	0.0460707i	$-0.0146700\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−40250.0	−1.27857	−0.639283	−	0.768972i	$-0.720769\pi$
$997$	−40250.0	−1.27857	−0.639283	+	0.768972i	$0.720769\pi$
$998$	0	0
$999$	0	0

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.4.f.b.129.2	✓	4
4.3	odd	2	inner	416.4.f.b.129.3	yes	4
8.3	odd	2		832.4.f.i.129.3		4
8.5	even	2		832.4.f.i.129.2		4
13.12	even	2	inner	416.4.f.b.129.3	yes	4
52.51	odd	2	CM	416.4.f.b.129.2	✓	4
104.51	odd	2		832.4.f.i.129.2		4
104.77	even	2		832.4.f.i.129.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.4.f.b.129.2	✓	4	1.1	even	1	trivial
416.4.f.b.129.2	✓	4	52.51	odd	2	CM
416.4.f.b.129.3	yes	4	4.3	odd	2	inner
416.4.f.b.129.3	yes	4	13.12	even	2	inner
832.4.f.i.129.2		4	8.5	even	2
832.4.f.i.129.2		4	104.51	odd	2
832.4.f.i.129.3		4	8.3	odd	2
832.4.f.i.129.3		4	104.77	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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	416.4.f.b.129.2

Level	$416$
Weight	$4$
Character	416.129
Analytic conductor	$24.545$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-52
Inner twists	$4$