LMFDB - Embedded newform 256.3.c.a.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,3,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$256 = 2^{8}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	256.c (of order $2$ , degree $1$ , not 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:	yes
Analytic conductor:	$6.97549476762$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			255.1
Character	$\chi$	$=$	256.255

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-8.00000 q^{5} +9.00000 q^{9} +24.0000 q^{13} +30.0000 q^{17} +39.0000 q^{25} -40.0000 q^{29} +24.0000 q^{37} -18.0000 q^{41} -72.0000 q^{45} +49.0000 q^{49} +56.0000 q^{53} +120.000 q^{61} -192.000 q^{65} -110.000 q^{73} +81.0000 q^{81} -240.000 q^{85} -78.0000 q^{89} -130.000 q^{97} +O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$ .

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	−8.00000	−1.60000	−0.800000	−	0.600000i	$-0.795167\pi$
$5$	−8.00000	−1.60000	−0.800000	+	0.600000i	$0.795167\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	9.00000	1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	24.0000	1.84615	0.923077	−	0.384615i	$-0.125666\pi$
$13$	24.0000	1.84615	0.923077	+	0.384615i	$0.125666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.0000	1.76471	0.882353	−	0.470588i	$-0.155958\pi$
$17$	30.0000	1.76471	0.882353	+	0.470588i	$0.155958\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	39.0000	1.56000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.0000	−1.37931	−0.689655	−	0.724138i	$-0.742238\pi$
$29$	−40.0000	−1.37931	−0.689655	+	0.724138i	$0.742238\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	24.0000	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
$37$	24.0000	0.648649	0.324324	+	0.945946i	$0.394863\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−18.0000	−0.439024	−0.219512	−	0.975610i	$-0.570447\pi$
$41$	−18.0000	−0.439024	−0.219512	+	0.975610i	$0.570447\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−72.0000	−1.60000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	56.0000	1.05660	0.528302	−	0.849057i	$-0.322829\pi$
$53$	56.0000	1.05660	0.528302	+	0.849057i	$0.322829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	120.000	1.96721	0.983607	−	0.180328i	$-0.0577159\pi$
$61$	120.000	1.96721	0.983607	+	0.180328i	$0.0577159\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−192.000	−2.95385
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−110.000	−1.50685	−0.753425	−	0.657534i	$-0.771599\pi$
$73$	−110.000	−1.50685	−0.753425	+	0.657534i	$0.771599\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−240.000	−2.82353
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−78.0000	−0.876404	−0.438202	−	0.898876i	$-0.644385\pi$
$89$	−78.0000	−0.876404	−0.438202	+	0.898876i	$0.644385\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−130.000	−1.34021	−0.670103	−	0.742268i	$-0.733750\pi$
$97$	−130.000	−1.34021	−0.670103	+	0.742268i	$0.733750\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−40.0000	−0.396040	−0.198020	−	0.980198i	$-0.563451\pi$
$101$	−40.0000	−0.396040	−0.198020	+	0.980198i	$0.563451\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	120.000	1.10092	0.550459	−	0.834862i	$-0.314453\pi$
$109$	120.000	1.10092	0.550459	+	0.834862i	$0.314453\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−30.0000	−0.265487	−0.132743	−	0.991150i	$-0.542379\pi$
$113$	−30.0000	−0.265487	−0.132743	+	0.991150i	$0.542379\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	216.000	1.84615
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−112.000	−0.896000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−210.000	−1.53285	−0.766423	−	0.642336i	$-0.777965\pi$
$137$	−210.000	−1.53285	−0.766423	+	0.642336i	$0.777965\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	320.000	2.20690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	280.000	1.87919	0.939597	−	0.342282i	$-0.111200\pi$
$149$	280.000	1.87919	0.939597	+	0.342282i	$0.111200\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	270.000	1.76471
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−264.000	−1.68153	−0.840764	−	0.541401i	$-0.817894\pi$
$157$	−264.000	−1.68153	−0.840764	+	0.541401i	$0.817894\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	407.000	2.40828
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−104.000	−0.601156	−0.300578	−	0.953757i	$-0.597180\pi$
$173$	−104.000	−0.601156	−0.300578	+	0.953757i	$0.597180\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−360.000	−1.98895	−0.994475	−	0.104972i	$-0.966525\pi$
$181$	−360.000	−1.98895	−0.994475	+	0.104972i	$0.966525\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−192.000	−1.03784
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	190.000	0.984456	0.492228	−	0.870466i	$-0.336183\pi$
$193$	190.000	0.984456	0.492228	+	0.870466i	$0.336183\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	56.0000	0.284264	0.142132	−	0.989848i	$-0.454604\pi$
$197$	56.0000	0.284264	0.142132	+	0.989848i	$0.454604\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	144.000	0.702439
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	720.000	3.25792
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	351.000	1.56000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	120.000	0.524017	0.262009	−	0.965066i	$-0.415615\pi$
$229$	120.000	0.524017	0.262009	+	0.965066i	$0.415615\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	210.000	0.901288	0.450644	−	0.892704i	$-0.351194\pi$
$233$	210.000	0.901288	0.450644	+	0.892704i	$0.351194\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−418.000	−1.73444	−0.867220	−	0.497925i	$-0.834095\pi$
$241$	−418.000	−1.73444	−0.867220	+	0.497925i	$0.834095\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−392.000	−1.60000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−510.000	−1.98444	−0.992218	−	0.124514i	$-0.960263\pi$
$257$	−510.000	−1.98444	−0.992218	+	0.124514i	$0.960263\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−360.000	−1.37931
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−448.000	−1.69057
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−520.000	−1.93309	−0.966543	−	0.256506i	$-0.917429\pi$
$269$	−520.000	−1.93309	−0.966543	+	0.256506i	$0.917429\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	504.000	1.81949	0.909747	−	0.415162i	$-0.136275\pi$
$277$	504.000	1.81949	0.909747	+	0.415162i	$0.136275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−462.000	−1.64413	−0.822064	−	0.569395i	$-0.807178\pi$
$281$	−462.000	−1.64413	−0.822064	+	0.569395i	$0.807178\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	611.000	2.11419
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−136.000	−0.464164	−0.232082	−	0.972696i	$-0.574554\pi$
$293$	−136.000	−0.464164	−0.232082	+	0.972696i	$0.574554\pi$
$294$	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
$305$	−960.000	−3.14754
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−50.0000	−0.159744	−0.0798722	−	0.996805i	$-0.525451\pi$
$313$	−50.0000	−0.159744	−0.0798722	+	0.996805i	$0.525451\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−616.000	−1.94322	−0.971609	−	0.236593i	$-0.923969\pi$
$317$	−616.000	−1.94322	−0.971609	+	0.236593i	$0.923969\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	936.000	2.88000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	216.000	0.648649
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−350.000	−1.03858	−0.519288	−	0.854599i	$-0.673803\pi$
$337$	−350.000	−1.03858	−0.519288	+	0.854599i	$0.673803\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−360.000	−1.03152	−0.515759	−	0.856734i	$-0.672490\pi$
$349$	−360.000	−1.03152	−0.515759	+	0.856734i	$0.672490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	450.000	1.27479	0.637394	−	0.770538i	$-0.280012\pi$
$353$	450.000	1.27479	0.637394	+	0.770538i	$0.280012\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	880.000	2.41096
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−162.000	−0.439024
$370$	0	0
$371$	0	0
$372$	0	0
$373$	504.000	1.35121	0.675603	−	0.737265i	$-0.263883\pi$
$373$	504.000	1.35121	0.675603	+	0.737265i	$0.263883\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−960.000	−2.54642
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−680.000	−1.74807	−0.874036	−	0.485861i	$-0.838506\pi$
$389$	−680.000	−1.74807	−0.874036	+	0.485861i	$0.838506\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−456.000	−1.14861	−0.574307	−	0.818640i	$-0.694729\pi$
$397$	−456.000	−1.14861	−0.574307	+	0.818640i	$0.694729\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	798.000	1.99002	0.995012	−	0.0997506i	$-0.0318045\pi$
$401$	798.000	1.99002	0.995012	+	0.0997506i	$0.0318045\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−648.000	−1.60000
$406$	0	0
$407$	0	0
$408$	0	0
$409$	782.000	1.91198	0.955990	−	0.293399i	$-0.0947863\pi$
$409$	782.000	1.91198	0.955990	+	0.293399i	$0.0947863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−840.000	−1.99525	−0.997625	−	0.0688836i	$-0.978056\pi$
$421$	−840.000	−1.99525	−0.997625	+	0.0688836i	$0.978056\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1170.00	2.75294
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−290.000	−0.669746	−0.334873	−	0.942263i	$-0.608693\pi$
$433$	−290.000	−0.669746	−0.334873	+	0.942263i	$0.608693\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	441.000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	624.000	1.40225
$446$	0	0
$447$	0	0
$448$	0	0
$449$	702.000	1.56347	0.781737	−	0.623608i	$-0.214334\pi$
$449$	702.000	1.56347	0.781737	+	0.623608i	$0.214334\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−850.000	−1.85996	−0.929978	−	0.367615i	$-0.880174\pi$
$457$	−850.000	−1.85996	−0.929978	+	0.367615i	$0.880174\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	760.000	1.64859	0.824295	−	0.566161i	$-0.191572\pi$
$461$	760.000	1.64859	0.824295	+	0.566161i	$0.191572\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	504.000	1.05660
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	576.000	1.19751
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1040.00	2.14433
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−1200.00	−2.43408
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	320.000	0.633663
$506$	0	0
$507$	0	0
$508$	0	0
$509$	440.000	0.864440	0.432220	−	0.901768i	$-0.357730\pi$
$509$	440.000	0.864440	0.432220	+	0.901768i	$0.357730\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	558.000	1.07102	0.535509	−	0.844530i	$-0.320120\pi$
$521$	558.000	1.07102	0.535509	+	0.844530i	$0.320120\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−432.000	−0.810507
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−840.000	−1.55268	−0.776340	−	0.630314i	$-0.782926\pi$
$541$	−840.000	−1.55268	−0.776340	+	0.630314i	$0.782926\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−960.000	−1.76147
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	1080.00	1.96721
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1064.00	−1.91023	−0.955117	−	0.296230i	$-0.904271\pi$
$557$	−1064.00	−1.91023	−0.955117	+	0.296230i	$0.904271\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	240.000	0.424779
$566$	0	0
$567$	0	0
$568$	0	0
$569$	462.000	0.811951	0.405975	−	0.913884i	$-0.366932\pi$
$569$	462.000	0.811951	0.405975	+	0.913884i	$0.366932\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1150.00	−1.99307	−0.996534	−	0.0831889i	$-0.973490\pi$
$577$	−1150.00	−1.99307	−0.996534	+	0.0831889i	$0.973490\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1728.00	−2.95385
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	930.000	1.56830	0.784148	−	0.620573i	$-0.213100\pi$
$593$	930.000	1.56830	0.784148	+	0.620573i	$0.213100\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1102.00	−1.83361	−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−1102.00	−1.83361	−0.916805	+	0.399334i	$0.869241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−968.000	−1.60000
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1224.00	−1.99674	−0.998369	−	0.0570962i	$-0.981816\pi$
$613$	−1224.00	−1.99674	−0.998369	+	0.0570962i	$0.981816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	210.000	0.340357	0.170178	−	0.985413i	$-0.445566\pi$
$617$	210.000	0.340357	0.170178	+	0.985413i	$0.445566\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−79.0000	−0.126400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	720.000	1.14467
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1176.00	1.84615
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1218.00	−1.90016	−0.950078	−	0.312012i	$-0.898997\pi$
$641$	−1218.00	−1.90016	−0.950078	+	0.312012i	$0.898997\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1144.00	1.75191	0.875957	−	0.482389i	$-0.160231\pi$
$653$	1144.00	1.75191	0.875957	+	0.482389i	$0.160231\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−990.000	−1.50685
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	600.000	0.907716	0.453858	−	0.891074i	$-0.350047\pi$
$661$	600.000	0.907716	0.453858	+	0.891074i	$0.350047\pi$
$662$	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
$671$	0	0
$672$	0	0
$673$	−770.000	−1.14413	−0.572065	−	0.820208i	$-0.693858\pi$
$673$	−770.000	−1.14413	−0.572065	+	0.820208i	$0.693858\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−104.000	−0.153619	−0.0768095	−	0.997046i	$-0.524473\pi$
$677$	−104.000	−0.153619	−0.0768095	+	0.997046i	$0.524473\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	1680.00	2.45255
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1344.00	1.95065
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−540.000	−0.774749
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−520.000	−0.741797	−0.370899	−	0.928673i	$-0.620950\pi$
$701$	−520.000	−0.741797	−0.370899	+	0.928673i	$0.620950\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1320.00	−1.86178	−0.930889	−	0.365303i	$-0.880965\pi$
$709$	−1320.00	−1.86178	−0.930889	+	0.365303i	$0.880965\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1560.00	−2.15172
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	216.000	0.294679	0.147340	−	0.989086i	$-0.452929\pi$
$733$	216.000	0.294679	0.147340	+	0.989086i	$0.452929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2240.00	−3.00671
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−936.000	−1.23646	−0.618230	−	0.785997i	$-0.712150\pi$
$757$	−936.000	−1.23646	−0.618230	+	0.785997i	$0.712150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	78.0000	0.102497	0.0512484	−	0.998686i	$-0.483680\pi$
$761$	78.0000	0.102497	0.0512484	+	0.998686i	$0.483680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2160.00	−2.82353
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−962.000	−1.25098	−0.625488	−	0.780234i	$-0.715100\pi$
$769$	−962.000	−1.25098	−0.625488	+	0.780234i	$0.715100\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1496.00	1.93532	0.967658	−	0.252264i	$-0.0811751\pi$
$773$	1496.00	1.93532	0.967658	+	0.252264i	$0.0811751\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2112.00	2.69045
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2880.00	3.63178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1144.00	1.43538	0.717691	−	0.696361i	$-0.245199\pi$
$797$	1144.00	1.43538	0.717691	+	0.696361i	$0.245199\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−702.000	−0.876404
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1518.00	1.87639	0.938195	−	0.346106i	$-0.112496\pi$
$809$	1518.00	1.87639	0.938195	+	0.346106i	$0.112496\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1400.00	1.70524	0.852619	−	0.522533i	$-0.175013\pi$
$821$	1400.00	1.70524	0.852619	+	0.522533i	$0.175013\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1080.00	1.30277	0.651387	−	0.758745i	$-0.274187\pi$
$829$	1080.00	1.30277	0.651387	+	0.758745i	$0.274187\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1470.00	1.76471
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	759.000	0.902497
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3256.00	−3.85325
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1656.00	1.94138	0.970692	−	0.240328i	$-0.0772551\pi$
$853$	1656.00	1.94138	0.970692	+	0.240328i	$0.0772551\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1650.00	−1.92532	−0.962660	−	0.270712i	$-0.912741\pi$
$857$	−1650.00	−1.92532	−0.962660	+	0.270712i	$0.912741\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	832.000	0.961850
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1170.00	−1.34021
$874$	0	0
$875$	0	0
$876$	0	0
$877$	696.000	0.793615	0.396807	−	0.917902i	$-0.370118\pi$
$877$	696.000	0.793615	0.396807	+	0.917902i	$0.370118\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	738.000	0.837684	0.418842	−	0.908059i	$-0.362436\pi$
$881$	738.000	0.837684	0.418842	+	0.908059i	$0.362436\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	1680.00	1.86459
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2880.00	3.18232
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−360.000	−0.396040
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	936.000	1.01189
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−258.000	−0.277718	−0.138859	−	0.990312i	$-0.544343\pi$
$929$	−258.000	−0.277718	−0.138859	+	0.990312i	$0.544343\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−430.000	−0.458911	−0.229456	−	0.973319i	$-0.573695\pi$
$937$	−430.000	−0.458911	−0.229456	+	0.973319i	$0.573695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1160.00	−1.23273	−0.616366	−	0.787460i	$-0.711396\pi$
$941$	−1160.00	−1.23273	−0.616366	+	0.787460i	$0.711396\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−2640.00	−2.78188
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1230.00	1.29066	0.645331	−	0.763903i	$-0.276720\pi$
$953$	1230.00	1.29066	0.645331	+	0.763903i	$0.276720\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1520.00	−1.57513
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1890.00	−1.93449	−0.967247	−	0.253838i	$-0.918307\pi$
$977$	−1890.00	−1.93449	−0.967247	+	0.253838i	$0.918307\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1080.00	1.10092
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−448.000	−0.454822
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−744.000	−0.746239	−0.373119	−	0.927783i	$-0.621712\pi$
$997$	−744.000	−0.746239	−0.373119	+	0.927783i	$0.621712\pi$
$998$	0	0
$999$	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	256.3.c.a.255.1		1
3.2	odd	2		2304.3.g.f.1279.1		1
4.3	odd	2	CM	256.3.c.a.255.1		1
8.3	odd	2		256.3.c.b.255.1		1
8.5	even	2		256.3.c.b.255.1		1
12.11	even	2		2304.3.g.f.1279.1		1
16.3	odd	4		128.3.d.a.63.2	yes	2
16.5	even	4		128.3.d.a.63.1	✓	2
16.11	odd	4		128.3.d.a.63.1	✓	2
16.13	even	4		128.3.d.a.63.2	yes	2
24.5	odd	2		2304.3.g.a.1279.1		1
24.11	even	2		2304.3.g.a.1279.1		1
48.5	odd	4		1152.3.b.a.703.2		2
48.11	even	4		1152.3.b.a.703.2		2
48.29	odd	4		1152.3.b.a.703.1		2
48.35	even	4		1152.3.b.a.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.d.a.63.1	✓	2	16.5	even	4
128.3.d.a.63.1	✓	2	16.11	odd	4
128.3.d.a.63.2	yes	2	16.3	odd	4
128.3.d.a.63.2	yes	2	16.13	even	4
256.3.c.a.255.1		1	1.1	even	1	trivial
256.3.c.a.255.1		1	4.3	odd	2	CM
256.3.c.b.255.1		1	8.3	odd	2
256.3.c.b.255.1		1	8.5	even	2
1152.3.b.a.703.1		2	48.29	odd	4
1152.3.b.a.703.1		2	48.35	even	4
1152.3.b.a.703.2		2	48.5	odd	4
1152.3.b.a.703.2		2	48.11	even	4
2304.3.g.a.1279.1		1	24.5	odd	2
2304.3.g.a.1279.1		1	24.11	even	2
2304.3.g.f.1279.1		1	3.2	odd	2
2304.3.g.f.1279.1		1	12.11	even	2

Overview	Random
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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	256.3.c.a.255.1

Level	$256$
Weight	$3$
Character	256.255
Self dual	yes
Analytic conductor	$6.975$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-4
Inner twists	$2$