LMFDB - Embedded newform 1152.3.b.b.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1152.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.b (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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			703.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.3.b.b.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.00000i q^{5} +O(q^{10})$	$q-6.00000i q^{5} -24.0000i q^{13} -16.0000 q^{17} -11.0000 q^{25} +42.0000i q^{29} -24.0000i q^{37} -80.0000 q^{41} +49.0000 q^{49} -90.0000i q^{53} +120.000i q^{61} -144.000 q^{65} -110.000 q^{73} +96.0000i q^{85} -160.000 q^{89} +130.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 32 q^{17} - 22 q^{25} - 160 q^{41} + 98 q^{49} - 288 q^{65} - 220 q^{73} - 320 q^{89} + 260 q^{97}+O(q^{100}) $ 2 * q - 32 * q^17 - 22 * q^25 - 160 * q^41 + 98 * q^49 - 288 * q^65 - 220 * q^73 - 320 * q^89 + 260 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$4$	0	0
$5$	− 6.00000i	− 1.20000i	−0.800000	−	0.600000i	$-0.795167\pi$
$5$	− 6.00000i	− 1.20000i	0.800000	−	0.600000i	$-0.204833\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 24.0000i	− 1.84615i	−0.384615	−	0.923077i	$-0.625666\pi$
$13$	− 24.0000i	− 1.84615i	0.384615	−	0.923077i	$-0.374334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16.0000	−0.941176	−0.470588	−	0.882353i	$-0.655958\pi$
$17$	−16.0000	−0.941176	−0.470588	+	0.882353i	$0.655958\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−11.0000	−0.440000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42.0000i	1.44828i	0.689655	+	0.724138i	$0.257762\pi$
$29$	42.0000i	1.44828i	−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
$37$	− 24.0000i	− 0.648649i	−0.945946	−	0.324324i	$-0.894863\pi$
$37$	− 24.0000i	− 0.648649i	0.945946	−	0.324324i	$-0.105137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−80.0000	−1.95122	−0.975610	−	0.219512i	$-0.929553\pi$
$41$	−80.0000	−1.95122	−0.975610	+	0.219512i	$0.929553\pi$
$42$	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
$47$	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
$53$	− 90.0000i	− 1.69811i	−0.528302	−	0.849057i	$-0.677171\pi$
$53$	− 90.0000i	− 1.69811i	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		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	120.000i	1.96721i	0.180328	+	0.983607i	$0.442284\pi$
$61$	120.000i	1.96721i	−0.180328	+	0.983607i	$0.557716\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−144.000	−2.21538
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	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
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	96.0000i	1.12941i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−160.000	−1.79775	−0.898876	−	0.438202i	$-0.855615\pi$
$89$	−160.000	−1.79775	−0.898876	+	0.438202i	$0.855615\pi$
$90$	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	$-0.266250\pi$
$97$	130.000	1.34021	0.670103	+	0.742268i	$0.266250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	198.000i	1.96040i	0.198020	+	0.980198i	$0.436549\pi$
$101$	198.000i	1.96040i	−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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 120.000i	− 1.10092i	−0.834862	−	0.550459i	$-0.814453\pi$
$109$	− 120.000i	− 1.10092i	0.834862	−	0.550459i	$-0.185547\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−224.000	−1.98230	−0.991150	−	0.132743i	$-0.957621\pi$
$113$	−224.000	−1.98230	−0.991150	+	0.132743i	$0.957621\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−121.000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 84.0000i	− 0.672000i
$126$	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		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−176.000	−1.28467	−0.642336	−	0.766423i	$-0.722035\pi$
$137$	−176.000	−1.28467	−0.642336	+	0.766423i	$0.722035\pi$
$138$	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
$143$	0	0
$144$	0	0
$145$	252.000	1.73793
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 102.000i	− 0.684564i	−0.939597	−	0.342282i	$-0.888800\pi$
$149$	− 102.000i	− 0.684564i	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
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 264.000i	− 1.68153i	−0.541401	−	0.840764i	$-0.682106\pi$
$157$	− 264.000i	− 1.68153i	0.541401	−	0.840764i	$-0.317894\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	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
$173$	− 330.000i	− 1.90751i	−0.300578	−	0.953757i	$-0.597180\pi$
$173$	− 330.000i	− 1.90751i	0.300578	−	0.953757i	$-0.402820\pi$
$174$	0	0
$175$	0	0
$176$	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
$181$	− 360.000i	− 1.98895i	−0.104972	−	0.994475i	$-0.533475\pi$
$181$	− 360.000i	− 1.98895i	0.104972	−	0.994475i	$-0.466525\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−144.000	−0.778378
$186$	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
$197$	390.000i	1.97970i	0.142132	+	0.989848i	$0.454604\pi$
$197$	390.000i	1.97970i	−0.142132	+	0.989848i	$0.545396\pi$
$198$	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
$205$	480.000i	2.34146i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	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
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	384.000i	1.73756i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	120.000i	0.524017i	0.965066	+	0.262009i	$0.0843849\pi$
$229$	120.000i	0.524017i	−0.965066	+	0.262009i	$0.915615\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	416.000	1.78541	0.892704	−	0.450644i	$-0.148806\pi$
$233$	416.000	1.78541	0.892704	+	0.450644i	$0.148806\pi$
$234$	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	$-0.165905\pi$
$241$	418.000	1.73444	0.867220	+	0.497925i	$0.165905\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 294.000i	− 1.20000i
$246$	0	0
$247$	0	0
$248$	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
$257$	64.0000	0.249027	0.124514	−	0.992218i	$-0.460263\pi$
$257$	64.0000	0.249027	0.124514	+	0.992218i	$0.460263\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−540.000	−2.03774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	138.000i	0.513011i	0.966543	+	0.256506i	$0.0825712\pi$
$269$	138.000i	0.513011i	−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
$277$	504.000i	1.81949i	0.415162	+	0.909747i	$0.363725\pi$
$277$	504.000i	1.81949i	−0.415162	+	0.909747i	$0.636275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	320.000	1.13879	0.569395	−	0.822064i	$-0.307178\pi$
$281$	320.000	1.13879	0.569395	+	0.822064i	$0.307178\pi$
$282$	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
$289$	−33.0000	−0.114187
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 570.000i	− 1.94539i	−0.232082	−	0.972696i	$-0.574554\pi$
$293$	− 570.000i	− 1.94539i	0.232082	−	0.972696i	$-0.425446\pi$
$294$	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
$305$	720.000	2.36066
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	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	$-0.474549\pi$
$313$	50.0000	0.159744	0.0798722	+	0.996805i	$0.474549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 150.000i	− 0.473186i	−0.971609	−	0.236593i	$-0.923969\pi$
$317$	− 150.000i	− 0.473186i	0.971609	−	0.236593i	$-0.0760308\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	264.000i	0.812308i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	350.000	1.03858	0.519288	−	0.854599i	$-0.326197\pi$
$337$	350.000	1.03858	0.519288	+	0.854599i	$0.326197\pi$
$338$	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		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	− 360.000i	− 1.03152i	−0.856734	−	0.515759i	$-0.827510\pi$
$349$	− 360.000i	− 1.03152i	0.856734	−	0.515759i	$-0.172490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	544.000	1.54108	0.770538	−	0.637394i	$-0.219988\pi$
$353$	544.000	1.54108	0.770538	+	0.637394i	$0.219988\pi$
$354$	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
$365$	660.000i	1.80822i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 504.000i	− 1.35121i	−0.737265	−	0.675603i	$-0.763883\pi$
$373$	− 504.000i	− 1.35121i	0.737265	−	0.675603i	$-0.236117\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1008.00	2.67374
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	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
$389$	378.000i	0.971722i	0.874036	+	0.485861i	$0.161494\pi$
$389$	378.000i	0.971722i	−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
$397$	− 456.000i	− 1.14861i	−0.818640	−	0.574307i	$-0.805271\pi$
$397$	− 456.000i	− 1.14861i	0.818640	−	0.574307i	$-0.194729\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	80.0000	0.199501	0.0997506	−	0.995012i	$-0.468195\pi$
$401$	80.0000	0.199501	0.0997506	+	0.995012i	$0.468195\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	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		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	− 840.000i	− 1.99525i	−0.0688836	−	0.997625i	$-0.521944\pi$
$421$	− 840.000i	− 1.99525i	0.0688836	−	0.997625i	$-0.478056\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	176.000	0.414118
$426$	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
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	960.000i	2.15730i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	560.000	1.24722	0.623608	−	0.781737i	$-0.285666\pi$
$449$	560.000	1.24722	0.623608	+	0.781737i	$0.285666\pi$
$450$	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
$461$	− 522.000i	− 1.13232i	−0.824295	−	0.566161i	$-0.808428\pi$
$461$	− 522.000i	− 1.13232i	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		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	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
$477$	0	0
$478$	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
$485$	− 780.000i	− 1.60825i
$486$	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		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	− 672.000i	− 1.36308i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1188.00	2.35248
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 918.000i	− 1.80354i	−0.432220	−	0.901768i	$-0.642270\pi$
$509$	− 918.000i	− 1.80354i	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
$521$	880.000	1.68906	0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000	1.68906	0.844530	+	0.535509i	$0.179880\pi$
$522$	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
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1920.00i	3.60225i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	840.000i	1.55268i	0.630314	+	0.776340i	$0.282926\pi$
$541$	840.000i	1.55268i	−0.630314	+	0.776340i	$0.717074\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−720.000	−1.32110
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 330.000i	− 0.592460i	−0.955117	−	0.296230i	$-0.904271\pi$
$557$	− 330.000i	− 0.592460i	0.955117	−	0.296230i	$-0.0957294\pi$
$558$	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
$565$	1344.00i	2.37876i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1040.00	−1.82777	−0.913884	−	0.405975i	$-0.866932\pi$
$569$	−1040.00	−1.82777	−0.913884	+	0.405975i	$0.866932\pi$
$570$	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
$577$	−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
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	736.000	1.24115	0.620573	−	0.784148i	$-0.286900\pi$
$593$	736.000	1.24115	0.620573	+	0.784148i	$0.286900\pi$
$594$	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	$-0.130759\pi$
$601$	1102.00	1.83361	0.916805	+	0.399334i	$0.130759\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	726.000i	1.20000i
$606$	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
$613$	1224.00i	1.99674i	0.0570962	+	0.998369i	$0.481816\pi$
$613$	1224.00i	1.99674i	−0.0570962	+	0.998369i	$0.518184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1216.00	−1.97083	−0.985413	−	0.170178i	$-0.945566\pi$
$617$	−1216.00	−1.97083	−0.985413	+	0.170178i	$0.945566\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	0	0
$628$	0	0
$629$	384.000i	0.610493i
$630$	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
$637$	− 1176.00i	− 1.84615i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−400.000	−0.624025	−0.312012	−	0.950078i	$-0.601003\pi$
$641$	−400.000	−0.624025	−0.312012	+	0.950078i	$0.601003\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	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
$653$	630.000i	0.964778i	0.875957	+	0.482389i	$0.160231\pi$
$653$	630.000i	0.964778i	−0.875957	+	0.482389i	$0.839769\pi$
$654$	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
$661$	− 600.000i	− 0.907716i	−0.891074	−	0.453858i	$-0.850047\pi$
$661$	− 600.000i	− 0.907716i	0.891074	−	0.453858i	$-0.149953\pi$
$662$	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
$677$	− 1350.00i	− 1.99409i	−0.0768095	−	0.997046i	$-0.524473\pi$
$677$	− 1350.00i	− 1.99409i	0.0768095	−	0.997046i	$-0.475527\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	1056.00i	1.54161i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2160.00	−3.13498
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1280.00	1.83644
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1302.00i	− 1.85735i	−0.370899	−	0.928673i	$-0.620950\pi$
$701$	− 1302.00i	− 1.85735i	0.370899	−	0.928673i	$-0.379050\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 1320.00i	− 1.86178i	−0.365303	−	0.930889i	$-0.619035\pi$
$709$	− 1320.00i	− 1.86178i	0.365303	−	0.930889i	$-0.380965\pi$
$710$	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
$725$	− 462.000i	− 0.637241i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 216.000i	− 0.294679i	−0.989086	−	0.147340i	$-0.952929\pi$
$733$	− 216.000i	− 0.294679i	0.989086	−	0.147340i	$-0.0470711\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−612.000	−0.821477
$746$	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
$757$	− 936.000i	− 1.23646i	−0.785997	−	0.618230i	$-0.787850\pi$
$757$	− 936.000i	− 1.23646i	0.785997	−	0.618230i	$-0.212150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1520.00	−1.99737	−0.998686	−	0.0512484i	$-0.983680\pi$
$761$	−1520.00	−1.99737	−0.998686	+	0.0512484i	$0.983680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	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
$773$	390.000i	0.504528i	0.967658	+	0.252264i	$0.0811751\pi$
$773$	390.000i	0.504528i	−0.967658	+	0.252264i	$0.918825\pi$
$774$	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
$785$	−1584.00	−2.01783
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	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
$797$	1110.00i	1.39272i	0.717691	+	0.696361i	$0.245199\pi$
$797$	1110.00i	1.39272i	−0.717691	+	0.696361i	$0.754801\pi$
$798$	0	0
$799$	0	0
$800$	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
$809$	−560.000	−0.692213	−0.346106	−	0.938195i	$-0.612496\pi$
$809$	−560.000	−0.692213	−0.346106	+	0.938195i	$0.612496\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	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
$821$	− 858.000i	− 1.04507i	−0.852619	−	0.522533i	$-0.824987\pi$
$821$	− 858.000i	− 1.04507i	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		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	− 1080.00i	− 1.30277i	−0.758745	−	0.651387i	$-0.774187\pi$
$829$	− 1080.00i	− 1.30277i	0.758745	−	0.651387i	$-0.225813\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−784.000	−0.941176
$834$	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
$841$	−923.000	−1.09750
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2442.00i	2.88994i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1656.00i	− 1.94138i	−0.240328	−	0.970692i	$-0.577255\pi$
$853$	− 1656.00i	− 1.94138i	0.240328	−	0.970692i	$-0.422745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−464.000	−0.541424	−0.270712	−	0.962660i	$-0.587259\pi$
$857$	−464.000	−0.541424	−0.270712	+	0.962660i	$0.587259\pi$
$858$	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
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−1980.00	−2.28902
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	696.000i	0.793615i	0.917902	+	0.396807i	$0.129882\pi$
$877$	696.000i	0.793615i	−0.917902	+	0.396807i	$0.870118\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1600.00	1.81612	0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00	1.81612	0.908059	+	0.418842i	$0.137564\pi$
$882$	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	1.00000			$0$
$887$	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
$901$	1440.00i	1.59822i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2160.00	−2.38674
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	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
$925$	264.000i	0.285405i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1840.00	−1.98062	−0.990312	−	0.138859i	$-0.955657\pi$
$929$	−1840.00	−1.98062	−0.990312	+	0.138859i	$0.955657\pi$
$930$	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	$-0.426305\pi$
$937$	430.000	0.458911	0.229456	+	0.973319i	$0.426305\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1482.00i	− 1.57492i	−0.616366	−	0.787460i	$-0.711396\pi$
$941$	− 1482.00i	− 1.57492i	0.616366	−	0.787460i	$-0.288604\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	2640.00i	2.78188i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1456.00	1.52781	0.763903	−	0.645331i	$-0.223280\pi$
$953$	1456.00	1.52781	0.763903	+	0.645331i	$0.223280\pi$
$954$	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
$965$	− 1140.00i	− 1.18135i
$966$	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		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−496.000	−0.507677	−0.253838	−	0.967247i	$-0.581693\pi$
$977$	−496.000	−0.507677	−0.253838	+	0.967247i	$0.581693\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	2340.00	2.37563
$986$	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
$997$	744.000i	0.746239i	0.927783	+	0.373119i	$0.121712\pi$
$997$	744.000i	0.746239i	−0.927783	+	0.373119i	$0.878288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.b.b.703.1	✓	2
3.2	odd	2		1152.3.b.d.703.2	yes	2
4.3	odd	2	CM	1152.3.b.b.703.1	✓	2
8.3	odd	2	inner	1152.3.b.b.703.2	yes	2
8.5	even	2	inner	1152.3.b.b.703.2	yes	2
12.11	even	2		1152.3.b.d.703.2	yes	2
16.3	odd	4		2304.3.g.c.1279.1		1
16.5	even	4		2304.3.g.d.1279.1		1
16.11	odd	4		2304.3.g.d.1279.1		1
16.13	even	4		2304.3.g.c.1279.1		1
24.5	odd	2		1152.3.b.d.703.1	yes	2
24.11	even	2		1152.3.b.d.703.1	yes	2
48.5	odd	4		2304.3.g.b.1279.1		1
48.11	even	4		2304.3.g.b.1279.1		1
48.29	odd	4		2304.3.g.e.1279.1		1
48.35	even	4		2304.3.g.e.1279.1		1

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.00000i q^{5} +O(q^{10})\)	\(q-6.00000i q^{5} -24.0000i q^{13} -16.0000 q^{17} -11.0000 q^{25} +42.0000i q^{29} -24.0000i q^{37} -80.0000 q^{41} +49.0000 q^{49} -90.0000i q^{53} +120.000i q^{61} -144.000 q^{65} -110.000 q^{73} +96.0000i q^{85} -160.000 q^{89} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 32 q^{17} - 22 q^{25} - 160 q^{41} + 98 q^{49} - 288 q^{65} - 220 q^{73} - 320 q^{89} + 260 q^{97}+O(q^{100}) \) 2 * q - 32 * q^17 - 22 * q^25 - 160 * q^41 + 98 * q^49 - 288 * q^65 - 220 * q^73 - 320 * q^89 + 260 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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