LMFDB - Embedded newform 1152.3.b.f.703.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			703.4
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.3.b.f.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +9.79796i q^{7} +O(q^{10})$	$q+2.00000i q^{5} +9.79796i q^{7} +19.5959 q^{11} +21.0000 q^{25} +50.0000i q^{29} -48.9898i q^{31} -19.5959 q^{35} -47.0000 q^{49} +94.0000i q^{53} +39.1918i q^{55} -117.576 q^{59} +50.0000 q^{73} +192.000i q^{77} +146.969i q^{79} +97.9796 q^{83} -190.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 84 q^{25} - 188 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100}) $ 4 * q + 84 * q^25 - 188 * q^49 + 200 * q^73 - 760 * 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
$5$	2.00000i	0.400000i	0.979796	+	0.200000i	$0.0640942\pi$
$5$	2.00000i	0.400000i	−0.979796	+	0.200000i	$0.935906\pi$
$6$	0	0
$7$	9.79796i	1.39971i	0.714286	+	0.699854i	$0.246752\pi$
$7$	9.79796i	1.39971i	−0.714286	+	0.699854i	$0.753248\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.5959	1.78145	0.890724	−	0.454545i	$-0.150198\pi$
$11$	19.5959	1.78145	0.890724	+	0.454545i	$0.150198\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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
$25$	21.0000	0.840000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	50.0000i	1.72414i	0.506791	+	0.862069i	$0.330832\pi$
$29$	50.0000i	1.72414i	−0.506791	+	0.862069i	$0.669168\pi$
$30$	0	0
$31$	− 48.9898i	− 1.58032i	−0.612903	−	0.790158i	$-0.709998\pi$
$31$	− 48.9898i	− 1.58032i	0.612903	−	0.790158i	$-0.290002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−19.5959	−0.559883
$36$	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		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	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
$49$	−47.0000	−0.959184
$50$	0	0
$51$	0	0
$52$	0	0
$53$	94.0000i	1.77358i	0.462168	+	0.886792i	$0.347072\pi$
$53$	94.0000i	1.77358i	−0.462168	+	0.886792i	$0.652928\pi$
$54$	0	0
$55$	39.1918i	0.712579i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−117.576	−1.99281	−0.996403	−	0.0847458i	$-0.972992\pi$
$59$	−117.576	−1.99281	−0.996403	+	0.0847458i	$0.972992\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	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
$73$	50.0000	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$73$	50.0000	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	192.000i	2.49351i
$78$	0	0
$79$	146.969i	1.86037i	0.367089	+	0.930186i	$0.380355\pi$
$79$	146.969i	1.86037i	−0.367089	+	0.930186i	$0.619645\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	97.9796	1.18048	0.590238	−	0.807229i	$-0.299034\pi$
$83$	97.9796	1.18048	0.590238	+	0.807229i	$0.299034\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−190.000	−1.95876	−0.979381	−	0.202020i	$-0.935249\pi$
$97$	−190.000	−1.95876	−0.979381	+	0.202020i	$0.935249\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	190.000i	1.88119i	0.339533	+	0.940594i	$0.389731\pi$
$101$	190.000i	1.88119i	−0.339533	+	0.940594i	$0.610269\pi$
$102$	0	0
$103$	− 205.757i	− 1.99764i	−0.0485437	−	0.998821i	$-0.515458\pi$
$103$	− 205.757i	− 1.99764i	0.0485437	−	0.998821i	$-0.484542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	195.959	1.83139	0.915697	−	0.401869i	$-0.131639\pi$
$107$	195.959	1.83139	0.915697	+	0.401869i	$0.131639\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	263.000	2.17355
$122$	0	0
$123$	0	0
$124$	0	0
$125$	92.0000i	0.736000i
$126$	0	0
$127$	− 107.778i	− 0.848642i	−0.905512	−	0.424321i	$-0.860513\pi$
$127$	− 107.778i	− 0.848642i	0.905512	−	0.424321i	$-0.139487\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−78.3837	−0.598349	−0.299174	−	0.954198i	$-0.596711\pi$
$131$	−78.3837	−0.598349	−0.299174	+	0.954198i	$0.596711\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	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
$145$	−100.000	−0.689655
$146$	0	0
$147$	0	0
$148$	0	0
$149$	290.000i	1.94631i	0.230153	+	0.973154i	$0.426077\pi$
$149$	290.000i	1.94631i	−0.230153	+	0.973154i	$0.573923\pi$
$150$	0	0
$151$	48.9898i	0.324436i	0.986755	+	0.162218i	$0.0518647\pi$
$151$	48.9898i	0.324436i	−0.986755	+	0.162218i	$0.948135\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	97.9796	0.632126
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	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
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	46.0000i	0.265896i	0.991123	+	0.132948i	$0.0424443\pi$
$173$	46.0000i	0.265896i	−0.991123	+	0.132948i	$0.957556\pi$
$174$	0	0
$175$	205.757i	1.17576i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−274.343	−1.53264	−0.766321	−	0.642458i	$-0.777915\pi$
$179$	−274.343	−1.53264	−0.766321	+	0.642458i	$0.777915\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	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
$193$	−290.000	−1.50259	−0.751295	−	0.659966i	$-0.770571\pi$
$193$	−290.000	−1.50259	−0.751295	+	0.659966i	$0.770571\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 194.000i	− 0.984772i	−0.870377	−	0.492386i	$-0.836125\pi$
$197$	− 194.000i	− 0.984772i	0.870377	−	0.492386i	$-0.163875\pi$
$198$	0	0
$199$	342.929i	1.72326i	0.507538	+	0.861630i	$0.330556\pi$
$199$	342.929i	1.72326i	−0.507538	+	0.861630i	$0.669444\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−489.898	−2.41329
$204$	0	0
$205$	0	0
$206$	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
$217$	480.000	2.21198
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 382.120i	− 1.71354i	−0.515695	−	0.856772i	$-0.672466\pi$
$223$	− 382.120i	− 1.71354i	0.515695	−	0.856772i	$-0.327534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	293.939	1.29488	0.647442	−	0.762115i	$-0.275839\pi$
$227$	293.939	1.29488	0.647442	+	0.762115i	$0.275839\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	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
$241$	−382.000	−1.58506	−0.792531	−	0.609831i	$-0.791237\pi$
$241$	−382.000	−1.58506	−0.792531	+	0.609831i	$0.791237\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 94.0000i	− 0.383673i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	176.363	0.702642	0.351321	−	0.936255i	$-0.385733\pi$
$251$	176.363	0.702642	0.351321	+	0.936255i	$0.385733\pi$
$252$	0	0
$253$	0	0
$254$	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
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−188.000	−0.709434
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 430.000i	− 1.59851i	−0.600990	−	0.799257i	$-0.705227\pi$
$269$	− 430.000i	− 1.59851i	0.600990	−	0.799257i	$-0.294773\pi$
$270$	0	0
$271$	− 538.888i	− 1.98852i	−0.107011	−	0.994258i	$-0.534128\pi$
$271$	− 538.888i	− 1.98852i	0.107011	−	0.994258i	$-0.465872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	411.514	1.49642
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	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
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 386.000i	− 1.31741i	−0.752403	−	0.658703i	$-0.771105\pi$
$293$	− 386.000i	− 1.31741i	0.752403	−	0.658703i	$-0.228895\pi$
$294$	0	0
$295$	− 235.151i	− 0.797122i
$296$	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
$307$	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
$313$	530.000	1.69329	0.846645	−	0.532158i	$-0.178619\pi$
$313$	530.000	1.69329	0.846645	+	0.532158i	$0.178619\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 334.000i	− 1.05363i	−0.849981	−	0.526814i	$-0.823386\pi$
$317$	− 334.000i	− 1.05363i	0.849981	−	0.526814i	$-0.176614\pi$
$318$	0	0
$319$	979.796i	3.07146i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	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
$337$	190.000	0.563798	0.281899	−	0.959444i	$-0.409036\pi$
$337$	190.000	0.563798	0.281899	+	0.959444i	$0.409036\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 960.000i	− 2.81525i
$342$	0	0
$343$	19.5959i	0.0571310i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	685.857	1.97653	0.988267	−	0.152738i	$-0.0488090\pi$
$347$	685.857	1.97653	0.988267	+	0.152738i	$0.0488090\pi$
$348$	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		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$365$	100.000i	0.273973i
$366$	0	0
$367$	186.161i	0.507251i	0.967302	+	0.253626i	$0.0816231\pi$
$367$	186.161i	0.507251i	−0.967302	+	0.253626i	$0.918377\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−921.008	−2.48250
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	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
$385$	−384.000	−0.997403
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 190.000i	− 0.488432i	−0.969721	−	0.244216i	$-0.921469\pi$
$389$	− 190.000i	− 0.488432i	0.969721	−	0.244216i	$-0.0785306\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−293.939	−0.744149
$396$	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		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	718.000	1.75550	0.877751	−	0.479118i	$-0.159043\pi$
$409$	718.000	1.75550	0.877751	+	0.479118i	$0.159043\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1152.00i	− 2.78935i
$414$	0	0
$415$	195.959i	0.472191i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−411.514	−0.982134	−0.491067	−	0.871122i	$-0.663393\pi$
$419$	−411.514	−0.982134	−0.491067	+	0.871122i	$0.663393\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	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
$433$	670.000	1.54734	0.773672	−	0.633586i	$-0.218418\pi$
$433$	670.000	1.54734	0.773672	+	0.633586i	$0.218418\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 832.827i	− 1.89710i	−0.316629	−	0.948550i	$-0.602551\pi$
$439$	− 832.827i	− 1.89710i	0.316629	−	0.948550i	$-0.397449\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−881.816	−1.99056	−0.995278	−	0.0970655i	$-0.969054\pi$
$443$	−881.816	−1.99056	−0.995278	+	0.0970655i	$0.969054\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−530.000	−1.15974	−0.579869	−	0.814710i	$-0.696896\pi$
$457$	−530.000	−1.15974	−0.579869	+	0.814710i	$0.696896\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 530.000i	− 1.14967i	−0.818268	−	0.574837i	$-0.805065\pi$
$461$	− 530.000i	− 1.14967i	0.818268	−	0.574837i	$-0.194935\pi$
$462$	0	0
$463$	− 891.614i	− 1.92573i	−0.269978	−	0.962866i	$-0.587017\pi$
$463$	− 891.614i	− 1.92573i	0.269978	−	0.962866i	$-0.412983\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	685.857	1.46864	0.734322	−	0.678801i	$-0.237500\pi$
$467$	685.857	1.46864	0.734322	+	0.678801i	$0.237500\pi$
$468$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 380.000i	− 0.783505i
$486$	0	0
$487$	88.1816i	0.181071i	0.995893	+	0.0905356i	$0.0288579\pi$
$487$	88.1816i	0.181071i	−0.995893	+	0.0905356i	$0.971142\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	862.220	1.75605	0.878025	−	0.478615i	$-0.158861\pi$
$491$	862.220	1.75605	0.878025	+	0.478615i	$0.158861\pi$
$492$	0	0
$493$	0	0
$494$	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
$505$	−380.000	−0.752475
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1010.00i	1.98428i	0.125121	+	0.992141i	$0.460068\pi$
$509$	1010.00i	1.98428i	−0.125121	+	0.992141i	$0.539932\pi$
$510$	0	0
$511$	489.898i	0.958704i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	411.514	0.799057
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	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
$533$	0	0
$534$	0	0
$535$	391.918i	0.732558i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−921.008	−1.70873
$540$	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
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1440.00	−2.60398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 914.000i	− 1.64093i	−0.571694	−	0.820467i	$-0.693714\pi$
$557$	− 914.000i	− 1.64093i	0.571694	−	0.820467i	$-0.306286\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1077.78	1.91434	0.957172	−	0.289520i	$-0.0934958\pi$
$563$	1077.78	1.91434	0.957172	+	0.289520i	$0.0934958\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	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
$577$	290.000	0.502600	0.251300	−	0.967909i	$-0.419142\pi$
$577$	290.000	0.502600	0.251300	+	0.967909i	$0.419142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	960.000i	1.65232i
$582$	0	0
$583$	1842.02i	3.15955i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	783.837	1.33533	0.667663	−	0.744463i	$-0.267295\pi$
$587$	783.837	1.33533	0.667663	+	0.744463i	$0.267295\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	1198.00	1.99334	0.996672	−	0.0815138i	$-0.0259755\pi$
$601$	1198.00	1.99334	0.996672	+	0.0815138i	$0.0259755\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	526.000i	0.869421i
$606$	0	0
$607$	− 969.998i	− 1.59802i	−0.601318	−	0.799010i	$-0.705357\pi$
$607$	− 969.998i	− 1.59802i	0.601318	−	0.799010i	$-0.294643\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	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		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	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
$625$	341.000	0.545600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	244.949i	0.388192i	0.980983	+	0.194096i	$0.0621773\pi$
$631$	244.949i	0.388192i	−0.980983	+	0.194096i	$0.937823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	215.555	0.339457
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	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
$649$	−2304.00	−3.55008
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1006.00i	1.54058i	0.637693	+	0.770291i	$0.279889\pi$
$653$	1006.00i	1.54058i	−0.637693	+	0.770291i	$0.720111\pi$
$654$	0	0
$655$	− 156.767i	− 0.239339i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1097.37	−1.66521	−0.832603	−	0.553869i	$-0.813151\pi$
$659$	−1097.37	−1.66521	−0.832603	+	0.553869i	$0.813151\pi$
$660$	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
$671$	0	0
$672$	0	0
$673$	190.000	0.282318	0.141159	−	0.989987i	$-0.454917\pi$
$673$	190.000	0.282318	0.141159	+	0.989987i	$0.454917\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1346.00i	1.98818i	0.108545	+	0.994092i	$0.465381\pi$
$677$	1346.00i	1.98818i	−0.108545	+	0.994092i	$0.534619\pi$
$678$	0	0
$679$	− 1861.61i	− 2.74170i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−293.939	−0.430364	−0.215182	−	0.976574i	$-0.569035\pi$
$683$	−293.939	−0.430364	−0.215182	+	0.976574i	$0.569035\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	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
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	50.0000i	0.0713267i	0.999364	+	0.0356633i	$0.0113544\pi$
$701$	50.0000i	0.0713267i	−0.999364	+	0.0356633i	$0.988646\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1861.61	−2.63311
$708$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	2016.00	2.79612
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1050.00i	1.44828i
$726$	0	0
$727$	− 107.778i	− 0.148250i	−0.997249	−	0.0741249i	$-0.976384\pi$
$727$	− 107.778i	− 0.148250i	0.997249	−	0.0741249i	$-0.0236163\pi$
$728$	0	0
$729$	0	0
$730$	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
$737$	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
$745$	−580.000	−0.778523
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1920.00i	2.56342i
$750$	0	0
$751$	538.888i	0.717560i	0.933422	+	0.358780i	$0.116807\pi$
$751$	538.888i	0.717560i	−0.933422	+	0.358780i	$0.883193\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−97.9796	−0.129774
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	862.000	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$769$	862.000	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1154.00i	− 1.49288i	−0.665450	−	0.746442i	$-0.731760\pi$
$773$	− 1154.00i	− 1.49288i	0.665450	−	0.746442i	$-0.268240\pi$
$774$	0	0
$775$	− 1028.79i	− 1.32747i
$776$	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
$785$	0	0
$786$	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
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1294.00i	1.62359i	0.583944	+	0.811794i	$0.301509\pi$
$797$	1294.00i	1.62359i	−0.583944	+	0.811794i	$0.698491\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	979.796	1.22017
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	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
$821$	670.000i	0.816078i	0.912965	+	0.408039i	$0.133787\pi$
$821$	670.000i	0.816078i	−0.912965	+	0.408039i	$0.866213\pi$
$822$	0	0
$823$	1577.47i	1.91673i	0.285541	+	0.958367i	$0.407827\pi$
$823$	1577.47i	1.91673i	−0.285541	+	0.958367i	$0.592173\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	587.878	0.710856	0.355428	−	0.934704i	$-0.384335\pi$
$827$	587.878	0.710856	0.355428	+	0.934704i	$0.384335\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	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
$841$	−1659.00	−1.97265
$842$	0	0
$843$	0	0
$844$	0	0
$845$	338.000i	0.400000i
$846$	0	0
$847$	2576.86i	3.04234i
$848$	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
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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
$865$	−92.0000	−0.106358
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2880.00i	3.31415i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−901.412	−1.03019
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	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
$889$	1056.00	1.18785
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 548.686i	− 0.613057i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2449.49	2.72468
$900$	0	0
$901$	0	0
$902$	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
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1920.00	2.10296
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 768.000i	− 0.837514i
$918$	0	0
$919$	− 1714.64i	− 1.86577i	−0.360174	−	0.932885i	$-0.617283\pi$
$919$	− 1714.64i	− 1.86577i	0.360174	−	0.932885i	$-0.382717\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1490.00	−1.59018	−0.795091	−	0.606491i	$-0.792577\pi$
$937$	−1490.00	−1.59018	−0.795091	+	0.606491i	$0.792577\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	430.000i	0.456961i	0.973549	+	0.228480i	$0.0733757\pi$
$941$	430.000i	0.456961i	−0.973549	+	0.228480i	$0.926624\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1371.71	1.44848	0.724242	−	0.689546i	$-0.242190\pi$
$947$	1371.71	1.44848	0.724242	+	0.689546i	$0.242190\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1439.00	−1.49740
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 580.000i	− 0.601036i
$966$	0	0
$967$	303.737i	0.314102i	0.987590	+	0.157051i	$0.0501987\pi$
$967$	303.737i	0.314102i	−0.987590	+	0.157051i	$0.949801\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−215.555	−0.221993	−0.110996	−	0.993821i	$-0.535404\pi$
$971$	−215.555	−0.221993	−0.110996	+	0.993821i	$0.535404\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	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
$985$	388.000	0.393909
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 1420.70i	− 1.43361i	−0.697275	−	0.716803i	$-0.745605\pi$
$991$	− 1420.70i	− 1.43361i	0.697275	−	0.716803i	$-0.254395\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−685.857	−0.689304
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.b.f.703.4	yes	4
3.2	odd	2	inner	1152.3.b.f.703.2	yes	4
4.3	odd	2	inner	1152.3.b.f.703.3	yes	4
8.3	odd	2	inner	1152.3.b.f.703.1	✓	4
8.5	even	2	inner	1152.3.b.f.703.2	yes	4
12.11	even	2	inner	1152.3.b.f.703.1	✓	4
16.3	odd	4		2304.3.g.l.1279.2		2
16.5	even	4		2304.3.g.i.1279.1		2
16.11	odd	4		2304.3.g.i.1279.2		2
16.13	even	4		2304.3.g.l.1279.1		2
24.5	odd	2	CM	1152.3.b.f.703.4	yes	4
24.11	even	2	inner	1152.3.b.f.703.3	yes	4
48.5	odd	4		2304.3.g.l.1279.1		2
48.11	even	4		2304.3.g.l.1279.2		2
48.29	odd	4		2304.3.g.i.1279.1		2
48.35	even	4		2304.3.g.i.1279.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.b.f.703.1	✓	4	8.3	odd	2	inner
1152.3.b.f.703.1	✓	4	12.11	even	2	inner
1152.3.b.f.703.2	yes	4	3.2	odd	2	inner
1152.3.b.f.703.2	yes	4	8.5	even	2	inner
1152.3.b.f.703.3	yes	4	4.3	odd	2	inner
1152.3.b.f.703.3	yes	4	24.11	even	2	inner
1152.3.b.f.703.4	yes	4	1.1	even	1	trivial
1152.3.b.f.703.4	yes	4	24.5	odd	2	CM
2304.3.g.i.1279.1		2	16.5	even	4
2304.3.g.i.1279.1		2	48.29	odd	4
2304.3.g.i.1279.2		2	16.11	odd	4
2304.3.g.i.1279.2		2	48.35	even	4
2304.3.g.l.1279.1		2	16.13	even	4
2304.3.g.l.1279.1		2	48.5	odd	4
2304.3.g.l.1279.2		2	16.3	odd	4
2304.3.g.l.1279.2		2	48.11	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +9.79796i q^{7} +O(q^{10})\)	\(q+2.00000i q^{5} +9.79796i q^{7} +19.5959 q^{11} +21.0000 q^{25} +50.0000i q^{29} -48.9898i q^{31} -19.5959 q^{35} -47.0000 q^{49} +94.0000i q^{53} +39.1918i q^{55} -117.576 q^{59} +50.0000 q^{73} +192.000i q^{77} +146.969i q^{79} +97.9796 q^{83} -190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 84 q^{25} - 188 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100}) \) 4 * q + 84 * q^25 - 188 * q^49 + 200 * q^73 - 760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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