LMFDB - Embedded newform 1152.2.f.b.575.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(575,1152)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1152.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.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:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			575.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.575
Dual form			1152.2.f.b.575.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	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	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264i	1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$	4.24264i	1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	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		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.89949	1.83829	0.919145	−	0.393919i	$-0.128881\pi$
$29$	9.89949	1.83829	0.919145	+	0.393919i	$0.128881\pi$
$30$	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
$37$	12.0000i	1.97279i	0.164399	+	0.986394i	$0.447432\pi$
$37$	12.0000i	1.97279i	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.7279i	− 1.98777i	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	− 12.7279i	− 1.98777i	0.110432	−	0.993884i	$-0.464777\pi$
$42$	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		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.07107	0.971286	0.485643	−	0.874157i	$-0.338586\pi$
$53$	7.07107	0.971286	0.485643	+	0.874157i	$0.338586\pi$
$54$	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
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.48528i	1.05247i
$66$	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		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	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	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.24264i	− 0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$	− 4.24264i	− 0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5563	1.54791	0.773957	−	0.633238i	$-0.218274\pi$
$101$	15.5563	1.54791	0.773957	+	0.633238i	$0.218274\pi$
$102$	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
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	21.2132i	1.99557i	0.0665190	+	0.997785i	$0.478811\pi$
$113$	21.2132i	1.99557i	−0.0665190	+	0.997785i	$0.521189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	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
$137$	− 21.2132i	− 1.81237i	−0.422885	−	0.906183i	$-0.638983\pi$
$137$	− 21.2132i	− 1.81237i	0.422885	−	0.906183i	$-0.361017\pi$
$138$	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
$145$	14.0000	1.16264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−24.0416	−1.96957	−0.984784	−	0.173785i	$-0.944400\pi$
$149$	−24.0416	−1.96957	−0.984784	+	0.173785i	$0.944400\pi$
$150$	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
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	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		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.5563	1.18273	0.591364	−	0.806405i	$-0.298590\pi$
$173$	15.5563	1.18273	0.591364	+	0.806405i	$0.298590\pi$
$174$	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
$181$	− 18.0000i	− 1.33793i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	− 18.0000i	− 1.33793i	0.743294	−	0.668965i	$-0.233262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9706i	1.24770i
$186$	0	0
$187$	0	0
$188$	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
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.3848	−1.30986	−0.654931	−	0.755689i	$-0.727302\pi$
$197$	−18.3848	−1.30986	−0.654931	+	0.755689i	$0.727302\pi$
$198$	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
$205$	− 18.0000i	− 1.25717i
$206$	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
$221$	−25.4558	−1.71235
$222$	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	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 30.0000i	− 1.98246i	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	− 30.0000i	− 1.98246i	0.132164	−	0.991228i	$-0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 29.6985i	− 1.94561i	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	− 29.6985i	− 1.94561i	0.231621	−	0.972806i	$-0.425597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.89949	0.632456
$246$	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
$257$	− 21.2132i	− 1.32324i	−0.749838	−	0.661622i	$-0.769869\pi$
$257$	− 21.2132i	− 1.32324i	0.749838	−	0.661622i	$-0.230131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−32.5269	−1.98320	−0.991600	−	0.129339i	$-0.958714\pi$
$269$	−32.5269	−1.98320	−0.991600	+	0.129339i	$0.958714\pi$
$270$	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
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.6985i	1.77166i	0.464007	+	0.885832i	$0.346411\pi$
$281$	29.6985i	1.77166i	−0.464007	+	0.885832i	$0.653589\pi$
$282$	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
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.8701	−1.56977	−0.784883	−	0.619644i	$-0.787277\pi$
$293$	−26.8701	−1.56977	−0.784883	+	0.619644i	$0.787277\pi$
$294$	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
$305$	16.9706i	0.971732i
$306$	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		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	35.3553	1.98575	0.992877	−	0.119145i	$-0.0380154\pi$
$317$	35.3553	1.98575	0.992877	+	0.119145i	$0.0380154\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 18.0000i	− 0.998460i
$326$	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
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	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
$349$	− 36.0000i	− 1.92704i	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	− 36.0000i	− 1.92704i	0.267644	−	0.963518i	$-0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 12.7279i	− 0.677439i	−0.940887	−	0.338719i	$-0.890006\pi$
$353$	− 12.7279i	− 0.677439i	0.940887	−	0.338719i	$-0.109994\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−22.6274	−1.18437
$366$	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
$373$	36.0000i	1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$	36.0000i	1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	59.3970i	3.05910i
$378$	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		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.89949	0.501924	0.250962	−	0.967997i	$-0.419253\pi$
$389$	9.89949	0.501924	0.250962	+	0.967997i	$0.419253\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0000i	0.602263i	0.953583	+	0.301131i	$0.0973643\pi$
$397$	12.0000i	0.602263i	−0.953583	+	0.301131i	$0.902636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 29.6985i	− 1.48307i	−0.670913	−	0.741536i	$-0.734098\pi$
$401$	− 29.6985i	− 1.48307i	0.670913	−	0.741536i	$-0.265902\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	40.0000	1.97787	0.988936	−	0.148340i	$-0.0473931\pi$
$409$	40.0000	1.97787	0.988936	+	0.148340i	$0.0473931\pi$
$410$	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
$421$	30.0000i	1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$	30.0000i	1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 12.7279i	− 0.617395i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	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	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 6.00000i	− 0.284427i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 38.1838i	− 1.80200i	−0.433816	−	0.901002i	$-0.642833\pi$
$449$	− 38.1838i	− 1.80200i	0.433816	−	0.901002i	$-0.357167\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.0122	1.91013	0.955064	−	0.296399i	$-0.0957859\pi$
$461$	41.0122	1.91013	0.955064	+	0.296399i	$0.0957859\pi$
$462$	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
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−72.0000	−3.28292
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.3137	0.513729
$486$	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
$493$	42.0000i	1.89158i
$494$	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		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	22.0000	0.978987
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.0416	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$509$	−24.0416	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$510$	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
$521$	12.7279i	0.557620i	0.960346	+	0.278810i	$0.0899400\pi$
$521$	12.7279i	0.557620i	−0.960346	+	0.278810i	$0.910060\pi$
$522$	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
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	76.3675	3.30785
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.0000i	1.80572i	0.429934	+	0.902861i	$0.358537\pi$
$541$	42.0000i	1.80572i	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.48528i	− 0.363470i
$546$	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
$557$	7.07107	0.299611	0.149805	−	0.988716i	$-0.452135\pi$
$557$	7.07107	0.299611	0.149805	+	0.988716i	$0.452135\pi$
$558$	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
$565$	30.0000i	1.26211i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.6690i	1.95647i	0.207504	+	0.978234i	$0.433466\pi$
$569$	46.6690i	1.95647i	−0.207504	+	0.978234i	$0.566534\pi$
$570$	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
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	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	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 21.2132i	− 0.871122i	−0.900159	−	0.435561i	$-0.856550\pi$
$593$	− 21.2132i	− 0.871122i	0.900159	−	0.435561i	$-0.143450\pi$
$594$	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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.5563	0.632456
$606$	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
$613$	36.0000i	1.45403i	0.686624	+	0.727013i	$0.259092\pi$
$613$	36.0000i	1.45403i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.24264i	− 0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$	− 4.24264i	− 0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$	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
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−50.9117	−2.02998
$630$	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
$637$	42.0000i	1.66410i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 29.6985i	− 1.17302i	−0.809942	−	0.586510i	$-0.800502\pi$
$641$	− 29.6985i	− 1.17302i	0.809942	−	0.586510i	$-0.199498\pi$
$642$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.4975	1.93699	0.968493	−	0.249041i	$-0.0801154\pi$
$653$	49.4975	1.93699	0.968493	+	0.249041i	$0.0801154\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	12.0000i	0.466746i	0.972387	+	0.233373i	$0.0749763\pi$
$661$	12.0000i	0.466746i	−0.972387	+	0.233373i	$0.925024\pi$
$662$	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
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.3553	1.35882	0.679408	−	0.733761i	$-0.262237\pi$
$677$	35.3553	1.35882	0.679408	+	0.733761i	$0.262237\pi$
$678$	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
$685$	− 30.0000i	− 1.14624i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	42.4264i	1.61632i
$690$	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
$697$	54.0000	2.04540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	43.8406	1.65584	0.827919	−	0.560848i	$-0.189525\pi$
$701$	43.8406	1.65584	0.827919	+	0.560848i	$0.189525\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 30.0000i	− 1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	− 30.0000i	− 1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$	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
$725$	−29.6985	−1.10297
$726$	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
$733$	54.0000i	1.99454i	0.0738717	+	0.997268i	$0.476464\pi$
$733$	54.0000i	1.99454i	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	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		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−34.0000	−1.24566
$746$	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
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	55.1543i	1.99934i	0.0256326	+	0.999671i	$0.491840\pi$
$761$	55.1543i	1.99934i	−0.0256326	+	0.999671i	$0.508160\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.07107	0.254329	0.127164	−	0.991882i	$-0.459412\pi$
$773$	7.07107	0.254329	0.127164	+	0.991882i	$0.459412\pi$
$774$	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
$785$	− 16.9706i	− 0.605705i
$786$	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
$793$	−72.0000	−2.55679
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.3259	−1.85348	−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−52.3259	−1.85348	−0.926739	+	0.375705i	$0.877401\pi$
$798$	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
$809$	− 46.6690i	− 1.64080i	−0.571793	−	0.820398i	$-0.693752\pi$
$809$	− 46.6690i	− 1.64080i	0.571793	−	0.820398i	$-0.306248\pi$
$810$	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
$821$	15.5563	0.542920	0.271460	−	0.962450i	$-0.412493\pi$
$821$	15.5563	0.542920	0.271460	+	0.962450i	$0.412493\pi$
$822$	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
$829$	− 54.0000i	− 1.87550i	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	− 54.0000i	− 1.87550i	0.347314	−	0.937749i	$-0.387094\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29.6985i	1.02899i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	69.0000	2.37931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−32.5269	−1.11896
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 36.0000i	− 1.23262i	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	− 36.0000i	− 1.23262i	0.787505	−	0.616308i	$-0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 46.6690i	− 1.59418i	−0.603858	−	0.797092i	$-0.706370\pi$
$857$	− 46.6690i	− 1.59418i	0.603858	−	0.797092i	$-0.293630\pi$
$858$	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		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	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
$877$	12.0000i	0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$	12.0000i	0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$	12.7279i	0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$	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
$891$	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
$901$	30.0000i	0.999445i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 25.4558i	− 0.846181i
$906$	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		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	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
$925$	− 36.0000i	− 1.18367i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.24264i	0.139197i	0.997575	+	0.0695983i	$0.0221717\pi$
$929$	4.24264i	0.139197i	−0.997575	+	0.0695983i	$0.977828\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.8701	−0.875939	−0.437969	−	0.898990i	$-0.644302\pi$
$941$	−26.8701	−0.875939	−0.437969	+	0.898990i	$0.644302\pi$
$942$	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
$949$	− 96.0000i	− 3.11629i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.2132i	− 0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$	− 21.2132i	− 0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.7990	−0.637352
$966$	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
$977$	− 38.1838i	− 1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$	− 38.1838i	− 1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	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
$997$	− 12.0000i	− 0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	− 12.0000i	− 0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.f.b.575.4	yes	4
3.2	odd	2	inner	1152.2.f.b.575.2	yes	4
4.3	odd	2	CM	1152.2.f.b.575.4	yes	4
8.3	odd	2	inner	1152.2.f.b.575.1	✓	4
8.5	even	2	inner	1152.2.f.b.575.1	✓	4
12.11	even	2	inner	1152.2.f.b.575.2	yes	4
16.3	odd	4		2304.2.c.d.2303.1		2
16.5	even	4		2304.2.c.f.2303.2		2
16.11	odd	4		2304.2.c.f.2303.2		2
16.13	even	4		2304.2.c.d.2303.1		2
24.5	odd	2	inner	1152.2.f.b.575.3	yes	4
24.11	even	2	inner	1152.2.f.b.575.3	yes	4
48.5	odd	4		2304.2.c.f.2303.1		2
48.11	even	4		2304.2.c.f.2303.1		2
48.29	odd	4		2304.2.c.d.2303.2		2
48.35	even	4		2304.2.c.d.2303.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.f.b.575.1	✓	4	8.3	odd	2	inner
1152.2.f.b.575.1	✓	4	8.5	even	2	inner
1152.2.f.b.575.2	yes	4	3.2	odd	2	inner
1152.2.f.b.575.2	yes	4	12.11	even	2	inner
1152.2.f.b.575.3	yes	4	24.5	odd	2	inner
1152.2.f.b.575.3	yes	4	24.11	even	2	inner
1152.2.f.b.575.4	yes	4	1.1	even	1	trivial
1152.2.f.b.575.4	yes	4	4.3	odd	2	CM
2304.2.c.d.2303.1		2	16.3	odd	4
2304.2.c.d.2303.1		2	16.13	even	4
2304.2.c.d.2303.2		2	48.29	odd	4
2304.2.c.d.2303.2		2	48.35	even	4
2304.2.c.f.2303.1		2	48.5	odd	4
2304.2.c.f.2303.1		2	48.11	even	4
2304.2.c.f.2303.2		2	16.5	even	4
2304.2.c.f.2303.2		2	16.11	odd	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{5} +O(q^{10})\)	\(q+1.41421 q^{5} +6.00000i q^{13} +4.24264i q^{17} -3.00000 q^{25} +9.89949 q^{29} +12.0000i q^{37} -12.7279i q^{41} +7.00000 q^{49} +7.07107 q^{53} +12.0000i q^{61} +8.48528i q^{65} -16.0000 q^{73} +6.00000i q^{85} -4.24264i q^{89} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{25} + 28 q^{49} - 64 q^{73} + 32 q^{97}+O(q^{100}) \) 4 * q - 12 * q^25 + 28 * q^49 - 64 * q^73 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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