LMFDB - Embedded newform 1152.5.h.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(449,1152)]

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([0, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1152.449");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.h (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:	$119.082197473$
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} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.449
Dual form			1152.5.h.b.449.1

$q$-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
$5$	−43.8406	−1.75362	−0.876812	−	0.480833i	$-0.840334\pi$
$5$	−43.8406	−1.75362	−0.876812	+	0.480833i	$0.840334\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	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
$13$	238.000i	1.40828i	0.710059	+	0.704142i	$0.248668\pi$
$13$	238.000i	1.40828i	−0.710059	+	0.704142i	$0.751332\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 111.723i	− 0.386584i	−0.981141	−	0.193292i	$-0.938083\pi$
$17$	− 111.723i	− 0.386584i	0.981141	−	0.193292i	$-0.0619165\pi$
$18$	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
$25$	1297.00	2.07520
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1129.96	1.34359	0.671793	−	0.740738i	$-0.265524\pi$
$29$	1129.96	1.34359	0.671793	+	0.740738i	$0.265524\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1680.00i	1.22717i	0.789627	+	0.613587i	$0.210274\pi$
$37$	1680.00i	1.22717i	−0.789627	+	0.613587i	$0.789726\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3166.42i	− 1.88366i	−0.336097	−	0.941828i	$-0.609107\pi$
$41$	− 3166.42i	− 1.88366i	0.336097	−	0.941828i	$-0.390893\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−2401.00	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5318.86	1.89351	0.946753	−	0.321961i	$-0.104342\pi$
$53$	5318.86	1.89351	0.946753	+	0.321961i	$0.104342\pi$
$54$	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
$61$	2640.00i	0.709487i	0.934964	+	0.354743i	$0.115432\pi$
$61$	2640.00i	0.709487i	−0.934964	+	0.354743i	$0.884568\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10434.1i	− 2.46960i
$66$	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
$73$	10560.0	1.98161	0.990805	−	0.135297i	$-0.0431990\pi$
$73$	10560.0	1.98161	0.990805	+	0.135297i	$0.0431990\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	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
$85$	4898.00i	0.677924i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15724.6i	1.98518i	0.121496	+	0.992592i	$0.461231\pi$
$89$	15724.6i	1.98518i	−0.121496	+	0.992592i	$0.538769\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18720.0	−1.98958	−0.994792	−	0.101924i	$-0.967500\pi$
$97$	−18720.0	−1.98958	−0.994792	+	0.101924i	$0.967500\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18895.3	−1.85230	−0.926150	−	0.377156i	$-0.876902\pi$
$101$	−18895.3	−1.85230	−0.926150	+	0.377156i	$0.876902\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	9362.00i	0.787981i	0.919115	+	0.393990i	$0.128906\pi$
$109$	9362.00i	0.787981i	−0.919115	+	0.393990i	$0.871094\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12669.9i	− 0.992242i	−0.868253	−	0.496121i	$-0.834757\pi$
$113$	− 12669.9i	− 0.992242i	0.868253	−	0.496121i	$-0.165243\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−14641.0	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−29460.9	−1.88550
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21494.6i	1.14522i	0.819828	+	0.572610i	$0.194069\pi$
$137$	21494.6i	1.14522i	−0.819828	+	0.572610i	$0.805931\pi$
$138$	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
$145$	−49538.0	−2.35615
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3845.25	0.173202	0.0866008	−	0.996243i	$-0.472400\pi$
$149$	3845.25	0.173202	0.0866008	+	0.996243i	$0.472400\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 44880.0i	− 1.82076i	−0.413769	−	0.910382i	$-0.635788\pi$
$157$	− 44880.0i	− 1.82076i	0.413769	−	0.910382i	$-0.364212\pi$
$158$	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
$169$	−28083.0	−0.983264
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−58945.8	−1.96952	−0.984761	−	0.173912i	$-0.944359\pi$
$173$	−58945.8	−1.96952	−0.984761	+	0.173912i	$0.944359\pi$
$174$	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
$181$	64078.0i	1.95592i	0.208785	+	0.977962i	$0.433049\pi$
$181$	64078.0i	1.95592i	−0.208785	+	0.977962i	$0.566951\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 73652.2i	− 2.15200i
$186$	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
$193$	−38398.0	−1.03085	−0.515423	−	0.856936i	$-0.672365\pi$
$193$	−38398.0	−1.03085	−0.515423	+	0.856936i	$0.672365\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−37223.5	−0.959146	−0.479573	−	0.877502i	$-0.659209\pi$
$197$	−37223.5	−0.959146	−0.479573	+	0.877502i	$0.659209\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	138818.i	3.30322i
$206$	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
$221$	26590.0	0.544421
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	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
$229$	90482.0i	1.72541i	0.505711	+	0.862703i	$0.331230\pi$
$229$	90482.0i	1.72541i	−0.505711	+	0.862703i	$0.668770\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 107366.i	− 1.97767i	−0.149017	−	0.988835i	$-0.547611\pi$
$233$	− 107366.i	− 1.97767i	0.149017	−	0.988835i	$-0.452389\pi$
$234$	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
$241$	100320.	1.72724	0.863621	−	0.504141i	$-0.168191\pi$
$241$	100320.	1.72724	0.863621	+	0.504141i	$0.168191\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	105261.	1.75362
$246$	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
$257$	− 67431.1i	− 1.02093i	−0.859900	−	0.510463i	$-0.829474\pi$
$257$	− 67431.1i	− 1.02093i	0.859900	−	0.510463i	$-0.170526\pi$
$258$	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
$265$	−233182.	−3.32050
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−139610.	−1.92935	−0.964675	−	0.263442i	$-0.915142\pi$
$269$	−139610.	−1.92935	−0.964675	+	0.263442i	$0.915142\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	100558.i	1.31056i	0.755386	+	0.655280i	$0.227449\pi$
$277$	100558.i	1.31056i	−0.755386	+	0.655280i	$0.772551\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 143799.i	− 1.82114i	−0.413360	−	0.910568i	$-0.635645\pi$
$281$	− 143799.i	− 1.82114i	0.413360	−	0.910568i	$-0.364355\pi$
$282$	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
$289$	71039.0	0.850553
$290$	0	0
$291$	0	0
$292$	0	0
$293$	53515.3	0.623365	0.311682	−	0.950186i	$-0.399107\pi$
$293$	53515.3	0.623365	0.311682	+	0.950186i	$0.399107\pi$
$294$	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
$305$	− 115739.i	− 1.24417i
$306$	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
$313$	−193438.	−1.97448	−0.987241	−	0.159234i	$-0.949098\pi$
$313$	−193438.	−1.97448	−0.987241	+	0.159234i	$0.949098\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−60866.3	−0.605701	−0.302851	−	0.953038i	$-0.597938\pi$
$317$	−60866.3	−0.605701	−0.302851	+	0.953038i	$0.597938\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	308686.i	2.92247i
$326$	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
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−201600.	−1.77513	−0.887566	−	0.460680i	$-0.847605\pi$
$337$	−201600.	−1.77513	−0.887566	+	0.460680i	$0.847605\pi$
$338$	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
$349$	− 215280.i	− 1.76747i	−0.467985	−	0.883737i	$-0.655020\pi$
$349$	− 215280.i	− 1.76747i	0.467985	−	0.883737i	$-0.344980\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	140065.i	1.12404i	0.827125	+	0.562018i	$0.189975\pi$
$353$	140065.i	1.12404i	−0.827125	+	0.562018i	$0.810025\pi$
$354$	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
$361$	130321.	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−462957.	−3.47500
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 277200.i	− 1.99240i	−0.0871206	−	0.996198i	$-0.527767\pi$
$373$	− 277200.i	− 1.99240i	0.0871206	−	0.996198i	$-0.472233\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	268930.i	1.89215i
$378$	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
$389$	−68788.8	−0.454588	−0.227294	−	0.973826i	$-0.572988\pi$
$389$	−68788.8	−0.454588	−0.227294	+	0.973826i	$0.572988\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	296400.i	1.88060i	0.340342	+	0.940302i	$0.389457\pi$
$397$	296400.i	1.88060i	−0.340342	+	0.940302i	$0.610543\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	268023.i	1.66680i	0.552670	+	0.833400i	$0.313609\pi$
$401$	268023.i	1.66680i	−0.552670	+	0.833400i	$0.686391\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−187680.	−1.12194	−0.560972	−	0.827835i	$-0.689573\pi$
$409$	−187680.	−1.12194	−0.560972	+	0.827835i	$0.689573\pi$
$410$	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
$421$	351118.i	1.98102i	0.137440	+	0.990510i	$0.456113\pi$
$421$	351118.i	1.98102i	−0.137440	+	0.990510i	$0.543887\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 144905.i	− 0.802240i
$426$	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
$433$	−290878.	−1.55144	−0.775720	−	0.631077i	$-0.782613\pi$
$433$	−290878.	−1.55144	−0.775720	+	0.631077i	$0.782613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	− 689378.i	− 3.48127i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 341336.i	− 1.69313i	−0.532288	−	0.846563i	$-0.678668\pi$
$449$	− 341336.i	− 1.69313i	0.532288	−	0.846563i	$-0.321332\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	285600.	1.36750	0.683748	−	0.729719i	$-0.260349\pi$
$457$	285600.	1.36750	0.683748	+	0.729719i	$0.260349\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−388398.	−1.82758	−0.913788	−	0.406192i	$-0.866856\pi$
$461$	−388398.	−1.82758	−0.913788	+	0.406192i	$0.866856\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	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
$481$	−399840.	−1.72821
$482$	0	0
$483$	0	0
$484$	0	0
$485$	820696.	3.48898
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	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
$493$	− 126242.i	− 0.519410i
$494$	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
$505$	828382.	3.24824
$506$	0	0
$507$	0	0
$508$	0	0
$509$	56114.6	0.216591	0.108295	−	0.994119i	$-0.465461\pi$
$509$	56114.6	0.216591	0.108295	+	0.994119i	$0.465461\pi$
$510$	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
$521$	510926.i	1.88227i	0.338029	+	0.941136i	$0.390240\pi$
$521$	510926.i	1.88227i	−0.338029	+	0.941136i	$0.609760\pi$
$522$	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
$529$	279841.	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	753609.	2.65272
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 120238.i	− 0.410816i	−0.978676	−	0.205408i	$-0.934148\pi$
$541$	− 120238.i	− 0.410816i	0.978676	−	0.205408i	$-0.0658521\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 410436.i	− 1.38182i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	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
$557$	−113475.	−0.365755	−0.182877	−	0.983136i	$-0.558541\pi$
$557$	−113475.	−0.365755	−0.182877	+	0.983136i	$0.558541\pi$
$558$	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
$565$	555458.i	1.74002i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 32811.2i	− 0.101344i	−0.998715	−	0.0506719i	$-0.983864\pi$
$569$	− 32811.2i	− 0.101344i	0.998715	−	0.0506719i	$-0.0161363\pi$
$570$	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
$577$	656642.	1.97232	0.986159	−	0.165801i	$-0.0530210\pi$
$577$	656642.	1.97232	0.986159	+	0.165801i	$0.0530210\pi$
$578$	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
$593$	− 598270.i	− 1.70133i	−0.525710	−	0.850664i	$-0.676200\pi$
$593$	− 598270.i	− 1.70133i	0.525710	−	0.850664i	$-0.323800\pi$
$594$	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
$601$	492002.	1.36213	0.681064	−	0.732224i	$-0.261518\pi$
$601$	492002.	1.36213	0.681064	+	0.732224i	$0.261518\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	641871.	1.75362
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	85680.0i	0.228012i	0.993480	+	0.114006i	$0.0363684\pi$
$613$	85680.0i	0.228012i	−0.993480	+	0.114006i	$0.963632\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	326625.i	0.857985i	0.903308	+	0.428992i	$0.141131\pi$
$617$	326625.i	0.857985i	−0.903308	+	0.428992i	$0.858869\pi$
$618$	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
$625$	480959.	1.23126
$626$	0	0
$627$	0	0
$628$	0	0
$629$	187694.	0.474406
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 571438.i	− 1.40828i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	812439.i	1.97731i	0.150206	+	0.988655i	$0.452006\pi$
$641$	812439.i	1.97731i	−0.150206	+	0.988655i	$0.547994\pi$
$642$	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
$653$	−187243.	−0.439117	−0.219558	−	0.975599i	$-0.570462\pi$
$653$	−187243.	−0.439117	−0.219558	+	0.975599i	$0.570462\pi$
$654$	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
$661$	− 706800.i	− 1.61768i	−0.588026	−	0.808842i	$-0.700095\pi$
$661$	− 706800.i	− 1.61768i	0.588026	−	0.808842i	$-0.299905\pi$
$662$	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
$673$	312958.	0.690965	0.345482	−	0.938425i	$-0.387715\pi$
$673$	312958.	0.690965	0.345482	+	0.938425i	$0.387715\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	541249.	1.18092	0.590459	−	0.807067i	$-0.298947\pi$
$677$	541249.	1.18092	0.590459	+	0.807067i	$0.298947\pi$
$678$	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
$685$	− 942338.i	− 2.00829i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.26589e6i	2.66659i
$690$	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
$697$	−353762.	−0.728192
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−982484.	−1.99935	−0.999676	−	0.0254423i	$-0.991901\pi$
$701$	−982484.	−1.99935	−0.999676	+	0.0254423i	$0.991901\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 737038.i	− 1.46621i	−0.680113	−	0.733107i	$-0.738069\pi$
$709$	− 737038.i	− 1.46621i	0.680113	−	0.733107i	$-0.261931\pi$
$710$	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
$725$	1.46555e6	2.78821
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1.02792e6i	− 1.91316i	−0.291463	−	0.956582i	$-0.594142\pi$
$733$	− 1.02792e6i	− 1.91316i	0.291463	−	0.956582i	$-0.405858\pi$
$734$	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
$745$	−168578.	−0.303730
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	270002.i	0.471167i	0.971854	+	0.235584i	$0.0757002\pi$
$757$	270002.i	0.471167i	−0.971854	+	0.235584i	$0.924300\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 730864.i	− 1.26202i	−0.775773	−	0.631012i	$-0.782640\pi$
$761$	− 730864.i	− 1.26202i	0.775773	−	0.631012i	$-0.217360\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	257278.	0.435061	0.217530	−	0.976054i	$-0.430200\pi$
$769$	257278.	0.435061	0.217530	+	0.976054i	$0.430200\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.15004e6	−1.92465	−0.962327	−	0.271893i	$-0.912350\pi$
$773$	−1.15004e6	−1.92465	−0.962327	+	0.271893i	$0.912350\pi$
$774$	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
$785$	1.96757e6i	3.19294i
$786$	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
$793$	−628320.	−0.999159
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−870818.	−1.37092	−0.685458	−	0.728113i	$-0.740398\pi$
$797$	−870818.	−1.37092	−0.685458	+	0.728113i	$0.740398\pi$
$798$	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
$809$	1.30492e6i	1.99383i	0.0784819	+	0.996916i	$0.474993\pi$
$809$	1.30492e6i	1.99383i	−0.0784819	+	0.996916i	$0.525007\pi$
$810$	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
$821$	−416685.	−0.618190	−0.309095	−	0.951031i	$-0.600026\pi$
$821$	−416685.	−0.618190	−0.309095	+	0.951031i	$0.600026\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	208082.i	0.302779i	0.988474	+	0.151389i	$0.0483747\pi$
$829$	208082.i	0.302779i	−0.988474	+	0.151389i	$0.951625\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	268247.i	0.386584i
$834$	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
$841$	569521.	0.805226
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.23118e6	1.72428
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 678960.i	− 0.933139i	−0.884485	−	0.466569i	$-0.845490\pi$
$853$	− 678960.i	− 0.933139i	0.884485	−	0.466569i	$-0.154510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	345070.i	0.469835i	0.972015	+	0.234917i	$0.0754819\pi$
$857$	345070.i	0.469835i	−0.972015	+	0.234917i	$0.924518\pi$
$858$	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
$865$	2.58422e6	3.45380
$866$	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
$877$	− 1.12056e6i	− 1.45692i	−0.685088	−	0.728460i	$-0.740236\pi$
$877$	− 1.12056e6i	− 1.45692i	0.685088	−	0.728460i	$-0.259764\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 122416.i	− 0.157720i	−0.996886	−	0.0788598i	$-0.974872\pi$
$881$	− 122416.i	− 0.157720i	0.996886	−	0.0788598i	$-0.0251279\pi$
$882$	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
$901$	− 594238.i	− 0.732000i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.80922e6i	− 3.42996i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	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		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.17896e6i	2.54663i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 837779.i	− 0.970729i	−0.874312	−	0.485364i	$-0.838687\pi$
$929$	− 837779.i	− 0.970729i	0.874312	−	0.485364i	$-0.161313\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.57104e6	−1.78940	−0.894700	−	0.446667i	$-0.852611\pi$
$937$	−1.57104e6	−1.78940	−0.894700	+	0.446667i	$0.852611\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.51638e6	1.71249	0.856245	−	0.516570i	$-0.172791\pi$
$941$	1.51638e6	1.71249	0.856245	+	0.516570i	$0.172791\pi$
$942$	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
$949$	2.51328e6i	2.79067i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.48096e6i	1.63064i	0.579010	+	0.815321i	$0.303439\pi$
$953$	1.48096e6i	1.63064i	−0.579010	+	0.815321i	$0.696561\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−923521.	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.68339e6	1.80772
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	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
$977$	− 1.83882e6i	− 1.92641i	−0.268760	−	0.963207i	$-0.586614\pi$
$977$	− 1.83882e6i	− 1.92641i	0.268760	−	0.963207i	$-0.413386\pi$
$978$	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
$985$	1.63190e6	1.68198
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.37640e6i	− 1.38470i	−0.721564	−	0.692348i	$-0.756576\pi$
$997$	− 1.37640e6i	− 1.38470i	0.721564	−	0.692348i	$-0.243424\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.5.h.b.449.2	yes	4
3.2	odd	2	inner	1152.5.h.b.449.4	yes	4
4.3	odd	2	CM	1152.5.h.b.449.2	yes	4
8.3	odd	2	inner	1152.5.h.b.449.3	yes	4
8.5	even	2	inner	1152.5.h.b.449.3	yes	4
12.11	even	2	inner	1152.5.h.b.449.4	yes	4
24.5	odd	2	inner	1152.5.h.b.449.1	✓	4
24.11	even	2	inner	1152.5.h.b.449.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.5.h.b.449.1	✓	4	24.5	odd	2	inner
1152.5.h.b.449.1	✓	4	24.11	even	2	inner
1152.5.h.b.449.2	yes	4	1.1	even	1	trivial
1152.5.h.b.449.2	yes	4	4.3	odd	2	CM
1152.5.h.b.449.3	yes	4	8.3	odd	2	inner
1152.5.h.b.449.3	yes	4	8.5	even	2	inner
1152.5.h.b.449.4	yes	4	3.2	odd	2	inner
1152.5.h.b.449.4	yes	4	12.11	even	2	inner

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-43.8406 q^{5} +238.000i q^{13} -111.723i q^{17} +1297.00 q^{25} +1129.96 q^{29} +1680.00i q^{37} -3166.42i q^{41} -2401.00 q^{49} +5318.86 q^{53} +2640.00i q^{61} -10434.1i q^{65} +10560.0 q^{73} +4898.00i q^{85} +15724.6i q^{89} -18720.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5188 q^{25} - 9604 q^{49} + 42240 q^{73} - 74880 q^{97}+O(q^{100}) \) 4 * q + 5188 * q^25 - 9604 * q^49 + 42240 * q^73 - 74880 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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