LMFDB - Embedded newform 1152.3.h.b.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,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, 3, names="a")

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

chi := DirichletCharacter("1152.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
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:	$31.3897264543$
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			449.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.449
Dual form			1152.3.h.b.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} +O(q^{10})$	$q-1.41421 q^{5} -10.0000i q^{13} +9.89949i q^{17} -23.0000 q^{25} +1.41421 q^{29} -24.0000i q^{37} +43.8406i q^{41} -49.0000 q^{49} -103.238 q^{53} -120.000i q^{61} +14.1421i q^{65} -96.0000 q^{73} -14.0000i q^{85} +57.9828i q^{89} -144.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 92 q^{25} - 196 q^{49} - 384 q^{73} - 576 q^{97}+O(q^{100}) $ 4 * q - 92 * q^25 - 196 * q^49 - 384 * q^73 - 576 * 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
$5$	−1.41421	−0.282843	−0.141421	−	0.989949i	$-0.545167\pi$
$5$	−1.41421	−0.282843	−0.141421	+	0.989949i	$0.545167\pi$
$6$	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
$13$	− 10.0000i	− 0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$	− 10.0000i	− 0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.89949i	0.582323i	0.956674	+	0.291162i	$0.0940417\pi$
$17$	9.89949i	0.582323i	−0.956674	+	0.291162i	$0.905958\pi$
$18$	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
$25$	−23.0000	−0.920000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41421	0.0487660	0.0243830	−	0.999703i	$-0.492238\pi$
$29$	1.41421	0.0487660	0.0243830	+	0.999703i	$0.492238\pi$
$30$	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
$37$	− 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
$41$	43.8406i	1.06928i	0.845079	+	0.534642i	$0.179553\pi$
$41$	43.8406i	1.06928i	−0.845079	+	0.534642i	$0.820447\pi$
$42$	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
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−103.238	−1.94788	−0.973940	−	0.226808i	$-0.927171\pi$
$53$	−103.238	−1.94788	−0.973940	+	0.226808i	$0.927171\pi$
$54$	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	$-0.557716\pi$
$61$	− 120.000i	− 1.96721i	0.180328	−	0.983607i	$-0.442284\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.1421i	0.217571i
$66$	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
$73$	−96.0000	−1.31507	−0.657534	−	0.753425i	$-0.728401\pi$
$73$	−96.0000	−1.31507	−0.657534	+	0.753425i	$0.728401\pi$
$74$	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
$85$	− 14.0000i	− 0.164706i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	57.9828i	0.651492i	0.945457	+	0.325746i	$0.105615\pi$
$89$	57.9828i	0.651492i	−0.945457	+	0.325746i	$0.894385\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−144.000	−1.48454	−0.742268	−	0.670103i	$-0.766250\pi$
$97$	−144.000	−1.48454	−0.742268	+	0.670103i	$0.766250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−168.291	−1.66625	−0.833126	−	0.553084i	$-0.813451\pi$
$101$	−168.291	−1.66625	−0.833126	+	0.553084i	$0.813451\pi$
$102$	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
$109$	− 182.000i	− 1.66972i	−0.550459	−	0.834862i	$-0.685547\pi$
$109$	− 182.000i	− 1.66972i	0.550459	−	0.834862i	$-0.314453\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	179.605i	1.58943i	0.606985	+	0.794713i	$0.292379\pi$
$113$	179.605i	1.58943i	−0.606985	+	0.794713i	$0.707621\pi$
$114$	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
$125$	67.8823	0.543058
$126$	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
$137$	24.0416i	0.175486i	0.996143	+	0.0877432i	$0.0279655\pi$
$137$	24.0416i	0.175486i	−0.996143	+	0.0877432i	$0.972035\pi$
$138$	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
$145$	−2.00000	−0.0137931
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−270.115	−1.81285	−0.906425	−	0.422366i	$-0.861200\pi$
$149$	−270.115	−1.81285	−0.906425	+	0.422366i	$0.861200\pi$
$150$	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
$157$	− 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	1.00000			$0$
$163$	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
$169$	69.0000	0.408284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	306.884	1.77390	0.886949	−	0.461867i	$-0.152820\pi$
$173$	306.884	1.77390	0.886949	+	0.461867i	$0.152820\pi$
$174$	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
$181$	38.0000i	0.209945i	0.994475	+	0.104972i	$0.0334754\pi$
$181$	38.0000i	0.209945i	−0.994475	+	0.104972i	$0.966525\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	33.9411i	0.183466i
$186$	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	$-0.663817\pi$
$193$	−190.000	−0.984456	−0.492228	+	0.870466i	$0.663817\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−236.174	−1.19885	−0.599426	−	0.800431i	$-0.704604\pi$
$197$	−236.174	−1.19885	−0.599426	+	0.800431i	$0.704604\pi$
$198$	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
$205$	− 62.0000i	− 0.302439i
$206$	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
$221$	98.9949	0.447941
$222$	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
$229$	442.000i	1.93013i	0.262009	+	0.965066i	$0.415615\pi$
$229$	442.000i	1.93013i	−0.262009	+	0.965066i	$0.584385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	145.664i	0.625167i	0.949890	+	0.312584i	$0.101194\pi$
$233$	145.664i	0.625167i	−0.949890	+	0.312584i	$0.898806\pi$
$234$	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
$241$	240.000	0.995851	0.497925	−	0.867220i	$-0.334095\pi$
$241$	240.000	0.995851	0.497925	+	0.867220i	$0.334095\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	69.2965	0.282843
$246$	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
$257$	− 315.370i	− 1.22712i	−0.789648	−	0.613560i	$-0.789737\pi$
$257$	− 315.370i	− 1.22712i	0.789648	−	0.613560i	$-0.210263\pi$
$258$	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
$265$	146.000	0.550943
$266$	0	0
$267$	0	0
$268$	0	0
$269$	270.115	1.00414	0.502072	−	0.864826i	$-0.332571\pi$
$269$	270.115	1.00414	0.502072	+	0.864826i	$0.332571\pi$
$270$	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
$277$	230.000i	0.830325i	0.909747	+	0.415162i	$0.136275\pi$
$277$	230.000i	0.830325i	−0.909747	+	0.415162i	$0.863725\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 552.958i	− 1.96782i	−0.178664	−	0.983910i	$-0.557178\pi$
$281$	− 552.958i	− 1.96782i	0.178664	−	0.983910i	$-0.442822\pi$
$282$	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
$289$	191.000	0.660900
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−306.884	−1.04739	−0.523693	−	0.851907i	$-0.675446\pi$
$293$	−306.884	−1.04739	−0.523693	+	0.851907i	$0.675446\pi$
$294$	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
$305$	169.706i	0.556412i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	50.0000	0.159744	0.0798722	−	0.996805i	$-0.474549\pi$
$313$	50.0000	0.159744	0.0798722	+	0.996805i	$0.474549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	541.644	1.70866	0.854328	−	0.519735i	$-0.173969\pi$
$317$	541.644	1.70866	0.854328	+	0.519735i	$0.173969\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	230.000i	0.707692i
$326$	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
$337$	576.000	1.70920	0.854599	−	0.519288i	$-0.173803\pi$
$337$	576.000	1.70920	0.854599	+	0.519288i	$0.173803\pi$
$338$	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	$0.172490\pi$
$349$	360.000i	1.03152i	−0.856734	+	0.515759i	$0.827510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 702.864i	− 1.99112i	−0.0941474	−	0.995558i	$-0.530012\pi$
$353$	− 702.864i	− 1.99112i	0.0941474	−	0.995558i	$-0.469988\pi$
$354$	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
$365$	135.765	0.371958
$366$	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
$373$	504.000i	1.35121i	0.737265	+	0.675603i	$0.236117\pi$
$373$	504.000i	1.35121i	−0.737265	+	0.675603i	$0.763883\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.1421i	− 0.0375123i
$378$	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
$389$	748.119	1.92319	0.961593	−	0.274481i	$-0.0885061\pi$
$389$	748.119	1.92319	0.961593	+	0.274481i	$0.0885061\pi$
$390$	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	$0.194729\pi$
$397$	456.000i	1.14861i	−0.818640	+	0.574307i	$0.805271\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	620.840i	1.54823i	0.633046	+	0.774114i	$0.281805\pi$
$401$	620.840i	1.54823i	−0.633046	+	0.774114i	$0.718195\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	240.000	0.586797	0.293399	−	0.955990i	$-0.405214\pi$
$409$	240.000	0.586797	0.293399	+	0.955990i	$0.405214\pi$
$410$	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
$421$	− 58.0000i	− 0.137767i	−0.997625	−	0.0688836i	$-0.978056\pi$
$421$	− 58.0000i	− 0.137767i	0.997625	−	0.0688836i	$-0.0219437\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 227.688i	− 0.535737i
$426$	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	$-0.391307\pi$
$433$	290.000	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$434$	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
$445$	− 82.0000i	− 0.184270i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 892.369i	− 1.98746i	−0.111814	−	0.993729i	$-0.535666\pi$
$449$	− 892.369i	− 1.98746i	0.111814	−	0.993729i	$-0.464334\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	336.000	0.735230	0.367615	−	0.929978i	$-0.380174\pi$
$457$	336.000	0.735230	0.367615	+	0.929978i	$0.380174\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	168.291	0.365057	0.182529	−	0.983201i	$-0.441572\pi$
$461$	168.291	0.365057	0.182529	+	0.983201i	$0.441572\pi$
$462$	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
$481$	−240.000	−0.498960
$482$	0	0
$483$	0	0
$484$	0	0
$485$	203.647	0.419890
$486$	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
$493$	14.0000i	0.0283976i
$494$	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
$505$	238.000	0.471287
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−337.997	−0.664041	−0.332021	−	0.943272i	$-0.607730\pi$
$509$	−337.997	−0.664041	−0.332021	+	0.943272i	$0.607730\pi$
$510$	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
$521$	227.688i	0.437022i	0.975835	+	0.218511i	$0.0701199\pi$
$521$	227.688i	0.437022i	−0.975835	+	0.218511i	$0.929880\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	438.406	0.822526
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	682.000i	1.26063i	0.776340	+	0.630314i	$0.217074\pi$
$541$	682.000i	1.26063i	−0.776340	+	0.630314i	$0.782926\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	257.387i	0.472269i
$546$	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
$557$	−985.707	−1.76967	−0.884836	−	0.465903i	$-0.845729\pi$
$557$	−985.707	−1.76967	−0.884836	+	0.465903i	$0.845729\pi$
$558$	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
$565$	− 254.000i	− 0.449558i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1062.07i	− 1.86656i	−0.359146	−	0.933282i	$-0.616932\pi$
$569$	− 1062.07i	− 1.86656i	0.359146	−	0.933282i	$-0.383068\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1150.00	−1.99307	−0.996534	−	0.0831889i	$-0.973490\pi$
$577$	−1150.00	−1.99307	−0.996534	+	0.0831889i	$0.973490\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	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
$593$	1178.04i	1.98658i	0.115665	+	0.993288i	$0.463100\pi$
$593$	1178.04i	1.98658i	−0.115665	+	0.993288i	$0.536900\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1102.00	−1.83361	−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−1102.00	−1.83361	−0.916805	+	0.399334i	$0.869241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	171.120	0.282843
$606$	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
$613$	− 1224.00i	− 1.99674i	−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	− 1224.00i	− 1.99674i	0.0570962	−	0.998369i	$-0.481816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1008.33i	1.63425i	0.576458	+	0.817127i	$0.304434\pi$
$617$	1008.33i	1.63425i	−0.576458	+	0.817127i	$0.695566\pi$
$618$	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
$625$	479.000	0.766400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	237.588	0.377723
$630$	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
$637$	490.000i	0.769231i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1144.10i	− 1.78487i	−0.451180	−	0.892433i	$-0.648997\pi$
$641$	− 1144.10i	− 1.78487i	0.451180	−	0.892433i	$-0.351003\pi$
$642$	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
$653$	−1254.41	−1.92099	−0.960496	−	0.278295i	$-0.910231\pi$
$653$	−1254.41	−1.92099	−0.960496	+	0.278295i	$0.910231\pi$
$654$	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
$677$	881.055	1.30141	0.650705	−	0.759330i	$-0.274473\pi$
$677$	881.055	1.30141	0.650705	+	0.759330i	$0.274473\pi$
$678$	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
$685$	− 34.0000i	− 0.0496350i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1032.38i	1.49837i
$690$	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
$697$	−434.000	−0.622669
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1288.35	−1.83787	−0.918936	−	0.394406i	$-0.870950\pi$
$701$	−1288.35	−1.83787	−0.918936	+	0.394406i	$0.870950\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 518.000i	− 0.730606i	−0.930889	−	0.365303i	$-0.880965\pi$
$709$	− 518.000i	− 0.730606i	0.930889	−	0.365303i	$-0.119035\pi$
$710$	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
$725$	−32.5269	−0.0448647
$726$	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
$733$	− 1450.00i	− 1.97817i	−0.147340	−	0.989086i	$-0.547071\pi$
$733$	− 1450.00i	− 1.97817i	0.147340	−	0.989086i	$-0.452929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	382.000	0.512752
$746$	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
$757$	− 1190.00i	− 1.57199i	−0.618230	−	0.785997i	$-0.712150\pi$
$757$	− 1190.00i	− 1.57199i	0.618230	−	0.785997i	$-0.287850\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1129.96i	1.48483i	0.669940	+	0.742416i	$0.266320\pi$
$761$	1129.96i	1.48483i	−0.669940	+	0.742416i	$0.733680\pi$
$762$	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
$773$	−782.060	−1.01172	−0.505860	−	0.862615i	$-0.668825\pi$
$773$	−782.060	−1.01172	−0.505860	+	0.862615i	$0.668825\pi$
$774$	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
$785$	373.352i	0.475608i
$786$	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
$793$	−1200.00	−1.51324
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1593.82	−1.99977	−0.999886	−	0.0150826i	$-0.995199\pi$
$797$	−1593.82	−1.99977	−0.999886	+	0.0150826i	$0.995199\pi$
$798$	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
$809$	1469.37i	1.81628i	0.418670	+	0.908138i	$0.362496\pi$
$809$	1469.37i	1.81628i	−0.418670	+	0.908138i	$0.637504\pi$
$810$	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
$821$	1596.65	1.94476	0.972379	−	0.233406i	$-0.0749870\pi$
$821$	1596.65	1.94476	0.972379	+	0.233406i	$0.0749870\pi$
$822$	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
$829$	1258.00i	1.51749i	0.651387	+	0.758745i	$0.274187\pi$
$829$	1258.00i	1.51749i	−0.651387	+	0.758745i	$0.725813\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 485.075i	− 0.582323i
$834$	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
$841$	−839.000	−0.997622
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−97.5807	−0.115480
$846$	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
$857$	− 838.629i	− 0.978563i	−0.872126	−	0.489282i	$-0.837259\pi$
$857$	− 838.629i	− 0.978563i	0.872126	−	0.489282i	$-0.162741\pi$
$858$	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
$865$	−434.000	−0.501734
$866$	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	$-0.870118\pi$
$877$	− 696.000i	− 0.793615i	0.917902	−	0.396807i	$-0.129882\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1653.22i	1.87652i	0.345929	+	0.938261i	$0.387564\pi$
$881$	1653.22i	1.87652i	−0.345929	+	0.938261i	$0.612436\pi$
$882$	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
$901$	− 1022.00i	− 1.13430i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 53.7401i	− 0.0593813i
$906$	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
$925$	552.000i	0.596757i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1483.51i	− 1.59689i	−0.602068	−	0.798445i	$-0.705657\pi$
$929$	− 1483.51i	− 1.59689i	0.602068	−	0.798445i	$-0.294343\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−430.000	−0.458911	−0.229456	−	0.973319i	$-0.573695\pi$
$937$	−430.000	−0.458911	−0.229456	+	0.973319i	$0.573695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1868.18	−1.98531	−0.992655	−	0.120982i	$-0.961396\pi$
$941$	−1868.18	−1.98531	−0.992655	+	0.120982i	$0.961396\pi$
$942$	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
$949$	960.000i	1.01159i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	159.806i	0.167687i	0.996479	+	0.0838437i	$0.0267197\pi$
$953$	159.806i	0.167687i	−0.996479	+	0.0838437i	$0.973280\pi$
$954$	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
$965$	268.701	0.278446
$966$	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
$977$	1687.16i	1.72687i	0.504456	+	0.863437i	$0.331693\pi$
$977$	1687.16i	1.72687i	−0.504456	+	0.863437i	$0.668307\pi$
$978$	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
$985$	334.000	0.339086
$986$	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
$997$	− 744.000i	− 0.746239i	−0.927783	−	0.373119i	$-0.878288\pi$
$997$	− 744.000i	− 0.746239i	0.927783	−	0.373119i	$-0.121712\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.h.b.449.1	✓	4
3.2	odd	2	inner	1152.3.h.b.449.3	yes	4
4.3	odd	2	CM	1152.3.h.b.449.1	✓	4
8.3	odd	2	inner	1152.3.h.b.449.4	yes	4
8.5	even	2	inner	1152.3.h.b.449.4	yes	4
12.11	even	2	inner	1152.3.h.b.449.3	yes	4
16.3	odd	4		2304.3.e.d.1025.2		2
16.5	even	4		2304.3.e.b.1025.1		2
16.11	odd	4		2304.3.e.b.1025.1		2
16.13	even	4		2304.3.e.d.1025.2		2
24.5	odd	2	inner	1152.3.h.b.449.2	yes	4
24.11	even	2	inner	1152.3.h.b.449.2	yes	4
48.5	odd	4		2304.3.e.b.1025.2		2
48.11	even	4		2304.3.e.b.1025.2		2
48.29	odd	4		2304.3.e.d.1025.1		2
48.35	even	4		2304.3.e.d.1025.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.b.449.1	✓	4	1.1	even	1	trivial
1152.3.h.b.449.1	✓	4	4.3	odd	2	CM
1152.3.h.b.449.2	yes	4	24.5	odd	2	inner
1152.3.h.b.449.2	yes	4	24.11	even	2	inner
1152.3.h.b.449.3	yes	4	3.2	odd	2	inner
1152.3.h.b.449.3	yes	4	12.11	even	2	inner
1152.3.h.b.449.4	yes	4	8.3	odd	2	inner
1152.3.h.b.449.4	yes	4	8.5	even	2	inner
2304.3.e.b.1025.1		2	16.5	even	4
2304.3.e.b.1025.1		2	16.11	odd	4
2304.3.e.b.1025.2		2	48.5	odd	4
2304.3.e.b.1025.2		2	48.11	even	4
2304.3.e.d.1025.1		2	48.29	odd	4
2304.3.e.d.1025.1		2	48.35	even	4
2304.3.e.d.1025.2		2	16.3	odd	4
2304.3.e.d.1025.2		2	16.13	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +O(q^{10})\)	\(q-1.41421 q^{5} -10.0000i q^{13} +9.89949i q^{17} -23.0000 q^{25} +1.41421 q^{29} -24.0000i q^{37} +43.8406i q^{41} -49.0000 q^{49} -103.238 q^{53} -120.000i q^{61} +14.1421i q^{65} -96.0000 q^{73} -14.0000i q^{85} +57.9828i q^{89} -144.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 92 q^{25} - 196 q^{49} - 384 q^{73} - 576 q^{97}+O(q^{100}) \) 4 * q - 92 * q^25 - 196 * q^49 - 384 * q^73 - 576 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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