LMFDB - Embedded newform 1152.4.d.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(577,1152)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.d (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:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			577.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1152.577
Dual form			1152.4.d.e.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q-70.7107i q^{11} +90.0000 q^{17} +127.279i q^{19} +125.000 q^{25} -522.000 q^{41} -483.661i q^{43} -343.000 q^{49} +325.269i q^{59} -1094.60i q^{67} +430.000 q^{73} -681.651i q^{83} +1026.00 q^{89} +1910.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 180 q^{17} + 250 q^{25} - 1044 q^{41} - 686 q^{49} + 860 q^{73} + 2052 q^{89} + 3820 q^{97}+O(q^{100}) $ 2 * q + 180 * q^17 + 250 * q^25 - 1044 * q^41 - 686 * q^49 + 860 * q^73 + 2052 * q^89 + 3820 * 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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	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
$11$	− 70.7107i	− 1.93819i	−0.246691	−	0.969094i	$-0.579343\pi$
$11$	− 70.7107i	− 1.93819i	0.246691	−	0.969094i	$-0.420657\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	90.0000	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$17$	90.0000	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$18$	0	0
$19$	127.279i	1.53683i	0.639949	+	0.768417i	$0.278955\pi$
$19$	127.279i	1.53683i	−0.639949	+	0.768417i	$0.721045\pi$
$20$	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
$25$	125.000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	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
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−522.000	−1.98836	−0.994179	−	0.107738i	$-0.965639\pi$
$41$	−522.000	−1.98836	−0.994179	+	0.107738i	$0.965639\pi$
$42$	0	0
$43$	− 483.661i	− 1.71529i	−0.514239	−	0.857647i	$-0.671926\pi$
$43$	− 483.661i	− 1.71529i	0.514239	−	0.857647i	$-0.328074\pi$
$44$	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
$49$	−343.000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	325.269i	0.717736i	0.933388	+	0.358868i	$0.116837\pi$
$59$	325.269i	0.717736i	−0.933388	+	0.358868i	$0.883163\pi$
$60$	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
$67$	− 1094.60i	− 1.99592i	−0.0638199	−	0.997961i	$-0.520328\pi$
$67$	− 1094.60i	− 1.99592i	0.0638199	−	0.997961i	$-0.479672\pi$
$68$	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
$73$	430.000	0.689420	0.344710	−	0.938709i	$-0.387977\pi$
$73$	430.000	0.689420	0.344710	+	0.938709i	$0.387977\pi$
$74$	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
$83$	− 681.651i	− 0.901457i	−0.892661	−	0.450728i	$-0.851164\pi$
$83$	− 681.651i	− 0.901457i	0.892661	−	0.450728i	$-0.148836\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1026.00	1.22198	0.610988	−	0.791640i	$-0.290773\pi$
$89$	1026.00	1.22198	0.610988	+	0.791640i	$0.290773\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1910.00	1.99929	0.999645	−	0.0266459i	$-0.00848265\pi$
$97$	1910.00	1.99929	0.999645	+	0.0266459i	$0.00848265\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	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
$107$	− 1405.73i	− 1.27006i	−0.772486	−	0.635032i	$-0.780987\pi$
$107$	− 1405.73i	− 1.27006i	0.772486	−	0.635032i	$-0.219013\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	270.000	0.224774	0.112387	−	0.993665i	$-0.464150\pi$
$113$	270.000	0.224774	0.112387	+	0.993665i	$0.464150\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3669.00	−2.75657
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	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
$131$	− 2729.43i	− 1.82039i	−0.414176	−	0.910197i	$-0.635930\pi$
$131$	− 2729.43i	− 1.82039i	0.414176	−	0.910197i	$-0.364070\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2250.00	−1.40314	−0.701571	−	0.712599i	$-0.747518\pi$
$137$	−2250.00	−1.40314	−0.701571	+	0.712599i	$0.747518\pi$
$138$	0	0
$139$	− 2927.42i	− 1.78634i	−0.449723	−	0.893168i	$-0.648477\pi$
$139$	− 2927.42i	− 1.78634i	0.449723	−	0.893168i	$-0.351523\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	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
$157$	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
$163$	− 4047.48i	− 1.94493i	−0.233056	−	0.972463i	$-0.574873\pi$
$163$	− 4047.48i	− 1.94493i	0.233056	−	0.972463i	$-0.425127\pi$
$164$	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
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2870.85i	1.19876i	0.800465	+	0.599379i	$0.204586\pi$
$179$	2870.85i	1.19876i	−0.800465	+	0.599379i	$0.795414\pi$
$180$	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
$187$	− 6363.96i	− 2.48866i
$188$	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
$193$	−2090.00	−0.779490	−0.389745	−	0.920923i	$-0.627437\pi$
$193$	−2090.00	−0.779490	−0.389745	+	0.920923i	$0.627437\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	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
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9000.00	2.97867
$210$	0	0
$211$	− 381.838i	− 0.124582i	−0.998058	−	0.0622910i	$-0.980159\pi$
$211$	− 381.838i	− 0.124582i	0.998058	−	0.0622910i	$-0.0198407\pi$
$212$	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
$221$	0	0
$222$	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
$227$	− 1903.53i	− 0.556572i	−0.960498	−	0.278286i	$-0.910234\pi$
$227$	− 1903.53i	− 0.556572i	0.960498	−	0.278286i	$-0.0897663\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6030.00	−1.69544	−0.847722	−	0.530441i	$-0.822026\pi$
$233$	−6030.00	−1.69544	−0.847722	+	0.530441i	$0.822026\pi$
$234$	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
$241$	1222.00	0.326622	0.163311	−	0.986575i	$-0.447783\pi$
$241$	1222.00	0.326622	0.163311	+	0.986575i	$0.447783\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6689.23i	− 1.68215i	−0.540916	−	0.841077i	$-0.681922\pi$
$251$	− 6689.23i	− 1.68215i	0.540916	−	0.841077i	$-0.318078\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3870.00	0.939315	0.469658	−	0.882849i	$-0.344377\pi$
$257$	3870.00	0.939315	0.469658	+	0.882849i	$0.344377\pi$
$258$	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
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	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
$275$	− 8838.83i	− 1.93819i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9342.00	−1.98326	−0.991632	−	0.129099i	$-0.958791\pi$
$281$	−9342.00	−1.98326	−0.991632	+	0.129099i	$0.958791\pi$
$282$	0	0
$283$	5116.62i	1.07474i	0.843346	+	0.537371i	$0.180582\pi$
$283$	5116.62i	1.07474i	−0.843346	+	0.537371i	$0.819418\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3187.00	0.648687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	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
$305$	0	0
$306$	0	0
$307$	− 7204.00i	− 1.33926i	−0.742693	−	0.669632i	$-0.766452\pi$
$307$	− 7204.00i	− 1.33926i	0.742693	−	0.669632i	$-0.233548\pi$
$308$	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
$313$	−8390.00	−1.51511	−0.757557	−	0.652769i	$-0.773607\pi$
$313$	−8390.00	−1.51511	−0.757557	+	0.652769i	$0.773607\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11455.1i	1.97331i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8782.27i	1.45836i	0.684322	+	0.729180i	$0.260098\pi$
$331$	8782.27i	1.45836i	−0.684322	+	0.729180i	$0.739902\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11410.0	1.84434	0.922170	−	0.386786i	$-0.126415\pi$
$337$	11410.0	1.84434	0.922170	+	0.386786i	$0.126415\pi$
$338$	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
$347$	11435.3i	1.76911i	0.466437	+	0.884554i	$0.345537\pi$
$347$	11435.3i	1.76911i	−0.466437	+	0.884554i	$0.654463\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4770.00	−0.719211	−0.359605	−	0.933104i	$-0.617089\pi$
$353$	−4770.00	−0.719211	−0.359605	+	0.933104i	$0.617089\pi$
$354$	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
$361$	−9341.00	−1.36186
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	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
$373$	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
$379$	− 9036.82i	− 1.22478i	−0.790557	−	0.612389i	$-0.790209\pi$
$379$	− 9036.82i	− 1.22478i	0.790557	−	0.612389i	$-0.209791\pi$
$380$	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
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7002.00	0.871978	0.435989	−	0.899952i	$-0.356399\pi$
$401$	7002.00	0.871978	0.435989	+	0.899952i	$0.356399\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16346.0	1.97618	0.988090	−	0.153877i	$-0.0491758\pi$
$409$	16346.0	1.97618	0.988090	+	0.153877i	$0.0491758\pi$
$410$	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
$419$	3493.11i	0.407278i	0.979046	+	0.203639i	$0.0652769\pi$
$419$	3493.11i	0.407278i	−0.979046	+	0.203639i	$0.934723\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11250.0	1.28401
$426$	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
$433$	5510.00	0.611533	0.305766	−	0.952107i	$-0.401087\pi$
$433$	5510.00	0.611533	0.305766	+	0.952107i	$0.401087\pi$
$434$	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
$443$	− 3736.35i	− 0.400721i	−0.979722	−	0.200361i	$-0.935789\pi$
$443$	− 3736.35i	− 0.400721i	0.979722	−	0.200361i	$-0.0642114\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17514.0	1.84084	0.920420	−	0.390932i	$-0.127847\pi$
$449$	17514.0	1.84084	0.920420	+	0.390932i	$0.127847\pi$
$450$	0	0
$451$	36911.0i	3.85381i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18070.0	−1.84963	−0.924813	−	0.380422i	$-0.875779\pi$
$457$	−18070.0	−1.84963	−0.924813	+	0.380422i	$0.875779\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	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
$467$	13471.8i	1.33490i	0.744653	+	0.667452i	$0.232615\pi$
$467$	13471.8i	1.33490i	−0.744653	+	0.667452i	$0.767385\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−34200.0	−3.32456
$474$	0	0
$475$	15909.9i	1.53683i
$476$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	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
$491$	− 18002.9i	− 1.65471i	−0.561681	−	0.827354i	$-0.689845\pi$
$491$	− 18002.9i	− 1.65471i	0.561681	−	0.827354i	$-0.310155\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12855.2i	1.15326i	0.817005	+	0.576631i	$0.195633\pi$
$499$	12855.2i	1.15326i	−0.817005	+	0.576631i	$0.804367\pi$
$500$	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
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	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
$521$	−9162.00	−0.770431	−0.385215	−	0.922827i	$-0.625873\pi$
$521$	−9162.00	−0.770431	−0.385215	+	0.922827i	$0.625873\pi$
$522$	0	0
$523$	23444.8i	1.96017i	0.198569	+	0.980087i	$0.436371\pi$
$523$	23444.8i	1.96017i	−0.198569	+	0.980087i	$0.563629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24253.8i	1.93819i
$540$	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
$547$	− 13313.4i	− 1.04066i	−0.853966	−	0.520329i	$-0.825809\pi$
$547$	− 13313.4i	− 1.04066i	0.853966	−	0.520329i	$-0.174191\pi$
$548$	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
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12391.3i	− 0.927589i	−0.885943	−	0.463795i	$-0.846488\pi$
$563$	− 12391.3i	− 0.927589i	0.885943	−	0.463795i	$-0.153512\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2394.00	−0.176383	−0.0881913	−	0.996104i	$-0.528109\pi$
$569$	−2394.00	−0.176383	−0.0881913	+	0.996104i	$0.528109\pi$
$570$	0	0
$571$	3691.10i	0.270521i	0.990810	+	0.135261i	$0.0431872\pi$
$571$	3691.10i	0.270521i	−0.990810	+	0.135261i	$0.956813\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19550.0	−1.41053	−0.705266	−	0.708943i	$-0.749173\pi$
$577$	−19550.0	−1.41053	−0.705266	+	0.708943i	$0.749173\pi$
$578$	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
$587$	26493.9i	1.86289i	0.363876	+	0.931447i	$0.381453\pi$
$587$	26493.9i	1.86289i	−0.363876	+	0.931447i	$0.618547\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26190.0	1.81365	0.906825	−	0.421507i	$-0.138499\pi$
$593$	26190.0	1.81365	0.906825	+	0.421507i	$0.138499\pi$
$594$	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
$601$	14398.0	0.977216	0.488608	−	0.872503i	$-0.337505\pi$
$601$	14398.0	0.977216	0.488608	+	0.872503i	$0.337505\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	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
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28530.0	1.86155	0.930774	−	0.365596i	$-0.119135\pi$
$617$	28530.0	1.86155	0.930774	+	0.365596i	$0.119135\pi$
$618$	0	0
$619$	− 2418.31i	− 0.157027i	−0.996913	−	0.0785136i	$-0.974983\pi$
$619$	− 2418.31i	− 0.157027i	0.996913	−	0.0785136i	$-0.0250174\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	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
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6678.00	−0.411490	−0.205745	−	0.978606i	$-0.565962\pi$
$641$	−6678.00	−0.411490	−0.205745	+	0.978606i	$0.565962\pi$
$642$	0	0
$643$	− 15757.2i	− 0.966411i	−0.875507	−	0.483205i	$-0.839472\pi$
$643$	− 15757.2i	− 0.966411i	0.875507	−	0.483205i	$-0.160528\pi$
$644$	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
$649$	23000.0	1.39111
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16107.9i	0.952161i	0.879402	+	0.476081i	$0.157943\pi$
$659$	16107.9i	0.952161i	−0.879402	+	0.476081i	$0.842057\pi$
$660$	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
$673$	19190.0	1.09914	0.549569	−	0.835448i	$-0.314792\pi$
$673$	19190.0	1.09914	0.549569	+	0.835448i	$0.314792\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33632.8i	1.88422i	0.335300	+	0.942112i	$0.391162\pi$
$683$	33632.8i	1.88422i	−0.335300	+	0.942112i	$0.608838\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	36274.6i	1.99703i	0.0544477	+	0.998517i	$0.482660\pi$
$691$	36274.6i	1.99703i	−0.0544477	+	0.998517i	$0.517340\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−46980.0	−2.55308
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	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		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	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
$731$	− 43529.5i	− 2.20246i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−77400.0	−3.86847
$738$	0	0
$739$	− 17691.8i	− 0.880655i	−0.897837	−	0.440327i	$-0.854862\pi$
$739$	− 17691.8i	− 0.880655i	0.897837	−	0.440327i	$-0.145138\pi$
$740$	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
$745$	0	0
$746$	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
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34182.0	1.62825	0.814124	−	0.580691i	$-0.197218\pi$
$761$	34182.0	1.62825	0.814124	+	0.580691i	$0.197218\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−40106.0	−1.88070	−0.940351	−	0.340207i	$-0.889503\pi$
$769$	−40106.0	−1.88070	−0.940351	+	0.340207i	$0.889503\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 66439.8i	− 3.05578i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43605.9i	1.97507i	0.157396	+	0.987536i	$0.449690\pi$
$787$	43605.9i	1.97507i	−0.157396	+	0.987536i	$0.550310\pi$
$788$	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
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 30405.6i	− 1.33623i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14346.0	−0.623459	−0.311730	−	0.950171i	$-0.600908\pi$
$809$	−14346.0	−0.623459	−0.311730	+	0.950171i	$0.600908\pi$
$810$	0	0
$811$	− 26855.9i	− 1.16281i	−0.813614	−	0.581405i	$-0.802503\pi$
$811$	− 26855.9i	− 1.16281i	0.813614	−	0.581405i	$-0.197497\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	61560.0	2.63612
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	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
$827$	41360.1i	1.73909i	0.493850	+	0.869547i	$0.335589\pi$
$827$	41360.1i	1.73909i	−0.493850	+	0.869547i	$0.664411\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−30870.0	−1.28401
$834$	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
$841$	24389.0	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	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
$857$	18630.0	0.742577	0.371289	−	0.928518i	$-0.378916\pi$
$857$	18630.0	0.742577	0.371289	+	0.928518i	$0.378916\pi$
$858$	0	0
$859$	− 21255.6i	− 0.844276i	−0.906532	−	0.422138i	$-0.861280\pi$
$859$	− 21255.6i	− 0.844276i	0.906532	−	0.422138i	$-0.138720\pi$
$860$	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
$865$	0	0
$866$	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
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41742.0	1.59628	0.798141	−	0.602471i	$-0.205817\pi$
$881$	41742.0	1.59628	0.798141	+	0.602471i	$0.205817\pi$
$882$	0	0
$883$	− 52209.9i	− 1.98981i	−0.100806	−	0.994906i	$-0.532142\pi$
$883$	− 52209.9i	− 1.98981i	0.100806	−	0.994906i	$-0.467858\pi$
$884$	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
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 47933.4i	− 1.75480i	−0.479762	−	0.877399i	$-0.659277\pi$
$907$	− 47933.4i	− 1.75480i	0.479762	−	0.877399i	$-0.340723\pi$
$908$	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
$913$	−48200.0	−1.74719
$914$	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
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6966.00	−0.246014	−0.123007	−	0.992406i	$-0.539254\pi$
$929$	−6966.00	−0.246014	−0.123007	+	0.992406i	$0.539254\pi$
$930$	0	0
$931$	− 43656.8i	− 1.53683i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	56270.0	1.96186	0.980929	−	0.194367i	$-0.0622652\pi$
$937$	56270.0	1.96186	0.980929	+	0.194367i	$0.0622652\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2525.79i	− 0.0866705i	−0.999061	−	0.0433353i	$-0.986202\pi$
$947$	− 2525.79i	− 0.0866705i	0.999061	−	0.0433353i	$-0.0137984\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45990.0	1.56323	0.781617	−	0.623759i	$-0.214395\pi$
$953$	45990.0	1.56323	0.781617	+	0.623759i	$0.214395\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	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
$971$	60514.2i	1.99999i	0.00267705	+	0.999996i	$0.499148\pi$
$971$	60514.2i	1.99999i	−0.00267705	+	0.999996i	$0.500852\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17370.0	0.568798	0.284399	−	0.958706i	$-0.408206\pi$
$977$	17370.0	0.568798	0.284399	+	0.958706i	$0.408206\pi$
$978$	0	0
$979$	− 72549.2i	− 2.36842i
$980$	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
$985$	0	0
$986$	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
$997$	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	1152.4.d.e.577.1		2
3.2	odd	2		128.4.b.b.65.2	yes	2
4.3	odd	2	inner	1152.4.d.e.577.2		2
8.3	odd	2	CM	1152.4.d.e.577.1		2
8.5	even	2	inner	1152.4.d.e.577.2		2
12.11	even	2		128.4.b.b.65.1	✓	2
16.3	odd	4		2304.4.a.bf.1.2		2
16.5	even	4		2304.4.a.bf.1.2		2
16.11	odd	4		2304.4.a.bf.1.1		2
16.13	even	4		2304.4.a.bf.1.1		2
24.5	odd	2		128.4.b.b.65.1	✓	2
24.11	even	2		128.4.b.b.65.2	yes	2
48.5	odd	4		256.4.a.k.1.2		2
48.11	even	4		256.4.a.k.1.1		2
48.29	odd	4		256.4.a.k.1.1		2
48.35	even	4		256.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.b.b.65.1	✓	2	12.11	even	2
128.4.b.b.65.1	✓	2	24.5	odd	2
128.4.b.b.65.2	yes	2	3.2	odd	2
128.4.b.b.65.2	yes	2	24.11	even	2
256.4.a.k.1.1		2	48.11	even	4
256.4.a.k.1.1		2	48.29	odd	4
256.4.a.k.1.2		2	48.5	odd	4
256.4.a.k.1.2		2	48.35	even	4
1152.4.d.e.577.1		2	1.1	even	1	trivial
1152.4.d.e.577.1		2	8.3	odd	2	CM
1152.4.d.e.577.2		2	4.3	odd	2	inner
1152.4.d.e.577.2		2	8.5	even	2	inner
2304.4.a.bf.1.1		2	16.11	odd	4
2304.4.a.bf.1.1		2	16.13	even	4
2304.4.a.bf.1.2		2	16.3	odd	4
2304.4.a.bf.1.2		2	16.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q-70.7107i q^{11} +90.0000 q^{17} +127.279i q^{19} +125.000 q^{25} -522.000 q^{41} -483.661i q^{43} -343.000 q^{49} +325.269i q^{59} -1094.60i q^{67} +430.000 q^{73} -681.651i q^{83} +1026.00 q^{89} +1910.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 180 q^{17} + 250 q^{25} - 1044 q^{41} - 686 q^{49} + 860 q^{73} + 2052 q^{89} + 3820 q^{97}+O(q^{100}) \) 2 * q + 180 * q^17 + 250 * q^25 - 1044 * q^41 - 686 * q^49 + 860 * q^73 + 2052 * q^89 + 3820 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more