LMFDB - Embedded newform 1472.2.h.a.735.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,2,Mod(735,1472)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.735");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.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:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.73358639104.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 18x^{4} - 96x^{2} + 256 $ x^8 - 6x^6 + 18x^4 - 96*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			735.2
Root			$-1.98720 + 0.225925i$ of defining polynomial
Character	$\chi$	$=$	1472.735
Dual form			1472.2.h.a.735.1

$q$-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 $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$-1$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−4.42625	−1.97948	−0.989739	−	0.142888i	$-0.954361\pi$
$5$	−4.42625	−1.97948	−0.989739	+	0.142888i	$0.954361\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	3.52255i	1.06209i	0.847344	+	0.531044i	$0.178200\pi$
$11$	3.52255i	1.06209i	−0.847344	+	0.531044i	$0.821800\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	5.32995i	1.22277i	0.791332	+	0.611387i	$0.209388\pi$
$19$	5.32995i	1.22277i	−0.791332	+	0.611387i	$0.790612\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.79583i	− 1.00000i
$23$	− 4.79583i	− 1.00000i
$24$	0	0
$25$	14.5917	2.91833
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 9.59166i	− 1.72271i	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	− 9.59166i	− 1.72271i	0.508001	−	0.861357i	$-0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.61885	−0.430536	−0.215268	−	0.976555i	$-0.569062\pi$
$37$	−2.61885	−0.430536	−0.215268	+	0.976555i	$0.569062\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.59166	1.49797	0.748983	−	0.662589i	$-0.230542\pi$
$41$	9.59166	1.49797	0.748983	+	0.662589i	$0.230542\pi$
$42$	0	0
$43$	− 12.3750i	− 1.88718i	−0.331122	−	0.943588i	$-0.607427\pi$
$43$	− 12.3750i	− 1.88718i	0.331122	−	0.943588i	$-0.392573\pi$
$44$	0	0
$45$	13.2787	1.97948
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.23365	−0.856257	−0.428129	−	0.903718i	$-0.640827\pi$
$53$	−6.23365	−0.856257	−0.428129	+	0.903718i	$0.640827\pi$
$54$	0	0
$55$	− 15.5917i	− 2.10238i
$56$	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
$61$	11.4713	1.46875	0.734377	−	0.678741i	$-0.237474\pi$
$61$	11.4713	1.46875	0.734377	+	0.678741i	$0.237474\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 14.1824i	− 1.73266i	−0.499471	−	0.866330i	$-0.666472\pi$
$67$	− 14.1824i	− 1.73266i	0.499471	−	0.866330i	$-0.333528\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0000i	1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$	10.0000i	1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$	0	0
$73$	9.59166	1.12262	0.561310	−	0.827606i	$-0.310298\pi$
$73$	9.59166	1.12262	0.561310	+	0.827606i	$0.310298\pi$
$74$	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
$81$	9.00000	1.00000
$82$	0	0
$83$	1.71515i	0.188262i	0.995560	+	0.0941309i	$0.0300072\pi$
$83$	1.71515i	0.188262i	−0.995560	+	0.0941309i	$0.969993\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 23.5917i	− 2.42045i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 10.5676i	− 1.06209i
$100$	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
$107$	7.13735i	0.689993i	0.938604	+	0.344997i	$0.112120\pi$
$107$	7.13735i	0.689993i	−0.938604	+	0.344997i	$0.887880\pi$
$108$	0	0
$109$	15.0861	1.44499	0.722495	−	0.691376i	$-0.242995\pi$
$109$	15.0861	1.44499	0.722495	+	0.691376i	$0.242995\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	21.2275i	1.97948i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.40834	−0.128031
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−42.4551	−3.79730
$126$	0	0
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	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
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	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
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.811447	−0.0664763	−0.0332381	−	0.999447i	$-0.510582\pi$
$149$	−0.811447	−0.0664763	−0.0332381	+	0.999447i	$0.510582\pi$
$150$	0	0
$151$	− 9.59166i	− 0.780558i	−0.920697	−	0.390279i	$-0.872378\pi$
$151$	− 9.59166i	− 0.780558i	0.920697	−	0.390279i	$-0.127622\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	42.4551i	3.41007i
$156$	0	0
$157$	9.66394	0.771266	0.385633	−	0.922652i	$-0.373983\pi$
$157$	9.66394	0.771266	0.385633	+	0.922652i	$0.373983\pi$
$158$	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
$167$	− 22.0000i	− 1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$	− 22.0000i	− 1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 15.9898i	− 1.22277i
$172$	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
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−8.04105	−0.597686	−0.298843	−	0.954302i	$-0.596601\pi$
$181$	−8.04105	−0.597686	−0.298843	+	0.954302i	$0.596601\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.5917	0.852236
$186$	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
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	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
$205$	−42.4551	−2.96519
$206$	0	0
$207$	14.3875i	1.00000i
$208$	0	0
$209$	−18.7750	−1.29869
$210$	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
$215$	54.7750i	3.73562i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 9.59166i	− 0.642305i	−0.947027	−	0.321153i	$-0.895930\pi$
$223$	− 9.59166i	− 0.642305i	0.947027	−	0.321153i	$-0.104070\pi$
$224$	0	0
$225$	−43.7750	−2.91833
$226$	0	0
$227$	− 30.0800i	− 1.99648i	−0.0592894	−	0.998241i	$-0.518883\pi$
$227$	− 30.0800i	− 1.99648i	0.0592894	−	0.998241i	$-0.481117\pi$
$228$	0	0
$229$	29.1763	1.92803	0.964013	−	0.265855i	$-0.0856542\pi$
$229$	29.1763	1.92803	0.964013	+	0.265855i	$0.0856542\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	− 8.85249i	− 0.577473i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.7750i	1.86130i	0.365911	+	0.930650i	$0.380757\pi$
$239$	28.7750i	1.86130i	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	30.9837	1.97948
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 28.2726i	− 1.78455i	−0.451490	−	0.892276i	$-0.649107\pi$
$251$	− 28.2726i	− 1.78455i	0.451490	−	0.892276i	$-0.350893\pi$
$252$	0	0
$253$	16.8935	1.06209
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.7750	−1.79493	−0.897467	−	0.441081i	$-0.854595\pi$
$257$	−28.7750	−1.79493	−0.897467	+	0.441081i	$0.854595\pi$
$258$	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
$265$	27.5917	1.69494
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 30.0000i	− 1.82237i	−0.411997	−	0.911185i	$-0.635169\pi$
$271$	− 30.0000i	− 1.82237i	0.411997	−	0.911185i	$-0.364831\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	51.3998i	3.09953i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	28.7750i	1.72271i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 31.8874i	− 1.89551i	−0.318998	−	0.947755i	$-0.603346\pi$
$283$	− 31.8874i	− 1.89551i	0.318998	−	0.947755i	$-0.396654\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.3689	1.59891	0.799455	−	0.600726i	$-0.205122\pi$
$293$	27.3689	1.59891	0.799455	+	0.600726i	$0.205122\pi$
$294$	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
$305$	−50.7750	−2.90737
$306$	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
$311$	− 9.59166i	− 0.543893i	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	− 9.59166i	− 0.543893i	0.962312	−	0.271947i	$-0.0876674\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	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
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	7.85654	0.430536
$334$	0	0
$335$	62.7750i	3.42976i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.7871	1.82967
$342$	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
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	− 44.2625i	− 2.34921i
$356$	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
$361$	−9.40834	−0.495176
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−42.4551	−2.22220
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−28.7750	−1.49797
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.0288	−1.96906	−0.984529	−	0.175222i	$-0.943936\pi$
$373$	−38.0288	−1.96906	−0.984529	+	0.175222i	$0.943936\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	38.9325i	1.99983i	0.0130611	+	0.999915i	$0.495842\pi$
$379$	38.9325i	1.99983i	−0.0130611	+	0.999915i	$0.504158\pi$
$380$	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
$387$	37.1251i	1.88718i
$388$	0	0
$389$	−9.84845	−0.499336	−0.249668	−	0.968331i	$-0.580322\pi$
$389$	−9.84845	−0.499336	−0.249668	+	0.968331i	$0.580322\pi$
$390$	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
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−39.8362	−1.97948
$406$	0	0
$407$	− 9.22501i	− 0.457267i
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 7.59166i	− 0.372660i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.6948i	− 1.64610i	−0.567969	−	0.823050i	$-0.692271\pi$
$419$	− 33.6948i	− 1.64610i	0.567969	−	0.823050i	$-0.307729\pi$
$420$	0	0
$421$	−36.2214	−1.76532	−0.882662	−	0.470008i	$-0.844251\pi$
$421$	−36.2214	−1.76532	−0.882662	+	0.470008i	$0.844251\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.5615	1.22277
$438$	0	0
$439$	10.0000i	0.477274i	0.971109	+	0.238637i	$0.0767006\pi$
$439$	10.0000i	0.477274i	−0.971109	+	0.238637i	$0.923299\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−28.7750	−1.35798	−0.678988	−	0.734150i	$-0.737581\pi$
$449$	−28.7750	−1.35798	−0.678988	+	0.734150i	$0.737581\pi$
$450$	0	0
$451$	33.7871i	1.59097i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.89965i	− 0.0879055i	−0.999034	−	0.0439527i	$-0.986005\pi$
$467$	− 1.89965i	− 0.0879055i	0.999034	−	0.0439527i	$-0.0139951\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	43.5917	2.00435
$474$	0	0
$475$	77.7728i	3.56846i
$476$	0	0
$477$	18.7009	0.856257
$478$	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
$487$	42.0000i	1.90320i	0.307337	+	0.951601i	$0.400562\pi$
$487$	42.0000i	1.90320i	−0.307337	+	0.951601i	$0.599438\pi$
$488$	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
$493$	0	0
$494$	0	0
$495$	46.7750i	2.10238i
$496$	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
$505$	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
$517$	−7.04509	−0.309843
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	10.7521i	0.470159i	0.971976	+	0.235079i	$0.0755350\pi$
$523$	10.7521i	0.470159i	−0.971976	+	0.235079i	$0.924465\pi$
$524$	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
$533$	0	0
$534$	0	0
$535$	− 31.5917i	− 1.36583i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 24.6578i	− 1.06209i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−66.7750	−2.86033
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−34.4140	−1.46875
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	46.8813	1.98643	0.993213	−	0.116313i	$-0.0371076\pi$
$557$	46.8813	1.98643	0.993213	+	0.116313i	$0.0371076\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 45.9776i	− 1.93773i	−0.247598	−	0.968863i	$-0.579641\pi$
$563$	− 45.9776i	− 1.93773i	0.247598	−	0.968863i	$-0.420359\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	− 47.7850i	− 1.99974i	−0.0161008	−	0.999870i	$-0.505125\pi$
$571$	− 47.7850i	− 1.99974i	0.0161008	−	0.999870i	$-0.494875\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 69.9792i	− 2.91833i
$576$	0	0
$577$	47.9583	1.99653	0.998265	−	0.0588745i	$-0.0187512\pi$
$577$	47.9583	1.99653	0.998265	+	0.0588745i	$0.0187512\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 21.9583i	− 0.909420i
$584$	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
$589$	51.1231	2.10649
$590$	0	0
$591$	0	0
$592$	0	0
$593$	47.9583	1.96941	0.984706	−	0.174224i	$-0.0557418\pi$
$593$	47.9583	1.96941	0.984706	+	0.174224i	$0.0557418\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.7750i	1.17571i	0.808965	+	0.587857i	$0.200028\pi$
$599$	28.7750i	1.17571i	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	9.59166	0.391252	0.195626	−	0.980679i	$-0.437326\pi$
$601$	9.59166	0.391252	0.195626	+	0.980679i	$0.437326\pi$
$602$	0	0
$603$	42.5473i	1.73266i
$604$	0	0
$605$	6.23365	0.253434
$606$	0	0
$607$	− 47.9583i	− 1.94657i	−0.229605	−	0.973284i	$-0.573743\pi$
$607$	− 47.9583i	− 1.94657i	0.229605	−	0.973284i	$-0.426257\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.80335	0.113226	0.0566132	−	0.998396i	$-0.481970\pi$
$613$	2.80335	0.113226	0.0566132	+	0.998396i	$0.481970\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 19.6046i	− 0.787977i	−0.919115	−	0.393989i	$-0.871095\pi$
$619$	− 19.6046i	− 0.787977i	0.919115	−	0.393989i	$-0.128905\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	114.958	4.59833
$626$	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
$635$	− 79.6724i	− 3.16170i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 30.0000i	− 1.18678i
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	33.5103i	1.32152i	0.750598	+	0.660759i	$0.229765\pi$
$643$	33.5103i	1.32152i	−0.750598	+	0.660759i	$0.770235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 47.9583i	− 1.88544i	−0.333591	−	0.942718i	$-0.608260\pi$
$647$	− 47.9583i	− 1.88544i	0.333591	−	0.942718i	$-0.391740\pi$
$648$	0	0
$649$	0	0
$650$	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
$657$	−28.7750	−1.12262
$658$	0	0
$659$	− 49.5924i	− 1.93185i	−0.258829	−	0.965923i	$-0.583337\pi$
$659$	− 49.5924i	− 1.93185i	0.258829	−	0.965923i	$-0.416663\pi$
$660$	0	0
$661$	23.7541	0.923929	0.461964	−	0.886899i	$-0.347145\pi$
$661$	23.7541	0.923929	0.461964	+	0.886899i	$0.347145\pi$
$662$	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
$671$	40.4083i	1.55995i
$672$	0	0
$673$	47.9583	1.84866	0.924329	−	0.381597i	$-0.124626\pi$
$673$	47.9583	1.84866	0.924329	+	0.381597i	$0.124626\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.6558	−0.447971	−0.223985	−	0.974593i	$-0.571907\pi$
$677$	−11.6558	−0.447971	−0.223985	+	0.974593i	$0.571907\pi$
$678$	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
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	43.2665	1.63415	0.817077	−	0.576529i	$-0.195593\pi$
$701$	43.2665	1.63415	0.817077	+	0.576529i	$0.195593\pi$
$702$	0	0
$703$	− 13.9583i	− 0.526448i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−43.4510	−1.63184	−0.815919	−	0.578167i	$-0.803768\pi$
$709$	−43.4510	−1.63184	−0.815919	+	0.578167i	$0.803768\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−46.0000	−1.72271
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28.7750i	1.07313i	0.843860	+	0.536563i	$0.180278\pi$
$719$	28.7750i	1.07313i	−0.843860	+	0.536563i	$0.819722\pi$
$720$	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
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	50.4961	1.86512	0.932558	−	0.361019i	$-0.117571\pi$
$733$	50.4961	1.86512	0.932558	+	0.361019i	$0.117571\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.9583	1.84024
$738$	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
$745$	3.59166	0.131588
$746$	0	0
$747$	− 5.14544i	− 0.188262i
$748$	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
$755$	42.4551i	1.54510i
$756$	0	0
$757$	−32.6066	−1.18511	−0.592554	−	0.805531i	$-0.701880\pi$
$757$	−32.6066	−1.18511	−0.592554	+	0.805531i	$0.701880\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.4059	1.30943	0.654715	−	0.755876i	$-0.272789\pi$
$773$	36.4059	1.30943	0.654715	+	0.755876i	$0.272789\pi$
$774$	0	0
$775$	− 139.958i	− 5.02745i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	51.1231i	1.83167i
$780$	0	0
$781$	−35.2255	−1.26047
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−42.7750	−1.52670
$786$	0	0
$787$	− 42.3628i	− 1.51007i	−0.655684	−	0.755036i	$-0.727620\pi$
$787$	− 42.3628i	− 1.51007i	0.655684	−	0.755036i	$-0.272380\pi$
$788$	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
$797$	−53.9264	−1.91017	−0.955086	−	0.296330i	$-0.904237\pi$
$797$	−53.9264	−1.91017	−0.955086	+	0.296330i	$0.904237\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	33.7871i	1.19232i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	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
$817$	65.9583	2.30759
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 9.59166i	− 0.334344i	−0.985928	−	0.167172i	$-0.946536\pi$
$823$	− 9.59166i	− 0.334344i	0.985928	−	0.167172i	$-0.0534636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 51.3998i	− 1.78735i	−0.448718	−	0.893673i	$-0.648119\pi$
$827$	− 51.3998i	− 1.78735i	0.448718	−	0.893673i	$-0.351881\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	97.3774i	3.36989i
$836$	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
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−57.5412	−1.97948
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.5595i	0.430536i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	70.7750i	2.42045i
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	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
$863$	− 47.9583i	− 1.63252i	−0.577685	−	0.816260i	$-0.696044\pi$
$863$	− 47.9583i	− 1.63252i	0.577685	−	0.816260i	$-0.303956\pi$
$864$	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
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	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
$887$	58.0000i	1.94745i	0.227728	+	0.973725i	$0.426870\pi$
$887$	58.0000i	1.94745i	−0.227728	+	0.973725i	$0.573130\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	31.7029i	1.06209i
$892$	0	0
$893$	−10.6599	−0.356720
$894$	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
$905$	35.5917	1.18311
$906$	0	0
$907$	58.4449i	1.94063i	0.241842	+	0.970316i	$0.422249\pi$
$907$	58.4449i	1.94063i	−0.241842	+	0.970316i	$0.577751\pi$
$908$	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
$913$	−6.04168	−0.199951
$914$	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
$925$	−38.2133	−1.25645
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.7750	−0.944077	−0.472038	−	0.881578i	$-0.656482\pi$
$929$	−28.7750	−0.944077	−0.472038	+	0.881578i	$0.656482\pi$
$930$	0	0
$931$	− 37.3096i	− 1.22277i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.3035	1.70505	0.852523	−	0.522690i	$-0.175071\pi$
$941$	52.3035	1.70505	0.852523	+	0.522690i	$0.175071\pi$
$942$	0	0
$943$	− 46.0000i	− 1.49797i
$944$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−61.0000	−1.96774
$962$	0	0
$963$	− 21.4120i	− 0.689993i
$964$	0	0
$965$	26.5575	0.854916
$966$	0	0
$967$	− 47.9583i	− 1.54224i	−0.636693	−	0.771118i	$-0.719698\pi$
$967$	− 47.9583i	− 1.54224i	0.636693	−	0.771118i	$-0.280302\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51.2153i	1.64358i	0.569792	+	0.821789i	$0.307024\pi$
$971$	51.2153i	1.64358i	−0.569792	+	0.821789i	$0.692976\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−45.2584	−1.44499
$982$	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
$989$	−59.3486	−1.88718
$990$	0	0
$991$	− 9.59166i	− 0.304689i	−0.988327	−	0.152345i	$-0.951318\pi$
$991$	− 9.59166i	− 0.304689i	0.988327	−	0.152345i	$-0.0486824\pi$
$992$	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	1472.2.h.a.735.2	yes	8
4.3	odd	2	inner	1472.2.h.a.735.1	✓	8
8.3	odd	2	inner	1472.2.h.a.735.8	yes	8
8.5	even	2	inner	1472.2.h.a.735.7	yes	8
23.22	odd	2	inner	1472.2.h.a.735.7	yes	8
92.91	even	2	inner	1472.2.h.a.735.8	yes	8
184.45	odd	2	CM	1472.2.h.a.735.2	yes	8
184.91	even	2	inner	1472.2.h.a.735.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.2.h.a.735.1	✓	8	4.3	odd	2	inner
1472.2.h.a.735.1	✓	8	184.91	even	2	inner
1472.2.h.a.735.2	yes	8	1.1	even	1	trivial
1472.2.h.a.735.2	yes	8	184.45	odd	2	CM
1472.2.h.a.735.7	yes	8	8.5	even	2	inner
1472.2.h.a.735.7	yes	8	23.22	odd	2	inner
1472.2.h.a.735.8	yes	8	8.3	odd	2	inner
1472.2.h.a.735.8	yes	8	92.91	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.42625 q^{5} -3.00000 q^{9} +3.52255i q^{11} +5.32995i q^{19} -4.79583i q^{23} +14.5917 q^{25} -9.59166i q^{31} -2.61885 q^{37} +9.59166 q^{41} -12.3750i q^{43} +13.2787 q^{45} +2.00000i q^{47} -7.00000 q^{49} -6.23365 q^{53} -15.5917i q^{55} +11.4713 q^{61} -14.1824i q^{67} +10.0000i q^{71} +9.59166 q^{73} +9.00000 q^{81} +1.71515i q^{83} -23.5917i q^{95} -10.5676i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81}+O(q^{100}) \) 8 * q - 24 * q^9 + 40 * q^25 - 56 * q^49 + 72 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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