LMFDB - Embedded newform 1472.2.h.a.735.3

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			735.3
Root			$1.24541 + 1.56491i$ of defining polynomial
Character	$\chi$	$=$	1472.735
Dual form			1472.2.h.a.735.4

$q$-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
$5$	−0.639012	−0.285775	−0.142888	−	0.989739i	$-0.545639\pi$
$5$	−0.639012	−0.285775	−0.142888	+	0.989739i	$0.545639\pi$
$6$	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
$11$	− 5.62065i	− 1.69469i	−0.531044	−	0.847344i	$-0.678200\pi$
$11$	− 5.62065i	− 1.69469i	0.531044	−	0.847344i	$-0.321800\pi$
$12$	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
$19$	6.89867i	1.58266i	0.611387	+	0.791332i	$0.290612\pi$
$19$	6.89867i	1.58266i	−0.611387	+	0.791332i	$0.709388\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.79583i	1.00000i
$23$	4.79583i	1.00000i
$24$	0	0
$25$	−4.59166	−0.918333
$26$	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	$0.330385\pi$
$31$	9.59166i	1.72271i	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.8803	1.95311	0.976555	−	0.215268i	$-0.0690625\pi$
$37$	11.8803	1.95311	0.976555	+	0.215268i	$0.0690625\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.59166	−1.49797	−0.748983	−	0.662589i	$-0.769458\pi$
$41$	−9.59166	−1.49797	−0.748983	+	0.662589i	$0.769458\pi$
$42$	0	0
$43$	4.34262i	0.662244i	0.943588	+	0.331122i	$0.107427\pi$
$43$	4.34262i	0.662244i	−0.943588	+	0.331122i	$0.892573\pi$
$44$	0	0
$45$	1.91704	0.285775
$46$	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
$53$	−13.1583	−1.80744	−0.903718	−	0.428129i	$-0.859173\pi$
$53$	−13.1583	−1.80744	−0.903718	+	0.428129i	$0.859173\pi$
$54$	0	0
$55$	3.59166i	0.484300i
$56$	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
$61$	−10.6023	−1.35748	−0.678741	−	0.734377i	$-0.737474\pi$
$61$	−10.6023	−1.35748	−0.678741	+	0.734377i	$0.737474\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.17670i	− 0.998942i	−0.866330	−	0.499471i	$-0.833528\pi$
$67$	− 8.17670i	− 0.998942i	0.866330	−	0.499471i	$-0.166472\pi$
$68$	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	$-0.689702\pi$
$73$	−9.59166	−1.12262	−0.561310	+	0.827606i	$0.689702\pi$
$74$	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
$83$	− 18.1400i	− 1.99112i	−0.0941309	−	0.995560i	$-0.530007\pi$
$83$	− 18.1400i	− 1.99112i	0.0941309	−	0.995560i	$-0.469993\pi$
$84$	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
$95$	− 4.40834i	− 0.452286i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	16.8619i	1.69469i
$100$	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
$107$	19.4180i	1.87721i	0.344997	+	0.938604i	$0.387880\pi$
$107$	19.4180i	1.87721i	−0.344997	+	0.938604i	$0.612120\pi$
$108$	0	0
$109$	14.4364	1.38275	0.691376	−	0.722495i	$-0.257005\pi$
$109$	14.4364	1.38275	0.691376	+	0.722495i	$0.257005\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 3.06460i	− 0.285775i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−20.5917	−1.87197
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.12919	0.548212
$126$	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
$149$	24.3996	1.99889	0.999447	−	0.0332381i	$-0.0105820\pi$
$149$	24.3996	1.99889	0.999447	+	0.0332381i	$0.0105820\pi$
$150$	0	0
$151$	9.59166i	0.780558i	0.920697	+	0.390279i	$0.127622\pi$
$151$	9.59166i	0.780558i	−0.920697	+	0.390279i	$0.872378\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6.12919i	− 0.492309i
$156$	0	0
$157$	−23.1216	−1.84530	−0.922652	−	0.385633i	$-0.873983\pi$
$157$	−23.1216	−1.84530	−0.922652	+	0.385633i	$0.873983\pi$
$158$	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
$171$	− 20.6960i	− 1.58266i
$172$	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
$181$	−25.6776	−1.90860	−0.954302	−	0.298843i	$-0.903399\pi$
$181$	−25.6776	−1.90860	−0.954302	+	0.298843i	$0.903399\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.59166	−0.558150
$186$	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
$205$	6.12919	0.428081
$206$	0	0
$207$	− 14.3875i	− 1.00000i
$208$	0	0
$209$	38.7750	2.68212
$210$	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
$215$	− 2.77499i	− 0.189253i
$216$	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	$0.104070\pi$
$223$	9.59166i	0.642305i	−0.947027	+	0.321153i	$0.895930\pi$
$224$	0	0
$225$	13.7750	0.918333
$226$	0	0
$227$	1.78657i	0.118579i	0.998241	+	0.0592894i	$0.0188835\pi$
$227$	1.78657i	0.118579i	−0.998241	+	0.0592894i	$0.981117\pi$
$228$	0	0
$229$	−8.04623	−0.531710	−0.265855	−	0.964013i	$-0.585654\pi$
$229$	−8.04623	−0.531710	−0.265855	+	0.964013i	$0.585654\pi$
$230$	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
$235$	− 1.27802i	− 0.0833692i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 28.7750i	− 1.86130i	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	− 28.7750i	− 1.86130i	0.365911	−	0.930650i	$-0.380757\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.47309	0.285775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.3059i	0.902980i	0.892276	+	0.451490i	$0.149107\pi$
$251$	14.3059i	0.902980i	−0.892276	+	0.451490i	$0.850893\pi$
$252$	0	0
$253$	26.9557	1.69469
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7750	1.79493	0.897467	−	0.441081i	$-0.145405\pi$
$257$	28.7750	1.79493	0.897467	+	0.441081i	$0.145405\pi$
$258$	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
$265$	8.40834	0.516520
$266$	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
$275$	25.8081i	1.55629i
$276$	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
$283$	− 10.7327i	− 0.637996i	−0.947755	−	0.318998i	$-0.896654\pi$
$283$	− 10.7327i	− 0.637996i	0.947755	−	0.318998i	$-0.103346\pi$
$284$	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
$293$	−20.5655	−1.20145	−0.600726	−	0.799455i	$-0.705122\pi$
$293$	−20.5655	−1.20145	−0.600726	+	0.799455i	$0.705122\pi$
$294$	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
$305$	6.77499	0.387935
$306$	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	$0.0876674\pi$
$311$	9.59166i	0.543893i	−0.962312	+	0.271947i	$0.912333\pi$
$312$	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
$333$	−35.6409	−1.95311
$334$	0	0
$335$	5.22501i	0.285473i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	53.9113	2.91946
$342$	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
$355$	− 6.39012i	− 0.339153i
$356$	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
$361$	−28.5917	−1.50482
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.12919	0.320817
$366$	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
$373$	6.76820	0.350444	0.175222	−	0.984529i	$-0.443936\pi$
$373$	6.76820	0.350444	0.175222	+	0.984529i	$0.443936\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 0.508546i	− 0.0261223i	−0.999915	−	0.0130611i	$-0.995842\pi$
$379$	− 0.508546i	− 0.0261223i	0.999915	−	0.0130611i	$-0.00415761\pi$
$380$	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
$387$	− 13.0279i	− 0.662244i
$388$	0	0
$389$	−38.1970	−1.93666	−0.968331	−	0.249668i	$-0.919678\pi$
$389$	−38.1970	−1.93666	−0.968331	+	0.249668i	$0.919678\pi$
$390$	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
$405$	−5.75111	−0.285775
$406$	0	0
$407$	− 66.7750i	− 3.30991i
$408$	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
$415$	11.5917i	0.569012i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 23.2521i	− 1.13594i	−0.823050	−	0.567969i	$-0.807729\pi$
$419$	− 23.2521i	− 1.13594i	0.823050	−	0.567969i	$-0.192271\pi$
$420$	0	0
$421$	19.2875	0.940016	0.470008	−	0.882662i	$-0.344251\pi$
$421$	19.2875	0.940016	0.470008	+	0.882662i	$0.344251\pi$
$422$	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
$437$	−33.0849	−1.58266
$438$	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	$-0.262419\pi$
$449$	28.7750	1.35798	0.678988	+	0.734150i	$0.262419\pi$
$450$	0	0
$451$	53.9113i	2.53859i
$452$	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
$467$	− 43.1786i	− 1.99807i	−0.0439527	−	0.999034i	$-0.513995\pi$
$467$	− 43.1786i	− 1.99807i	0.0439527	−	0.999034i	$-0.486005\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	24.4083	1.12230
$474$	0	0
$475$	− 31.6764i	− 1.45341i
$476$	0	0
$477$	39.4750	1.80744
$478$	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
$495$	− 10.7750i	− 0.484300i
$496$	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
$517$	11.2413	0.494391
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	44.4566i	1.94395i	0.235079	+	0.971976i	$0.424465\pi$
$523$	44.4566i	1.94395i	−0.235079	+	0.971976i	$0.575535\pi$
$524$	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
$535$	− 12.4083i	− 0.536459i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	39.3445i	1.69469i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.22501	−0.395156
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	31.8068	1.35748
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.49018	−0.232626	−0.116313	−	0.993213i	$-0.537108\pi$
$557$	−5.49018	−0.232626	−0.116313	+	0.993213i	$0.537108\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.7498i	0.495197i	0.968863	+	0.247598i	$0.0796413\pi$
$563$	11.7498i	0.495197i	−0.968863	+	0.247598i	$0.920359\pi$
$564$	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
$571$	− 0.769479i	− 0.0322017i	−0.999870	−	0.0161008i	$-0.994875\pi$
$571$	− 0.769479i	− 0.0322017i	0.999870	−	0.0161008i	$-0.00512528\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 22.0208i	− 0.918333i
$576$	0	0
$577$	−47.9583	−1.99653	−0.998265	−	0.0588745i	$-0.981249\pi$
$577$	−47.9583	−1.99653	−0.998265	+	0.0588745i	$0.981249\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	73.9583i	3.06304i
$584$	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
$589$	−66.1697	−2.72648
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−47.9583	−1.96941	−0.984706	−	0.174224i	$-0.944258\pi$
$593$	−47.9583	−1.96941	−0.984706	+	0.174224i	$0.944258\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 28.7750i	− 1.17571i	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	− 28.7750i	− 1.17571i	0.808965	−	0.587857i	$-0.200028\pi$
$600$	0	0
$601$	−9.59166	−0.391252	−0.195626	−	0.980679i	$-0.562674\pi$
$601$	−9.59166	−0.391252	−0.195626	+	0.980679i	$0.562674\pi$
$602$	0	0
$603$	24.5301i	0.998942i
$604$	0	0
$605$	13.1583	0.534962
$606$	0	0
$607$	47.9583i	1.94657i	0.229605	+	0.973284i	$0.426257\pi$
$607$	47.9583i	1.94657i	−0.229605	+	0.973284i	$0.573743\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	49.4383	1.99679	0.998396	−	0.0566132i	$-0.0180302\pi$
$613$	49.4383	1.99679	0.998396	+	0.0566132i	$0.0180302\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 45.7346i	− 1.83823i	−0.393989	−	0.919115i	$-0.628905\pi$
$619$	− 45.7346i	− 1.83823i	0.393989	−	0.919115i	$-0.371095\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.0417	0.761667
$626$	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
$635$	− 11.5022i	− 0.456452i
$636$	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
$643$	− 38.0665i	− 1.50120i	−0.660759	−	0.750598i	$-0.729765\pi$
$643$	− 38.0665i	− 1.50120i	0.660759	−	0.750598i	$-0.270235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.9583i	1.88544i	0.333591	+	0.942718i	$0.391740\pi$
$647$	47.9583i	1.88544i	−0.333591	+	0.942718i	$0.608260\pi$
$648$	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
$659$	− 13.2888i	− 0.517658i	−0.965923	−	0.258829i	$-0.916663\pi$
$659$	− 13.2888i	− 0.517658i	0.965923	−	0.258829i	$-0.0833366\pi$
$660$	0	0
$661$	−45.6042	−1.77380	−0.886899	−	0.461964i	$-0.847145\pi$
$661$	−45.6042	−1.77380	−0.886899	+	0.461964i	$0.847145\pi$
$662$	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
$671$	59.5917i	2.30051i
$672$	0	0
$673$	−47.9583	−1.84866	−0.924329	−	0.381597i	$-0.875374\pi$
$673$	−47.9583	−1.84866	−0.924329	+	0.381597i	$0.875374\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−50.7163	−1.94919	−0.974593	−	0.223985i	$-0.928093\pi$
$677$	−50.7163	−1.94919	−0.974593	+	0.223985i	$0.928093\pi$
$678$	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
$701$	−30.5288	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$701$	−30.5288	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$702$	0	0
$703$	81.9583i	3.09112i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30.7897	−1.15633	−0.578167	−	0.815919i	$-0.696232\pi$
$709$	−30.7897	−1.15633	−0.578167	+	0.815919i	$0.696232\pi$
$710$	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	$-0.819722\pi$
$719$	− 28.7750i	− 1.07313i	0.843860	−	0.536563i	$-0.180278\pi$
$720$	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
$733$	19.5485	0.722039	0.361019	−	0.932558i	$-0.382429\pi$
$733$	19.5485	0.722039	0.361019	+	0.932558i	$0.382429\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−45.9583	−1.69290
$738$	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
$745$	−15.5917	−0.571234
$746$	0	0
$747$	54.4199i	1.99112i
$748$	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
$755$	− 6.12919i	− 0.223064i
$756$	0	0
$757$	44.3262	1.61106	0.805531	−	0.592554i	$-0.201880\pi$
$757$	44.3262	1.61106	0.805531	+	0.592554i	$0.201880\pi$
$758$	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
$773$	42.0310	1.51175	0.755876	−	0.654715i	$-0.227211\pi$
$773$	42.0310	1.51175	0.755876	+	0.654715i	$0.227211\pi$
$774$	0	0
$775$	− 44.0417i	− 1.58202i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 66.1697i	− 2.37078i
$780$	0	0
$781$	56.2065	2.01123
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.7750	0.527342
$786$	0	0
$787$	36.7885i	1.31137i	0.755036	+	0.655684i	$0.227620\pi$
$787$	36.7885i	1.31137i	−0.755036	+	0.655684i	$0.772380\pi$
$788$	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
$797$	16.7315	0.592659	0.296330	−	0.955086i	$-0.404237\pi$
$797$	16.7315	0.592659	0.296330	+	0.955086i	$0.404237\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	53.9113i	1.90249i
$804$	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
$817$	−29.9583	−1.04811
$818$	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	$0.0534636\pi$
$823$	9.59166i	0.334344i	−0.985928	+	0.167172i	$0.946536\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.8081i	− 0.897436i	−0.893673	−	0.448718i	$-0.851881\pi$
$827$	− 25.8081i	− 0.897436i	0.893673	−	0.448718i	$-0.148119\pi$
$828$	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
$835$	14.0583i	0.486507i
$836$	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
$845$	−8.30716	−0.285775
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	56.9759i	1.95311i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	13.2250i	0.452286i
$856$	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	$0.303956\pi$
$863$	47.9583i	1.63252i	−0.577685	+	0.816260i	$0.696044\pi$
$864$	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
$891$	− 50.5858i	− 1.69469i
$892$	0	0
$893$	−13.7973	−0.461710
$894$	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
$905$	16.4083	0.545432
$906$	0	0
$907$	14.5668i	0.483683i	0.970316	+	0.241842i	$0.0777515\pi$
$907$	14.5668i	0.483683i	−0.970316	+	0.241842i	$0.922249\pi$
$908$	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
$913$	−101.958	−3.37433
$914$	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
$925$	−54.5504	−1.79360
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.7750	0.944077	0.472038	−	0.881578i	$-0.343518\pi$
$929$	28.7750	0.944077	0.472038	+	0.881578i	$0.343518\pi$
$930$	0	0
$931$	− 48.2907i	− 1.58266i
$932$	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
$941$	32.0678	1.04538	0.522690	−	0.852523i	$-0.324929\pi$
$941$	32.0678	1.04538	0.522690	+	0.852523i	$0.324929\pi$
$942$	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
$963$	− 58.2540i	− 1.87721i
$964$	0	0
$965$	3.83407	0.123423
$966$	0	0
$967$	47.9583i	1.54224i	0.636693	+	0.771118i	$0.280302\pi$
$967$	47.9583i	1.54224i	−0.636693	+	0.771118i	$0.719698\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 35.5104i	− 1.13958i	−0.821789	−	0.569792i	$-0.807024\pi$
$971$	− 35.5104i	− 1.13958i	0.821789	−	0.569792i	$-0.192976\pi$
$972$	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
$981$	−43.3091	−1.38275
$982$	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
$989$	−20.8265	−0.662244
$990$	0	0
$991$	9.59166i	0.304689i	0.988327	+	0.152345i	$0.0486824\pi$
$991$	9.59166i	0.304689i	−0.988327	+	0.152345i	$0.951318\pi$
$992$	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.735.3	✓	8
4.3	odd	2	inner	1472.2.h.a.735.4	yes	8
8.3	odd	2	inner	1472.2.h.a.735.5	yes	8
8.5	even	2	inner	1472.2.h.a.735.6	yes	8
23.22	odd	2	inner	1472.2.h.a.735.6	yes	8
92.91	even	2	inner	1472.2.h.a.735.5	yes	8
184.45	odd	2	CM	1472.2.h.a.735.3	✓	8
184.91	even	2	inner	1472.2.h.a.735.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.2.h.a.735.3	✓	8	1.1	even	1	trivial
1472.2.h.a.735.3	✓	8	184.45	odd	2	CM
1472.2.h.a.735.4	yes	8	4.3	odd	2	inner
1472.2.h.a.735.4	yes	8	184.91	even	2	inner
1472.2.h.a.735.5	yes	8	8.3	odd	2	inner
1472.2.h.a.735.5	yes	8	92.91	even	2	inner
1472.2.h.a.735.6	yes	8	8.5	even	2	inner
1472.2.h.a.735.6	yes	8	23.22	odd	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)

\(f(q)\)	\(=\)	\(q-0.639012 q^{5} -3.00000 q^{9} -5.62065i q^{11} +6.89867i q^{19} +4.79583i q^{23} -4.59166 q^{25} +9.59166i q^{31} +11.8803 q^{37} -9.59166 q^{41} +4.34262i q^{43} +1.91704 q^{45} +2.00000i q^{47} -7.00000 q^{49} -13.1583 q^{53} +3.59166i q^{55} -10.6023 q^{61} -8.17670i q^{67} +10.0000i q^{71} -9.59166 q^{73} +9.00000 q^{81} -18.1400i q^{83} -4.40834i q^{95} +16.8619i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81}+O(q^{100}) \) 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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