LMFDB - Embedded newform 5472.2.e.c.5167.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(5167,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5472.5167");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.e (of order $2$, degree $1$, not 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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4919453024256.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 96x^{4} + 248x^{2} + 900 $ x^8 - 12x^6 + 96x^4 + 248*x^2 + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			5167.3
Root			$-0.800688 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	5472.5167
Dual form			5472.2.e.c.5167.5

$q$-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 $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$	− 1.13234i	− 0.506399i	−0.967414	−	0.253200i	$-0.918517\pi$
$5$	− 1.13234i	− 0.506399i	0.967414	−	0.253200i	$-0.0814830\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	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
$13$	−5.89218	−1.63420	−0.817099	−	0.576498i	$-0.804419\pi$
$13$	−5.89218	−1.63420	−0.817099	+	0.576498i	$0.804419\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.35890	−1.00000
$19$	−4.35890	−1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 9.46515i	− 1.97362i	−0.161885	−	0.986810i	$-0.551757\pi$
$23$	− 9.46515i	− 1.97362i	0.161885	−	0.986810i	$-0.448243\pi$
$24$	0	0
$25$	3.71780	0.743560
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.29081	−0.770651	−0.385326	−	0.922781i	$-0.625911\pi$
$31$	−4.29081	−0.770651	−0.385326	+	0.922781i	$0.625911\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.09493	1.49520	0.747599	−	0.664151i	$-0.231207\pi$
$37$	9.09493	1.49520	0.747599	+	0.664151i	$0.231207\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.3288i	1.92544i	0.270501	+	0.962720i	$0.412811\pi$
$41$	12.3288i	1.92544i	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−8.71780	−1.32945	−0.664726	−	0.747087i	$-0.731452\pi$
$43$	−8.71780	−1.32945	−0.664726	+	0.747087i	$0.731452\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7298i	1.71097i	0.517826	+	0.855486i	$0.326741\pi$
$47$	11.7298i	1.71097i	−0.517826	+	0.855486i	$0.673259\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.82843i	− 0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$	− 2.82843i	− 0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.67197i	0.827556i
$66$	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		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	8.71780	1.02034	0.510171	−	0.860073i	$-0.329582\pi$
$73$	8.71780	1.02034	0.510171	+	0.860073i	$0.329582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.6963	1.20343	0.601714	−	0.798711i	$-0.294485\pi$
$79$	10.6963	1.20343	0.601714	+	0.798711i	$0.294485\pi$
$80$	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
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.3137i	1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$	11.3137i	1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.93577i	0.506399i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	20.0626i	1.99631i	0.0607431	+	0.998153i	$0.480653\pi$
$101$	20.0626i	1.99631i	−0.0607431	+	0.998153i	$0.519347\pi$
$102$	0	0
$103$	−19.2779	−1.89951	−0.949755	−	0.312995i	$-0.898668\pi$
$103$	−19.2779	−1.89951	−0.949755	+	0.312995i	$0.898668\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	−20.8793	−1.99987	−0.999937	−	0.0112369i	$-0.996423\pi$
$109$	−20.8793	−1.99987	−0.999937	+	0.0112369i	$0.996423\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.3288i	1.15980i	0.814688	+	0.579899i	$0.196908\pi$
$113$	12.3288i	1.15980i	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	−10.7178	−0.999440
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.87154i	− 0.882938i
$126$	0	0
$127$	1.08805	0.0965492	0.0482746	−	0.998834i	$-0.484628\pi$
$127$	1.08805	0.0965492	0.0482746	+	0.998834i	$0.484628\pi$
$128$	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		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	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
$149$	− 22.3273i	− 1.82913i	−0.404444	−	0.914563i	$-0.632535\pi$
$149$	− 22.3273i	− 1.82913i	0.404444	−	0.914563i	$-0.367465\pi$
$150$	0	0
$151$	16.0752	1.30818	0.654089	−	0.756417i	$-0.273052\pi$
$151$	16.0752	1.30818	0.654089	+	0.756417i	$0.273052\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.85867i	0.390257i
$156$	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
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	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
$169$	21.7178	1.67060
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.6577i	1.84300i	0.388379	+	0.921500i	$0.373035\pi$
$179$	24.6577i	1.84300i	−0.388379	+	0.921500i	$0.626965\pi$
$180$	0	0
$181$	24.0820	1.79000	0.895002	−	0.446062i	$-0.147174\pi$
$181$	24.0820	1.79000	0.895002	+	0.446062i	$0.147174\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 10.2986i	− 0.757167i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.67108i	− 0.193273i	−0.995320	−	0.0966364i	$-0.969192\pi$
$191$	− 2.67108i	− 0.193273i	0.995320	−	0.0966364i	$-0.0308084\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2686i	0.945347i	0.881238	+	0.472673i	$0.156711\pi$
$197$	13.2686i	0.945347i	−0.881238	+	0.472673i	$0.843289\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	13.9605	0.975042
$206$	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
$215$	9.87154i	0.673234i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.8991	−0.930750	−0.465375	−	0.885114i	$-0.654081\pi$
$223$	−13.8991	−0.930750	−0.465375	+	0.885114i	$0.654081\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.6577i	1.63659i	0.574801	+	0.818293i	$0.305079\pi$
$227$	24.6577i	1.63659i	−0.574801	+	0.818293i	$0.694921\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	13.2822	0.866435
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 30.6601i	− 1.98324i	−0.129194	−	0.991619i	$-0.541239\pi$
$239$	− 30.6601i	− 1.98324i	0.129194	−	0.991619i	$-0.458761\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 7.92641i	− 0.506399i
$246$	0	0
$247$	25.6834	1.63420
$248$	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
$257$	28.2843i	1.76432i	0.470946	+	0.882162i	$0.343913\pi$
$257$	28.2843i	1.76432i	−0.470946	+	0.882162i	$0.656087\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 23.8661i	− 1.47164i	−0.677175	−	0.735822i	$-0.736796\pi$
$263$	− 23.8661i	− 1.47164i	0.677175	−	0.735822i	$-0.263204\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.3288i	0.735476i	0.929929	+	0.367738i	$0.119868\pi$
$281$	12.3288i	0.735476i	−0.929929	+	0.367738i	$0.880132\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	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
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−3.20275	−0.186471
$296$	0	0
$297$	0	0
$298$	0	0
$299$	55.7704i	3.22528i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.9248i	1.86699i	0.358584	+	0.933497i	$0.383260\pi$
$311$	32.9248i	1.86699i	−0.358584	+	0.933497i	$0.616740\pi$
$312$	0	0
$313$	8.71780	0.492759	0.246380	−	0.969173i	$-0.420759\pi$
$313$	8.71780	0.492759	0.246380	+	0.969173i	$0.420759\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−21.9059	−1.21512
$326$	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
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	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
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.5239i	0.977654i	0.872381	+	0.488827i	$0.162575\pi$
$359$	18.5239i	0.977654i	−0.872381	+	0.488827i	$0.837425\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 9.87154i	− 0.516700i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.2710	−0.583592	−0.291796	−	0.956481i	$-0.594253\pi$
$373$	−11.2710	−0.583592	−0.291796	+	0.956481i	$0.594253\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	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		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.4636i	1.74737i	0.486491	+	0.873686i	$0.338277\pi$
$389$	34.4636i	1.74737i	−0.486491	+	0.873686i	$0.661723\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 12.1119i	− 0.609416i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.3288i	0.615672i	0.951439	+	0.307836i	$0.0996049\pi$
$401$	12.3288i	0.615672i	−0.951439	+	0.307836i	$0.900395\pi$
$402$	0	0
$403$	25.2822	1.25940
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	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
$421$	3.71607	0.181110	0.0905552	−	0.995891i	$-0.471136\pi$
$421$	3.71607	0.181110	0.0905552	+	0.995891i	$0.471136\pi$
$422$	0	0
$423$	0	0
$424$	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
$437$	41.2576i	1.97362i
$438$	0	0
$439$	−28.8862	−1.37866	−0.689331	−	0.724447i	$-0.742095\pi$
$439$	−28.8862	−1.37866	−0.689331	+	0.724447i	$0.742095\pi$
$440$	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
$445$	12.8110	0.607300
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 39.5980i	− 1.86874i	−0.356299	−	0.934372i	$-0.615961\pi$
$449$	− 39.5980i	− 1.86874i	0.356299	−	0.934372i	$-0.384039\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 8.73920i	− 0.407025i	−0.979072	−	0.203513i	$-0.934764\pi$
$461$	− 8.73920i	− 0.407025i	0.979072	−	0.203513i	$-0.0652358\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	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
$475$	−16.2055	−0.743560
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.3367i	0.883516i	0.897134	+	0.441758i	$0.145645\pi$
$479$	19.3367i	0.883516i	−0.897134	+	0.441758i	$0.854355\pi$
$480$	0	0
$481$	−53.5890	−2.44345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.66966	−0.438174	−0.219087	−	0.975705i	$-0.570308\pi$
$487$	−9.66966	−0.438174	−0.219087	+	0.975705i	$0.570308\pi$
$488$	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
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	43.5890	1.95131	0.975656	−	0.219308i	$-0.0703801\pi$
$499$	43.5890	1.95131	0.975656	+	0.219308i	$0.0703801\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 1.85829i	− 0.0828571i	−0.999141	−	0.0414285i	$-0.986809\pi$
$503$	− 1.85829i	− 0.0828571i	0.999141	−	0.0414285i	$-0.0131909\pi$
$504$	0	0
$505$	22.7178	1.01093
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.8292i	0.961911i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.65685i	− 0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$	− 5.65685i	− 0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$	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
$529$	−66.5890	−2.89517
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 72.6437i	− 3.14655i
$534$	0	0
$535$	16.0138	0.692335
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	23.6425i	1.01273i
$546$	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
$557$	− 29.9342i	− 1.26835i	−0.773189	−	0.634176i	$-0.781339\pi$
$557$	− 29.9342i	− 1.26835i	0.773189	−	0.634176i	$-0.218661\pi$
$558$	0	0
$559$	51.3668	2.17259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.1421i	0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$	14.1421i	0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$	0	0
$565$	13.9605	0.587321
$566$	0	0
$567$	0	0
$568$	0	0
$569$	45.2548i	1.89718i	0.316506	+	0.948591i	$0.397490\pi$
$569$	45.2548i	1.89718i	−0.316506	+	0.948591i	$0.602510\pi$
$570$	0	0
$571$	43.5890	1.82414	0.912071	−	0.410032i	$-0.134482\pi$
$571$	43.5890	1.82414	0.912071	+	0.410032i	$0.134482\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 35.1895i	− 1.46750i
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	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
$589$	18.7032	0.770651
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.4558i	0.506399i
$606$	0	0
$607$	−49.2521	−1.99908	−0.999541	−	0.0302794i	$-0.990360\pi$
$607$	−49.2521	−1.99908	−0.999541	+	0.0302794i	$0.990360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 69.1143i	− 2.79607i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−8.71780	−0.350398	−0.175199	−	0.984533i	$-0.556057\pi$
$619$	−8.71780	−0.350398	−0.175199	+	0.984533i	$0.556057\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	7.41101	0.296440
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.23205i	− 0.0488924i
$636$	0	0
$637$	−41.2453	−1.63420
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 22.6274i	− 0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$	− 22.6274i	− 0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$	0	0
$643$	46.0000	1.81406	0.907031	−	0.421063i	$-0.138343\pi$
$643$	46.0000	1.81406	0.907031	+	0.421063i	$0.138343\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 17.0720i	− 0.671170i	−0.942010	−	0.335585i	$-0.891066\pi$
$647$	− 17.0720i	− 0.671170i	0.942010	−	0.335585i	$-0.108934\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.47451i	0.253367i	0.991943	+	0.126684i	$0.0404333\pi$
$653$	6.47451i	0.253367i	−0.991943	+	0.126684i	$0.959567\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.6577i	0.960526i	0.877125	+	0.480263i	$0.159459\pi$
$659$	24.6577i	0.960526i	−0.877125	+	0.480263i	$0.840541\pi$
$660$	0	0
$661$	−50.8535	−1.97797	−0.988986	−	0.148007i	$-0.952714\pi$
$661$	−50.8535	−1.97797	−0.988986	+	0.148007i	$0.952714\pi$
$662$	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
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 49.3153i	− 1.88700i	−0.331375	−	0.943499i	$-0.607513\pi$
$683$	− 49.3153i	− 1.88700i	0.331375	−	0.943499i	$-0.392487\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	43.5890	1.65820	0.829102	−	0.559098i	$-0.188852\pi$
$691$	43.5890	1.65820	0.829102	+	0.559098i	$0.188852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 11.3234i	− 0.429522i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.6695i	1.04506i	0.852620	+	0.522531i	$0.175012\pi$
$701$	27.6695i	1.04506i	−0.852620	+	0.522531i	$0.824988\pi$
$702$	0	0
$703$	−39.6439	−1.49520
$704$	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
$713$	40.6131i	1.52097i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.12298i	0.153761i	0.997040	+	0.0768806i	$0.0244960\pi$
$719$	4.12298i	0.153761i	−0.997040	+	0.0768806i	$0.975504\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	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
$745$	−25.2822	−0.926268
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.3046	0.740924	0.370462	−	0.928848i	$-0.379199\pi$
$751$	20.3046	0.740924	0.370462	+	0.928848i	$0.379199\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 18.2026i	− 0.662461i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.6656i	0.601760i
$768$	0	0
$769$	−43.5890	−1.57186	−0.785930	−	0.618316i	$-0.787815\pi$
$769$	−43.5890	−1.57186	−0.785930	+	0.618316i	$0.787815\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−15.9523	−0.573025
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 53.7401i	− 1.92544i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	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
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	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
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.8528i	0.555300i
$816$	0	0
$817$	38.0000	1.32945
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.6585i	1.94250i	0.238070	+	0.971248i	$0.423485\pi$
$821$	55.6585i	1.94250i	−0.238070	+	0.971248i	$0.576515\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.6577i	0.857431i	0.903440	+	0.428715i	$0.141034\pi$
$827$	24.6577i	0.857431i	−0.903440	+	0.428715i	$0.858966\pi$
$828$	0	0
$829$	54.0563	1.87745	0.938726	−	0.344664i	$-0.112007\pi$
$829$	54.0563	1.87745	0.938726	+	0.344664i	$0.112007\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	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
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 24.5920i	− 0.845991i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 86.0849i	− 2.95095i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.5685i	− 1.93234i	−0.257897	−	0.966172i	$-0.583030\pi$
$857$	− 56.5685i	− 1.93234i	0.257897	−	0.966172i	$-0.416970\pi$
$858$	0	0
$859$	43.5890	1.48724	0.743619	−	0.668604i	$-0.233108\pi$
$859$	43.5890	1.48724	0.743619	+	0.668604i	$0.233108\pi$
$860$	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
$877$	33.6903	1.13764	0.568820	−	0.822462i	$-0.307400\pi$
$877$	33.6903	1.13764	0.568820	+	0.822462i	$0.307400\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−8.71780	−0.293377	−0.146689	−	0.989183i	$-0.546862\pi$
$883$	−8.71780	−0.293377	−0.146689	+	0.989183i	$0.546862\pi$
$884$	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
$893$	− 51.1292i	− 1.71097i
$894$	0	0
$895$	27.9209	0.933294
$896$	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
$905$	− 27.2692i	− 0.906457i
$906$	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		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	33.8131	1.11177
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−30.5123	−1.00000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	61.0246	1.99359	0.996793	−	0.0800213i	$-0.0254988\pi$
$937$	61.0246	1.99359	0.996793	+	0.0800213i	$0.0254988\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	116.694	3.80008
$944$	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
$949$	−51.3668	−1.66744
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 61.6441i	− 1.99685i	−0.0561066	−	0.998425i	$-0.517869\pi$
$953$	− 61.6441i	− 1.99685i	0.0561066	−	0.998425i	$-0.482131\pi$
$954$	0	0
$955$	−3.02459	−0.0978733
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12.5890	−0.406096
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 49.3153i	− 1.58260i	−0.611426	−	0.791302i	$-0.709404\pi$
$971$	− 49.3153i	− 1.58260i	0.611426	−	0.791302i	$-0.290596\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 61.6441i	− 1.97217i	−0.166240	−	0.986085i	$-0.553163\pi$
$977$	− 61.6441i	− 1.97217i	0.166240	−	0.986085i	$-0.446837\pi$
$978$	0	0
$979$	0	0
$980$	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
$985$	15.0246	0.478723
$986$	0	0
$987$	0	0
$988$	0	0
$989$	82.5152i	2.62383i
$990$	0	0
$991$	−54.6310	−1.73541	−0.867706	−	0.497079i	$-0.834406\pi$
$991$	−54.6310	−1.73541	−0.867706	+	0.497079i	$0.834406\pi$
$992$	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	5472.2.e.c.5167.3		8
3.2	odd	2	inner	5472.2.e.c.5167.5		8
4.3	odd	2		1368.2.e.c.379.6	yes	8
8.3	odd	2	inner	5472.2.e.c.5167.6		8
8.5	even	2		1368.2.e.c.379.7	yes	8
12.11	even	2		1368.2.e.c.379.3	yes	8
19.18	odd	2	inner	5472.2.e.c.5167.4		8
24.5	odd	2		1368.2.e.c.379.2	✓	8
24.11	even	2	inner	5472.2.e.c.5167.4		8
57.56	even	2	inner	5472.2.e.c.5167.6		8
76.75	even	2		1368.2.e.c.379.2	✓	8
152.37	odd	2		1368.2.e.c.379.3	yes	8
152.75	even	2	inner	5472.2.e.c.5167.5		8
228.227	odd	2		1368.2.e.c.379.7	yes	8
456.227	odd	2	CM	5472.2.e.c.5167.3		8
456.341	even	2		1368.2.e.c.379.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.e.c.379.2	✓	8	24.5	odd	2
1368.2.e.c.379.2	✓	8	76.75	even	2
1368.2.e.c.379.3	yes	8	12.11	even	2
1368.2.e.c.379.3	yes	8	152.37	odd	2
1368.2.e.c.379.6	yes	8	4.3	odd	2
1368.2.e.c.379.6	yes	8	456.341	even	2
1368.2.e.c.379.7	yes	8	8.5	even	2
1368.2.e.c.379.7	yes	8	228.227	odd	2
5472.2.e.c.5167.3		8	1.1	even	1	trivial
5472.2.e.c.5167.3		8	456.227	odd	2	CM
5472.2.e.c.5167.4		8	19.18	odd	2	inner
5472.2.e.c.5167.4		8	24.11	even	2	inner
5472.2.e.c.5167.5		8	3.2	odd	2	inner
5472.2.e.c.5167.5		8	152.75	even	2	inner
5472.2.e.c.5167.6		8	8.3	odd	2	inner
5472.2.e.c.5167.6		8	57.56	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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.13234i q^{5} -5.89218 q^{13} -4.35890 q^{19} -9.46515i q^{23} +3.71780 q^{25} -4.29081 q^{31} +9.09493 q^{37} +12.3288i q^{41} -8.71780 q^{43} +11.7298i q^{47} +7.00000 q^{49} -2.82843i q^{59} +6.67197i q^{65} +8.71780 q^{73} +10.6963 q^{79} +11.3137i q^{89} +4.93577i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{25} + 56 q^{49}+O(q^{100}) \) 8 * q - 40 * q^25 + 56 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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