LMFDB - Embedded newform 441.4.e.i.226.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(226,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.e (of order $3$, degree $2$, 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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			226.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.226
Dual form			441.4.e.i.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.00000 + 6.92820i) q^{4} +O(q^{10})$	$q+(4.00000 + 6.92820i) q^{4} -70.0000 q^{13} +(-32.0000 + 55.4256i) q^{16} +(-28.0000 + 48.4974i) q^{19} +(62.5000 + 108.253i) q^{25} +(-154.000 - 266.736i) q^{31} +(-55.0000 + 95.2628i) q^{37} -520.000 q^{43} +(-280.000 - 484.974i) q^{52} +(-91.0000 + 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-595.000 - 1030.57i) q^{73} -448.000 q^{76} +(-442.000 + 765.566i) q^{79} -1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4}+O(q^{10}) $ 2 * q + 8 * q^4	$ 2 q + 8 q^{4} - 140 q^{13} - 64 q^{16} - 56 q^{19} + 125 q^{25} - 308 q^{31} - 110 q^{37} - 1040 q^{43} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} - 1190 q^{73} - 896 q^{76} - 884 q^{79} - 2660 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 - 140 * q^13 - 64 * q^16 - 56 * q^19 + 125 * q^25 - 308 * q^31 - 110 * q^37 - 1040 * q^43 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 - 1190 * q^73 - 896 * q^76 - 884 * q^79 - 2660 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	4.00000	+	6.92820i	0.500000	+	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−70.0000			−1.49342			−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000			−1.49342			−0.746712	+	0.665148i	$0.768369\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−28.0000	+	48.4974i	−0.338086	+	0.585583i	−0.984073	−	0.177766i	$-0.943113\pi$
$19$	−28.0000	+	48.4974i	−0.338086	+	0.585583i	0.645986	+	0.763349i	$0.276446\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	62.5000	+	108.253i	0.500000	+	0.866025i
$26$		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
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.837192	−	0.546908i	$-0.815805\pi$
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.0550403	−	0.998484i	$-0.517529\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−55.0000	+	95.2628i	−0.244377	+	0.423273i	−0.961956	−	0.273204i	$-0.911917\pi$
$37$	−55.0000	+	95.2628i	−0.244377	+	0.423273i	0.717579	+	0.696477i	$0.245250\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−520.000			−1.84417			−0.922084	−	0.386989i	$-0.873515\pi$
$43$	−520.000			−1.84417			−0.922084	+	0.386989i	$0.873515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$	−280.000	−	484.974i	−0.746712	−	1.29334i
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	0.754578	+	0.656210i	$0.227842\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$		0			0
$67$	440.000	+	762.102i	0.802307	+	1.38964i	0.918094	+	0.396362i	$0.129728\pi$
$67$	440.000	+	762.102i	0.802307	+	1.38964i	−0.115787	+	0.993274i	$0.536939\pi$
$68$		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
$73$	−595.000	−	1030.57i	−0.953966	−	1.65232i	−0.736718	−	0.676200i	$-0.763625\pi$
$73$	−595.000	−	1030.57i	−0.953966	−	1.65232i	−0.217248	−	0.976117i	$-0.569708\pi$
$74$		0			0
$75$		0			0
$76$	−448.000			−0.676173
$77$		0			0
$78$		0			0
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	0.358177	+	0.933654i	$0.383399\pi$
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	−0.987656	+	0.156637i	$0.949935\pi$
$80$		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
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1330.00			−1.39218			−0.696088	−	0.717957i	$-0.745078\pi$
$97$	−1330.00			−1.39218			−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$	−500.000	+	866.025i	−0.500000	+	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−910.000	+	1576.17i	−0.870534	+	1.50781i	−0.00908799	+	0.999959i	$0.502893\pi$
$103$	−910.000	+	1576.17i	−0.870534	+	1.50781i	−0.861446	+	0.507850i	$0.830440\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	323.000	+	559.452i	0.283833	+	0.491613i	0.972325	−	0.233630i	$-0.0750606\pi$
$109$	323.000	+	559.452i	0.283833	+	0.491613i	−0.688493	+	0.725243i	$0.741727\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	665.500	−	1152.68i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	1232.00	−	2133.89i	0.892233	−	1.54539i
$125$		0			0
$126$		0			0
$127$	380.000			0.265508			0.132754	−	0.991149i	$-0.457618\pi$
$127$	380.000			0.265508			0.132754	+	0.991149i	$0.457618\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	2576.00			1.57190			0.785948	−	0.618293i	$-0.212175\pi$
$139$	2576.00			1.57190			0.785948	+	0.618293i	$0.212175\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−880.000			−0.488754
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−874.000	−	1513.81i	−0.471027	−	0.815843i	0.528424	−	0.848981i	$-0.322783\pi$
$151$	−874.000	−	1513.81i	−0.471027	−	0.815843i	−0.999451	+	0.0331378i	$0.989450\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.667699	+	0.744432i	$0.267279\pi$
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.310847	+	0.950460i	$0.399387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1700.00	−	2944.49i	0.816897	−	1.41491i	−0.0910600	−	0.995845i	$-0.529026\pi$
$163$	1700.00	−	2944.49i	0.816897	−	1.41491i	0.907957	−	0.419062i	$-0.137641\pi$
$164$		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
$169$	2703.00			1.23031
$170$		0			0
$171$		0			0
$172$	−2080.00	−	3602.67i	−0.922084	−	1.59710i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	3458.00			1.42006			0.710031	−	0.704171i	$-0.248681\pi$
$181$	3458.00			1.42006			0.710031	+	0.704171i	$0.248681\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	575.000	+	995.929i	0.214453	+	0.371443i	0.953103	−	0.302646i	$-0.0978698\pi$
$193$	575.000	+	995.929i	0.214453	+	0.371443i	−0.738650	+	0.674089i	$0.764536\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	2618.00	+	4534.51i	0.932588	+	1.61529i	0.778879	+	0.627175i	$0.215789\pi$
$199$	2618.00	+	4534.51i	0.932588	+	1.61529i	0.153710	+	0.988116i	$0.450878\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2240.00	−	3879.79i	0.746712	−	1.29334i
$209$		0			0
$210$		0			0
$211$	6032.00			1.96806			0.984028	−	0.178011i	$-0.0569664\pi$
$211$	6032.00			1.96806			0.984028	+	0.178011i	$0.0569664\pi$
$212$		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
$223$	−3220.00			−0.966938			−0.483469	−	0.875362i	$-0.660623\pi$
$223$	−3220.00			−0.966938			−0.483469	+	0.875362i	$0.660623\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−2233.00	+	3867.67i	−0.644370	+	1.11608i	0.340076	+	0.940398i	$0.389547\pi$
$229$	−2233.00	+	3867.67i	−0.644370	+	1.11608i	−0.984447	+	0.175684i	$0.943786\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		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
$241$	3689.00	+	6389.54i	0.986014	+	1.70783i	0.637341	+	0.770582i	$0.280034\pi$
$241$	3689.00	+	6389.54i	0.986014	+	1.70783i	0.348673	+	0.937244i	$0.386632\pi$
$242$		0			0
$243$		0			0
$244$	−1456.00			−0.382012
$245$		0			0
$246$		0			0
$247$	1960.00	−	3394.82i	0.504906	−	0.874523i
$248$		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
$256$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3520.00	+	6096.82i	−0.802307	+	1.38964i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−406.000	+	703.213i	−0.0910064	+	0.157628i	−0.907935	−	0.419111i	$-0.862342\pi$
$271$	−406.000	+	703.213i	−0.0910064	+	0.157628i	0.816928	+	0.576739i	$0.195675\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2015.00	+	3490.08i	0.437074	+	0.757035i	0.997462	−	0.0711951i	$-0.0226813\pi$
$277$	2015.00	+	3490.08i	0.437074	+	0.757035i	−0.560388	+	0.828230i	$0.689348\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	−0.994476	−	0.104961i	$-0.966528\pi$
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	0.406340	−	0.913722i	$-0.366805\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2456.50	−	4254.78i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	4760.00	−	8244.56i	0.953966	−	1.65232i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		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
$304$	−1792.00	−	3103.84i	−0.338086	−	0.585583i
$305$		0			0
$306$		0			0
$307$	10640.0			1.97804			0.989018	−	0.147797i	$-0.0472182\pi$
$307$	10640.0			1.97804			0.989018	+	0.147797i	$0.0472182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−5005.00	+	8668.91i	−0.903832	+	1.56548i	−0.0813539	+	0.996685i	$0.525924\pi$
$313$	−5005.00	+	8668.91i	−0.903832	+	1.56548i	−0.822478	+	0.568797i	$0.807409\pi$
$314$		0			0
$315$		0			0
$316$	−7072.00			−1.25896
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4375.00	−	7577.72i	−0.746712	−	1.29334i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−496.000	+	859.097i	−0.0823644	+	0.142659i	−0.904265	−	0.426971i	$-0.859580\pi$
$331$	−496.000	+	859.097i	−0.0823644	+	0.142659i	0.821901	+	0.569631i	$0.192914\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4930.00			−0.796897			−0.398448	−	0.917191i	$-0.630451\pi$
$337$	−4930.00			−0.796897			−0.398448	+	0.917191i	$0.630451\pi$
$338$		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		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−11914.0			−1.82734			−0.913670	−	0.406456i	$-0.866764\pi$
$349$	−11914.0			−1.82734			−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	1861.50	+	3224.21i	0.271395	+	0.470070i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2170.00	−	3758.55i	−0.308646	−	0.534591i	0.669420	−	0.742884i	$-0.266543\pi$
$367$	−2170.00	−	3758.55i	−0.308646	−	0.534591i	−0.978066	+	0.208293i	$0.933209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6175.00	+	10695.4i	−0.857183	+	1.48469i	0.0174213	+	0.999848i	$0.494454\pi$
$373$	−6175.00	+	10695.4i	−0.857183	+	1.48469i	−0.874605	+	0.484837i	$0.838879\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8584.00			−1.16340			−0.581702	−	0.813402i	$-0.697613\pi$
$379$	−8584.00			−1.16340			−0.581702	+	0.813402i	$0.697613\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−5320.00	−	9214.51i	−0.696088	−	1.20566i
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−595.000	+	1030.57i	−0.0752196	+	0.130284i	−0.901182	−	0.433441i	$-0.857299\pi$
$397$	−595.000	+	1030.57i	−0.0752196	+	0.130284i	0.825962	+	0.563726i	$0.190632\pi$
$398$		0			0
$399$		0			0
$400$	−8000.00			−1.00000
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	10780.0	+	18671.5i	1.33248	+	2.30793i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4123.00	−	7141.25i	−0.498458	−	0.863354i	0.501541	−	0.865134i	$-0.332767\pi$
$409$	−4123.00	−	7141.25i	−0.498458	−	0.863354i	−0.999998	+	0.00177990i	$0.999433\pi$
$410$		0			0
$411$		0			0
$412$	−14560.0			−1.74107
$413$		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
$421$	17138.0			1.98398			0.991989	−	0.126322i	$-0.0403172\pi$
$421$	17138.0			1.98398			0.991989	+	0.126322i	$0.0403172\pi$
$422$		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		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	−2590.00			−0.287454			−0.143727	−	0.989617i	$-0.545909\pi$
$433$	−2590.00			−0.287454			−0.143727	+	0.989617i	$0.545909\pi$
$434$		0			0
$435$		0			0
$436$	−2584.00	+	4475.62i	−0.283833	+	0.491613i
$437$		0			0
$438$		0			0
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	0.100728	+	0.994914i	$0.467883\pi$
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	−0.911985	+	0.410224i	$0.865450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	0.332513	+	0.943099i	$0.392103\pi$
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	−0.983004	+	0.183585i	$0.941230\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−19780.0			−1.98543			−0.992716	−	0.120482i	$-0.961556\pi$
$463$	−19780.0			−1.98543			−0.992716	+	0.120482i	$0.961556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−7000.00			−0.676173
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	3850.00	−	6668.40i	0.364958	−	0.632126i
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	−10450.0	−	18099.9i	−0.972351	−	1.68416i	−0.688415	−	0.725317i	$-0.741693\pi$
$487$	−10450.0	−	18099.9i	−0.972351	−	1.68416i	−0.283936	−	0.958843i	$-0.591640\pi$
$488$		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
$496$	19712.0			1.78447
$497$		0			0
$498$		0			0
$499$	7568.00	−	13108.2i	0.678938	−	1.17596i	−0.296363	−	0.955075i	$-0.595774\pi$
$499$	7568.00	−	13108.2i	0.678938	−	1.17596i	0.975301	−	0.220880i	$-0.0708930\pi$
$500$		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
$508$	1520.00	+	2632.72i	0.132754	+	0.229937i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		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
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	6020.00	−	10426.9i	0.503320	−	0.871775i	−0.496673	−	0.867938i	$-0.665445\pi$
$523$	6020.00	−	10426.9i	0.503320	−	0.871775i	0.999993	−	0.00383755i	$-0.00122153\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6083.50	+	10536.9i	0.500000	+	0.866025i
$530$		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
$539$		0			0
$540$		0			0
$541$	11339.0	−	19639.7i	0.901112	−	1.56077i	0.0750596	−	0.997179i	$-0.476085\pi$
$541$	11339.0	−	19639.7i	0.901112	−	1.56077i	0.826053	−	0.563593i	$-0.190581\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	10304.0	+	17847.1i	0.785948	+	1.36130i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$	36400.0			2.75413
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	−0.877320	−	0.479905i	$-0.840671\pi$
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	0.0230498	−	0.999734i	$-0.492662\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8855.00	+	15337.3i	0.638888	+	1.10659i	0.985677	+	0.168644i	$0.0539387\pi$
$577$	8855.00	+	15337.3i	0.638888	+	1.10659i	−0.346789	+	0.937943i	$0.612728\pi$
$578$		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
$589$	17248.0			1.20661
$590$		0			0
$591$		0			0
$592$	−3520.00	−	6096.82i	−0.244377	−	0.423273i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−29302.0			−1.98877			−0.994387	−	0.105801i	$-0.966259\pi$
$601$	−29302.0			−1.98877			−0.994387	+	0.105801i	$0.966259\pi$
$602$		0			0
$603$		0			0
$604$	6992.00	−	12110.5i	0.471027	−	0.815843i
$605$		0			0
$606$		0			0
$607$	14210.0	−	24612.4i	0.950191	−	1.64578i	0.205184	−	0.978723i	$-0.434221\pi$
$607$	14210.0	−	24612.4i	0.950191	−	1.64578i	0.745007	−	0.667056i	$-0.232446\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	−0.996266	−	0.0863334i	$-0.972485\pi$
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	0.423366	−	0.905959i	$-0.360848\pi$
$614$		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
$619$	13328.0	+	23084.8i	0.865424	+	1.49896i	0.866625	+	0.498959i	$0.166284\pi$
$619$	13328.0	+	23084.8i	0.865424	+	1.49896i	−0.00120126	+	0.999999i	$0.500382\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	+	13531.6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−15400.0	+	26673.6i	−0.978546	+	1.69489i
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		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
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	13160.0			0.807122			0.403561	−	0.914953i	$-0.367772\pi$
$643$	13160.0			0.807122			0.403561	+	0.914953i	$0.367772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	27200.0			1.63379
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	10241.0	+	17737.9i	0.602615	+	1.04376i	0.992423	+	0.122864i	$0.0392080\pi$
$661$	10241.0	+	17737.9i	0.602615	+	1.04376i	−0.389808	+	0.920896i	$0.627459\pi$
$662$		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
$673$	24050.0			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$673$	24050.0			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$674$		0			0
$675$		0			0
$676$	10812.0	+	18726.9i	0.615157	+	1.06548i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	16640.0	−	28821.3i	0.922084	−	1.59710i
$689$		0			0
$690$		0			0
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	−0.555460	−	0.831543i	$-0.687458\pi$
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	0.997868	+	0.0652705i	$0.0207910\pi$
$692$		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
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−3080.00	−	5334.72i	−0.165241	−	0.286206i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18073.0	+	31303.4i	−0.957328	+	1.65814i	−0.228381	+	0.973572i	$0.573343\pi$
$709$	−18073.0	+	31303.4i	−0.957328	+	1.65814i	−0.728948	+	0.684569i	$0.759990\pi$
$710$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	13832.0	+	23957.7i	0.710031	+	1.22981i
$725$		0			0
$726$		0			0
$727$	−10780.0			−0.549942			−0.274971	−	0.961452i	$-0.588668\pi$
$727$	−10780.0			−0.549942			−0.274971	+	0.961452i	$0.588668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−7525.00	+	13033.7i	−0.379184	+	0.656767i	−0.990944	−	0.134277i	$-0.957129\pi$
$733$	−7525.00	+	13033.7i	−0.379184	+	0.656767i	0.611759	+	0.791044i	$0.290462\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−15688.0	−	27172.4i	−0.780910	−	1.35258i	−0.931412	−	0.363966i	$-0.881422\pi$
$739$	−15688.0	−	27172.4i	−0.780910	−	1.35258i	0.150502	−	0.988610i	$-0.451911\pi$
$740$		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
$751$	11726.0	−	20310.0i	0.569757	−	0.986849i	−0.426832	−	0.904331i	$-0.640370\pi$
$751$	11726.0	−	20310.0i	0.569757	−	0.986849i	0.996590	−	0.0825179i	$-0.0262962\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−41470.0			−1.99109			−0.995543	−	0.0943039i	$-0.969937\pi$
$757$	−41470.0			−1.99109			−0.995543	+	0.0943039i	$0.969937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−4606.00			−0.215990			−0.107995	−	0.994151i	$-0.534443\pi$
$769$	−4606.00			−0.215990			−0.107995	+	0.994151i	$0.534443\pi$
$770$		0			0
$771$		0			0
$772$	−4600.00	+	7967.43i	−0.214453	+	0.371443i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	19250.0	−	33342.0i	0.892233	−	1.54539i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21700.0	−	37585.5i	−0.982874	−	1.70239i	−0.651029	−	0.759053i	$-0.725662\pi$
$787$	−21700.0	−	37585.5i	−0.982874	−	1.70239i	−0.331844	−	0.943334i	$-0.607671\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6370.00	−	11033.2i	0.285253	−	0.494072i
$794$		0			0
$795$		0			0
$796$	−20944.0	+	36276.1i	−0.932588	+	1.61529i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 + 6.92820i) q^{4} +O(q^{10})\)	\(q+(4.00000 + 6.92820i) q^{4} -70.0000 q^{13} +(-32.0000 + 55.4256i) q^{16} +(-28.0000 + 48.4974i) q^{19} +(62.5000 + 108.253i) q^{25} +(-154.000 - 266.736i) q^{31} +(-55.0000 + 95.2628i) q^{37} -520.000 q^{43} +(-280.000 - 484.974i) q^{52} +(-91.0000 + 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-595.000 - 1030.57i) q^{73} -448.000 q^{76} +(-442.000 + 765.566i) q^{79} -1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4}+O(q^{10}) \) 2 * q + 8 * q^4	\( 2 q + 8 q^{4} - 140 q^{13} - 64 q^{16} - 56 q^{19} + 125 q^{25} - 308 q^{31} - 110 q^{37} - 1040 q^{43} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} - 1190 q^{73} - 896 q^{76} - 884 q^{79} - 2660 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 - 140 * q^13 - 64 * q^16 - 56 * q^19 + 125 * q^25 - 308 * q^31 - 110 * q^37 - 1040 * q^43 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 - 1190 * q^73 - 896 * q^76 - 884 * q^79 - 2660 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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