LMFDB - Embedded newform 441.4.e.t.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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			226.1
Root			$-1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	441.226
Dual form			441.4.e.t.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 + 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} -23.8118 q^{8} +O(q^{10})$	$q+(-1.32288 + 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} -23.8118 q^{8} +(-13.2288 - 22.9129i) q^{11} +(27.5000 - 47.6314i) q^{16} +70.0000 q^{22} +(108.476 - 187.886i) q^{23} +(62.5000 + 108.253i) q^{25} -264.575 q^{29} +(-22.4889 - 38.9519i) q^{32} +(225.000 - 389.711i) q^{37} +180.000 q^{43} +(13.2288 - 22.9129i) q^{44} +(287.000 + 497.099i) q^{46} -330.719 q^{50} +(-248.701 - 430.762i) q^{53} +(350.000 - 606.218i) q^{58} +559.000 q^{64} +(370.000 + 640.859i) q^{67} +978.928 q^{71} +(595.294 + 1031.08i) q^{74} +(692.000 - 1198.58i) q^{79} +(-238.118 + 412.432i) q^{86} +(315.000 + 545.596i) q^{88} +216.952 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} + 110 q^{16} + 280 q^{22} + 250 q^{25} + 900 q^{37} + 720 q^{43} + 1148 q^{46} + 1400 q^{58} + 2236 q^{64} + 1480 q^{67} + 2768 q^{79} + 1260 q^{88}+O(q^{100}) $ 4 * q + 2 * q^4 + 110 * q^16 + 280 * q^22 + 250 * q^25 + 900 * q^37 + 720 * q^43 + 1148 * q^46 + 1400 * q^58 + 2236 * q^64 + 1480 * q^67 + 2768 * q^79 + 1260 * q^88

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 $

$2$	−1.32288	+	2.29129i	−0.467707	+	0.810093i	−0.999319	−	0.0368954i	$-0.988253\pi$
$2$	−1.32288	+	2.29129i	−0.467707	+	0.810093i	0.531612	+	0.846988i	$0.321586\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.0625000	+	0.108253i
$5$		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
$8$	−23.8118			−1.05234
$9$		0			0
$10$		0			0
$11$	−13.2288	−	22.9129i	−0.362602	−	0.628045i	0.625786	−	0.779994i	$-0.284778\pi$
$11$	−13.2288	−	22.9129i	−0.362602	−	0.628045i	−0.988388	+	0.151950i	$0.951445\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	27.5000	−	47.6314i	0.429688	−	0.744241i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$	70.0000			0.678366
$23$	108.476	−	187.886i	0.983425	−	1.70334i	0.334687	−	0.942329i	$-0.391369\pi$
$23$	108.476	−	187.886i	0.983425	−	1.70334i	0.648737	−	0.761012i	$-0.275297\pi$
$24$		0			0
$25$	62.5000	+	108.253i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−264.575			−1.69415			−0.847075	−	0.531473i	$-0.821639\pi$
$29$	−264.575			−1.69415			−0.847075	+	0.531473i	$0.821639\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−22.4889	−	38.9519i	−0.124235	−	0.215181i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	225.000	−	389.711i	0.999724	−	1.73157i	0.479500	−	0.877542i	$-0.340818\pi$
$37$	225.000	−	389.711i	0.999724	−	1.73157i	0.520223	−	0.854030i	$-0.325849\pi$
$38$		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
$43$	180.000			0.638366			0.319183	−	0.947693i	$-0.396592\pi$
$43$	180.000			0.638366			0.319183	+	0.947693i	$0.396592\pi$
$44$	13.2288	−	22.9129i	0.0453252	−	0.0785056i
$45$		0			0
$46$	287.000	+	497.099i	0.919910	+	1.59333i
$47$		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
$50$	−330.719			−0.935414
$51$		0			0
$52$		0			0
$53$	−248.701	−	430.762i	−0.644560	−	1.11641i	−0.984403	−	0.175928i	$-0.943707\pi$
$53$	−248.701	−	430.762i	−0.644560	−	1.11641i	0.339843	−	0.940482i	$-0.389626\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	350.000	−	606.218i	0.792366	−	1.37242i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	559.000			1.09180
$65$		0			0
$66$		0			0
$67$	370.000	+	640.859i	0.674667	+	1.16856i	0.976566	+	0.215218i	$0.0690461\pi$
$67$	370.000	+	640.859i	0.674667	+	1.16856i	−0.301899	+	0.953340i	$0.597621\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	978.928			1.63630			0.818151	−	0.575004i	$-0.195000\pi$
$71$	978.928			1.63630			0.818151	+	0.575004i	$0.195000\pi$
$72$		0			0
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$	595.294	+	1031.08i	0.935156	+	1.61974i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	692.000	−	1198.58i	0.985520	−	1.70697i	0.345918	−	0.938265i	$-0.387568\pi$
$79$	692.000	−	1198.58i	0.985520	−	1.70697i	0.639602	−	0.768706i	$-0.279099\pi$
$80$		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
$86$	−238.118	+	412.432i	−0.298568	+	0.517136i
$87$		0			0
$88$	315.000	+	545.596i	0.381581	+	0.660917i
$89$		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
$92$	216.952			0.245856
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	−62.5000	+	108.253i	−0.0625000	+	0.108253i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	1316.00			1.20586
$107$	−775.205	+	1342.69i	−0.700392	+	1.21311i	0.267937	+	0.963436i	$0.413658\pi$
$107$	−775.205	+	1342.69i	−0.700392	+	1.21311i	−0.968329	+	0.249678i	$0.919675\pi$
$108$		0			0
$109$	−27.0000	−	46.7654i	−0.0237260	−	0.0410946i	0.853919	−	0.520407i	$-0.174220\pi$
$109$	−27.0000	−	46.7654i	−0.0237260	−	0.0410946i	−0.877645	+	0.479312i	$0.840886\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2307.10			1.92065			0.960324	−	0.278886i	$-0.0899653\pi$
$113$	2307.10			1.92065			0.960324	+	0.278886i	$0.0899653\pi$
$114$		0			0
$115$		0			0
$116$	−132.288	−	229.129i	−0.105884	−	0.183397i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	315.500	−	546.462i	0.237040	−	0.410565i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2000.00			−1.39741			−0.698706	−	0.715409i	$-0.746240\pi$
$127$	−2000.00			−1.39741			−0.698706	+	0.715409i	$0.746240\pi$
$128$	−559.576	+	969.215i	−0.386407	+	0.669276i
$129$		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
$134$	−1957.86			−1.26219
$135$		0			0
$136$		0			0
$137$	−391.571	−	678.221i	−0.244191	−	0.422951i	0.717713	−	0.696339i	$-0.245189\pi$
$137$	−391.571	−	678.221i	−0.244191	−	0.422951i	−0.961904	+	0.273388i	$0.911856\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−1295.00	+	2243.01i	−0.765310	+	1.32556i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	450.000			0.249931
$149$	1772.65	−	3070.33i	0.974640	−	1.68813i	0.293524	−	0.955952i	$-0.405172\pi$
$149$	1772.65	−	3070.33i	0.974640	−	1.68813i	0.681117	−	0.732175i	$-0.261495\pi$
$150$		0			0
$151$	−1476.00	−	2556.51i	−0.795465	−	1.37779i	−0.922544	−	0.385893i	$-0.873893\pi$
$151$	−1476.00	−	2556.51i	−0.795465	−	1.37779i	0.127079	−	0.991893i	$-0.459440\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	1830.86	+	3171.14i	0.921870	+	1.59672i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−890.000	+	1541.53i	−0.427670	+	0.740746i	−0.996666	−	0.0815946i	$-0.973999\pi$
$163$	−890.000	+	1541.53i	−0.427670	+	0.740746i	0.568996	+	0.822340i	$0.307332\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−2197.00			−1.00000
$170$		0			0
$171$		0			0
$172$	90.0000	+	155.885i	0.0398979	+	0.0691052i
$173$		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
$176$	−1455.16			−0.623222
$177$		0			0
$178$		0			0
$179$	−2156.29	−	3734.80i	−0.900383	−	1.55951i	−0.826998	−	0.562205i	$-0.809953\pi$
$179$	−2156.29	−	3734.80i	−0.900383	−	1.55951i	−0.0733844	−	0.997304i	$-0.523380\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−2583.00	+	4473.89i	−1.03490	+	1.79250i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1680.05	−	2909.94i	0.636462	−	1.10239i	−0.349741	−	0.936846i	$-0.613730\pi$
$191$	1680.05	−	2909.94i	0.636462	−	1.10239i	0.986203	−	0.165539i	$-0.0529363\pi$
$192$		0			0
$193$	−2295.00	−	3975.06i	−0.855947	−	1.48254i	−0.875764	−	0.482740i	$-0.839642\pi$
$193$	−2295.00	−	3975.06i	−0.855947	−	1.48254i	0.0198172	−	0.999804i	$-0.493692\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5069.26			−1.83335			−0.916675	−	0.399634i	$-0.869137\pi$
$197$	−5069.26			−1.83335			−0.916675	+	0.399634i	$0.869137\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−1488.24	−	2577.70i	−0.526171	−	0.911354i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−5868.00			−1.91455			−0.957274	−	0.289181i	$-0.906617\pi$
$211$	−5868.00			−1.91455			−0.957274	+	0.289181i	$0.906617\pi$
$212$	248.701	−	430.762i	0.0805699	−	0.139551i
$213$		0			0
$214$	−2051.00	−	3552.44i	−0.655156	−	1.13476i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	142.871			0.0443872
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−3052.00	+	5286.22i	−0.898301	+	1.55590i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	6300.00			1.78282
$233$	−2656.33	+	4600.91i	−0.746877	+	1.29363i	0.202436	+	0.979295i	$0.435114\pi$
$233$	−2656.33	+	4600.91i	−0.746877	+	1.29363i	−0.949313	+	0.314333i	$0.898219\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−449.778			−0.121731			−0.0608655	−	0.998146i	$-0.519386\pi$
$239$	−449.778			−0.121731			−0.0608655	+	0.998146i	$0.519386\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	834.735	+	1445.80i	0.221730	+	0.384048i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−5740.00			−1.42637
$254$	2645.75	−	4582.58i	0.653580	−	1.13203i
$255$		0			0
$256$	755.500	+	1308.56i	0.184448	+	0.319474i
$257$		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
$263$	2013.42	+	3487.34i	0.472063	+	0.817637i	0.999489	−	0.0319637i	$-0.0101761\pi$
$263$	2013.42	+	3487.34i	0.472063	+	0.817637i	−0.527426	+	0.849601i	$0.676843\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−370.000	+	640.859i	−0.0843334	+	0.146070i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	2072.00			0.456840
$275$	1653.59	−	2864.11i	0.362602	−	0.628045i
$276$		0			0
$277$	−3655.00	−	6330.65i	−0.792807	−	1.37318i	−0.924222	−	0.381855i	$-0.875285\pi$
$277$	−3655.00	−	6330.65i	−0.792807	−	1.37318i	0.131415	−	0.991327i	$-0.458048\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8360.57			1.77491			0.887456	−	0.460893i	$-0.152471\pi$
$281$	8360.57			1.77491			0.887456	+	0.460893i	$0.152471\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$	489.464	+	847.777i	0.102269	+	0.177135i
$285$		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
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	−5357.65	+	9279.72i	−1.05205	+	1.82220i
$297$		0			0
$298$	4690.00	+	8123.32i	0.911693	+	1.57910i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	7810.26			1.48818
$303$		0			0
$304$		0			0
$305$		0			0
$306$		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
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$	1384.00			0.246380
$317$	−4439.57	+	7689.56i	−0.786597	+	1.36243i	0.141444	+	0.989946i	$0.454825\pi$
$317$	−4439.57	+	7689.56i	−0.786597	+	1.36243i	−0.928041	+	0.372479i	$0.878508\pi$
$318$		0			0
$319$	3500.00	+	6062.18i	0.614302	+	1.06400i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−2354.72	−	4078.49i	−0.400048	−	0.692904i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5454.00	−	9446.61i	0.905677	−	1.56868i	0.0856702	−	0.996324i	$-0.472697\pi$
$331$	5454.00	−	9446.61i	0.905677	−	1.56868i	0.820006	−	0.572354i	$-0.193970\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3330.00			0.538269			0.269135	−	0.963103i	$-0.413262\pi$
$337$	3330.00			0.538269			0.269135	+	0.963103i	$0.413262\pi$
$338$	2906.36	−	5033.96i	0.467707	−	0.810093i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−4286.12			−0.671779
$345$		0			0
$346$		0			0
$347$	6130.21	+	10617.8i	0.948377	+	1.64264i	0.748845	+	0.662745i	$0.230609\pi$
$347$	6130.21	+	10617.8i	0.948377	+	1.64264i	0.199532	+	0.979891i	$0.436058\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−595.000	+	1030.57i	−0.0900955	+	0.156050i
$353$		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
$358$	11410.0			1.68446
$359$	−5463.48	+	9463.02i	−0.803207	+	1.39120i	0.114288	+	0.993448i	$0.463541\pi$
$359$	−5463.48	+	9463.02i	−0.803207	+	1.39120i	−0.917495	+	0.397747i	$0.869792\pi$
$360$		0			0
$361$	3429.50	+	5940.07i	0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−5966.17	−	10333.7i	−0.845131	−	1.46381i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6985.00	−	12098.4i	0.969624	−	1.67944i	0.272980	−	0.962020i	$-0.411991\pi$
$373$	6985.00	−	12098.4i	0.969624	−	1.67944i	0.696643	−	0.717418i	$-0.254676\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11916.0			−1.61500			−0.807498	−	0.589870i	$-0.799179\pi$
$379$	−11916.0			−1.61500			−0.807498	+	0.589870i	$0.799179\pi$
$380$		0			0
$381$		0			0
$382$	4445.00	+	7698.97i	0.595356	+	1.03119i
$383$		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
$386$	12144.0			1.60133
$387$		0			0
$388$		0			0
$389$	5582.54	+	9669.23i	0.727624	+	1.26028i	0.957885	+	0.287153i	$0.0927088\pi$
$389$	5582.54	+	9669.23i	0.727624	+	1.26028i	−0.230261	+	0.973129i	$0.573958\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	6706.00	−	11615.1i	0.857471	−	1.48518i
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	6875.00			0.859375
$401$	7990.17	−	13839.4i	0.995037	−	1.72346i	0.411348	−	0.911478i	$-0.365058\pi$
$401$	7990.17	−	13839.4i	0.995037	−	1.72346i	0.583690	−	0.811977i	$-0.301608\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11905.9			−1.45001
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		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
$421$	15262.0			1.76680			0.883402	−	0.468616i	$-0.155247\pi$
$421$	15262.0			1.76680			0.883402	+	0.468616i	$0.155247\pi$
$422$	7762.63	−	13445.3i	0.895448	−	1.55096i
$423$		0			0
$424$	5922.00	+	10257.2i	0.678297	+	1.17484i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1550.41			−0.175098
$429$		0			0
$430$		0			0
$431$	−7844.65	−	13587.3i	−0.876714	−	1.51851i	−0.854926	−	0.518750i	$-0.826398\pi$
$431$	−7844.65	−	13587.3i	−0.876714	−	1.51851i	−0.0217878	−	0.999763i	$-0.506936\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	27.0000	−	46.7654i	0.00296575	−	0.00513682i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	796.371	−	1379.36i	0.0854102	−	0.147935i	−0.820156	−	0.572140i	$-0.806113\pi$
$443$	796.371	−	1379.36i	0.0854102	−	0.147935i	0.905566	+	0.424205i	$0.139447\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18837.7			−1.97997			−0.989987	−	0.141158i	$-0.954917\pi$
$449$	−18837.7			−1.97997			−0.989987	+	0.141158i	$0.954917\pi$
$450$		0			0
$451$		0			0
$452$	1153.55	+	1998.00i	0.120041	+	0.207916i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4005.00	−	6936.86i	0.409947	−	0.710050i	−0.584936	−	0.811079i	$-0.698880\pi$
$457$	4005.00	−	6936.86i	0.409947	−	0.710050i	0.994883	+	0.101030i	$0.0322137\pi$
$458$		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
$463$	−8440.00			−0.847171			−0.423585	−	0.905856i	$-0.639229\pi$
$463$	−8440.00			−0.847171			−0.423585	+	0.905856i	$0.639229\pi$
$464$	−7275.82	+	12602.1i	−0.727955	+	1.26086i
$465$		0			0
$466$	−7028.00	−	12172.9i	−0.698639	−	1.21008i
$467$		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
$473$	−2381.18	−	4124.32i	−0.231473	−	0.400923i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	595.000	−	1030.57i	0.0569344	−	0.0986134i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	631.000			0.0592600
$485$		0			0
$486$		0			0
$487$	10620.0	+	18394.4i	0.988169	+	1.71156i	0.626908	+	0.779094i	$0.284320\pi$
$487$	10620.0	+	18394.4i	0.988169	+	1.71156i	0.361261	+	0.932465i	$0.382346\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7646.22			−0.702788			−0.351394	−	0.936228i	$-0.614292\pi$
$491$	−7646.22			−0.702788			−0.351394	+	0.936228i	$0.614292\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3618.00	+	6266.56i	−0.324577	+	0.562184i	−0.981427	−	0.191838i	$-0.938555\pi$
$499$	−3618.00	+	6266.56i	−0.324577	+	0.562184i	0.656850	+	0.754022i	$0.271889\pi$
$500$		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
$506$	7593.31	−	13152.0i	0.667122	−	1.15549i
$507$		0			0
$508$	−1000.00	−	1732.05i	−0.0873382	−	0.151274i
$509$		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
$512$	−12951.0			−1.11788
$513$		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
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$	−10654.0			−0.883149
$527$		0			0
$528$		0			0
$529$	−17450.5	−	30225.2i	−1.43425	−	2.48419i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−8810.35	−	15260.0i	−0.709980	−	1.22972i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7939.00	+	13750.8i	−0.630914	+	1.09277i	0.356452	+	0.934314i	$0.383986\pi$
$541$	−7939.00	+	13750.8i	−0.630914	+	1.09277i	−0.987365	+	0.158461i	$0.949347\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12980.0			1.01460			0.507299	−	0.861770i	$-0.330644\pi$
$547$	12980.0			1.01460			0.507299	+	0.861770i	$0.330644\pi$
$548$	391.571	−	678.221i	0.0305239	−	0.0528689i
$549$		0			0
$550$	4375.00	+	7577.72i	0.339183	+	0.587482i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	19340.4			1.48321
$555$		0			0
$556$		0			0
$557$	8249.45	+	14288.5i	0.627541	+	1.08693i	0.988044	+	0.154175i	$0.0492720\pi$
$557$	8249.45	+	14288.5i	0.627541	+	1.08693i	−0.360502	+	0.932758i	$0.617395\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−11060.0	+	19156.5i	−0.830139	+	1.43784i
$563$		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
$568$	−23310.0			−1.72195
$569$	−1799.11	+	3116.15i	−0.132553	+	0.229589i	−0.924660	−	0.380794i	$-0.875651\pi$
$569$	−1799.11	+	3116.15i	−0.132553	+	0.229589i	0.792107	+	0.610382i	$0.208984\pi$
$570$		0			0
$571$	3394.00	+	5878.58i	0.248747	+	0.430842i	0.963178	−	0.268863i	$-0.0866479\pi$
$571$	3394.00	+	5878.58i	0.248747	+	0.430842i	−0.714431	+	0.699705i	$0.753315\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	27119.0			1.96685
$576$		0			0
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	6499.29	+	11257.1i	0.467707	+	0.810093i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6580.00	+	11396.9i	−0.467437	+	0.809625i
$584$		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
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−12375.0	−	21434.1i	−0.859137	−	1.48807i
$593$		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
$596$	3545.31			0.243660
$597$		0			0
$598$		0			0
$599$	7871.11	+	13633.2i	0.536903	+	0.929943i	0.999069	+	0.0431495i	$0.0137392\pi$
$599$	7871.11	+	13633.2i	0.536903	+	0.929943i	−0.462166	+	0.886794i	$0.652927\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$	1476.00	−	2556.51i	0.0994331	−	0.172223i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7505.00	−	12999.0i	−0.494493	−	0.856487i	0.505487	−	0.862834i	$-0.331313\pi$
$613$	−7505.00	−	12999.0i	−0.494493	−	0.856487i	−0.999980	+	0.00634752i	$0.997980\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2497.59			−0.162965			−0.0814823	−	0.996675i	$-0.525965\pi$
$617$	−2497.59			−0.162965			−0.0814823	+	0.996675i	$0.525965\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		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
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−26192.0			−1.65244			−0.826218	−	0.563351i	$-0.809512\pi$
$631$	−26192.0			−1.65244			−0.826218	+	0.563351i	$0.809512\pi$
$632$	−16477.7	+	28540.3i	−1.03710	+	1.79632i
$633$		0			0
$634$	−11746.0	−	20344.7i	−0.735794	−	1.27443i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−18520.3			−1.14925
$639$		0			0
$640$		0			0
$641$	−15609.9	−	27037.2i	−0.961865	−	1.66600i	−0.717813	−	0.696236i	$-0.754857\pi$
$641$	−15609.9	−	27037.2i	−0.961865	−	1.66600i	−0.244052	−	0.969762i	$-0.578477\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		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
$652$	−1780.00			−0.106917
$653$	−9773.41	+	16928.0i	−0.585701	+	1.01446i	0.409086	+	0.912496i	$0.365847\pi$
$653$	−9773.41	+	16928.0i	−0.585701	+	1.01446i	−0.994788	+	0.101969i	$0.967486\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	33786.2			1.99716			0.998578	−	0.0533186i	$-0.0169799\pi$
$659$	33786.2			1.99716			0.998578	+	0.0533186i	$0.0169799\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	14429.9	+	24993.4i	0.847183	+	1.46736i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−28700.0	+	49709.9i	−1.66607	+	2.88572i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	33570.0			1.92278			0.961388	−	0.275196i	$-0.0887428\pi$
$673$	33570.0			1.92278			0.961388	+	0.275196i	$0.0887428\pi$
$674$	−4405.18	+	7629.99i	−0.251752	+	0.436048i
$675$		0			0
$676$	−1098.50	−	1902.66i	−0.0625000	−	0.108253i
$677$		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
$683$	5347.06	+	9261.39i	0.299560	+	0.518854i	0.976035	−	0.217612i	$-0.0698267\pi$
$683$	5347.06	+	9261.39i	0.299560	+	0.518854i	−0.676475	+	0.736466i	$0.736493\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	4950.00	−	8573.65i	0.274298	−	0.475098i
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	−32438.0			−1.77425
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36881.8			1.98717			0.993584	−	0.113093i	$-0.0360758\pi$
$701$	36881.8			1.98717			0.993584	+	0.113093i	$0.0360758\pi$
$702$		0			0
$703$		0			0
$704$	−7394.87	−	12808.3i	−0.395888	−	0.685697i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6273.00	−	10865.2i	0.332281	−	0.575528i	−0.650677	−	0.759354i	$-0.725515\pi$
$709$	6273.00	−	10865.2i	0.332281	−	0.575528i	0.982959	+	0.183826i	$0.0588483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	2156.29	−	3734.80i	0.112548	−	0.194939i
$717$		0			0
$718$	−14455.0	−	25036.8i	−0.751331	−	1.30134i
$719$		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
$722$	−18147.2			−0.935414
$723$		0			0
$724$		0			0
$725$	−16535.9	−	28641.1i	−0.847075	−	1.46718i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$	−9758.00			−0.488702
$737$	9789.28	−	16955.5i	0.489271	−	0.847442i
$738$		0			0
$739$	12662.0	+	21931.2i	0.630283	+	1.09168i	0.987494	+	0.157658i	$0.0503945\pi$
$739$	12662.0	+	21931.2i	0.630283	+	1.09168i	−0.357211	+	0.934024i	$0.616272\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−31743.7			−1.56738			−0.783691	−	0.621151i	$-0.786665\pi$
$743$	−31743.7			−1.56738			−0.783691	+	0.621151i	$0.786665\pi$
$744$		0			0
$745$		0			0
$746$	18480.6	+	32009.3i	0.907000	+	1.57097i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1224.00	+	2120.03i	−0.0594732	+	0.103011i	−0.894229	−	0.447610i	$-0.852275\pi$
$751$	−1224.00	+	2120.03i	−0.0594732	+	0.103011i	0.834756	+	0.550620i	$0.185609\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34830.0			1.67228			0.836141	−	0.548514i	$-0.184806\pi$
$757$	34830.0			1.67228			0.836141	+	0.548514i	$0.184806\pi$
$758$	15763.4	−	27303.0i	0.755346	−	1.30830i
$759$		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
$764$	3360.10			0.159116
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	2295.00	−	3975.06i	0.106993	−	0.185318i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−29540.0			−1.36126
$779$		0			0
$780$		0			0
$781$	−12950.0	−	22430.1i	−0.593326	−	1.02767i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	−2534.63	−	4390.11i	−0.114584	−	0.198466i
$789$		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
$800$	2811.11	−	4868.99i	0.124235	−	0.215181i
$801$		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)

\(f(q)\)	\(=\)	\(q+(-1.32288 + 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} -23.8118 q^{8} +O(q^{10})\)	\(q+(-1.32288 + 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} -23.8118 q^{8} +(-13.2288 - 22.9129i) q^{11} +(27.5000 - 47.6314i) q^{16} +70.0000 q^{22} +(108.476 - 187.886i) q^{23} +(62.5000 + 108.253i) q^{25} -264.575 q^{29} +(-22.4889 - 38.9519i) q^{32} +(225.000 - 389.711i) q^{37} +180.000 q^{43} +(13.2288 - 22.9129i) q^{44} +(287.000 + 497.099i) q^{46} -330.719 q^{50} +(-248.701 - 430.762i) q^{53} +(350.000 - 606.218i) q^{58} +559.000 q^{64} +(370.000 + 640.859i) q^{67} +978.928 q^{71} +(595.294 + 1031.08i) q^{74} +(692.000 - 1198.58i) q^{79} +(-238.118 + 412.432i) q^{86} +(315.000 + 545.596i) q^{88} +216.952 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} + 110 q^{16} + 280 q^{22} + 250 q^{25} + 900 q^{37} + 720 q^{43} + 1148 q^{46} + 1400 q^{58} + 2236 q^{64} + 1480 q^{67} + 2768 q^{79} + 1260 q^{88}+O(q^{100}) \) 4 * q + 2 * q^4 + 110 * q^16 + 280 * q^22 + 250 * q^25 + 900 * q^37 + 720 * q^43 + 1148 * q^46 + 1400 * q^58 + 2236 * q^64 + 1480 * q^67 + 2768 * q^79 + 1260 * q^88

Introduction

L-functions

Modular forms

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Database

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