LMFDB - Embedded newform 441.2.p.b.215.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(80,441)]

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([3, 1]))

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

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

chi := DirichletCharacter("441.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.p (of order $6$, 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:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			215.3
Root			$1.00781 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	441.215
Dual form			441.2.p.b.80.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00781 + 0.581861i) q^{2} +(-0.322876 - 0.559237i) q^{4} -3.07892i q^{8} +O(q^{10})$	$q+(1.00781 + 0.581861i) q^{2} +(-0.322876 - 0.559237i) q^{4} -3.07892i q^{8} +(5.68986 - 3.28504i) q^{11} +(1.14575 - 1.98450i) q^{16} +7.64575 q^{22} +(1.65861 + 0.957598i) q^{23} +(2.50000 + 4.33013i) q^{25} -8.89753i q^{29} +(-3.02344 + 1.74558i) q^{32} +(-5.29150 + 9.16515i) q^{37} -5.29150 q^{43} +(-3.67423 - 2.12132i) q^{44} +(1.11438 + 1.93016i) q^{46} +5.81861i q^{50} +(-0.357016 + 0.206123i) q^{53} +(5.17712 - 8.96704i) q^{58} -8.64575 q^{64} +(2.00000 + 3.46410i) q^{67} +15.0554i q^{71} +(-10.6657 + 6.15784i) q^{74} +(-4.00000 + 6.92820i) q^{79} +(-5.33284 - 3.07892i) q^{86} +(-10.1144 - 17.5186i) q^{88} -1.23674i q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4}+O(q^{10}) $ 8 * q + 8 * q^4	$ 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) $ 8 * q + 8 * q^4 - 12 * q^16 + 40 * q^22 + 20 * q^25 - 44 * q^46 + 52 * q^58 - 48 * q^64 + 16 * q^67 - 32 * q^79 - 28 * q^88

Display 100 coefficients 10 coefficients

Character values

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

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

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 $

$2$	1.00781	+	0.581861i	0.712631	+	0.411438i	0.812035	−	0.583609i	$-0.198360\pi$
$2$	1.00781	+	0.581861i	0.712631	+	0.411438i	−0.0994033	+	0.995047i	$0.531693\pi$
$3$		0			0
$3$		0			0
$4$	−0.322876	−	0.559237i	−0.161438	−	0.279619i
$5$		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
$8$		−	3.07892i		−	1.08856i
$9$		0			0
$10$		0			0
$11$	5.68986	−	3.28504i	1.71556	−	0.990478i	0.788941	−	0.614468i	$-0.210630\pi$
$11$	5.68986	−	3.28504i	1.71556	−	0.990478i	0.926616	−	0.376009i	$-0.122704\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	1.14575	−	1.98450i	0.286438	−	0.496125i
$17$		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		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$	7.64575			1.63008
$23$	1.65861	+	0.957598i	0.345844	+	0.199673i	0.662853	−	0.748749i	$-0.269345\pi$
$23$	1.65861	+	0.957598i	0.345844	+	0.199673i	−0.317009	+	0.948422i	$0.602679\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.89753i		−	1.65223i	−0.563502	−	0.826115i	$-0.690546\pi$
$29$		−	8.89753i		−	1.65223i	0.563502	−	0.826115i	$-0.309454\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−3.02344	+	1.74558i	−0.534473	+	0.308578i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.29150	+	9.16515i	−0.869918	+	1.50674i	−0.00783774	+	0.999969i	$0.502495\pi$
$37$	−5.29150	+	9.16515i	−0.869918	+	1.50674i	−0.862080	+	0.506772i	$0.830838\pi$
$38$		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
$43$	−5.29150			−0.806947			−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150			−0.806947			−0.403473	+	0.914991i	$0.632197\pi$
$44$	−3.67423	−	2.12132i	−0.553912	−	0.319801i
$45$		0			0
$46$	1.11438	+	1.93016i	0.164306	+	0.284587i
$47$		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
$50$			5.81861i			0.822876i
$51$		0			0
$52$		0			0
$53$	−0.357016	+	0.206123i	−0.0490400	+	0.0283132i	−0.524320	−	0.851522i	$-0.675680\pi$
$53$	−0.357016	+	0.206123i	−0.0490400	+	0.0283132i	0.475280	+	0.879835i	$0.342347\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	5.17712	−	8.96704i	0.679790	−	1.17743i
$59$		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		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−8.64575			−1.08072
$65$		0			0
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			15.0554i			1.78674i	0.449319	+	0.893372i	$0.351667\pi$
$71$			15.0554i			1.78674i	−0.449319	+	0.893372i	$0.648333\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	−10.6657	+	6.15784i	−1.23986	+	0.715834i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		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
$86$	−5.33284	−	3.07892i	−0.575055	−	0.332008i
$87$		0			0
$88$	−10.1144	−	17.5186i	−1.07820	−	1.86749i
$89$		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
$92$		−	1.23674i		−	0.128939i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	1.61438	−	2.79619i	0.161438	−	0.279619i
$101$		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		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−0.479741			−0.0465965
$107$	9.00708	+	5.20024i	0.870747	+	0.502726i	0.867596	−	0.497269i	$-0.165664\pi$
$107$	9.00708	+	5.20024i	0.870747	+	0.502726i	0.00315068	+	0.999995i	$0.498997\pi$
$108$		0			0
$109$	−5.29150	−	9.16515i	−0.506834	−	0.877862i	−0.999969	−	0.00790932i	$-0.997482\pi$
$109$	−5.29150	−	9.16515i	−0.506834	−	0.877862i	0.493135	−	0.869953i	$-0.335851\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.5524i			1.27490i	0.770490	+	0.637452i	$0.220012\pi$
$113$			13.5524i			1.27490i	−0.770490	+	0.637452i	$0.779988\pi$
$114$		0			0
$115$		0			0
$116$	−4.97583	+	2.87280i	−0.461994	+	0.266732i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	16.0830	−	27.8566i	1.46209	−	2.53242i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−2.66642	−	1.53946i	−0.235681	−	0.136070i
$129$		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
$134$			4.65489i			0.402121i
$135$		0			0
$136$		0			0
$137$	19.0852	−	11.0188i	1.63056	−	0.941404i	0.646639	−	0.762796i	$-0.276174\pi$
$137$	19.0852	−	11.0188i	1.63056	−	0.941404i	0.983920	−	0.178607i	$-0.0571592\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−8.76013	+	15.1730i	−0.735134	+	1.27329i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	6.83399			0.561750
$149$	−6.99145	−	4.03652i	−0.572762	−	0.330684i	0.185490	−	0.982646i	$-0.440613\pi$
$149$	−6.99145	−	4.03652i	−0.572762	−	0.330684i	−0.758252	+	0.651962i	$0.773946\pi$
$150$		0			0
$151$	2.64575	+	4.58258i	0.215308	+	0.372925i	0.953368	−	0.301811i	$-0.0975911\pi$
$151$	2.64575	+	4.58258i	0.215308	+	0.372925i	−0.738060	+	0.674735i	$0.764258\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−8.06250	+	4.65489i	−0.641418	+	0.370323i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	0.146772	+	0.989170i	$0.453112\pi$
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	−0.930033	+	0.367477i	$0.880222\pi$
$164$		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
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$	1.70850	+	2.95920i	0.130272	+	0.225637i
$173$		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
$176$		−	15.0554i		−	1.13484i
$177$		0			0
$178$		0			0
$179$	−13.7524	+	7.93993i	−1.02790	+	0.593458i	−0.916382	−	0.400304i	$-0.868904\pi$
$179$	−13.7524	+	7.93993i	−1.02790	+	0.593458i	−0.111518	+	0.993762i	$0.535571\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	2.94837	−	5.10672i	0.217357	−	0.376473i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.1008	+	12.1826i	1.52680	+	0.881500i	0.999493	+	0.0318264i	$0.0101324\pi$
$191$	21.1008	+	12.1826i	1.52680	+	0.881500i	0.527309	+	0.849674i	$0.323201\pi$
$192$		0			0
$193$	10.5830	+	18.3303i	0.761781	+	1.31944i	0.941932	+	0.335805i	$0.109008\pi$
$193$	10.5830	+	18.3303i	0.761781	+	1.31944i	−0.180150	+	0.983639i	$0.557658\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	25.8681i		−	1.84303i	−0.388348	−	0.921513i	$-0.626954\pi$
$197$		−	25.8681i		−	1.84303i	0.388348	−	0.921513i	$-0.373046\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	13.3321	−	7.69730i	0.942722	−	0.544281i
$201$		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
$211$	26.4575			1.82141			0.910705	−	0.413057i	$-0.135539\pi$
$211$	26.4575			1.82141			0.910705	+	0.413057i	$0.135539\pi$
$212$	0.230544	+	0.133105i	0.0158338	+	0.00914166i
$213$		0			0
$214$	6.05163	+	10.4817i	0.413681	+	0.716517i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	12.3157i		−	0.834123i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−7.88562	+	13.6583i	−0.524544	+	0.908536i
$227$		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		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−27.3948			−1.79855
$233$	−26.4337	−	15.2615i	−1.73173	−	0.999813i	−0.875522	−	0.483178i	$-0.839482\pi$
$233$	−26.4337	−	15.2615i	−1.73173	−	0.999813i	−0.856205	−	0.516636i	$-0.827184\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.39458i		−	0.478316i	−0.970981	−	0.239158i	$-0.923129\pi$
$239$		−	7.39458i		−	0.478316i	0.970981	−	0.239158i	$-0.0768713\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	32.4173	−	18.7161i	2.08386	−	1.20312i
$243$		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
$253$	12.5830			0.791087
$254$	−16.1250	−	9.30978i	−1.01177	−	0.584147i
$255$		0			0
$256$	6.85425	+	11.8719i	0.428391	+	0.741994i
$257$		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
$263$	−16.3555	+	9.44288i	−1.00853	+	0.582273i	−0.910760	−	0.412936i	$-0.864503\pi$
$263$	−16.3555	+	9.44288i	−1.00853	+	0.582273i	−0.0977664	+	0.995209i	$0.531170\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.29150	−	2.23695i	0.0788911	−	0.136643i
$269$		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		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	25.6458			1.54932
$275$	28.4493	+	16.4252i	1.71556	+	0.990478i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.41815i		−	0.203910i	−0.994789	−	0.101955i	$-0.967490\pi$
$281$		−	3.41815i		−	0.203910i	0.994789	−	0.101955i	$-0.0325097\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	8.41952	−	4.86101i	0.499606	−	0.288448i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	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
$296$	28.2188	+	16.2921i	1.64018	+	0.946959i
$297$		0			0
$298$	−4.69738	−	8.13611i	−0.272112	−	0.471312i
$299$		0			0
$300$		0			0
$301$		0			0
$302$			6.15784i			0.354344i
$303$		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
$311$		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		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$	5.16601			0.290611
$317$	−14.3399	−	8.27916i	−0.805410	−	0.465004i	0.0399492	−	0.999202i	$-0.487280\pi$
$317$	−14.3399	−	8.27916i	−0.805410	−	0.465004i	−0.845359	+	0.534198i	$0.820614\pi$
$318$		0			0
$319$	−29.2288	−	50.6257i	−1.63650	−	2.83449i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−20.1563	+	11.6372i	−1.11635	+	0.644526i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.64575	−	4.58258i	0.145424	−	0.251881i	−0.784107	−	0.620625i	$-0.786879\pi$
$331$	2.64575	−	4.58258i	0.145424	−	0.251881i	0.929531	+	0.368744i	$0.120212\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−21.1660			−1.15299			−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660			−1.15299			−0.576493	+	0.817102i	$0.695579\pi$
$338$	13.1016	+	7.56419i	0.712631	+	0.411438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$			16.2921i			0.878412i
$345$		0			0
$346$		0			0
$347$	25.1321	−	14.5100i	1.34916	−	0.778938i	0.361030	−	0.932554i	$-0.382425\pi$
$347$	25.1321	−	14.5100i	1.34916	−	0.778938i	0.988131	+	0.153616i	$0.0490918\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−11.4686	+	19.8642i	−0.611280	+	1.05877i
$353$		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
$358$	−18.4797			−0.976685
$359$	−17.7836	−	10.2674i	−0.938583	−	0.541891i	−0.0490673	−	0.998795i	$-0.515625\pi$
$359$	−17.7836	−	10.2674i	−0.938583	−	0.541891i	−0.889516	+	0.456904i	$0.848958\pi$
$360$		0			0
$361$	−9.50000	−	16.4545i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	3.80071	−	2.19434i	0.198126	−	0.114388i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	0.996610	−	0.0822766i	$-0.0262191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−37.0405			−1.90264			−0.951322	−	0.308199i	$-0.900274\pi$
$379$	−37.0405			−1.90264			−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	14.1771	+	24.5555i	0.725365	+	1.25637i
$383$		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
$386$			24.6314i			1.25370i
$387$		0			0
$388$		0			0
$389$	−29.7509	+	17.1767i	−1.50843	+	0.870893i	−0.508478	+	0.861075i	$0.669792\pi$
$389$	−29.7509	+	17.1767i	−1.50843	+	0.870893i	−0.999952	+	0.00981780i	$0.996875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	15.0516	−	26.0702i	0.758290	−	1.31340i
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	11.4575			0.572876
$401$	−33.7821	−	19.5041i	−1.68700	−	0.973990i	−0.956796	−	0.290760i	$-0.906092\pi$
$401$	−33.7821	−	19.5041i	−1.68700	−	0.973990i	−0.730204	−	0.683229i	$-0.760575\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			69.5312i			3.44654i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		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
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$	26.6642	+	15.3946i	1.29799	+	0.749397i
$423$		0			0
$424$	0.634637	+	1.09922i	0.0308207	+	0.0533831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	6.71612i		−	0.324636i
$429$		0			0
$430$		0			0
$431$	−35.7978	+	20.6679i	−1.72432	+	0.995535i	−0.814964	+	0.579511i	$0.803244\pi$
$431$	−35.7978	+	20.6679i	−1.72432	+	0.995535i	−0.909353	+	0.416024i	$0.863423\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−3.41699	+	5.91841i	−0.163644	+	0.283440i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.4351	−	6.02473i	−0.495789	−	0.286244i	0.231184	−	0.972910i	$-0.425740\pi$
$443$	−10.4351	−	6.02473i	−0.495789	−	0.286244i	−0.726973	+	0.686666i	$0.759073\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.3475i		−	1.47938i	−0.672948	−	0.739689i	$-0.734972\pi$
$449$		−	31.3475i		−	1.47938i	0.672948	−	0.739689i	$-0.265028\pi$
$450$		0			0
$451$		0			0
$452$	7.57901	−	4.37575i	0.356487	−	0.205818i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−21.1660	+	36.6606i	−0.990104	+	1.71491i	−0.373519	+	0.927622i	$0.621849\pi$
$457$	−21.1660	+	36.6606i	−0.990104	+	1.71491i	−0.616585	+	0.787288i	$0.711484\pi$
$458$		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
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$	−17.6571	−	10.1944i	−0.819712	−	0.473261i
$465$		0			0
$466$	−17.7601	−	30.7614i	−0.822722	−	1.42500i
$467$		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
$473$	−30.1079	+	17.3828i	−1.38436	+	0.799262i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	4.30262	−	7.45235i	0.196797	−	0.340863i
$479$		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
$484$	−20.7712			−0.944147
$485$		0			0
$486$		0			0
$487$	18.5203	+	32.0780i	0.839233	+	1.45359i	0.890537	+	0.454911i	$0.150329\pi$
$487$	18.5203	+	32.0780i	0.839233	+	1.45359i	−0.0513038	+	0.998683i	$0.516338\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	27.3710i		−	1.23524i	−0.786478	−	0.617619i	$-0.788097\pi$
$491$		−	27.3710i		−	1.23524i	0.786478	−	0.617619i	$-0.211903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.2288	+	22.9129i	−0.592200	+	1.02572i	0.401735	+	0.915756i	$0.368407\pi$
$499$	−13.2288	+	22.9129i	−0.592200	+	1.02572i	−0.993935	+	0.109965i	$0.964926\pi$
$500$		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
$506$	12.6813	+	7.32156i	0.563753	+	0.325483i
$507$		0			0
$508$	5.16601	+	8.94779i	0.229205	+	0.396994i
$509$		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
$512$			22.1107i			0.977165i
$513$		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		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	−21.9778			−0.958276
$527$		0			0
$528$		0			0
$529$	−9.66601	−	16.7420i	−0.420261	−	0.727914i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	10.6657	−	6.15784i	0.460688	−	0.265978i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000	−	29.4449i	0.730887	−	1.26593i	−0.225617	−	0.974216i	$-0.572440\pi$
$541$	17.0000	−	29.4449i	0.730887	−	1.26593i	0.956504	−	0.291718i	$-0.0942267\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$	−12.3243	−	7.11544i	−0.526468	−	0.303956i
$549$		0			0
$550$	19.1144	+	33.1071i	0.815040	+	1.41169i
$551$		0			0
$552$		0			0
$553$		0			0
$554$			11.6372i			0.494418i
$555$		0			0
$556$		0			0
$557$	21.6884	−	12.5218i	0.918967	−	0.530566i	0.0356614	−	0.999364i	$-0.488646\pi$
$557$	21.6884	−	12.5218i	0.918967	−	0.530566i	0.883305	+	0.468798i	$0.155313\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.98889	−	3.44485i	0.0838961	−	0.145312i
$563$		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
$568$	46.3542			1.94498
$569$	12.4508	+	7.18845i	0.521963	+	0.301356i	0.737738	−	0.675088i	$-0.235894\pi$
$569$	12.4508	+	7.18845i	0.521963	+	0.301356i	−0.215774	+	0.976443i	$0.569228\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			9.57598i			0.399346i
$576$		0			0
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	17.1328	−	9.89164i	0.712631	−	0.411438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.35425	+	2.34563i	−0.0560872	+	0.0971460i
$584$		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
$592$	12.1255	+	21.0020i	0.498355	+	0.863176i
$593$		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
$596$			5.21317i			0.213540i
$597$		0			0
$598$		0			0
$599$	3.08667	−	1.78209i	0.126118	−	0.0728143i	−0.435614	−	0.900134i	$-0.643469\pi$
$599$	3.08667	−	1.78209i	0.126118	−	0.0728143i	0.561732	+	0.827319i	$0.310135\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	1.70850	−	2.95920i	0.0695178	−	0.120408i
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	−0.938967	−	0.344008i	$-0.888215\pi$
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	0.171564	−	0.985173i	$-0.445118\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	48.3180i		−	1.94521i	−0.232462	−	0.972605i	$-0.574678\pi$
$617$		−	48.3180i		−	1.94521i	0.232462	−	0.972605i	$-0.425322\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	21.3314	+	12.3157i	0.848517	+	0.489891i
$633$		0			0
$634$	−9.63464	−	16.6877i	−0.382640	−	0.662753i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		−	68.0283i		−	2.69327i
$639$		0			0
$640$		0			0
$641$	41.1306	−	23.7468i	1.62456	−	0.937941i	0.638883	−	0.769304i	$-0.279397\pi$
$641$	41.1306	−	23.7468i	1.62456	−	0.937941i	0.985678	−	0.168637i	$-0.0539367\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		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
$652$	12.9150			0.505791
$653$	37.0994	+	21.4193i	1.45181	+	0.838203i	0.998584	−	0.0531926i	$-0.0169397\pi$
$653$	37.0994	+	21.4193i	1.45181	+	0.838203i	0.453226	+	0.891396i	$0.350273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	49.8210i		−	1.94075i	−0.241604	−	0.970375i	$-0.577673\pi$
$659$		−	49.8210i		−	1.94075i	0.241604	−	0.970375i	$-0.422327\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	5.33284	−	3.07892i	0.207267	−	0.119666i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.52026	−	14.7575i	0.329906	−	0.571414i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	42.3320			1.63178			0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320			1.63178			0.815890	+	0.578208i	$0.196248\pi$
$674$	−21.3314	−	12.3157i	−0.821654	−	0.474382i
$675$		0			0
$676$	−4.19738	−	7.27008i	−0.161438	−	0.279619i
$677$		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
$683$	35.0837	−	20.2556i	1.34244	−	0.775059i	0.355277	−	0.934761i	$-0.384387\pi$
$683$	35.0837	−	20.2556i	1.34244	−	0.775059i	0.987165	+	0.159702i	$0.0510533\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−6.06275	+	10.5010i	−0.231140	+	0.400346i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	33.7712			1.28194
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			36.0024i			1.35979i	0.733309	+	0.679895i	$0.237975\pi$
$701$			36.0024i			1.35979i	−0.733309	+	0.679895i	$0.762025\pi$
$702$		0			0
$703$		0			0
$704$	−49.1931	+	28.4017i	−1.85404	+	1.07043i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.4575	−	45.8258i	0.993633	−	1.72102i	0.399244	−	0.916845i	$-0.369273\pi$
$709$	26.4575	−	45.8258i	0.993633	−	1.72102i	0.594389	−	0.804178i	$-0.297394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	8.88061	+	5.12722i	0.331884	+	0.191613i
$717$		0			0
$718$	−11.9484	−	20.6952i	−0.445909	−	0.772337i
$719$		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
$722$		−	22.1107i		−	0.822876i
$723$		0			0
$724$		0			0
$725$	38.5274	−	22.2438i	1.43087	−	0.826115i
$726$		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		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$	−6.68627			−0.246459
$737$	22.7594	+	13.1402i	0.838355	+	0.484024i
$738$		0			0
$739$	26.0000	+	45.0333i	0.956425	+	1.65658i	0.731072	+	0.682300i	$0.239020\pi$
$739$	26.0000	+	45.0333i	0.956425	+	1.65658i	0.225354	+	0.974277i	$0.427646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			54.4759i			1.99853i	0.0383863	+	0.999263i	$0.487778\pi$
$743$			54.4759i			1.99853i	−0.0383863	+	0.999263i	$0.512222\pi$
$744$		0			0
$745$		0			0
$746$	22.1719	−	12.8009i	0.811770	−	0.468676i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−13.2288	+	22.9129i	−0.482724	+	0.836103i	−0.999803	−	0.0198348i	$-0.993686\pi$
$751$	−13.2288	+	22.9129i	−0.482724	+	0.836103i	0.517079	+	0.855938i	$0.327019\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	10.5830			0.384646			0.192323	−	0.981332i	$-0.438398\pi$
$757$	10.5830			0.384646			0.192323	+	0.981332i	$0.438398\pi$
$758$	−37.3299	−	21.5524i	−1.35588	−	0.782820i
$759$		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
$764$		−	15.7338i		−	0.569230i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	6.83399	−	11.8368i	0.245961	−	0.426016i
$773$		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
$778$	−39.9778			−1.43327
$779$		0			0
$780$		0			0
$781$	49.4575	+	85.6629i	1.76973	+	3.06526i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−14.4664	+	8.35218i	−0.515344	+	0.297534i
$789$		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
$800$	−15.1172	−	8.72791i	−0.534473	−	0.308578i
$801$		0			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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00781 + 0.581861i) q^{2} +(-0.322876 - 0.559237i) q^{4} -3.07892i q^{8} +O(q^{10})\)	\(q+(1.00781 + 0.581861i) q^{2} +(-0.322876 - 0.559237i) q^{4} -3.07892i q^{8} +(5.68986 - 3.28504i) q^{11} +(1.14575 - 1.98450i) q^{16} +7.64575 q^{22} +(1.65861 + 0.957598i) q^{23} +(2.50000 + 4.33013i) q^{25} -8.89753i q^{29} +(-3.02344 + 1.74558i) q^{32} +(-5.29150 + 9.16515i) q^{37} -5.29150 q^{43} +(-3.67423 - 2.12132i) q^{44} +(1.11438 + 1.93016i) q^{46} +5.81861i q^{50} +(-0.357016 + 0.206123i) q^{53} +(5.17712 - 8.96704i) q^{58} -8.64575 q^{64} +(2.00000 + 3.46410i) q^{67} +15.0554i q^{71} +(-10.6657 + 6.15784i) q^{74} +(-4.00000 + 6.92820i) q^{79} +(-5.33284 - 3.07892i) q^{86} +(-10.1144 - 17.5186i) q^{88} -1.23674i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 8 * q + 8 * q^4	\( 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) 8 * q + 8 * q^4 - 12 * q^16 + 40 * q^22 + 20 * q^25 - 44 * q^46 + 52 * q^58 - 48 * q^64 + 16 * q^67 - 32 * q^79 - 28 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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