LMFDB - Embedded newform 441.2.p.b.80.1

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			80.1
Root			$-2.23256 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	441.80
Dual form			441.2.p.b.215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23256 + 1.28897i) q^{2} +(2.32288 - 4.02334i) q^{4} +6.82058i q^{8} +O(q^{10})$	$q+(-2.23256 + 1.28897i) q^{2} +(2.32288 - 4.02334i) q^{4} +6.82058i q^{8} +(-0.790881 - 0.456615i) q^{11} +(-4.14575 - 7.18065i) q^{16} +2.35425 q^{22} +(8.13935 - 4.69926i) q^{23} +(2.50000 - 4.33013i) q^{25} -6.06910i q^{29} +(6.69767 + 3.86690i) q^{32} +(5.29150 + 9.16515i) q^{37} +5.29150 q^{43} +(-3.67423 + 2.12132i) q^{44} +(-12.1144 + 20.9827i) q^{46} +12.8897i q^{50} +(12.6045 + 7.27719i) q^{53} +(7.82288 + 13.5496i) q^{58} -3.35425 q^{64} +(2.00000 - 3.46410i) q^{67} -7.57205i q^{71} +(-23.6272 - 13.6412i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-11.8136 + 6.82058i) q^{86} +(3.11438 - 5.39426i) q^{88} -43.6631i q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

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

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 $

$2$	−2.23256	+	1.28897i	−1.57866	+	0.911438i	−0.583609	+	0.812035i	$0.698360\pi$
$2$	−2.23256	+	1.28897i	−1.57866	+	0.911438i	−0.995047	+	0.0994033i	$0.968307\pi$
$3$		0			0
$3$		0			0
$4$	2.32288	−	4.02334i	1.16144	−	2.01167i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$			6.82058i			2.41144i
$9$		0			0
$10$		0			0
$11$	−0.790881	−	0.456615i	−0.238459	−	0.137675i	0.376009	−	0.926616i	$-0.377296\pi$
$11$	−0.790881	−	0.456615i	−0.238459	−	0.137675i	−0.614468	+	0.788941i	$0.710630\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−4.14575	−	7.18065i	−1.03644	−	1.79516i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$		0			0
$22$	2.35425			0.501928
$23$	8.13935	−	4.69926i	1.69717	−	0.979863i	0.748749	−	0.662853i	$-0.230655\pi$
$23$	8.13935	−	4.69926i	1.69717	−	0.979863i	0.948422	−	0.317009i	$-0.102679\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.06910i		−	1.12700i	−0.826115	−	0.563502i	$-0.809454\pi$
$29$		−	6.06910i		−	1.12700i	0.826115	−	0.563502i	$-0.190546\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	6.69767	+	3.86690i	1.18399	+	0.683578i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.29150	+	9.16515i	0.869918	+	1.50674i	0.862080	+	0.506772i	$0.169162\pi$
$37$	5.29150	+	9.16515i	0.869918	+	1.50674i	0.00783774	+	0.999969i	$0.497505\pi$
$38$		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	$-0.367803\pi$
$43$	5.29150			0.806947			0.403473	+	0.914991i	$0.367803\pi$
$44$	−3.67423	+	2.12132i	−0.553912	+	0.319801i
$45$		0			0
$46$	−12.1144	+	20.9827i	−1.78617	+	3.09373i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$			12.8897i			1.82288i
$51$		0			0
$52$		0			0
$53$	12.6045	+	7.27719i	1.73136	+	0.999599i	0.879835	+	0.475280i	$0.157653\pi$
$53$	12.6045	+	7.27719i	1.73136	+	0.999599i	0.851522	+	0.524320i	$0.175680\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	7.82288	+	13.5496i	1.02719	+	1.77915i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−3.35425			−0.419281
$65$		0			0
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.57205i		−	0.898637i	−0.893372	−	0.449319i	$-0.851667\pi$
$71$		−	7.57205i		−	0.898637i	0.893372	−	0.449319i	$-0.148333\pi$
$72$		0			0
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−23.6272	−	13.6412i	−2.74660	−	1.58575i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		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
$86$	−11.8136	+	6.82058i	−1.27389	+	0.735482i
$87$		0			0
$88$	3.11438	−	5.39426i	0.331994	−	0.575030i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		−	43.6631i		−	4.55220i
$93$		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
$100$	−11.6144	−	20.1167i	−1.16144	−	2.01167i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	−37.5203			−3.64429
$107$	15.4878	−	8.94190i	1.49726	−	0.864446i	0.497269	−	0.867596i	$-0.334336\pi$
$107$	15.4878	−	8.94190i	1.49726	−	0.864446i	0.999995	+	0.00315068i	$0.00100290\pi$
$108$		0			0
$109$	5.29150	−	9.16515i	0.506834	−	0.877862i	−0.493135	−	0.869953i	$-0.664149\pi$
$109$	5.29150	−	9.16515i	0.506834	−	0.877862i	0.999969	−	0.00790932i	$-0.00251764\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.3808i			1.54098i	0.637452	+	0.770490i	$0.279988\pi$
$113$			16.3808i			1.54098i	−0.637452	+	0.770490i	$0.720012\pi$
$114$		0			0
$115$		0			0
$116$	−24.4180	−	14.0978i	−2.26716	−	1.30894i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.08301	−	8.80402i	−0.462091	−	0.800366i
$122$		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	$0.751253\pi$
$128$	−5.90679	+	3.41029i	−0.522092	+	0.301430i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$			10.3117i			0.890799i
$135$		0			0
$136$		0			0
$137$	−6.83776	−	3.94778i	−0.584189	−	0.337282i	0.178607	−	0.983920i	$-0.442841\pi$
$137$	−6.83776	−	3.94778i	−0.584189	−	0.337282i	−0.762796	+	0.646639i	$0.776174\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	9.76013	+	16.9050i	0.819052	+	1.41864i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	49.1660			4.04142
$149$	−19.9529	+	11.5198i	−1.63461	+	0.943741i	−0.651962	+	0.758252i	$0.726054\pi$
$149$	−19.9529	+	11.5198i	−1.63461	+	0.943741i	−0.982646	+	0.185490i	$0.940613\pi$
$150$		0			0
$151$	−2.64575	+	4.58258i	−0.215308	+	0.372925i	−0.953368	−	0.301811i	$-0.902409\pi$
$151$	−2.64575	+	4.58258i	−0.215308	+	0.372925i	0.738060	+	0.674735i	$0.235742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	17.8605	+	10.3117i	1.42090	+	0.820358i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	−0.930033	−	0.367477i	$-0.880222\pi$
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	0.146772	−	0.989170i	$-0.453112\pi$
$164$		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
$172$	12.2915	−	21.2895i	0.937218	−	1.62331i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$			7.57205i			0.570765i
$177$		0			0
$178$		0			0
$179$	18.6513	+	10.7684i	1.39407	+	0.804865i	0.993762	−	0.111518i	$-0.0355711\pi$
$179$	18.6513	+	10.7684i	1.39407	+	0.804865i	0.400304	+	0.916382i	$0.368904\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	32.0516	+	55.5151i	2.36288	+	4.09262i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.3029	+	6.52572i	−0.817847	+	0.472184i	−0.849674	−	0.527309i	$-0.823201\pi$
$191$	−11.3029	+	6.52572i	−0.817847	+	0.472184i	0.0318264	+	0.999493i	$0.489868\pi$
$192$		0			0
$193$	−10.5830	+	18.3303i	−0.761781	+	1.31944i	0.180150	+	0.983639i	$0.442342\pi$
$193$	−10.5830	+	18.3303i	−0.761781	+	1.31944i	−0.941932	+	0.335805i	$0.890992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.9015i			0.776697i	0.921513	+	0.388348i	$0.126954\pi$
$197$			10.9015i			0.776697i	−0.921513	+	0.388348i	$0.873046\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	29.5340	+	17.0514i	2.08837	+	1.20572i
$201$		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	$-0.864461\pi$
$211$	−26.4575			−1.82141			−0.910705	+	0.413057i	$0.864461\pi$
$212$	58.5572	−	33.8080i	4.02173	−	2.32194i
$213$		0			0
$214$	−23.0516	+	39.9266i	−1.57578	+	2.72933i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			27.2823i			1.84779i
$219$		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
$226$	−21.1144	−	36.5712i	−1.40451	−	2.43268i
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	41.3948			2.71770
$233$	−0.510712	+	0.294860i	−0.0334579	+	0.0193169i	−0.516636	−	0.856205i	$-0.672816\pi$
$233$	−0.510712	+	0.294860i	−0.0334579	+	0.0193169i	0.483178	+	0.875522i	$0.339482\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	30.0220i		−	1.94196i	−0.239158	−	0.970981i	$-0.576871\pi$
$239$		−	30.0220i		−	1.94196i	0.239158	−	0.970981i	$-0.423129\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	22.6962	+	13.1037i	1.45897	+	0.842335i
$243$		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
$253$	−8.58301			−0.539609
$254$	35.7209	−	20.6235i	2.24133	−	1.29403i
$255$		0			0
$256$	12.1458	−	21.0371i	0.759109	−	1.31482i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.8363	−	13.1845i	−1.40815	−	0.812993i	−0.412936	−	0.910760i	$-0.635497\pi$
$263$	−22.8363	−	13.1845i	−1.40815	−	0.812993i	−0.995209	+	0.0977664i	$0.968830\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−9.29150	−	16.0934i	−0.567569	−	0.983058i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	20.3542			1.22965
$275$	−3.95440	+	2.28308i	−0.238459	+	0.137675i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			33.3514i			1.98958i	0.101955	+	0.994789i	$0.467490\pi$
$281$			33.3514i			1.98958i	−0.101955	+	0.994789i	$0.532510\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	−30.4649	−	17.5889i	−1.80776	−	1.04371i
$285$		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
$296$	−62.5116	+	36.0911i	−3.63341	+	2.09775i
$297$		0			0
$298$	29.6974	−	51.4374i	1.72032	−	2.97969i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		−	13.6412i		−	0.784960i
$303$		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	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	−37.1660			−2.09075
$317$	−27.3014	+	15.7625i	−1.53340	+	0.885309i	−0.534198	+	0.845359i	$0.679386\pi$
$317$	−27.3014	+	15.7625i	−1.53340	+	0.885309i	−0.999202	+	0.0399492i	$0.987280\pi$
$318$		0			0
$319$	−2.77124	+	4.79993i	−0.155160	+	0.268745i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	44.6512	+	25.7794i	2.47300	+	1.42779i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.64575	−	4.58258i	−0.145424	−	0.251881i	0.784107	−	0.620625i	$-0.213121\pi$
$331$	−2.64575	−	4.58258i	−0.145424	−	0.251881i	−0.929531	+	0.368744i	$0.879788\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	21.1660			1.15299			0.576493	−	0.817102i	$-0.304421\pi$
$337$	21.1660			1.15299			0.576493	+	0.817102i	$0.304421\pi$
$338$	−29.0232	+	16.7566i	−1.57866	+	0.911438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$			36.0911i			1.94590i
$345$		0			0
$346$		0			0
$347$	−20.2331	−	11.6816i	−1.08617	−	0.627100i	−0.153616	−	0.988131i	$-0.549092\pi$
$347$	−20.2331	−	11.6816i	−1.08617	−	0.627100i	−0.932554	+	0.361030i	$0.882425\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−3.53137	−	6.11652i	−0.188223	−	0.326011i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−55.5203			−2.93434
$359$	27.5816	−	15.9242i	1.45570	−	0.840449i	0.456904	−	0.889516i	$-0.348958\pi$
$359$	27.5816	−	15.9242i	1.45570	−	0.840449i	0.998795	+	0.0490673i	$0.0156249\pi$
$360$		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	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−67.4874	−	38.9639i	−3.51803	−	2.03113i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	37.0405			1.90264			0.951322	−	0.308199i	$-0.0997264\pi$
$379$	37.0405			1.90264			0.951322	+	0.308199i	$0.0997264\pi$
$380$		0			0
$381$		0			0
$382$	16.8229	−	29.1381i	0.860733	−	1.49083i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		−	54.5646i		−	2.77727i
$387$		0			0
$388$		0			0
$389$	−16.7894	−	9.69337i	−0.851257	−	0.491473i	0.00981780	−	0.999952i	$-0.496875\pi$
$389$	−16.7894	−	9.69337i	−0.851257	−	0.491473i	−0.861075	+	0.508478i	$0.830208\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−14.0516	−	24.3381i	−0.707911	−	1.22614i
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	−41.4575			−2.07288
$401$	−7.85918	+	4.53750i	−0.392469	+	0.226592i	−0.683229	−	0.730204i	$-0.739425\pi$
$401$	−7.85918	+	4.53750i	−0.392469	+	0.226592i	0.290760	+	0.956796i	$0.406092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.66472i		−	0.479062i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		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	$0.281584\pi$
$422$	59.0679	−	34.1029i	2.87538	−	1.66010i
$423$		0			0
$424$	−49.6346	+	85.9697i	−2.41047	+	4.17506i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	83.0837i		−	4.01600i
$429$		0			0
$430$		0			0
$431$	−3.39407	−	1.95957i	−0.163486	−	0.0943889i	0.416024	−	0.909353i	$-0.363423\pi$
$431$	−3.39407	−	1.95957i	−0.163486	−	0.0943889i	−0.579511	+	0.814964i	$0.696756\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−24.5830	−	42.5790i	−1.17731	−	2.03916i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	34.9300	−	20.1669i	1.65958	−	0.958157i	0.686666	−	0.726973i	$-0.259073\pi$
$443$	34.9300	−	20.1669i	1.65958	−	0.958157i	0.972910	−	0.231184i	$-0.0742600\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	28.5190i		−	1.34590i	−0.739689	−	0.672948i	$-0.765028\pi$
$449$		−	28.5190i		−	1.34590i	0.739689	−	0.672948i	$-0.234972\pi$
$450$		0			0
$451$		0			0
$452$	65.9057	+	38.0507i	3.09994	+	1.78975i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	21.1660	+	36.6606i	0.990104	+	1.71491i	0.616585	+	0.787288i	$0.288516\pi$
$457$	21.1660	+	36.6606i	0.990104	+	1.71491i	0.373519	+	0.927622i	$0.378151\pi$
$458$		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	$0.879743\pi$
$464$	−43.5801	+	25.1610i	−2.02316	+	1.16807i
$465$		0			0
$466$	0.760130	−	1.31658i	0.0352123	−	0.0609895i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.18495	−	2.41618i	−0.192424	−	0.111096i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	38.6974	+	67.0258i	1.76998	+	3.06569i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−47.2288			−2.14676
$485$		0			0
$486$		0			0
$487$	−18.5203	+	32.0780i	−0.839233	+	1.45359i	0.0513038	+	0.998683i	$0.483662\pi$
$487$	−18.5203	+	32.0780i	−0.839233	+	1.45359i	−0.890537	+	0.454911i	$0.849671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			34.8544i			1.57296i	0.617619	+	0.786478i	$0.288097\pi$
$491$			34.8544i			1.57296i	−0.617619	+	0.786478i	$0.711903\pi$
$492$		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	0.993935	+	0.109965i	$0.0350740\pi$
$499$	13.2288	+	22.9129i	0.592200	+	1.02572i	−0.401735	+	0.915756i	$0.631593\pi$
$500$		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
$506$	19.1621	−	11.0632i	0.851857	−	0.491820i
$507$		0			0
$508$	−37.1660	+	64.3734i	−1.64898	+	2.85611i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$			48.9808i			2.16466i
$513$		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	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	67.9778			2.96397
$527$		0			0
$528$		0			0
$529$	32.6660	−	56.5792i	1.42026	−	2.45996i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	23.6272	+	13.6412i	1.02054	+	0.589208i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$		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	$0.110215\pi$
$548$	−31.7665	+	18.3404i	−1.35700	+	0.783463i
$549$		0			0
$550$	5.88562	−	10.1942i	0.250964	−	0.434682i
$551$		0			0
$552$		0			0
$553$		0			0
$554$			25.7794i			1.09526i
$555$		0			0
$556$		0			0
$557$	34.6499	+	20.0051i	1.46816	+	0.847644i	0.999364	−	0.0356614i	$-0.0113538\pi$
$557$	34.6499	+	20.0051i	1.46816	+	0.847644i	0.468798	+	0.883305i	$0.344687\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−42.9889	−	74.4589i	−1.81338	−	3.14086i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	51.6458			2.16701
$569$	−39.3952	+	22.7448i	−1.65153	+	0.953512i	−0.675088	+	0.737738i	$0.735894\pi$
$569$	−39.3952	+	22.7448i	−1.65153	+	0.953512i	−0.976443	+	0.215774i	$0.930772\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	46.9926i		−	1.95973i
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−37.9535	−	21.9125i	−1.57866	−	0.911438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.64575	−	11.5108i	−0.275239	−	0.476728i
$584$		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
$592$	43.8745	−	75.9929i	1.80323	−	3.12329i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$			107.037i			4.38439i
$597$		0			0
$598$		0			0
$599$	−42.2785	−	24.4095i	−1.72745	−	0.997346i	−0.900134	−	0.435614i	$-0.856531\pi$
$599$	−42.2785	−	24.4095i	−1.72745	−	0.997346i	−0.827319	−	0.561732i	$-0.810135\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	12.2915	+	21.2895i	0.500134	+	0.866258i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	0.171564	+	0.985173i	$0.445118\pi$
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	−0.938967	+	0.344008i	$0.888215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11.5485i		−	0.464924i	−0.972605	−	0.232462i	$-0.925322\pi$
$617$		−	11.5485i		−	0.464924i	0.972605	−	0.232462i	$-0.0746782\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		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	$0.603171\pi$
$632$	47.2543	−	27.2823i	1.87968	−	1.08523i
$633$		0			0
$634$	40.6346	−	70.3813i	1.61381	−	2.79520i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		−	14.2882i		−	0.565674i
$639$		0			0
$640$		0			0
$641$	15.2077	+	8.78014i	0.600666	+	0.346795i	0.769304	−	0.638883i	$-0.220603\pi$
$641$	15.2077	+	8.78014i	0.600666	+	0.346795i	−0.168637	+	0.985678i	$0.553937\pi$
$642$		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	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−92.9150			−3.63883
$653$	24.1379	−	13.9360i	0.944588	−	0.545358i	0.0531926	−	0.998584i	$-0.483060\pi$
$653$	24.1379	−	13.9360i	0.944588	−	0.545358i	0.891396	+	0.453226i	$0.149727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			12.4044i			0.483207i	0.970375	+	0.241604i	$0.0776734\pi$
$659$			12.4044i			0.483207i	−0.970375	+	0.241604i	$0.922327\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	11.8136	+	6.82058i	0.459148	+	0.265089i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−28.5203	−	49.3985i	−1.10431	−	1.91272i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−42.3320			−1.63178			−0.815890	−	0.578208i	$-0.803752\pi$
$673$	−42.3320			−1.63178			−0.815890	+	0.578208i	$0.803752\pi$
$674$	−47.2543	+	27.2823i	−1.82017	+	1.05088i
$675$		0			0
$676$	30.1974	−	52.3034i	1.16144	−	2.01167i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.6030	+	16.5139i	1.09446	+	0.631889i	0.934761	−	0.355277i	$-0.115613\pi$
$683$	28.6030	+	16.5139i	1.09446	+	0.631889i	0.159702	+	0.987165i	$0.448947\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−21.9373	−	37.9964i	−0.836350	−	1.44860i
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$	60.2288			2.28625
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			38.8308i			1.46662i	0.679895	+	0.733309i	$0.262025\pi$
$701$			38.8308i			1.46662i	−0.679895	+	0.733309i	$0.737975\pi$
$702$		0			0
$703$		0			0
$704$	2.65281	+	1.53160i	0.0999815	+	0.0577244i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−26.4575	−	45.8258i	−0.993633	−	1.72102i	−0.594389	−	0.804178i	$-0.702606\pi$
$709$	−26.4575	−	45.8258i	−0.993633	−	1.72102i	−0.399244	−	0.916845i	$-0.630727\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	86.6495	−	50.0271i	3.23824	−	1.86960i
$717$		0			0
$718$	−41.0516	+	71.1035i	−1.53203	+	2.65356i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		−	48.9808i		−	1.82288i
$723$		0			0
$724$		0			0
$725$	−26.2800	−	15.1728i	−0.976014	−	0.563502i
$726$		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	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	72.6863			2.67925
$737$	−3.16352	+	1.82646i	−0.116530	+	0.0672785i
$738$		0			0
$739$	26.0000	−	45.0333i	0.956425	−	1.65658i	0.225354	−	0.974277i	$-0.427646\pi$
$739$	26.0000	−	45.0333i	0.956425	−	1.65658i	0.731072	−	0.682300i	$-0.239020\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.09267i		−	0.0767726i	−0.999263	−	0.0383863i	$-0.987778\pi$
$743$		−	2.09267i		−	0.0767726i	0.999263	−	0.0383863i	$-0.0122217\pi$
$744$		0			0
$745$		0			0
$746$	−49.1163	−	28.3573i	−1.79827	−	1.03823i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	13.2288	+	22.9129i	0.482724	+	0.836103i	0.999803	−	0.0198348i	$-0.00631403\pi$
$751$	13.2288	+	22.9129i	0.482724	+	0.836103i	−0.517079	+	0.855938i	$0.672981\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.5830			−0.384646			−0.192323	−	0.981332i	$-0.561602\pi$
$757$	−10.5830			−0.384646			−0.192323	+	0.981332i	$0.561602\pi$
$758$	−82.6951	+	47.7440i	−3.00362	+	1.73414i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$			60.6337i			2.19365i
$765$		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
$772$	49.1660	+	85.1580i	1.76952	+	3.06490i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	49.9778			1.79179
$779$		0			0
$780$		0			0
$781$	−3.45751	+	5.98859i	−0.123720	+	0.214289i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	43.8603	+	25.3227i	1.56246	+	0.902085i
$789$		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
$800$	33.4884	−	19.3345i	1.18399	−	0.683578i
$801$		0			0
$802$	11.6974	−	20.2605i	0.413049	−	0.715422i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.65281	+	1.53160i	0.0932678	+	0.0538482i	0.545908	−	0.837845i	$-0.316185\pi$
$809$	2.65281	+	1.53160i	0.0932678	+	0.0538482i	−0.452641	+	0.891693i	$0.649518\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	12.4575	+	21.5770i	0.436636	+	0.756275i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.9315	−	10.9301i	0.660714	−	0.381464i	−0.131835	−	0.991272i	$-0.542087\pi$
$821$	18.9315	−	10.9301i	0.660714	−	0.381464i	0.792549	+	0.609808i	$0.208754\pi$
$822$		0			0
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	0.439961	+	0.898017i	$0.354992\pi$
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			57.3043i			1.99267i	0.0855616	+	0.996333i	$0.472732\pi$
$827$			57.3043i			1.99267i	−0.0855616	+	0.996333i	$0.527268\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−7.83399			−0.270138
$842$	−58.0465	+	33.5132i	−2.00041	+	1.15494i
$843$		0			0
$844$	−61.4575	+	106.448i	−2.11545	+	3.66408i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		−	120.678i		−	4.14409i
$849$		0			0
$850$		0			0
$851$	86.1388	+	49.7322i	2.95280	+	1.70480i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	60.9889	+	105.636i	2.08456	+	3.61056i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	10.1033			0.344119
$863$	−30.7451	+	17.7507i	−1.04658	+	0.604240i	−0.921689	−	0.387931i	$-0.873190\pi$
$863$	−30.7451	+	17.7507i	−1.04658	+	0.604240i	−0.124887	+	0.992171i	$0.539857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7.30584i			0.247834i
$870$		0			0
$871$		0			0
$872$	62.5116	+	36.0911i	2.11691	+	1.22220i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	−0.886323	−	0.463068i	$-0.846749\pi$
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	0.0421327	−	0.999112i	$-0.486585\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−58.2065			−1.95881			−0.979403	−	0.201916i	$-0.935283\pi$
$883$	−58.2065			−1.95881			−0.979403	+	0.201916i	$0.935283\pi$
$884$		0			0
$885$		0			0
$886$	−51.9889	+	90.0474i	−1.74660	+	3.02520i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	36.7601	+	63.6704i	1.22670	+	2.12471i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−111.727			−3.71598
$905$		0			0
$906$		0			0
$907$	−2.64575	+	4.58258i	−0.0878507	+	0.152162i	−0.906602	−	0.421986i	$-0.861333\pi$
$907$	−2.64575	+	4.58258i	−0.0878507	+	0.152162i	0.818752	+	0.574148i	$0.194667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	52.4719i		−	1.73847i	−0.494397	−	0.869236i	$-0.664611\pi$
$911$		−	52.4719i		−	1.73847i	0.494397	−	0.869236i	$-0.335389\pi$
$912$		0			0
$913$		0			0
$914$	−94.5087	−	54.5646i	−3.12607	−	1.80484i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−18.5203	−	32.0780i	−0.610927	−	1.05816i	−0.991084	−	0.133235i	$-0.957464\pi$
$919$	−18.5203	−	32.0780i	−0.610927	−	1.05816i	0.380158	−	0.924922i	$-0.375870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	52.9150			1.73984
$926$	89.3023	−	51.5587i	2.93466	−	1.69432i
$927$		0			0
$928$	23.4686	−	40.6489i	0.770395	−	1.33436i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$			2.73969i			0.0897416i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	12.4575			0.405029
$947$	−23.3966	+	13.5080i	−0.760288	+	0.438953i	−0.829399	−	0.558656i	$-0.811317\pi$
$947$	−23.3966	+	13.5080i	−0.760288	+	0.438953i	0.0691110	+	0.997609i	$0.477984\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.0456i		−	0.843699i	−0.906666	−	0.421849i	$-0.861381\pi$
$953$		−	26.0456i		−	0.843699i	0.906666	−	0.421849i	$-0.138619\pi$
$954$		0			0
$955$		0			0
$956$	−120.789	−	69.7374i	−3.90658	−	2.25547i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−40.0000			−1.28631			−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000			−1.28631			−0.643157	+	0.765735i	$0.722376\pi$
$968$	60.0485	−	34.6690i	1.93003	−	1.11430i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$		−	95.4881i		−	3.05963i
$975$		0			0
$976$		0			0
$977$	54.0921	+	31.2301i	1.73056	+	0.999139i	0.886024	+	0.463639i	$0.153456\pi$
$977$	54.0921	+	31.2301i	1.73056	+	0.999139i	0.844535	+	0.535500i	$0.179877\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−44.9261	−	77.8144i	−1.43365	−	2.48316i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43.0694	−	24.8661i	1.36953	−	0.790697i
$990$		0			0
$991$	29.1033	−	50.4083i	0.924496	−	1.60127i	0.132125	−	0.991233i	$-0.457820\pi$
$991$	29.1033	−	50.4083i	0.924496	−	1.60127i	0.792370	−	0.610040i	$-0.208847\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−59.0679	−	34.1029i	−1.86976	−	1.07951i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.p.b.80.1		8
3.2	odd	2	inner	441.2.p.b.80.4		8
7.2	even	3	inner	441.2.p.b.215.4		8
7.3	odd	6		63.2.c.a.62.1	✓	4
7.4	even	3		63.2.c.a.62.1	✓	4
7.5	odd	6	inner	441.2.p.b.215.4		8
7.6	odd	2	CM	441.2.p.b.80.1		8
21.2	odd	6	inner	441.2.p.b.215.1		8
21.5	even	6	inner	441.2.p.b.215.1		8
21.11	odd	6		63.2.c.a.62.4	yes	4
21.17	even	6		63.2.c.a.62.4	yes	4
21.20	even	2	inner	441.2.p.b.80.4		8
28.3	even	6		1008.2.k.a.881.1		4
28.11	odd	6		1008.2.k.a.881.1		4
35.3	even	12		1575.2.g.d.1574.1		8
35.4	even	6		1575.2.b.a.251.4		4
35.17	even	12		1575.2.g.d.1574.8		8
35.18	odd	12		1575.2.g.d.1574.1		8
35.24	odd	6		1575.2.b.a.251.4		4
35.32	odd	12		1575.2.g.d.1574.8		8
56.3	even	6		4032.2.k.b.3905.2		4
56.11	odd	6		4032.2.k.b.3905.2		4
56.45	odd	6		4032.2.k.c.3905.3		4
56.53	even	6		4032.2.k.c.3905.3		4
63.4	even	3		567.2.o.f.188.4		8
63.11	odd	6		567.2.o.f.377.4		8
63.25	even	3		567.2.o.f.377.1		8
63.31	odd	6		567.2.o.f.188.4		8
63.32	odd	6		567.2.o.f.188.1		8
63.38	even	6		567.2.o.f.377.4		8
63.52	odd	6		567.2.o.f.377.1		8
63.59	even	6		567.2.o.f.188.1		8
84.11	even	6		1008.2.k.a.881.2		4
84.59	odd	6		1008.2.k.a.881.2		4
105.17	odd	12		1575.2.g.d.1574.2		8
105.32	even	12		1575.2.g.d.1574.2		8
105.38	odd	12		1575.2.g.d.1574.7		8
105.53	even	12		1575.2.g.d.1574.7		8
105.59	even	6		1575.2.b.a.251.1		4
105.74	odd	6		1575.2.b.a.251.1		4
168.11	even	6		4032.2.k.b.3905.1		4
168.53	odd	6		4032.2.k.c.3905.4		4
168.59	odd	6		4032.2.k.b.3905.1		4
168.101	even	6		4032.2.k.c.3905.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.c.a.62.1	✓	4	7.3	odd	6
63.2.c.a.62.1	✓	4	7.4	even	3
63.2.c.a.62.4	yes	4	21.11	odd	6
63.2.c.a.62.4	yes	4	21.17	even	6
441.2.p.b.80.1		8	1.1	even	1	trivial
441.2.p.b.80.1		8	7.6	odd	2	CM
441.2.p.b.80.4		8	3.2	odd	2	inner
441.2.p.b.80.4		8	21.20	even	2	inner
441.2.p.b.215.1		8	21.2	odd	6	inner
441.2.p.b.215.1		8	21.5	even	6	inner
441.2.p.b.215.4		8	7.2	even	3	inner
441.2.p.b.215.4		8	7.5	odd	6	inner
567.2.o.f.188.1		8	63.32	odd	6
567.2.o.f.188.1		8	63.59	even	6
567.2.o.f.188.4		8	63.4	even	3
567.2.o.f.188.4		8	63.31	odd	6
567.2.o.f.377.1		8	63.25	even	3
567.2.o.f.377.1		8	63.52	odd	6
567.2.o.f.377.4		8	63.11	odd	6
567.2.o.f.377.4		8	63.38	even	6
1008.2.k.a.881.1		4	28.3	even	6
1008.2.k.a.881.1		4	28.11	odd	6
1008.2.k.a.881.2		4	84.11	even	6
1008.2.k.a.881.2		4	84.59	odd	6
1575.2.b.a.251.1		4	105.59	even	6
1575.2.b.a.251.1		4	105.74	odd	6
1575.2.b.a.251.4		4	35.4	even	6
1575.2.b.a.251.4		4	35.24	odd	6
1575.2.g.d.1574.1		8	35.3	even	12
1575.2.g.d.1574.1		8	35.18	odd	12
1575.2.g.d.1574.2		8	105.17	odd	12
1575.2.g.d.1574.2		8	105.32	even	12
1575.2.g.d.1574.7		8	105.38	odd	12
1575.2.g.d.1574.7		8	105.53	even	12
1575.2.g.d.1574.8		8	35.17	even	12
1575.2.g.d.1574.8		8	35.32	odd	12
4032.2.k.b.3905.1		4	168.11	even	6
4032.2.k.b.3905.1		4	168.59	odd	6
4032.2.k.b.3905.2		4	56.3	even	6
4032.2.k.b.3905.2		4	56.11	odd	6
4032.2.k.c.3905.3		4	56.45	odd	6
4032.2.k.c.3905.3		4	56.53	even	6
4032.2.k.c.3905.4		4	168.53	odd	6
4032.2.k.c.3905.4		4	168.101	even	6

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)

\(f(q)\)	\(=\)	\(q+(-2.23256 + 1.28897i) q^{2} +(2.32288 - 4.02334i) q^{4} +6.82058i q^{8} +O(q^{10})\)	\(q+(-2.23256 + 1.28897i) q^{2} +(2.32288 - 4.02334i) q^{4} +6.82058i q^{8} +(-0.790881 - 0.456615i) q^{11} +(-4.14575 - 7.18065i) q^{16} +2.35425 q^{22} +(8.13935 - 4.69926i) q^{23} +(2.50000 - 4.33013i) q^{25} -6.06910i q^{29} +(6.69767 + 3.86690i) q^{32} +(5.29150 + 9.16515i) q^{37} +5.29150 q^{43} +(-3.67423 + 2.12132i) q^{44} +(-12.1144 + 20.9827i) q^{46} +12.8897i q^{50} +(12.6045 + 7.27719i) q^{53} +(7.82288 + 13.5496i) q^{58} -3.35425 q^{64} +(2.00000 - 3.46410i) q^{67} -7.57205i q^{71} +(-23.6272 - 13.6412i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-11.8136 + 6.82058i) q^{86} +(3.11438 - 5.39426i) q^{88} -43.6631i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 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

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Database

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