LMFDB - Embedded newform 804.2.s.a.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(5,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 5]))

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

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

chi := DirichletCharacter("804.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.s (of order $22$, degree $10$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			53.2
Root			$-0.327068 - 0.945001i$ of defining polynomial
Character	$\chi$	$=$	804.53
Dual form			804.2.s.a.713.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.936417 - 1.45709i) q^{3} +(4.81332 - 2.19817i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})$	$q+(0.936417 - 1.45709i) q^{3} +(4.81332 - 2.19817i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(-0.298303 - 1.01593i) q^{13} +(-2.01775 + 4.41826i) q^{19} +(1.30434 - 9.07186i) q^{21} +(4.79746 - 1.40866i) q^{25} +(-5.14326 - 0.739490i) q^{27} +(-1.53214 + 5.21798i) q^{31} -7.64303 q^{37} +(-1.75964 - 0.516676i) q^{39} +(5.83248 - 5.05388i) q^{43} +(13.7521 - 15.8707i) q^{49} +(4.54836 + 7.07739i) q^{57} +(1.72288 + 0.247713i) q^{61} +(-11.9972 - 10.3956i) q^{63} +(-8.16392 + 0.591970i) q^{67} +(-1.74937 + 12.1671i) q^{73} +(2.43988 - 8.30945i) q^{75} +(-4.36459 - 14.8644i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(-3.66900 - 4.23426i) q^{91} +(6.16836 + 7.11867i) q^{93} +19.2257i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^9	$ 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) $ 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{19}{22}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.936417	−	1.45709i	0.540641	−	0.841254i
$3$	0.936417	−	1.45709i	0.540641	−	0.841254i
$4$		0			0
$5$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$5$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$6$		0			0
$7$	4.81332	−	2.19817i	1.81926	−	0.830830i	0.909518	−	0.415664i	$-0.136451\pi$
$7$	4.81332	−	2.19817i	1.81926	−	0.830830i	0.909746	−	0.415166i	$-0.136276\pi$
$8$		0			0
$9$	−1.24625	−	2.72890i	−0.415415	−	0.909632i
$10$		0			0
$11$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$11$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$12$		0			0
$13$	−0.298303	−	1.01593i	−0.0827343	−	0.281767i	0.907726	−	0.419563i	$-0.137817\pi$
$13$	−0.298303	−	1.01593i	−0.0827343	−	0.281767i	−0.990461	+	0.137795i	$0.955998\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$17$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$18$		0			0
$19$	−2.01775	+	4.41826i	−0.462904	+	1.01362i	0.523912	+	0.851772i	$0.324472\pi$
$19$	−2.01775	+	4.41826i	−0.462904	+	1.01362i	−0.986816	+	0.161846i	$0.948255\pi$
$20$		0			0
$21$	1.30434	−	9.07186i	0.284630	−	1.97964i
$22$		0			0
$23$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$23$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$24$		0			0
$25$	4.79746	−	1.40866i	0.959493	−	0.281733i
$26$		0			0
$27$	−5.14326	−	0.739490i	−0.989821	−	0.142315i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.53214	+	5.21798i	−0.275180	+	0.937176i	0.699699	+	0.714438i	$0.253318\pi$
$31$	−1.53214	+	5.21798i	−0.275180	+	0.937176i	−0.974878	+	0.222738i	$0.928501\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.64303			−1.25651			−0.628253	−	0.778009i	$-0.716230\pi$
$37$	−7.64303			−1.25651			−0.628253	+	0.778009i	$0.716230\pi$
$38$		0			0
$39$	−1.75964	−	0.516676i	−0.281767	−	0.0827343i
$40$		0			0
$41$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$41$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$42$		0			0
$43$	5.83248	−	5.05388i	0.889446	−	0.770709i	−0.0847529	−	0.996402i	$-0.527010\pi$
$43$	5.83248	−	5.05388i	0.889446	−	0.770709i	0.974198	+	0.225693i	$0.0724646\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$47$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$48$		0			0
$49$	13.7521	−	15.8707i	1.96458	−	2.26725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$53$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.54836	+	7.07739i	0.602445	+	0.937423i
$58$		0			0
$59$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$59$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$60$		0			0
$61$	1.72288	+	0.247713i	0.220592	+	0.0317164i	0.251725	−	0.967799i	$-0.419002\pi$
$61$	1.72288	+	0.247713i	0.220592	+	0.0317164i	−0.0311325	+	0.999515i	$0.509911\pi$
$62$		0			0
$63$	−11.9972	−	10.3956i	−1.51150	−	1.30972i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.16392	+	0.591970i	−0.997381	+	0.0723206i
$67$	−8.16392	+	0.591970i	−0.997381	+	0.0723206i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$71$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$72$		0			0
$73$	−1.74937	+	12.1671i	−0.204748	+	1.42405i	0.585206	+	0.810885i	$0.301014\pi$
$73$	−1.74937	+	12.1671i	−0.204748	+	1.42405i	−0.789953	+	0.613167i	$0.789895\pi$
$74$		0			0
$75$	2.43988	−	8.30945i	0.281733	−	0.959493i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.36459	−	14.8644i	−0.491055	−	1.67238i	−0.716073	−	0.698026i	$-0.754062\pi$
$79$	−4.36459	−	14.8644i	−0.491055	−	1.67238i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$	−5.89375	+	6.80175i	−0.654861	+	0.755750i
$82$		0			0
$83$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$83$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$89$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$90$		0			0
$91$	−3.66900	−	4.23426i	−0.384616	−	0.443871i
$92$		0			0
$93$	6.16836	+	7.11867i	0.639629	+	0.738172i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.2257i			1.95208i	0.217599	+	0.976038i	$0.430177\pi$
$97$			19.2257i			1.95208i	−0.217599	+	0.976038i	$0.569823\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$101$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$102$		0			0
$103$	4.94834	+	1.45296i	0.487574	+	0.143165i	0.516274	−	0.856423i	$-0.327319\pi$
$103$	4.94834	+	1.45296i	0.487574	+	0.143165i	−0.0286999	+	0.999588i	$0.509137\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$107$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$108$		0			0
$109$	0.275623	+	0.938687i	0.0263999	+	0.0899099i	0.971639	−	0.236468i	$-0.0759899\pi$
$109$	0.275623	+	0.938687i	0.0263999	+	0.0899099i	−0.945239	+	0.326378i	$0.894172\pi$
$110$		0			0
$111$	−7.15707	+	11.1366i	−0.679319	+	1.05704i
$112$		0			0
$113$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$113$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.40060	+	2.08013i	−0.221935	+	0.192308i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5544	−	3.09906i	0.959493	−	0.281733i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.18803	+	20.1190i	0.815306	+	1.78527i	0.582675	+	0.812705i	$0.302006\pi$
$127$	9.18803	+	20.1190i	0.815306	+	1.78527i	0.232631	+	0.972565i	$0.425267\pi$
$128$		0			0
$129$	−1.90233	−	13.2310i	−0.167491	−	1.16493i
$130$		0			0
$131$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$131$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$132$		0			0
$133$			25.7019i			2.22863i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$137$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$138$		0			0
$139$	−5.86042	+	0.842601i	−0.497074	+	0.0714685i	−0.386293	−	0.922376i	$-0.626245\pi$
$139$	−5.86042	+	0.842601i	−0.497074	+	0.0714685i	−0.110782	+	0.993845i	$0.535335\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−10.2475	−	34.8997i	−0.845198	−	2.87848i
$148$		0			0
$149$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$149$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$150$		0			0
$151$	15.3451	+	17.7092i	1.24877	+	1.44115i	0.852256	+	0.523125i	$0.175234\pi$
$151$	15.3451	+	17.7092i	1.24877	+	1.44115i	0.396511	+	0.918030i	$0.370221\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	20.7976	+	13.3658i	1.65983	+	1.06671i	0.918381	+	0.395698i	$0.129497\pi$
$157$	20.7976	+	13.3658i	1.65983	+	1.06671i	0.741448	+	0.671010i	$0.234139\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−23.5648			−1.84574			−0.922870	−	0.385111i	$-0.874163\pi$
$163$	−23.5648			−1.84574			−0.922870	+	0.385111i	$0.874163\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$167$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$168$		0			0
$169$	9.99317	−	6.42222i	0.768706	−	0.494017i
$170$		0			0
$171$	14.5716			1.11432
$172$		0			0
$173$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$173$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$174$		0			0
$175$	19.9953	−	17.3260i	1.51150	−	1.30972i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$179$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$180$		0			0
$181$	21.5515	+	13.8503i	1.60191	+	1.02948i	0.966282	+	0.257485i	$0.0828937\pi$
$181$	21.5515	+	13.8503i	1.60191	+	1.02948i	0.635624	+	0.771998i	$0.280743\pi$
$182$		0			0
$183$	1.97427	−	2.27843i	0.145943	−	0.168427i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−26.3817	+	7.74636i	−1.91899	+	0.563465i
$190$		0			0
$191$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$191$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$192$		0			0
$193$	−24.9845	−	7.33612i	−1.79842	−	0.528065i	−0.800927	−	0.598762i	$-0.795660\pi$
$193$	−24.9845	−	7.33612i	−1.79842	−	0.528065i	−0.997498	+	0.0706968i	$0.977478\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$197$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$198$		0			0
$199$	−6.52954	−	14.2977i	−0.462867	−	1.01354i	−0.986825	−	0.161793i	$-0.948272\pi$
$199$	−6.52954	−	14.2977i	−0.462867	−	1.01354i	0.523958	−	0.851744i	$-0.324455\pi$
$200$		0			0
$201$	−6.78228	+	12.4499i	−0.478385	+	0.878150i
$202$		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
$211$	−22.1072	+	14.2075i	−1.52193	+	0.978082i	−0.530460	+	0.847710i	$0.677981\pi$
$211$	−22.1072	+	14.2075i	−1.52193	+	0.978082i	−0.991465	+	0.130372i	$0.958383\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.09534	+	28.4837i	0.278010	+	1.93360i
$218$		0			0
$219$	16.0905	+	13.9425i	1.08729	+	0.942145i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−15.0892	+	9.69726i	−1.01045	+	0.649376i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−15.0892	+	9.69726i	−1.01045	+	0.649376i	−0.0729398	+	0.997336i	$0.523238\pi$
$224$		0			0
$225$	−9.82291	−	11.3362i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$227$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$228$		0			0
$229$	−7.36472	+	25.0819i	−0.486674	+	1.65746i	0.240226	+	0.970717i	$0.422778\pi$
$229$	−7.36472	+	25.0819i	−0.486674	+	1.65746i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$233$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−25.7460	−	7.55970i	−1.67238	−	0.491055i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	3.02284	−	21.0243i	0.194718	−	1.35429i	−0.624595	−	0.780949i	$-0.714736\pi$
$241$	3.02284	−	21.0243i	0.194718	−	1.35429i	0.819313	−	0.573346i	$-0.194355\pi$
$242$		0			0
$243$	4.39178	+	14.9570i	0.281733	+	0.959493i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.09053	+	0.731907i	0.323902	+	0.0465701i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$251$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$257$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$258$		0			0
$259$	−36.7883	+	16.8007i	−2.28592	+	1.04394i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$263$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	9.87994	−	15.3735i	0.600164	−	0.933873i	−0.399688	−	0.916651i	$-0.630882\pi$
$271$	9.87994	−	15.3735i	0.600164	−	0.933873i	0.999852	−	0.0172215i	$-0.00548205\pi$
$272$		0			0
$273$	−9.60543	+	1.38105i	−0.581347	+	0.0835851i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	13.5983	+	29.7762i	0.817045	+	1.78908i	0.573537	+	0.819180i	$0.305571\pi$
$277$	13.5983	+	29.7762i	0.817045	+	1.78908i	0.243508	+	0.969899i	$0.421702\pi$
$278$		0			0
$279$	16.1487	−	2.32184i	0.966799	−	0.139005i
$280$		0			0
$281$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$281$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$282$		0			0
$283$	2.38135	−	5.21442i	0.141556	−	0.309965i	−0.825554	−	0.564324i	$-0.809137\pi$
$283$	2.38135	−	5.21442i	0.141556	−	0.309965i	0.967110	+	0.254358i	$0.0818643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.41935	+	16.8270i	−0.142315	+	0.989821i
$290$		0			0
$291$	28.0137	+	18.0033i	1.64219	+	1.05537i
$292$		0			0
$293$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$293$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	16.9643	−	37.1467i	0.977808	−	2.14110i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−32.6609	−	9.59009i	−1.86405	−	0.547336i	−0.998954	−	0.0457370i	$-0.985436\pi$
$307$	−32.6609	−	9.59009i	−1.86405	−	0.547336i	−0.865100	−	0.501599i	$-0.832745\pi$
$308$		0			0
$309$	6.75081	−	5.84961i	0.384040	−	0.332773i
$310$		0			0
$311$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$311$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$312$		0			0
$313$	−17.0433	−	26.5199i	−0.963343	−	1.49899i	−0.863724	−	0.503966i	$-0.831874\pi$
$313$	−17.0433	−	26.5199i	−0.963343	−	1.49899i	−0.0996196	−	0.995026i	$-0.531763\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$317$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.86219	−	4.45366i	−0.158766	−	0.247045i
$326$		0			0
$327$	1.62585	+	0.477394i	0.0899099	+	0.0263999i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−24.0868	−	20.8714i	−1.32393	−	1.14719i	−0.977924	−	0.208962i	$-0.932991\pi$
$331$	−24.0868	−	20.8714i	−1.32393	−	1.14719i	−0.346008	−	0.938231i	$-0.612463\pi$
$332$		0			0
$333$	9.52509	+	20.8570i	0.521972	+	1.14296i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.8070	−	9.50222i	1.13343	−	0.517619i	0.241771	−	0.970333i	$-0.422272\pi$
$337$	20.8070	−	9.50222i	1.13343	−	0.517619i	0.891656	+	0.452715i	$0.149544\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	20.8710	−	71.0802i	1.12693	−	3.83797i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$347$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$348$		0			0
$349$	24.4436	−	28.2094i	1.30844	−	1.51001i	0.620730	−	0.784024i	$-0.286836\pi$
$349$	24.4436	−	28.2094i	1.30844	−	1.51001i	0.687705	−	0.725991i	$-0.258618\pi$
$350$		0			0
$351$	0.782983	+	5.44577i	0.0417925	+	0.290674i
$352$		0			0
$353$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$353$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$359$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$360$		0			0
$361$	−3.00734	−	3.47065i	−0.158281	−	0.182666i
$362$		0			0
$363$	5.36773	−	18.2808i	0.281733	−	0.959493i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.791302	−	1.23129i	−0.0413056	−	0.0642728i	0.819987	−	0.572383i	$-0.193981\pi$
$367$	−0.791302	−	1.23129i	−0.0413056	−	0.0642728i	−0.861292	+	0.508110i	$0.830344\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	31.6879i		−	1.64074i	−0.571834	−	0.820370i	$-0.693768\pi$
$373$		−	31.6879i		−	1.64074i	0.571834	−	0.820370i	$-0.306232\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−4.62377	+	7.19472i	−0.237507	+	0.369568i	−0.939462	−	0.342653i	$-0.888674\pi$
$379$	−4.62377	+	7.19472i	−0.237507	+	0.369568i	0.701955	+	0.712221i	$0.252311\pi$
$380$		0			0
$381$	37.9191	+	5.45194i	1.94265	+	0.279311i
$382$		0			0
$383$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$383$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−21.0602	−	9.61787i	−1.07055	−	0.488904i
$388$		0			0
$389$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$389$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.933202	+	6.49056i	0.0468361	+	0.325752i	0.999747	+	0.0225011i	$0.00716292\pi$
$397$	0.933202	+	6.49056i	0.0468361	+	0.325752i	−0.952911	+	0.303251i	$0.901928\pi$
$398$		0			0
$399$	37.4500	+	24.0677i	1.87485	+	1.20489i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	5.75812			0.286832
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.4204	−	14.8059i	1.60309	−	0.732105i	0.605137	−	0.796121i	$-0.293118\pi$
$409$	32.4204	−	14.8059i	1.60309	−	0.732105i	0.997949	+	0.0640160i	$0.0203909\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.26005	+	9.32821i	−0.208616	+	0.456804i
$418$		0			0
$419$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$419$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$420$		0			0
$421$	−12.8232	+	28.0788i	−0.624963	+	1.36848i	0.286890	+	0.957963i	$0.407378\pi$
$421$	−12.8232	+	28.0788i	−0.624963	+	1.36848i	−0.911854	+	0.410515i	$0.865349\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.83728	−	2.59486i	0.427666	−	0.125574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	11.0783	−	37.7291i	0.532387	−	1.81314i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	11.0783	−	37.7291i	0.532387	−	1.81314i	0.580444	−	0.814300i	$-0.302879\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	5.22584			0.249416			0.124708	−	0.992194i	$-0.460201\pi$
$439$	5.22584			0.249416			0.124708	+	0.992194i	$0.460201\pi$
$440$		0			0
$441$	−60.4481	−	17.7491i	−2.87848	−	0.845198i
$442$		0			0
$443$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$443$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$449$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	40.1734	−	5.77606i	1.88751	−	0.271383i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.2951	+	9.48271i	−1.51070	+	0.443582i	−0.929081	−	0.369877i	$-0.879400\pi$
$457$	−32.2951	+	9.48271i	−1.51070	+	0.443582i	−0.581622	+	0.813459i	$0.697582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$461$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$462$		0			0
$463$	−2.33915	−	0.336319i	−0.108709	−	0.0156301i	0.0877454	−	0.996143i	$-0.472034\pi$
$463$	−2.33915	−	0.336319i	−0.108709	−	0.0156301i	−0.196455	+	0.980513i	$0.562943\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$467$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$468$		0			0
$469$	−37.9943	+	20.7950i	−1.75441	+	0.960225i
$470$		0			0
$471$	38.9505	−	17.7881i	1.79474	−	0.819632i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−3.45625	+	24.0388i	−0.158584	+	1.10297i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$479$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$480$		0			0
$481$	2.27994	+	7.76475i	0.103956	+	0.354042i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−26.8453	−	23.2616i	−1.21648	−	1.05408i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−26.8453	−	23.2616i	−1.21648	−	1.05408i	−0.219564	−	0.975598i	$-0.570463\pi$
$488$		0			0
$489$	−22.0665	+	34.3362i	−0.997883	+	1.55274i
$490$		0			0
$491$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$491$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	38.9295i		−	1.74272i	−0.490642	−	0.871361i	$-0.663238\pi$
$499$		−	38.9295i		−	1.74272i	0.490642	−	0.871361i	$-0.336762\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$503$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	20.5749i		−	0.913762i
$508$		0			0
$509$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$509$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$510$		0			0
$511$	18.3251	+	62.4096i	0.810655	+	2.76084i
$512$		0			0
$513$	13.6451	−	21.2322i	0.602445	−	0.937423i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$521$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$522$		0			0
$523$	42.4931	−	12.4771i	1.85809	−	0.545586i	0.858640	−	0.512579i	$-0.171310\pi$
$523$	42.4931	−	12.4771i	1.85809	−	0.545586i	0.999455	−	0.0330064i	$-0.0105082\pi$
$524$		0			0
$525$	−6.52168	−	45.3593i	−0.284630	−	1.97964i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.55455	+	20.9215i	0.415415	+	0.909632i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	29.9198	−	4.30182i	1.28635	−	0.184950i	0.535001	−	0.844851i	$-0.320311\pi$
$541$	29.9198	−	4.30182i	1.28635	−	0.184950i	0.751352	+	0.659902i	$0.229402\pi$
$542$		0			0
$543$	40.3623	−	18.4328i	1.73211	−	0.791029i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−45.6402	+	6.56207i	−1.95143	+	0.280574i	−0.999656	−	0.0262168i	$-0.991654\pi$
$547$	−45.6402	+	6.56207i	−1.95143	+	0.280574i	−0.951777	+	0.306791i	$0.900745\pi$
$548$		0			0
$549$	−1.47115	−	5.01027i	−0.0627871	−	0.213833i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−53.6827	−	61.9532i	−2.28282	−	2.63452i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$557$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$558$		0			0
$559$	−6.87421	−	4.41779i	−0.290748	−	0.186853i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$563$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−13.4171	+	45.6944i	−0.563465	+	1.91899i
$568$		0			0
$569$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$569$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$570$		0			0
$571$	−31.8052	+	20.4400i	−1.33101	+	0.855386i	−0.996216	−	0.0869073i	$-0.972302\pi$
$571$	−31.8052	+	20.4400i	−1.33101	+	0.855386i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	32.9203	−	28.5256i	1.37049	−	1.18754i	0.409054	−	0.912510i	$-0.365859\pi$
$577$	32.9203	−	28.5256i	1.37049	−	1.18754i	0.961437	−	0.275027i	$-0.0886866\pi$
$578$		0			0
$579$	−34.0853	+	29.5351i	−1.41654	+	1.22744i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$587$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$588$		0			0
$589$	−19.9629	−	17.2980i	−0.822557	−	0.712750i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$593$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−26.9475	−	3.87446i	−1.10289	−	0.158571i
$598$		0			0
$599$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$599$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$600$		0			0
$601$	−15.5556	−	34.0620i	−0.634526	−	1.38942i	−0.904468	−	0.426542i	$-0.859732\pi$
$601$	−15.5556	−	34.0620i	−0.634526	−	1.38942i	0.269942	−	0.962877i	$-0.412995\pi$
$602$		0			0
$603$	11.7897	+	21.5407i	0.480112	+	0.877207i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−32.2663	+	37.2373i	−1.30965	+	1.51142i	−0.646928	+	0.762551i	$0.723947\pi$
$607$	−32.2663	+	37.2373i	−1.30965	+	1.51142i	−0.662722	+	0.748866i	$0.730599\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	36.3426	−	23.3560i	1.46786	−	0.943338i	0.469696	−	0.882828i	$-0.344364\pi$
$613$	36.3426	−	23.3560i	1.46786	−	0.943338i	0.998168	−	0.0605099i	$-0.0192727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$617$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$	6.01584	+	41.8411i	0.241797	+	1.68173i	0.643094	+	0.765787i	$0.277650\pi$
$619$	6.01584	+	41.8411i	0.241797	+	1.68173i	−0.401297	+	0.915948i	$0.631441\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.0313	−	13.5160i	0.841254	−	0.540641i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.954758	−	3.25161i	0.0380083	−	0.129444i	−0.938303	−	0.345813i	$-0.887603\pi$
$631$	0.954758	−	3.25161i	0.0380083	−	0.129444i	0.976312	+	0.216369i	$0.0694213\pi$
$632$		0			0
$633$			45.5164i			1.80912i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−20.2258	−	9.23681i	−0.801375	−	0.365976i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	0.374642	−	2.60569i	0.0147745	−	0.102759i	−0.981100	−	0.193502i	$-0.938015\pi$
$643$	0.374642	−	2.60569i	0.0147745	−	0.102759i	0.995874	+	0.0907437i	$0.0289244\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$647$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	45.3384	+	20.7053i	1.77695	+	0.811506i
$652$		0			0
$653$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$653$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	35.3829	−	10.3894i	1.38042	−	0.405327i
$658$		0			0
$659$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$659$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$660$		0			0
$661$	−46.7583	+	21.3538i	−1.81869	+	0.830566i	−0.898879	+	0.438196i	$0.855618\pi$
$661$	−46.7583	+	21.3538i	−1.81869	+	0.830566i	−0.919807	+	0.392370i	$0.871655\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			31.0671i			1.20112i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−27.8230	+	43.2935i	−1.07250	+	1.66884i	−0.429243	+	0.903189i	$0.641220\pi$
$673$	−27.8230	+	43.2935i	−1.07250	+	1.66884i	−0.643255	+	0.765652i	$0.722417\pi$
$674$		0			0
$675$	−25.7163	+	3.69745i	−0.989821	+	0.142315i
$676$		0			0
$677$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$677$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$678$		0			0
$679$	42.2614	+	92.5396i	1.62184	+	3.55134i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$683$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	29.6503	+	34.2182i	1.13123	+	1.30551i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.990982	−	6.89243i	0.0376987	−	0.262201i	−0.962252	−	0.272161i	$-0.912262\pi$
$691$	0.990982	−	6.89243i	0.0376987	−	0.262201i	0.999950	+	0.00996078i	$0.00317067\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$701$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$702$		0			0
$703$	15.4217	−	33.7689i	0.581642	−	1.27362i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−42.9155	−	12.6011i	−1.61173	−	0.473245i	−0.652948	−	0.757403i	$-0.726468\pi$
$709$	−42.9155	−	12.6011i	−1.61173	−	0.473245i	−0.958778	+	0.284158i	$0.908286\pi$
$710$		0			0
$711$	−35.1242	+	30.4353i	−1.31726	+	1.14141i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$719$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$720$		0			0
$721$	27.0118	−	3.88371i	1.00597	−	0.144637i
$722$		0			0
$723$	−27.8037	−	24.0921i	−1.03403	−	0.895994i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	29.0159	+	45.1497i	1.07614	+	1.67451i	0.620002	+	0.784600i	$0.287132\pi$
$727$	29.0159	+	45.1497i	1.07614	+	1.67451i	0.456140	+	0.889908i	$0.349232\pi$
$728$		0			0
$729$	25.9063	+	7.60678i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−40.8980	−	35.4383i	−1.51060	−	1.30894i	−0.777849	−	0.628451i	$-0.783689\pi$
$733$	−40.8980	−	35.4383i	−1.51060	−	1.30894i	−0.732754	−	0.680494i	$-0.761765\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−19.9533	+	9.11236i	−0.733994	+	0.335204i	−0.747101	−	0.664710i	$-0.768555\pi$
$739$	−19.9533	+	9.11236i	−0.733994	+	0.335204i	0.0131073	+	0.999914i	$0.495828\pi$
$740$		0			0
$741$	5.83331	−	6.73200i	0.214292	−	0.247306i
$742$		0			0
$743$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$743$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−13.7277	+	15.8426i	−0.500931	+	0.578105i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−13.7277	+	15.8426i	−0.500931	+	0.578105i	0.447822	+	0.894123i	$0.352200\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	10.1821	−	15.8437i	0.370076	−	0.575849i	−0.605409	−	0.795914i	$-0.706991\pi$
$757$	10.1821	−	15.8437i	0.370076	−	0.575849i	0.975485	+	0.220065i	$0.0706269\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$761$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$762$		0			0
$763$	3.39006	+	3.91233i	0.122728	+	0.141636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−29.0562	−	45.2123i	−1.04779	−	1.63040i	−0.731643	−	0.681688i	$-0.761246\pi$
$769$	−29.0562	−	45.2123i	−1.04779	−	1.63040i	−0.316150	−	0.948709i	$-0.602390\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$773$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$774$		0			0
$775$			27.1913i			0.976741i
$776$		0			0
$777$	−9.96909	+	69.3365i	−0.357639	+	2.48743i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	9.41153	−	8.15514i	0.335485	−	0.290699i	−0.470784	−	0.882248i	$-0.656029\pi$
$787$	9.41153	−	8.15514i	0.335485	−	0.290699i	0.806269	+	0.591549i	$0.201483\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.262282	−	1.82421i	−0.00931391	−	0.0647796i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$797$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$809$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$810$		0			0
$811$	5.28103	−	2.41177i	0.185442	−	0.0846886i	−0.320530	−	0.947238i	$-0.603861\pi$
$811$	5.28103	−	2.41177i	0.185442	−	0.0846886i	0.505972	+	0.862550i	$0.331134\pi$
$812$		0			0
$813$	−13.1489	−	28.7920i	−0.461151	−	1.00978i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	10.5608	+	35.9669i	0.369477	+	1.25832i
$818$		0			0
$819$	−6.98237	+	15.2893i	−0.243984	+	0.534250i
$820$		0			0
$821$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$821$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$822$		0			0
$823$	18.1366	−	39.7136i	0.632202	−	1.38433i	−0.274101	−	0.961701i	$-0.588380\pi$
$823$	18.1366	−	39.7136i	0.632202	−	1.38433i	0.906303	−	0.422628i	$-0.138892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$827$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$828$		0			0
$829$	3.78391	−	1.11106i	0.131421	−	0.0385886i	−0.215361	−	0.976535i	$-0.569093\pi$
$829$	3.78391	−	1.11106i	0.131421	−	0.0385886i	0.346781	+	0.937946i	$0.387275\pi$
$830$		0			0
$831$	56.1204	+	8.06890i	1.94680	+	0.279907i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	11.7388	−	25.7044i	0.405753	−	0.888475i
$838$		0			0
$839$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$839$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	43.9896	−	38.1172i	1.51150	−	1.30972i
$848$		0			0
$849$	−5.36797	−	8.35272i	−0.184228	−	0.286665i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−37.7216	+	43.5331i	−1.29156	+	1.49054i	−0.520373	+	0.853939i	$0.674207\pi$
$853$	−37.7216	+	43.5331i	−1.29156	+	1.49054i	−0.771191	+	0.636604i	$0.780338\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$857$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$858$		0			0
$859$	46.8730	−	13.7632i	1.59929	−	0.469593i	0.643939	−	0.765077i	$-0.277299\pi$
$859$	46.8730	−	13.7632i	1.59929	−	0.469593i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$863$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	22.2529	+	19.2823i	0.755750	+	0.654861i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	3.03672	+	8.11735i	0.102895	+	0.275046i
$872$		0			0
$873$	52.4650	−	23.9600i	1.77567	−	0.810922i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.26812	−	29.6854i	0.144124	−	1.00241i	−0.781485	−	0.623924i	$-0.785537\pi$
$877$	4.26812	−	29.6854i	0.144124	−	1.00241i	0.925609	−	0.378481i	$-0.123554\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$881$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$882$		0			0
$883$	−15.1759	−	51.6844i	−0.510710	−	1.73932i	−0.660720	−	0.750633i	$-0.729749\pi$
$883$	−15.1759	−	51.6844i	−0.510710	−	1.73932i	0.150010	−	0.988684i	$-0.452069\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$887$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$888$		0			0
$889$	88.4498	+	76.6422i	2.96651	+	2.57050i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−38.2406	−	59.5035i	−1.27257	−	1.98015i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−48.2314	−	14.1620i	−1.60150	−	0.470242i	−0.645534	−	0.763732i	$-0.723365\pi$
$907$	−48.2314	−	14.1620i	−1.60150	−	0.470242i	−0.955962	+	0.293490i	$0.905183\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$911$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−42.7834	−	19.5385i	−1.41130	−	0.644517i	−0.443504	−	0.896272i	$-0.646265\pi$
$919$	−42.7834	−	19.5385i	−1.41130	−	0.644517i	−0.967791	+	0.251755i	$0.918992\pi$
$920$		0			0
$921$	−44.5579	+	38.6096i	−1.46823	+	1.27223i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−36.6672	+	10.7665i	−1.20561	+	0.353999i
$926$		0			0
$927$	−2.20186	−	15.3142i	−0.0723185	−	0.502986i
$928$		0			0
$929$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$929$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$930$		0			0
$931$	42.3728	+	92.7834i	1.38871	+	3.04085i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			16.6466i			0.543820i	0.962323	+	0.271910i	$0.0876553\pi$
$937$			16.6466i			0.543820i	−0.962323	+	0.271910i	$0.912345\pi$
$938$		0			0
$939$	−54.6016			−1.78185
$940$		0			0
$941$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$941$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$947$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$948$		0			0
$949$	12.8827	−	1.85226i	0.418191	−	0.0601268i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$953$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.19902	+	0.770561i	0.0386780	+	0.0248568i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−48.6633			−1.56490			−0.782452	−	0.622710i	$-0.786032\pi$
$967$	−48.6633			−1.56490			−0.782452	+	0.622710i	$0.786032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$971$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$972$		0			0
$973$	−26.3559	+	16.9379i	−0.844931	+	0.543004i
$974$		0			0
$975$	−9.16961			−0.293663
$976$		0			0
$977$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$977$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.21808	−	1.92198i	0.0708180	−	0.0613641i
$982$		0			0
$983$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$983$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	37.7580	+	32.7175i	1.19942	+	1.03931i	0.998213	+	0.0597587i	$0.0190331\pi$
$991$	37.7580	+	32.7175i	1.19942	+	1.03931i	0.201211	+	0.979548i	$0.435512\pi$
$992$		0			0
$993$	−52.9669	+	15.5525i	−1.68085	+	0.493543i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	24.2737	+	7.12741i	0.768757	+	0.225727i	0.642516	−	0.766272i	$-0.277891\pi$
$997$	24.2737	+	7.12741i	0.768757	+	0.225727i	0.126241	+	0.992000i	$0.459709\pi$
$998$		0			0
$999$	39.3101	+	5.65194i	1.24372	+	0.178819i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.53.2	✓	20
3.2	odd	2	CM	804.2.s.a.53.2	✓	20
67.43	odd	22	inner	804.2.s.a.713.2	yes	20
201.110	even	22	inner	804.2.s.a.713.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.53.2	✓	20	1.1	even	1	trivial
804.2.s.a.53.2	✓	20	3.2	odd	2	CM
804.2.s.a.713.2	yes	20	67.43	odd	22	inner
804.2.s.a.713.2	yes	20	201.110	even	22	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.s (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(0.936417 - 1.45709i) q^{3} +(4.81332 - 2.19817i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})\)	\(q+(0.936417 - 1.45709i) q^{3} +(4.81332 - 2.19817i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(-0.298303 - 1.01593i) q^{13} +(-2.01775 + 4.41826i) q^{19} +(1.30434 - 9.07186i) q^{21} +(4.79746 - 1.40866i) q^{25} +(-5.14326 - 0.739490i) q^{27} +(-1.53214 + 5.21798i) q^{31} -7.64303 q^{37} +(-1.75964 - 0.516676i) q^{39} +(5.83248 - 5.05388i) q^{43} +(13.7521 - 15.8707i) q^{49} +(4.54836 + 7.07739i) q^{57} +(1.72288 + 0.247713i) q^{61} +(-11.9972 - 10.3956i) q^{63} +(-8.16392 + 0.591970i) q^{67} +(-1.74937 + 12.1671i) q^{73} +(2.43988 - 8.30945i) q^{75} +(-4.36459 - 14.8644i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(-3.66900 - 4.23426i) q^{91} +(6.16836 + 7.11867i) q^{93} +19.2257i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^9	\( 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) \) 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more