LMFDB - Embedded newform 804.2.ba.a.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(41,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 53]))

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

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

chi := DirichletCharacter("804.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.ba (of order $66$, degree $20$, 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:	$20$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			593.1
Root			$0.928368 + 0.371662i$ of defining polynomial
Character	$\chi$	$=$	804.593
Dual form			804.2.ba.a.221.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71442 + 0.246497i) q^{3} +(4.93365 + 1.19689i) q^{7} +(2.87848 - 0.845198i) q^{9} +O(q^{10})$	$q+(-1.71442 + 0.246497i) q^{3} +(4.93365 + 1.19689i) q^{7} +(2.87848 - 0.845198i) q^{9} +(-3.38028 - 1.16993i) q^{13} +(-0.539964 - 2.22576i) q^{19} +(-8.75337 - 0.835846i) q^{21} +(3.27430 + 3.77875i) q^{25} +(-4.72659 + 2.15856i) q^{27} +(8.77569 - 3.03729i) q^{31} +(5.89230 + 10.2058i) q^{37} +(6.08360 + 1.17252i) q^{39} +(4.69843 - 7.31091i) q^{43} +(16.6865 + 8.60247i) q^{49} +(1.47437 + 3.68280i) q^{57} +(1.00083 + 0.712685i) q^{61} +(15.2130 - 0.724685i) q^{63} +(8.18322 + 0.186740i) q^{67} +(-6.53832 + 9.18178i) q^{73} +(-6.54498 - 5.67126i) q^{75} +(0.356735 + 1.85092i) q^{79} +(7.57128 - 4.86577i) q^{81} +(-15.2768 - 9.81781i) q^{91} +(-14.2965 + 7.37038i) q^{93} +(-1.93786 + 1.11882i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^7 + 6 * q^9	$ 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) $ 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{49}{66}\right)$	$1$

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
$3$	−1.71442	+	0.246497i	−0.989821	+	0.142315i
$3$	−1.71442	+	0.246497i	−0.989821	+	0.142315i
$4$		0			0
$5$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$5$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$6$		0			0
$7$	4.93365	+	1.19689i	1.86474	+	0.452382i	0.998854	−	0.0478556i	$-0.0152387\pi$
$7$	4.93365	+	1.19689i	1.86474	+	0.452382i	0.865888	+	0.500237i	$0.166754\pi$
$8$		0			0
$9$	2.87848	−	0.845198i	0.959493	−	0.281733i
$10$		0			0
$11$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$11$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$12$		0			0
$13$	−3.38028	−	1.16993i	−0.937520	−	0.324479i	−0.184767	−	0.982782i	$-0.559153\pi$
$13$	−3.38028	−	1.16993i	−0.937520	−	0.324479i	−0.752753	+	0.658303i	$0.771274\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$17$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$18$		0			0
$19$	−0.539964	−	2.22576i	−0.123876	−	0.510625i	−0.999613	−	0.0278351i	$-0.991139\pi$
$19$	−0.539964	−	2.22576i	−0.123876	−	0.510625i	0.875736	−	0.482790i	$-0.160376\pi$
$20$		0			0
$21$	−8.75337	−	0.835846i	−1.91014	−	0.182397i
$22$		0			0
$23$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$23$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$24$		0			0
$25$	3.27430	+	3.77875i	0.654861	+	0.755750i
$26$		0			0
$27$	−4.72659	+	2.15856i	−0.909632	+	0.415415i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	8.77569	−	3.03729i	1.57616	−	0.545514i	0.607589	−	0.794252i	$-0.292137\pi$
$31$	8.77569	−	3.03729i	1.57616	−	0.545514i	0.968571	+	0.248738i	$0.0800157\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.89230	+	10.2058i	0.968688	+	1.67782i	0.699363	+	0.714767i	$0.253467\pi$
$37$	5.89230	+	10.2058i	0.968688	+	1.67782i	0.269325	+	0.963049i	$0.413199\pi$
$38$		0			0
$39$	6.08360	+	1.17252i	0.974156	+	0.187753i
$40$		0			0
$41$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$41$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$42$		0			0
$43$	4.69843	−	7.31091i	0.716505	−	1.11490i	−0.271792	−	0.962356i	$-0.587616\pi$
$43$	4.69843	−	7.31091i	0.716505	−	1.11490i	0.988296	−	0.152547i	$-0.0487475\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$47$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$48$		0			0
$49$	16.6865	+	8.60247i	2.38378	+	1.22892i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$53$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.47437	+	3.68280i	0.195285	+	0.487798i
$58$		0			0
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$	1.00083	+	0.712685i	0.128143	+	0.0912500i	0.642331	−	0.766427i	$-0.277967\pi$
$61$	1.00083	+	0.712685i	0.128143	+	0.0912500i	−0.514188	+	0.857677i	$0.671907\pi$
$62$		0			0
$63$	15.2130	−	0.724685i	1.91666	−	0.0913017i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.18322	+	0.186740i	0.999740	+	0.0228140i
$67$	8.18322	+	0.186740i	0.999740	+	0.0228140i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$71$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$72$		0			0
$73$	−6.53832	+	9.18178i	−0.765252	+	1.07465i	0.229598	+	0.973286i	$0.426259\pi$
$73$	−6.53832	+	9.18178i	−0.765252	+	1.07465i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−6.54498	−	5.67126i	−0.755750	−	0.654861i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.356735	+	1.85092i	0.0401358	+	0.208244i	0.996461	−	0.0840621i	$-0.0267894\pi$
$79$	0.356735	+	1.85092i	0.0401358	+	0.208244i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	7.57128	−	4.86577i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$83$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$89$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$90$		0			0
$91$	−15.2768	−	9.81781i	−1.60145	−	1.02919i
$92$		0			0
$93$	−14.2965	+	7.37038i	−1.48248	+	0.764273i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.93786	+	1.11882i	−0.196760	+	0.113599i	−0.595143	−	0.803620i	$-0.702905\pi$
$97$	−1.93786	+	1.11882i	−0.196760	+	0.113599i	0.398384	+	0.917219i	$0.369571\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$101$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$102$		0			0
$103$	−6.59559	−	19.0567i	−0.649883	−	1.87771i	−0.432528	−	0.901620i	$-0.642378\pi$
$103$	−6.59559	−	19.0567i	−0.649883	−	1.87771i	−0.217355	−	0.976093i	$-0.569743\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$		0			0
$109$	−15.7805	+	13.6739i	−1.51150	+	1.30972i	−0.755271	+	0.655412i	$0.772495\pi$
$109$	−15.7805	+	13.6739i	−1.51150	+	1.30972i	−0.756227	+	0.654309i	$0.772960\pi$
$110$		0			0
$111$	−12.6176	−	16.0445i	−1.19761	−	1.52288i
$112$		0			0
$113$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$113$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−10.7189	−	0.510603i	−0.990960	−	0.0472052i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.59775	−	10.3950i	0.327068	−	0.945001i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.09365	+	20.9963i	−0.451988	+	1.86312i	0.0507955	+	0.998709i	$0.483824\pi$
$127$	−5.09365	+	20.9963i	−0.451988	+	1.86312i	−0.502784	+	0.864412i	$0.667691\pi$
$128$		0			0
$129$	−6.25298	+	13.6921i	−0.550544	+	1.20552i
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		−	11.6274i		−	1.00822i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$137$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$138$		0			0
$139$	21.3812	+	9.76449i	1.81353	+	0.828213i	0.939712	+	0.341967i	$0.111093\pi$
$139$	21.3812	+	9.76449i	1.81353	+	0.828213i	0.873822	+	0.486246i	$0.161634\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−30.7281	−	10.6351i	−2.53441	−	0.877168i
$148$		0			0
$149$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$149$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$150$		0			0
$151$	0.829457	−	17.4124i	0.0675003	−	1.41700i	−0.670352	−	0.742043i	$-0.733857\pi$
$151$	0.829457	−	17.4124i	0.0675003	−	1.41700i	0.737853	−	0.674962i	$-0.235840\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.95608	−	4.68391i	0.475347	−	0.373817i	−0.351551	−	0.936169i	$-0.614346\pi$
$157$	5.95608	−	4.68391i	0.475347	−	0.373817i	0.826898	+	0.562352i	$0.190103\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.9203	−	20.6466i	0.933670	−	1.61716i	0.156680	−	0.987649i	$-0.449921\pi$
$163$	11.9203	−	20.6466i	0.933670	−	1.61716i	0.776989	−	0.629514i	$-0.216746\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$167$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$168$		0			0
$169$	−0.161146	−	0.126727i	−0.0123959	−	0.00974822i
$170$		0			0
$171$	−3.43549	−	5.95043i	−0.262718	−	0.455041i
$172$		0			0
$173$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$173$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$174$		0			0
$175$	11.6315	+	22.5620i	0.879259	+	1.70553i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$179$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$180$		0			0
$181$	−14.9975	−	6.00408i	−1.11475	−	0.446280i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−14.9975	−	6.00408i	−1.11475	−	0.446280i	−0.854599	+	0.519288i	$0.826197\pi$
$182$		0			0
$183$	−1.89151	−	0.975143i	−0.139825	−	0.0720846i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−25.9029	+	4.99237i	−1.88416	+	0.363141i
$190$		0			0
$191$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$191$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$192$		0			0
$193$	−13.4715	+	15.5469i	−0.969697	+	1.11909i	0.0231538	+	0.999732i	$0.492629\pi$
$193$	−13.4715	+	15.5469i	−0.969697	+	1.11909i	−0.992851	+	0.119359i	$0.961916\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$197$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$198$		0			0
$199$	−19.1607	−	18.2697i	−1.35826	−	1.29510i	−0.917642	−	0.397409i	$-0.869909\pi$
$199$	−19.1607	−	18.2697i	−1.35826	−	1.29510i	−0.440622	−	0.897693i	$-0.645242\pi$
$200$		0			0
$201$	−14.0755	+	1.69698i	−0.992811	+	0.119696i
$202$		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
$211$	−12.6468	+	5.06300i	−0.870638	+	0.348551i	−0.763589	−	0.645703i	$-0.776565\pi$
$211$	−12.6468	+	5.06300i	−0.870638	+	0.348551i	−0.107049	+	0.994254i	$0.534140\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	46.9314	−	4.48141i	3.18591	−	0.304218i
$218$		0			0
$219$	8.94615	−	17.3531i	0.604525	−	1.17261i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.04652	+	28.1442i	−0.270975	+	1.88467i	0.167412	+	0.985887i	$0.446459\pi$
$223$	−4.04652	+	28.1442i	−0.270975	+	1.88467i	−0.438387	+	0.898786i	$0.644450\pi$
$224$		0			0
$225$	12.6188	+	8.10961i	0.841254	+	0.540641i
$226$		0			0
$227$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$227$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$228$		0			0
$229$	5.22106	−	27.0894i	0.345018	−	1.79012i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	5.22106	−	27.0894i	0.345018	−	1.79012i	0.576304	−	0.817235i	$-0.304494\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$233$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.06784	−	3.08532i	−0.0693635	−	0.200413i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−12.7813	−	27.9872i	−0.823317	−	1.80281i	−0.533712	−	0.845666i	$-0.679203\pi$
$241$	−12.7813	−	27.9872i	−0.823317	−	1.80281i	−0.289605	−	0.957146i	$-0.593524\pi$
$242$		0			0
$243$	−11.7810	+	10.2083i	−0.755750	+	0.654861i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.778747	+	8.15541i	−0.0495505	+	0.518916i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$251$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$257$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$258$		0			0
$259$	16.8553	+	57.4040i	1.04734	+	3.56691i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		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
$271$	−6.71769	+	0.965858i	−0.408071	+	0.0586717i	−0.343294	−	0.939228i	$-0.611543\pi$
$271$	−6.71769	+	0.965858i	−0.408071	+	0.0586717i	−0.0647769	+	0.997900i	$0.520634\pi$
$272$		0			0
$273$	28.6109	+	13.0662i	1.73161	+	0.790801i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18.7331	+	5.50053i	−1.12556	+	0.330495i	−0.790962	−	0.611866i	$-0.790419\pi$
$277$	−18.7331	+	5.50053i	−1.12556	+	0.330495i	−0.334600	+	0.942360i	$0.608601\pi$
$278$		0			0
$279$	22.6935	−	16.1600i	1.35863	−	0.967473i
$280$		0			0
$281$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$281$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$282$		0			0
$283$	−21.8251	−	6.40842i	−1.29737	−	0.380941i	−0.441091	−	0.897462i	$-0.645408\pi$
$283$	−21.8251	−	6.40842i	−1.29737	−	0.380941i	−0.856274	+	0.516522i	$0.827227\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.9230	−	1.61595i	−0.995472	−	0.0950560i
$290$		0			0
$291$	3.04652	−	2.39581i	0.178590	−	0.140445i
$292$		0			0
$293$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$293$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	31.9308	−	30.4459i	1.84046	−	1.75487i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.0635523	−	0.0122487i	−0.00362712	−	0.000699071i	0.187437	−	0.982277i	$-0.439982\pi$
$307$	−0.0635523	−	0.0122487i	−0.00362712	−	0.000699071i	−0.191064	+	0.981577i	$0.561194\pi$
$308$		0			0
$309$	16.0050	+	31.0454i	0.910494	+	1.76611i
$310$		0			0
$311$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$311$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$312$		0			0
$313$	−20.1846	−	2.90210i	−1.14090	−	0.164037i	−0.454146	−	0.890927i	$-0.650056\pi$
$313$	−20.1846	−	2.90210i	−1.14090	−	0.164037i	−0.686753	+	0.726891i	$0.740965\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$317$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−6.64720	−	16.6039i	−0.368720	−	0.921019i
$326$		0			0
$327$	23.6839	−	27.3326i	1.30972	−	1.51150i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−27.9413	+	1.33101i	−1.53579	+	0.0731589i	−0.798327	−	0.602224i	$-0.794281\pi$
$331$	−27.9413	+	1.33101i	−1.53579	+	0.0731589i	−0.737467	+	0.675383i	$0.763978\pi$
$332$		0			0
$333$	25.5867	+	24.3969i	1.40214	+	1.33694i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−13.8296	+	14.5040i	−0.753344	+	0.790085i	−0.983082	−	0.183168i	$-0.941365\pi$
$337$	−13.8296	+	14.5040i	−0.753344	+	0.790085i	0.229737	+	0.973253i	$0.426213\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	45.1716	+	39.1414i	2.43904	+	2.11344i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$347$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$348$		0			0
$349$	25.5438	−	16.4160i	1.36733	−	0.878727i	0.368620	−	0.929580i	$-0.379831\pi$
$349$	25.5438	−	16.4160i	1.36733	−	0.878727i	0.998706	+	0.0508535i	$0.0161942\pi$
$350$		0			0
$351$	18.5025	−	1.76678i	0.987592	−	0.0943036i
$352$		0			0
$353$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$353$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$359$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$360$		0			0
$361$	12.2254	−	6.30264i	0.643443	−	0.331718i
$362$		0			0
$363$	−3.60572	+	18.7083i	−0.189251	+	0.981929i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	12.0343	−	15.3029i	0.628185	−	0.798803i	−0.362929	−	0.931817i	$-0.618223\pi$
$367$	12.0343	−	15.3029i	0.628185	−	0.798803i	0.991114	+	0.133014i	$0.0424656\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	18.3674	+	10.6044i	0.951027	+	0.549076i	0.893400	−	0.449262i	$-0.148313\pi$
$373$	18.3674	+	10.6044i	0.951027	+	0.549076i	0.0576272	+	0.998338i	$0.481647\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−13.6349	−	17.3382i	−0.700380	−	0.890606i	0.297564	−	0.954702i	$-0.403826\pi$
$379$	−13.6349	−	17.3382i	−0.700380	−	0.890606i	−0.997944	+	0.0640964i	$0.979583\pi$
$380$		0			0
$381$	3.55714	−	37.2521i	0.182238	−	1.90848i
$382$		0			0
$383$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$383$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.34518	−	25.0154i	0.373377	−	1.27160i
$388$		0			0
$389$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$389$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.5104	+	36.1528i	−0.828636	+	1.81446i	−0.348249	+	0.937402i	$0.613224\pi$
$397$	−16.5104	+	36.1528i	−0.828636	+	1.81446i	−0.480387	+	0.877057i	$0.659504\pi$
$398$		0			0
$399$	2.86611	+	19.9343i	0.143485	+	0.997961i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−33.2177			−1.65469
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−39.1856	−	9.50631i	−1.93760	−	0.470057i	−0.987345	−	0.158587i	$-0.949306\pi$
$409$	−39.1856	−	9.50631i	−1.93760	−	0.470057i	−0.950256	−	0.311470i	$-0.899179\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−39.0634	−	11.4700i	−1.91294	−	0.561690i
$418$		0			0
$419$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$419$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$420$		0			0
$421$	6.63857	+	27.3645i	0.323544	+	1.33367i	0.868703	+	0.495334i	$0.164954\pi$
$421$	6.63857	+	27.3645i	0.323544	+	1.33367i	−0.545159	+	0.838333i	$0.683531\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.08472	+	4.71402i	0.197673	+	0.228127i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	27.1532	−	9.39782i	1.30490	−	0.451631i	0.415848	−	0.909434i	$-0.363485\pi$
$433$	27.1532	−	9.39782i	1.30490	−	0.451631i	0.889053	+	0.457804i	$0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−16.2369	−	28.1232i	−0.774947	−	1.34225i	−0.934824	−	0.355111i	$-0.884443\pi$
$439$	−16.2369	−	28.1232i	−0.774947	−	1.34225i	0.159877	−	0.987137i	$-0.448890\pi$
$440$		0			0
$441$	55.3024	+	10.6587i	2.63345	+	0.507556i
$442$		0			0
$443$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$443$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$449$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.87007	+	30.0567i	0.134848	+	1.41219i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−41.9824	+	8.09144i	−1.96385	+	0.378502i	−0.981563	+	0.191139i	$0.938782\pi$
$457$	−41.9824	+	8.09144i	−1.96385	+	0.378502i	−0.982291	+	0.187363i	$0.940006\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	34.3720	+	24.4762i	1.59740	+	1.13751i	0.912657	+	0.408726i	$0.134027\pi$
$463$	34.3720	+	24.4762i	1.59740	+	1.13751i	0.684747	+	0.728781i	$0.259913\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$467$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$468$		0			0
$469$	40.1496	+	10.7157i	1.85394	+	0.494806i
$470$		0			0
$471$	−9.05666	+	9.49836i	−0.417309	+	0.437661i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.64259	−	9.32821i	0.304783	−	0.428008i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$479$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$480$		0			0
$481$	−7.97762	−	41.3918i	−0.363748	−	1.88730i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	20.0636	−	38.9180i	0.909169	−	1.76354i	0.342739	−	0.939430i	$-0.388645\pi$
$487$	20.0636	−	38.9180i	0.909169	−	1.76354i	0.566429	−	0.824110i	$-0.308325\pi$
$488$		0			0
$489$	−15.3471	+	38.3352i	−0.694020	+	1.73358i
$490$		0			0
$491$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$491$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.26271	−	1.88372i	0.146059	−	0.0843271i	−0.425190	−	0.905104i	$-0.639793\pi$
$499$	3.26271	−	1.88372i	0.146059	−	0.0843271i	0.571248	+	0.820777i	$0.306459\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$503$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.307510	+	0.177541i	0.0136570	+	0.00788488i
$508$		0			0
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$	−43.2473	+	37.4740i	−1.91315	+	1.65775i
$512$		0			0
$513$	7.35663	+	9.35472i	0.324803	+	0.413021i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$521$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$522$		0			0
$523$	1.63907	−	4.73579i	0.0716716	−	0.207081i	−0.903476	−	0.428640i	$-0.858993\pi$
$523$	1.63907	−	4.73579i	0.0716716	−	0.207081i	0.975147	+	0.221558i	$0.0711142\pi$
$524$		0			0
$525$	−25.5028	−	35.8136i	−1.11303	−	1.56303i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5.42246	−	22.3517i	0.235759	−	0.971812i
$530$		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
$541$	42.0955	+	19.2244i	1.80983	+	0.826521i	0.947168	+	0.320739i	$0.103931\pi$
$541$	42.0955	+	19.2244i	1.80983	+	0.826521i	0.862661	+	0.505782i	$0.168796\pi$
$542$		0			0
$543$	27.1919	+	6.59669i	1.16692	+	0.283091i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	31.6063	−	22.5067i	1.35139	−	0.962318i	0.351730	−	0.936101i	$-0.385593\pi$
$547$	31.6063	−	22.5067i	1.35139	−	0.962318i	0.999656	−	0.0262168i	$-0.00834602\pi$
$548$		0			0
$549$	3.48322	+	1.20555i	0.148660	+	0.0514518i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.455339	+	9.55874i	−0.0193630	+	0.406479i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$557$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$558$		0			0
$559$	−24.4352	+	19.2161i	−1.03350	+	0.812753i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$563$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	43.1778	−	14.9440i	1.81330	−	0.627588i
$568$		0			0
$569$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$569$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$570$		0			0
$571$	24.8323	+	19.5283i	1.03920	+	0.817234i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	24.8323	+	19.5283i	1.03920	+	0.817234i	0.0557537	+	0.998445i	$0.482244\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.32425	+	16.1468i	0.346543	+	0.672200i	0.996111	−	0.0881033i	$-0.0280806\pi$
$577$	8.32425	+	16.1468i	0.346543	+	0.672200i	−0.649568	+	0.760303i	$0.725050\pi$
$578$		0			0
$579$	19.2635	−	29.9746i	0.800564	−	1.24570i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$587$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$588$		0			0
$589$	−11.4989	−	17.8926i	−0.473802	−	0.737250i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$593$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	37.3529	+	26.5989i	1.52875	+	1.08862i
$598$		0			0
$599$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$599$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$600$		0			0
$601$	13.5916	+	12.9596i	0.554412	+	0.528631i	0.914659	−	0.404226i	$-0.132459\pi$
$601$	13.5916	+	12.9596i	0.554412	+	0.528631i	−0.360247	+	0.932857i	$0.617308\pi$
$602$		0			0
$603$	23.7131	−	6.37891i	0.965671	−	0.259769i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.16741	+	45.4996i	0.0879727	+	1.84677i	0.424897	+	0.905242i	$0.360310\pi$
$607$	2.16741	+	45.4996i	0.0879727	+	1.84677i	−0.336925	+	0.941532i	$0.609387\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.3793	+	4.95591i	−0.499994	+	0.200168i	−0.607930	−	0.793990i	$-0.708000\pi$
$613$	−12.3793	+	4.95591i	−0.499994	+	0.200168i	0.107936	+	0.994158i	$0.465576\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$617$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$	−49.3767	+	4.71491i	−1.98462	+	0.189508i	−0.984738	+	0.174042i	$0.944317\pi$
$619$	−49.3767	+	4.71491i	−1.98462	+	0.189508i	−0.999880	+	0.0154656i	$0.995077\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.55787	+	24.7455i	−0.142315	+	0.989821i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	7.58920	−	39.3765i	0.302121	−	1.56755i	−0.440444	−	0.897780i	$-0.645179\pi$
$631$	7.58920	−	39.3765i	0.302121	−	1.56755i	0.742565	−	0.669774i	$-0.233609\pi$
$632$		0			0
$633$	20.4338	−	11.7975i	0.812172	−	0.468908i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−46.3406	−	48.6006i	−1.83608	−	1.92563i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	6.97674	+	15.2769i	0.275136	+	0.602463i	0.995874	−	0.0907437i	$-0.0289244\pi$
$643$	6.97674	+	15.2769i	0.275136	+	0.602463i	−0.720738	+	0.693207i	$0.756197\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$647$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−79.3556	+	19.2515i	−3.11019	+	0.754524i
$652$		0			0
$653$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$653$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−11.0600	+	31.9557i	−0.431491	+	1.24671i
$658$		0			0
$659$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$659$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$660$		0			0
$661$	−10.2293	−	34.8379i	−0.397875	−	1.35504i	−0.878345	−	0.478027i	$-0.841352\pi$
$661$	−10.2293	−	34.8379i	−0.397875	−	1.35504i	0.480470	−	0.877011i	$-0.340466\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	49.2484i		−	1.90405i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−48.2699	+	6.94017i	−1.86067	+	0.267524i	−0.978927	−	0.204209i	$-0.934538\pi$
$673$	−48.2699	+	6.94017i	−1.86067	+	0.267524i	−0.881742	+	0.471732i	$0.843629\pi$
$674$		0			0
$675$	−23.6329	−	10.7928i	−0.909632	−	0.415415i
$676$		0			0
$677$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$677$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$678$		0			0
$679$	−10.8998	+	3.20047i	−0.418296	+	0.122823i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$683$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−2.27365	+	47.7297i	−0.0867450	+	1.82100i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−44.3072	−	4.23083i	−1.68553	−	0.160948i	−0.792171	−	0.610299i	$-0.791049\pi$
$691$	−44.3072	−	4.23083i	−1.68553	−	0.160948i	−0.893356	+	0.449350i	$0.851656\pi$
$692$		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		0.945001	−	0.327068i	$-0.106061\pi$
$701$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$702$		0			0
$703$	19.5340	−	18.6256i	0.736737	−	0.702478i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−50.2106	−	9.67731i	−1.88570	−	0.363439i	−0.889991	−	0.455978i	$-0.849290\pi$
$709$	−50.2106	−	9.67731i	−1.88570	−	0.363439i	−0.995709	+	0.0925387i	$0.970502\pi$
$710$		0			0
$711$	2.59124	+	5.02631i	0.0971792	+	0.188501i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$719$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$720$		0			0
$721$	−9.73153	−	101.913i	−0.362421	−	3.79545i
$722$		0			0
$723$	28.8113	+	44.8313i	1.07150	+	1.66729i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.80575	+	7.00842i	0.104059	+	0.259928i	0.971256	−	0.238039i	$-0.0765045\pi$
$727$	2.80575	+	7.00842i	0.104059	+	0.259928i	−0.867196	+	0.497966i	$0.834080\pi$
$728$		0			0
$729$	17.6812	−	20.4052i	0.654861	−	0.755750i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.84943	−	0.278643i	0.216053	−	0.0102919i	0.0607239	−	0.998155i	$-0.480659\pi$
$733$	5.84943	−	0.278643i	0.216053	−	0.0102919i	0.155330	+	0.987863i	$0.450356\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−36.9309	+	38.7320i	−1.35852	+	1.42478i	−0.551621	+	0.834095i	$0.685990\pi$
$739$	−36.9309	+	38.7320i	−1.35852	+	1.42478i	−0.806903	+	0.590684i	$0.798858\pi$
$740$		0			0
$741$	−0.675180	−	14.1738i	−0.0248033	−	0.520686i
$742$		0			0
$743$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$743$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.9231	+	18.5878i	−1.05542	+	0.678278i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−28.9231	+	18.5878i	−1.05542	+	0.678278i	−0.106667	+	0.994295i	$0.534018\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.58391	−	6.45428i	0.0939136	−	0.234585i	−0.873972	−	0.485977i	$-0.838464\pi$
$757$	2.58391	−	6.45428i	0.0939136	−	0.234585i	0.967885	+	0.251392i	$0.0808882\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$	−94.2216	+	48.5746i	−3.41105	+	1.75852i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	33.8767	−	43.0777i	1.22162	−	1.55342i	0.489981	−	0.871733i	$-0.337004\pi$
$769$	33.8767	−	43.0777i	1.22162	−	1.55342i	0.731643	−	0.681688i	$-0.238754\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$773$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$774$		0			0
$775$	40.2114	+	23.2161i	1.44444	+	0.833946i
$776$		0			0
$777$	−43.0470	−	94.2599i	−1.54430	−	3.38155i
$778$		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
$787$	−44.5400	−	2.12170i	−1.58768	−	0.0756305i	−0.764959	−	0.644078i	$-0.777241\pi$
$787$	−44.5400	−	2.12170i	−1.58768	−	0.0756305i	−0.822719	+	0.568448i	$0.807544\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.54928	−	3.57997i	−0.0905277	−	0.127128i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$797$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$798$		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		−0.909632	−	0.415415i	$-0.863636\pi$
$809$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$810$		0			0
$811$	−48.1251	−	11.6750i	−1.68990	−	0.409966i	−0.728485	−	0.685062i	$-0.759775\pi$
$811$	−48.1251	−	11.6750i	−1.68990	−	0.409966i	−0.961417	+	0.275096i	$0.911290\pi$
$812$		0			0
$813$	11.2789	−	3.31178i	0.395567	−	0.116149i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−18.8093	−	6.50997i	−0.658055	−	0.227755i
$818$		0			0
$819$	−52.2720	−	15.3484i	−1.82653	−	0.536318i
$820$		0			0
$821$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$821$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$822$		0			0
$823$	12.8177	+	52.8352i	0.446796	+	1.84172i	0.533940	+	0.845522i	$0.320711\pi$
$823$	12.8177	+	52.8352i	0.446796	+	1.84172i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$827$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$828$		0			0
$829$	−24.0927	−	27.8044i	−0.836774	−	0.965688i	0.163007	−	0.986625i	$-0.447881\pi$
$829$	−24.0927	−	27.8044i	−0.836774	−	0.965688i	−0.999781	+	0.0209367i	$0.993335\pi$
$830$		0			0
$831$	30.7605	−	14.0479i	1.06707	−	0.487315i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−34.9229	+	33.2989i	−1.20711	+	1.15098i
$838$		0			0
$839$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$839$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	30.1917	−	46.9792i	1.03740	−	1.61422i
$848$		0			0
$849$	38.9970	+	5.60692i	1.33837	+	0.192429i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	51.7863	+	26.6977i	1.77313	+	0.914111i	0.919304	+	0.393548i	$0.128752\pi$
$853$	51.7863	+	26.6977i	1.77313	+	0.914111i	0.853823	+	0.520563i	$0.174278\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$857$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$858$		0			0
$859$	50.7756	−	9.78620i	1.73244	−	0.333901i	0.777094	−	0.629385i	$-0.216693\pi$
$859$	50.7756	−	9.78620i	1.73244	−	0.333901i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$863$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.4115	−	1.40104i	0.998867	−	0.0475819i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−27.4431	−	10.2050i	−0.929873	−	0.345783i
$872$		0			0
$873$	−4.63246	+	4.85838i	−0.156785	+	0.164431i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.9879	−	44.9207i	1.08016	−	1.51687i	0.242901	−	0.970051i	$-0.421901\pi$
$877$	31.9879	−	44.9207i	1.08016	−	1.51687i	0.837254	−	0.546814i	$-0.184160\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$881$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$882$		0			0
$883$	−3.08469	−	16.0049i	−0.103808	−	0.538608i	−0.996181	−	0.0873094i	$-0.972173\pi$
$883$	−3.08469	−	16.0049i	−0.103808	−	0.538608i	0.892373	−	0.451298i	$-0.149039\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$887$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$888$		0			0
$889$	−50.2605	+	97.4918i	−1.68568	+	3.26977i
$890$		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
$903$	−47.2380	+	60.0679i	−1.57198	+	1.99894i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−7.96317	−	23.0081i	−0.264413	−	0.763971i	−0.996564	−	0.0828283i	$-0.973605\pi$
$907$	−7.96317	−	23.0081i	−0.264413	−	0.763971i	0.732151	−	0.681142i	$-0.238517\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$911$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	58.5538	−	14.2050i	1.93151	−	0.468580i	0.939478	−	0.342610i	$-0.111311\pi$
$919$	58.5538	−	14.2050i	1.93151	−	0.468580i	0.992034	−	0.125970i	$-0.0402043\pi$
$920$		0			0
$921$	0.111975	+	0.00533401i	0.00368969	+	0.000175762i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−19.2718	+	55.6823i	−0.633653	+	1.83082i
$926$		0			0
$927$	−35.0920	−	49.2798i	−1.15257	−	1.61856i
$928$		0			0
$929$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$929$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$930$		0			0
$931$	10.1370	−	41.7851i	0.332226	−	1.36945i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			61.1782i			1.99860i	0.0373691	+	0.999302i	$0.488102\pi$
$937$			61.1782i			1.99860i	−0.0373691	+	0.999302i	$0.511898\pi$
$938$		0			0
$939$	35.3202			1.15263
$940$		0			0
$941$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$941$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$947$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$948$		0			0
$949$	32.8433	−	23.3876i	1.06614	−	0.759194i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$953$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	43.4199	−	34.1458i	1.40064	−	1.10148i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.3411	−	17.9113i	0.332547	−	0.575988i	−0.650464	−	0.759537i	$-0.725425\pi$
$967$	10.3411	−	17.9113i	0.332547	−	0.575988i	0.983010	+	0.183550i	$0.0587588\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$971$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$972$		0			0
$973$	93.8005	+	73.7655i	3.00711	+	2.36481i
$974$		0			0
$975$	15.4889	+	26.8276i	0.496042	+	0.859170i
$976$		0			0
$977$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$977$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−33.8667	+	52.6977i	−1.08128	+	1.68251i
$982$		0			0
$983$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$983$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−11.5245	−	17.9324i	−0.366086	−	0.569642i	0.608528	−	0.793533i	$-0.291760\pi$
$991$	−11.5245	−	17.9324i	−0.366086	−	0.569642i	−0.974614	+	0.223891i	$0.928124\pi$
$992$		0			0
$993$	47.5751	−	9.16935i	1.50975	−	0.290980i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−39.4855	+	45.5687i	−1.25052	+	1.44317i	−0.400606	+	0.916251i	$0.631200\pi$
$997$	−39.4855	+	45.5687i	−1.25052	+	1.44317i	−0.849912	+	0.526924i	$0.823345\pi$
$998$		0			0
$999$	−49.8802	−	35.5195i	−1.57814	−	1.12379i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.593.1	yes	20
3.2	odd	2	CM	804.2.ba.a.593.1	yes	20
67.20	odd	66	inner	804.2.ba.a.221.1	✓	20
201.20	even	66	inner	804.2.ba.a.221.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.221.1	✓	20	67.20	odd	66	inner
804.2.ba.a.221.1	✓	20	201.20	even	66	inner
804.2.ba.a.593.1	yes	20	1.1	even	1	trivial
804.2.ba.a.593.1	yes	20	3.2	odd	2	CM

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.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.71442 + 0.246497i) q^{3} +(4.93365 + 1.19689i) q^{7} +(2.87848 - 0.845198i) q^{9} +O(q^{10})\)	\(q+(-1.71442 + 0.246497i) q^{3} +(4.93365 + 1.19689i) q^{7} +(2.87848 - 0.845198i) q^{9} +(-3.38028 - 1.16993i) q^{13} +(-0.539964 - 2.22576i) q^{19} +(-8.75337 - 0.835846i) q^{21} +(3.27430 + 3.77875i) q^{25} +(-4.72659 + 2.15856i) q^{27} +(8.77569 - 3.03729i) q^{31} +(5.89230 + 10.2058i) q^{37} +(6.08360 + 1.17252i) q^{39} +(4.69843 - 7.31091i) q^{43} +(16.6865 + 8.60247i) q^{49} +(1.47437 + 3.68280i) q^{57} +(1.00083 + 0.712685i) q^{61} +(15.2130 - 0.724685i) q^{63} +(8.18322 + 0.186740i) q^{67} +(-6.53832 + 9.18178i) q^{73} +(-6.54498 - 5.67126i) q^{75} +(0.356735 + 1.85092i) q^{79} +(7.57128 - 4.86577i) q^{81} +(-15.2768 - 9.81781i) q^{91} +(-14.2965 + 7.37038i) q^{93} +(-1.93786 + 1.11882i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^7 + 6 * q^9	\( 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) \) 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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