LMFDB - Embedded newform 804.2.g.c.401.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(401,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("804.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.g (of order $2$, degree $1$, 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - x^{12} - 27x^{10} + 88x^{8} - 243x^{6} - 81x^{4} - 729x^{2} + 6561 $ x^16 - x^14 - x^12 - 27x^10 + 88x^8 - 243x^6 - 81x^4 - 729*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.13
Root			$1.49351 - 0.877170i$ of defining polynomial
Character	$\chi$	$=$	804.401
Dual form			804.2.g.c.401.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.49351 - 0.877170i) q^{3} -1.18740 q^{5} -0.405325i q^{7} +(1.46115 - 2.62012i) q^{9} +O(q^{10})$	\(q+(1.49351 - 0.877170i) q^{3} -1.18740 q^{5} -0.405325i q^{7} +(1.46115 - 2.62012i) q^{9} +5.79333 q^{11} +4.82750i q^{13} +(-1.77339 + 1.04155i) q^{15} -0.491931i q^{17} +3.95671 q^{19} +(-0.355539 - 0.605358i) q^{21} -3.05632i q^{23} -3.59008 q^{25} +(-0.116058 - 5.19486i) q^{27} -8.30722i q^{29} -5.44181i q^{31} +(8.65240 - 5.08173i) q^{33} +0.481284i q^{35} +0.956710 q^{37} +(4.23454 + 7.20993i) q^{39} +6.72993 q^{41} +7.64593i q^{43} +(-1.73497 + 3.11114i) q^{45} -4.74832i q^{47} +6.83571 q^{49} +(-0.431507 - 0.734705i) q^{51} -10.1999 q^{53} -6.87900 q^{55} +(5.90939 - 3.47071i) q^{57} -10.8716i q^{59} +13.2539i q^{61} +(-1.06200 - 0.592240i) q^{63} -5.73218i q^{65} +(2.92229 + 7.64593i) q^{67} +(-2.68091 - 4.56464i) q^{69} +0.718784i q^{71} -9.46908 q^{73} +(-5.36182 + 3.14911i) q^{75} -2.34818i q^{77} +14.1074i q^{79} +(-4.73010 - 7.65677i) q^{81} +15.8850i q^{83} +0.584120i q^{85} +(-7.28684 - 12.4069i) q^{87} +10.5439i q^{89} +1.95671 q^{91} +(-4.77339 - 8.12740i) q^{93} -4.69820 q^{95} -12.0694i q^{97} +(8.46490 - 15.1792i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{9}+O(q^{10}) $ 16 * q + 2 * q^9	$ 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) $ 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.49351	−	0.877170i	0.862279	−	0.506434i
$3$	1.49351	−	0.877170i	0.862279	−	0.506434i
$4$		0			0
$5$	−1.18740			−0.531022			−0.265511	−	0.964108i	$-0.585541\pi$
$5$	−1.18740			−0.531022			−0.265511	+	0.964108i	$0.585541\pi$
$6$		0			0
$7$		−	0.405325i		−	0.153199i	−0.997062	−	0.0765993i	$-0.975594\pi$
$7$		−	0.405325i		−	0.153199i	0.997062	−	0.0765993i	$-0.0244062\pi$
$8$		0			0
$9$	1.46115	−	2.62012i	0.487049	−	0.873375i
$10$		0			0
$11$	5.79333			1.74675			0.873377	−	0.487044i	$-0.161925\pi$
$11$	5.79333			1.74675			0.873377	+	0.487044i	$0.161925\pi$
$12$		0			0
$13$			4.82750i			1.33891i	0.742853	+	0.669454i	$0.233472\pi$
$13$			4.82750i			1.33891i	−0.742853	+	0.669454i	$0.766528\pi$
$14$		0			0
$15$	−1.77339	+	1.04155i	−0.457889	+	0.268928i
$16$		0			0
$17$		−	0.491931i		−	0.119311i	−0.998219	−	0.0596555i	$-0.981000\pi$
$17$		−	0.491931i		−	0.119311i	0.998219	−	0.0596555i	$-0.0190002\pi$
$18$		0			0
$19$	3.95671			0.907731			0.453866	−	0.891070i	$-0.350045\pi$
$19$	3.95671			0.907731			0.453866	+	0.891070i	$0.350045\pi$
$20$		0			0
$21$	−0.355539	−	0.605358i	−0.0775850	−	0.132100i
$22$		0			0
$23$		−	3.05632i		−	0.637287i	−0.947875	−	0.318643i	$-0.896773\pi$
$23$		−	3.05632i		−	0.637287i	0.947875	−	0.318643i	$-0.103227\pi$
$24$		0			0
$25$	−3.59008			−0.718016
$26$		0			0
$27$	−0.116058	−	5.19486i	−0.0223353	−	0.999751i
$28$		0			0
$29$		−	8.30722i		−	1.54261i	−0.636465	−	0.771306i	$-0.719604\pi$
$29$		−	8.30722i		−	1.54261i	0.636465	−	0.771306i	$-0.280396\pi$
$30$		0			0
$31$		−	5.44181i		−	0.977379i	−0.872458	−	0.488689i	$-0.837475\pi$
$31$		−	5.44181i		−	0.977379i	0.872458	−	0.488689i	$-0.162525\pi$
$32$		0			0
$33$	8.65240	−	5.08173i	1.50619	−	0.884616i
$34$		0			0
$35$			0.481284i			0.0813518i
$36$		0			0
$37$	0.956710			0.157282			0.0786410	−	0.996903i	$-0.474942\pi$
$37$	0.956710			0.157282			0.0786410	+	0.996903i	$0.474942\pi$
$38$		0			0
$39$	4.23454	+	7.20993i	0.678069	+	1.15451i
$40$		0			0
$41$	6.72993			1.05104			0.525520	−	0.850782i	$-0.323871\pi$
$41$	6.72993			1.05104			0.525520	+	0.850782i	$0.323871\pi$
$42$		0			0
$43$			7.64593i			1.16599i	0.812475	+	0.582997i	$0.198120\pi$
$43$			7.64593i			1.16599i	−0.812475	+	0.582997i	$0.801880\pi$
$44$		0			0
$45$	−1.73497	+	3.11114i	−0.258633	+	0.463781i
$46$		0			0
$47$		−	4.74832i		−	0.692613i	−0.938121	−	0.346307i	$-0.887436\pi$
$47$		−	4.74832i		−	0.692613i	0.938121	−	0.346307i	$-0.112564\pi$
$48$		0			0
$49$	6.83571			0.976530
$50$		0			0
$51$	−0.431507	−	0.734705i	−0.0604231	−	0.102879i
$52$		0			0
$53$	−10.1999			−1.40106			−0.700530	−	0.713623i	$-0.747053\pi$
$53$	−10.1999			−1.40106			−0.700530	+	0.713623i	$0.747053\pi$
$54$		0			0
$55$	−6.87900			−0.927564
$56$		0			0
$57$	5.90939	−	3.47071i	0.782717	−	0.459706i
$58$		0			0
$59$		−	10.8716i		−	1.41536i	−0.706532	−	0.707681i	$-0.749741\pi$
$59$		−	10.8716i		−	1.41536i	0.706532	−	0.707681i	$-0.250259\pi$
$60$		0			0
$61$			13.2539i			1.69699i	0.529206	+	0.848493i	$0.322490\pi$
$61$			13.2539i			1.69699i	−0.529206	+	0.848493i	$0.677510\pi$
$62$		0			0
$63$	−1.06200	−	0.592240i	−0.133800	−	0.0746152i
$64$		0			0
$65$		−	5.73218i		−	0.710989i
$66$		0			0
$67$	2.92229	+	7.64593i	0.357015	+	0.934099i
$67$	2.92229	+	7.64593i	0.357015	+	0.934099i
$68$		0			0
$69$	−2.68091	−	4.56464i	−0.322744	−	0.549519i
$70$		0			0
$71$			0.718784i			0.0853039i	0.999090	+	0.0426520i	$0.0135807\pi$
$71$			0.718784i			0.0853039i	−0.999090	+	0.0426520i	$0.986419\pi$
$72$		0			0
$73$	−9.46908			−1.10827			−0.554136	−	0.832426i	$-0.686951\pi$
$73$	−9.46908			−1.10827			−0.554136	+	0.832426i	$0.686951\pi$
$74$		0			0
$75$	−5.36182	+	3.14911i	−0.619130	+	0.363628i
$76$		0			0
$77$		−	2.34818i		−	0.267600i
$78$		0			0
$79$			14.1074i			1.58720i	0.608437	+	0.793602i	$0.291797\pi$
$79$			14.1074i			1.58720i	−0.608437	+	0.793602i	$0.708203\pi$
$80$		0			0
$81$	−4.73010	−	7.65677i	−0.525567	−	0.850752i
$82$		0			0
$83$			15.8850i			1.74361i	0.489857	+	0.871803i	$0.337049\pi$
$83$			15.8850i			1.74361i	−0.489857	+	0.871803i	$0.662951\pi$
$84$		0			0
$85$			0.584120i			0.0633567i
$86$		0			0
$87$	−7.28684	−	12.4069i	−0.781231	−	1.33016i
$88$		0			0
$89$			10.5439i			1.11765i	0.829284	+	0.558827i	$0.188748\pi$
$89$			10.5439i			1.11765i	−0.829284	+	0.558827i	$0.811252\pi$
$90$		0			0
$91$	1.95671			0.205119
$92$		0			0
$93$	−4.77339	−	8.12740i	−0.494978	−	0.842773i
$94$		0			0
$95$	−4.69820			−0.482025
$96$		0			0
$97$		−	12.0694i		−	1.22546i	−0.790291	−	0.612732i	$-0.790071\pi$
$97$		−	12.0694i		−	1.22546i	0.790291	−	0.612732i	$-0.209929\pi$
$98$		0			0
$99$	8.46490	−	15.1792i	0.850754	−	1.52557i
$100$		0			0
$101$	−6.88846			−0.685427			−0.342714	−	0.939440i	$-0.611346\pi$
$101$	−6.88846			−0.685427			−0.342714	+	0.939440i	$0.611346\pi$
$102$		0			0
$103$	−5.24563			−0.516867			−0.258434	−	0.966029i	$-0.583206\pi$
$103$	−5.24563			−0.516867			−0.258434	+	0.966029i	$0.583206\pi$
$104$		0			0
$105$	0.422167	+	0.718802i	0.0411993	+	0.0701479i
$106$		0			0
$107$			12.9654i			1.25341i	0.779257	+	0.626704i	$0.215597\pi$
$107$			12.9654i			1.25341i	−0.779257	+	0.626704i	$0.784403\pi$
$108$		0			0
$109$		−	3.80787i		−	0.364728i	−0.983231	−	0.182364i	$-0.941625\pi$
$109$		−	3.80787i		−	0.364728i	0.983231	−	0.182364i	$-0.0583749\pi$
$110$		0			0
$111$	1.42886	−	0.839197i	0.135621	−	0.0796530i
$112$		0			0
$113$	6.04412			0.568583			0.284292	−	0.958738i	$-0.408242\pi$
$113$	6.04412			0.568583			0.284292	+	0.958738i	$0.408242\pi$
$114$		0			0
$115$			3.62907i			0.338413i
$116$		0			0
$117$	12.6487	+	7.05369i	1.16937	+	0.652114i
$118$		0			0
$119$	−0.199392			−0.0182783
$120$		0			0
$121$	22.5627			2.05115
$122$		0			0
$123$	10.0512	−	5.90330i	0.906289	−	0.532282i
$124$		0			0
$125$	10.1999			0.912304
$126$		0			0
$127$	−16.1838			−1.43608			−0.718040	−	0.696002i	$-0.754960\pi$
$127$	−16.1838			−1.43608			−0.718040	+	0.696002i	$0.754960\pi$
$128$		0			0
$129$	6.70678	+	11.4193i	0.590499	+	1.00541i
$130$		0			0
$131$		−	2.09375i		−	0.182932i	−0.995808	−	0.0914660i	$-0.970845\pi$
$131$		−	2.09375i		−	0.182932i	0.995808	−	0.0914660i	$-0.0291553\pi$
$132$		0			0
$133$		−	1.60375i		−	0.139063i
$134$		0			0
$135$	0.137807	+	6.16837i	0.0118605	+	0.530889i
$136$		0			0
$137$	−0.988008			−0.0844112			−0.0422056	−	0.999109i	$-0.513438\pi$
$137$	−0.988008			−0.0844112			−0.0422056	+	0.999109i	$0.513438\pi$
$138$		0			0
$139$		−	5.60796i		−	0.475661i	−0.971307	−	0.237831i	$-0.923564\pi$
$139$		−	5.60796i		−	0.475661i	0.971307	−	0.237831i	$-0.0764363\pi$
$140$		0			0
$141$	−4.16508	−	7.09166i	−0.350763	−	0.597226i
$142$		0			0
$143$			27.9673i			2.33874i
$144$		0			0
$145$			9.86399i			0.819160i
$146$		0			0
$147$	10.2092	−	5.99608i	0.842041	−	0.494548i
$148$		0			0
$149$		−	1.09924i		−	0.0900533i	−0.998986	−	0.0450267i	$-0.985663\pi$
$149$		−	1.09924i		−	0.0900533i	0.998986	−	0.0450267i	$-0.0143373\pi$
$150$		0			0
$151$	−14.7236			−1.19819			−0.599094	−	0.800679i	$-0.704473\pi$
$151$	−14.7236			−1.19819			−0.599094	+	0.800679i	$0.704473\pi$
$152$		0			0
$153$	−1.28892	−	0.718784i	−0.104203	−	0.0581102i
$154$		0			0
$155$			6.46161i			0.519009i
$156$		0			0
$157$	−0.633370			−0.0505485			−0.0252742	−	0.999681i	$-0.508046\pi$
$157$	−0.633370			−0.0505485			−0.0252742	+	0.999681i	$0.508046\pi$
$158$		0			0
$159$	−15.2336	+	8.94701i	−1.20810	+	0.709544i
$160$		0			0
$161$	−1.23880			−0.0976314
$162$		0			0
$163$	−15.9692			−1.25081			−0.625403	−	0.780302i	$-0.715065\pi$
$163$	−15.9692			−1.25081			−0.625403	+	0.780302i	$0.715065\pi$
$164$		0			0
$165$	−10.2739	+	6.03405i	−0.799819	+	0.469750i
$166$		0			0
$167$			6.23476i			0.482460i	0.970468	+	0.241230i	$0.0775509\pi$
$167$			6.23476i			0.482460i	−0.970468	+	0.241230i	$0.922449\pi$
$168$		0			0
$169$	−10.3048			−0.792676
$170$		0			0
$171$	5.78133	−	10.3671i	0.442109	−	0.792790i
$172$		0			0
$173$		−	6.92160i		−	0.526239i	−0.964763	−	0.263120i	$-0.915249\pi$
$173$		−	6.92160i		−	0.526239i	0.964763	−	0.263120i	$-0.0847514\pi$
$174$		0			0
$175$			1.45515i			0.109999i
$176$		0			0
$177$	−9.53624	−	16.2369i	−0.716788	−	1.22044i
$178$		0			0
$179$	−6.63766			−0.496122			−0.248061	−	0.968744i	$-0.579793\pi$
$179$	−6.63766			−0.496122			−0.248061	+	0.968744i	$0.579793\pi$
$180$		0			0
$181$	22.5194			1.67385			0.836926	−	0.547316i	$-0.184351\pi$
$181$	22.5194			1.67385			0.836926	+	0.547316i	$0.184351\pi$
$182$		0			0
$183$	11.6259	+	19.7948i	0.859412	+	1.46328i
$184$		0			0
$185$	−1.13600			−0.0835202
$186$		0			0
$187$		−	2.84992i		−	0.208407i
$188$		0			0
$189$	−2.10561	+	0.0470411i	−0.153160	+	0.00342174i
$190$		0			0
$191$	−17.0778			−1.23571			−0.617853	−	0.786294i	$-0.711997\pi$
$191$	−17.0778			−1.23571			−0.617853	+	0.786294i	$0.711997\pi$
$192$		0			0
$193$	−20.7739			−1.49534			−0.747668	−	0.664073i	$-0.768827\pi$
$193$	−20.7739			−1.49534			−0.747668	+	0.664073i	$0.768827\pi$
$194$		0			0
$195$	−5.02810	−	8.56107i	−0.360069	−	0.613071i
$196$		0			0
$197$	10.1999			0.726710			0.363355	−	0.931651i	$-0.381631\pi$
$197$	10.1999			0.726710			0.363355	+	0.931651i	$0.381631\pi$
$198$		0			0
$199$	0.968946			0.0686868			0.0343434	−	0.999410i	$-0.489066\pi$
$199$	0.968946			0.0686868			0.0343434	+	0.999410i	$0.489066\pi$
$200$		0			0
$201$	11.0713	+	8.85593i	0.780906	+	0.624649i
$202$		0			0
$203$	−3.36713			−0.236326
$204$		0			0
$205$	−7.99113			−0.558125
$206$		0			0
$207$	−8.00794	−	4.46573i	−0.556590	−	0.310390i
$208$		0			0
$209$	22.9225			1.58558
$210$		0			0
$211$	−14.2579			−0.981552			−0.490776	−	0.871286i	$-0.663287\pi$
$211$	−14.2579			−0.981552			−0.490776	+	0.871286i	$0.663287\pi$
$212$		0			0
$213$	0.630495	+	1.07351i	0.0432008	+	0.0735558i
$214$		0			0
$215$		−	9.07878i		−	0.619168i
$216$		0			0
$217$	−2.20570			−0.149733
$218$		0			0
$219$	−14.1422	+	8.30599i	−0.955639	+	0.561267i
$220$		0			0
$221$	2.37480			0.159746
$222$		0			0
$223$	14.4292			0.966248			0.483124	−	0.875552i	$-0.339502\pi$
$223$	14.4292			0.966248			0.483124	+	0.875552i	$0.339502\pi$
$224$		0			0
$225$	−5.24563	+	9.40646i	−0.349709	+	0.627097i
$226$		0			0
$227$		−	14.6956i		−	0.975380i	−0.873017	−	0.487690i	$-0.837840\pi$
$227$		−	14.6956i		−	0.975380i	0.873017	−	0.487690i	$-0.162160\pi$
$228$		0			0
$229$		−	0.373828i		−	0.0247033i	−0.999924	−	0.0123516i	$-0.996068\pi$
$229$		−	0.373828i		−	0.0247033i	0.999924	−	0.0123516i	$-0.00393175\pi$
$230$		0			0
$231$	−2.05976	−	3.50704i	−0.135522	−	0.230746i
$232$		0			0
$233$	−7.18012			−0.470385			−0.235193	−	0.971949i	$-0.575572\pi$
$233$	−7.18012			−0.470385			−0.235193	+	0.971949i	$0.575572\pi$
$234$		0			0
$235$			5.63815i			0.367793i
$236$		0			0
$237$	12.3746	+	21.0695i	0.803814	+	1.36861i
$238$		0			0
$239$	3.46993			0.224451			0.112226	−	0.993683i	$-0.464202\pi$
$239$	3.46993			0.224451			0.112226	+	0.993683i	$0.464202\pi$
$240$		0			0
$241$	−10.3100			−0.664127			−0.332064	−	0.943257i	$-0.607745\pi$
$241$	−10.3100			−0.664127			−0.332064	+	0.943257i	$0.607745\pi$
$242$		0			0
$243$	−13.7807	−	7.28636i	−0.884035	−	0.467420i
$244$		0			0
$245$	−8.11673			−0.518559
$246$		0			0
$247$			19.1010i			1.21537i
$248$		0			0
$249$	13.9338	+	23.7244i	0.883021	+	1.50347i
$250$		0			0
$251$	−20.3997			−1.28762			−0.643810	−	0.765186i	$-0.722647\pi$
$251$	−20.3997			−1.28762			−0.643810	+	0.765186i	$0.722647\pi$
$252$		0			0
$253$		−	17.7063i		−	1.11318i
$254$		0			0
$255$	0.512372	+	0.872389i	0.0320860	+	0.0546311i
$256$		0			0
$257$			0.735715i			0.0458926i	0.999737	+	0.0229463i	$0.00730468\pi$
$257$			0.735715i			0.0458926i	−0.999737	+	0.0229463i	$0.992695\pi$
$258$		0			0
$259$		−	0.387779i		−	0.0240954i
$260$		0			0
$261$	−21.7659	−	12.1381i	−1.34728	−	0.751327i
$262$		0			0
$263$			15.8850i			0.979511i	0.871860	+	0.489756i	$0.162914\pi$
$263$			15.8850i			0.979511i	−0.871860	+	0.489756i	$0.837086\pi$
$264$		0			0
$265$	12.1113			0.743993
$266$		0			0
$267$	9.24881	+	15.7475i	0.566018	+	0.963729i
$268$		0			0
$269$		−	16.0323i		−	0.977508i	−0.872422	−	0.488754i	$-0.837452\pi$
$269$		−	16.0323i		−	0.977508i	0.872422	−	0.488754i	$-0.162548\pi$
$270$		0			0
$271$			21.9510i			1.33343i	0.745315	+	0.666713i	$0.232299\pi$
$271$			21.9510i			1.33343i	−0.745315	+	0.666713i	$0.767701\pi$
$272$		0			0
$273$	2.92237	−	1.71637i	0.176870	−	0.103879i
$274$		0			0
$275$	−20.7985			−1.25420
$276$		0			0
$277$	28.3207			1.70162			0.850812	−	0.525470i	$-0.176111\pi$
$277$	28.3207			1.70162			0.850812	+	0.525470i	$0.176111\pi$
$278$		0			0
$279$	−14.2582	−	7.95128i	−0.853618	−	0.476031i
$280$		0			0
$281$	−13.1685			−0.785568			−0.392784	−	0.919631i	$-0.628488\pi$
$281$	−13.1685			−0.785568			−0.392784	+	0.919631i	$0.628488\pi$
$282$		0			0
$283$	−14.0592			−0.835730			−0.417865	−	0.908509i	$-0.637222\pi$
$283$	−14.0592			−0.835730			−0.417865	+	0.908509i	$0.637222\pi$
$284$		0			0
$285$	−7.01681	+	4.12112i	−0.415640	+	0.244114i
$286$		0			0
$287$		−	2.72781i		−	0.161018i
$288$		0			0
$289$	16.7580			0.985765
$290$		0			0
$291$	−10.5869	−	18.0258i	−0.620617	−	1.05669i
$292$		0			0
$293$			8.96257i			0.523599i	0.965122	+	0.261800i	$0.0843159\pi$
$293$			8.96257i			0.523599i	−0.965122	+	0.261800i	$0.915684\pi$
$294$		0			0
$295$			12.9089i			0.751588i
$296$		0			0
$297$	−0.672360	−	30.0955i	−0.0390143	−	1.74632i
$298$		0			0
$299$	14.7544			0.853268
$300$		0			0
$301$	3.09909			0.178628
$302$		0			0
$303$	−10.2880	+	6.04235i	−0.591029	+	0.347124i
$304$		0			0
$305$		−	15.7377i		−	0.901137i
$306$		0			0
$307$	17.5848			1.00362			0.501810	−	0.864978i	$-0.332668\pi$
$307$	17.5848			1.00362			0.501810	+	0.864978i	$0.332668\pi$
$308$		0			0
$309$	−7.83440	+	4.60131i	−0.445684	+	0.261759i
$310$		0			0
$311$	9.69860			0.549957			0.274978	−	0.961450i	$-0.411329\pi$
$311$	9.69860			0.549957			0.274978	+	0.961450i	$0.411329\pi$
$312$		0			0
$313$			11.6326i			0.657513i	0.944415	+	0.328756i	$0.106630\pi$
$313$			11.6326i			0.657513i	−0.944415	+	0.328756i	$0.893370\pi$
$314$		0			0
$315$	1.26102	+	0.703225i	0.0710506	+	0.0396223i
$316$		0			0
$317$			21.4781i			1.20633i	0.797616	+	0.603166i	$0.206094\pi$
$317$			21.4781i			1.20633i	−0.797616	+	0.603166i	$0.793906\pi$
$318$		0			0
$319$		−	48.1264i		−	2.69456i
$320$		0			0
$321$	11.3728	+	19.3639i	0.634769	+	1.08079i
$322$		0			0
$323$		−	1.94643i		−	0.108302i
$324$		0			0
$325$		−	17.3311i		−	0.961358i
$326$		0			0
$327$	−3.34015	−	5.68709i	−0.184711	−	0.314497i
$328$		0			0
$329$	−1.92461			−0.106107
$330$		0			0
$331$			12.0379i			0.661664i	0.943690	+	0.330832i	$0.107329\pi$
$331$			12.0379i			0.661664i	−0.943690	+	0.330832i	$0.892671\pi$
$332$		0			0
$333$	1.39789	−	2.50670i	0.0766040	−	0.137366i
$334$		0			0
$335$	−3.46993	−	9.07878i	−0.189583	−	0.496027i
$336$		0			0
$337$		−	24.3816i		−	1.32815i	−0.747666	−	0.664075i	$-0.768826\pi$
$337$		−	24.3816i		−	1.32815i	0.747666	−	0.664075i	$-0.231174\pi$
$338$		0			0
$339$	9.02696	−	5.30172i	0.490277	−	0.287950i
$340$		0			0
$341$		−	31.5262i		−	1.70724i
$342$		0			0
$343$		−	5.60796i		−	0.302802i
$344$		0			0
$345$	3.18331	+	5.42006i	0.171384	+	0.291806i
$346$		0			0
$347$	33.1267			1.77833			0.889167	−	0.457582i	$-0.151284\pi$
$347$	33.1267			1.77833			0.889167	+	0.457582i	$0.151284\pi$
$348$		0			0
$349$	−14.3082			−0.765898			−0.382949	−	0.923769i	$-0.625092\pi$
$349$	−14.3082			−0.765898			−0.382949	+	0.923769i	$0.625092\pi$
$350$		0			0
$351$	25.0782	−	0.560268i	1.33857	−	0.0299049i
$352$		0			0
$353$	2.96866			0.158006			0.0790030	−	0.996874i	$-0.474826\pi$
$353$	2.96866			0.158006			0.0790030	+	0.996874i	$0.474826\pi$
$354$		0			0
$355$		−	0.853484i		−	0.0452982i
$356$		0			0
$357$	−0.297794	+	0.174901i	−0.0157610	+	0.00925674i
$358$		0			0
$359$			24.5727i			1.29690i	0.761259	+	0.648448i	$0.224582\pi$
$359$			24.5727i			1.29690i	−0.761259	+	0.648448i	$0.775418\pi$
$360$		0			0
$361$	−3.34445			−0.176024
$362$		0			0
$363$	33.6976	−	19.7913i	1.76866	−	1.03877i
$364$		0			0
$365$	11.2436			0.588517
$366$		0			0
$367$			28.2148i			1.47280i	0.676547	+	0.736399i	$0.263475\pi$
$367$			28.2148i			1.47280i	−0.676547	+	0.736399i	$0.736525\pi$
$368$		0			0
$369$	9.83342	−	17.6333i	0.511907	−	0.917951i
$370$		0			0
$371$			4.13426i			0.214640i
$372$		0			0
$373$		−	30.3409i		−	1.57099i	−0.618865	−	0.785497i	$-0.712407\pi$
$373$		−	30.3409i		−	1.57099i	0.618865	−	0.785497i	$-0.287593\pi$
$374$		0			0
$375$	15.2336	−	8.94701i	0.786660	−	0.462022i
$376$		0			0
$377$	40.1031			2.06542
$378$		0			0
$379$		−	5.51974i		−	0.283530i	−0.989900	−	0.141765i	$-0.954722\pi$
$379$		−	5.51974i		−	0.283530i	0.989900	−	0.141765i	$-0.0452777\pi$
$380$		0			0
$381$	−24.1707	+	14.1959i	−1.23830	+	0.727280i
$382$		0			0
$383$	9.98994			0.510462			0.255231	−	0.966880i	$-0.417848\pi$
$383$	9.98994			0.510462			0.255231	+	0.966880i	$0.417848\pi$
$384$		0			0
$385$			2.78823i			0.142102i
$386$		0			0
$387$	20.0333	+	11.1718i	1.01835	+	0.567895i
$388$		0			0
$389$			24.9638i			1.26571i	0.774269	+	0.632857i	$0.218118\pi$
$389$			24.9638i			1.26571i	−0.774269	+	0.632857i	$0.781882\pi$
$390$		0			0
$391$	−1.50350			−0.0760352
$392$		0			0
$393$	−1.83658	−	3.12704i	−0.0926430	−	0.157738i
$394$		0			0
$395$		−	16.7511i		−	0.842840i
$396$		0			0
$397$	11.4313			0.573721			0.286860	−	0.957972i	$-0.407388\pi$
$397$	11.4313			0.573721			0.286860	+	0.957972i	$0.407388\pi$
$398$		0			0
$399$	−1.40677	−	2.39522i	−0.0704264	−	0.119911i
$400$		0			0
$401$	−35.4972			−1.77264			−0.886322	−	0.463069i	$-0.846748\pi$
$401$	−35.4972			−1.77264			−0.886322	+	0.463069i	$0.846748\pi$
$402$		0			0
$403$	26.2704			1.30862
$404$		0			0
$405$	5.61653	+	9.09165i	0.279088	+	0.451768i
$406$		0			0
$407$	5.54253			0.274733
$408$		0			0
$409$			11.2159i			0.554592i	0.960785	+	0.277296i	$0.0894383\pi$
$409$			11.2159i			0.554592i	−0.960785	+	0.277296i	$0.910562\pi$
$410$		0			0
$411$	−1.47560	+	0.866651i	−0.0727860	+	0.0427487i
$412$		0			0
$413$	−4.40654			−0.216831
$414$		0			0
$415$		−	18.8619i		−	0.925892i
$416$		0			0
$417$	−4.91914	−	8.37555i	−0.240891	−	0.410153i
$418$		0			0
$419$		−	33.2871i		−	1.62618i	−0.582137	−	0.813091i	$-0.697783\pi$
$419$		−	33.2871i		−	1.62618i	0.582137	−	0.813091i	$-0.302217\pi$
$420$		0			0
$421$	−18.6185			−0.907408			−0.453704	−	0.891152i	$-0.649898\pi$
$421$	−18.6185			−0.907408			−0.453704	+	0.891152i	$0.649898\pi$
$422$		0			0
$423$	−12.4412	−	6.93799i	−0.604911	−	0.337336i
$424$		0			0
$425$			1.76607i			0.0856671i
$426$		0			0
$427$	5.37214			0.259976
$428$		0			0
$429$	24.5321	+	41.7695i	1.18442	+	2.01665i
$430$		0			0
$431$		−	6.17133i		−	0.297263i	−0.988893	−	0.148631i	$-0.952513\pi$
$431$		−	6.17133i		−	0.297263i	0.988893	−	0.148631i	$-0.0474868\pi$
$432$		0			0
$433$		−	0.164844i		−	0.00792190i	−0.999992	−	0.00396095i	$-0.998739\pi$
$433$		−	0.164844i		−	0.00792190i	0.999992	−	0.00396095i	$-0.00126081\pi$
$434$		0			0
$435$	8.65240	+	14.7320i	0.414851	+	0.706344i
$436$		0			0
$437$		−	12.0930i		−	0.578485i
$438$		0			0
$439$	17.9981			0.859004			0.429502	−	0.903066i	$-0.358689\pi$
$439$	17.9981			0.859004			0.429502	+	0.903066i	$0.358689\pi$
$440$		0			0
$441$	9.98797	−	17.9104i	0.475618	−	0.852877i
$442$		0			0
$443$	5.50598			0.261597			0.130799	−	0.991409i	$-0.458246\pi$
$443$	5.50598			0.261597			0.130799	+	0.991409i	$0.458246\pi$
$444$		0			0
$445$		−	12.5199i		−	0.593498i
$446$		0			0
$447$	−0.964221	−	1.64173i	−0.0456061	−	0.0776511i
$448$		0			0
$449$		−	23.5147i		−	1.10973i	−0.831941	−	0.554864i	$-0.812770\pi$
$449$		−	23.5147i		−	1.10973i	0.831941	−	0.554864i	$-0.187230\pi$
$450$		0			0
$451$	38.9887			1.83591
$452$		0			0
$453$	−21.9898	+	12.9151i	−1.03317	+	0.606804i
$454$		0			0
$455$	−2.32340			−0.108923
$456$		0			0
$457$	−0.108761			−0.00508765			−0.00254382	−	0.999997i	$-0.500810\pi$
$457$	−0.108761			−0.00508765			−0.00254382	+	0.999997i	$0.500810\pi$
$458$		0			0
$459$	−2.55551	+	0.0570924i	−0.119281	+	0.00266484i
$460$		0			0
$461$		−	33.5711i		−	1.56356i	−0.623553	−	0.781781i	$-0.714312\pi$
$461$		−	33.5711i		−	1.56356i	0.623553	−	0.781781i	$-0.285688\pi$
$462$		0			0
$463$		−	6.79244i		−	0.315672i	−0.987465	−	0.157836i	$-0.949548\pi$
$463$		−	6.79244i		−	0.315672i	0.987465	−	0.157836i	$-0.0504517\pi$
$464$		0			0
$465$	5.66793	+	9.65048i	0.262844	+	0.447530i
$466$		0			0
$467$		−	24.6314i		−	1.13980i	−0.821713	−	0.569902i	$-0.806981\pi$
$467$		−	24.6314i		−	1.13980i	0.821713	−	0.569902i	$-0.193019\pi$
$468$		0			0
$469$	3.09909	−	1.18448i	0.143103	−	0.0546942i
$470$		0			0
$471$	−0.945945	+	0.555573i	−0.0435869	+	0.0255995i
$472$		0			0
$473$			44.2954i			2.03670i
$474$		0			0
$475$	−14.2049			−0.651766
$476$		0			0
$477$	−14.9035	+	26.7249i	−0.682384	+	1.22365i
$478$		0			0
$479$			11.3466i			0.518440i	0.965818	+	0.259220i	$0.0834654\pi$
$479$			11.3466i			0.518440i	−0.965818	+	0.259220i	$0.916535\pi$
$480$		0			0
$481$			4.61852i			0.210586i
$482$		0			0
$483$	−1.85017	+	1.08664i	−0.0841855	+	0.0494439i
$484$		0			0
$485$			14.3312i			0.650747i
$486$		0			0
$487$		−	17.9467i		−	0.813245i	−0.913596	−	0.406622i	$-0.866707\pi$
$487$		−	17.9467i		−	0.813245i	0.913596	−	0.406622i	$-0.133293\pi$
$488$		0			0
$489$	−23.8502	+	14.0077i	−1.07854	+	0.633451i
$490$		0			0
$491$		−	27.4608i		−	1.23929i	−0.784882	−	0.619645i	$-0.787277\pi$
$491$		−	27.4608i		−	1.23929i	0.784882	−	0.619645i	$-0.212723\pi$
$492$		0			0
$493$	−4.08658			−0.184050
$494$		0			0
$495$	−10.0512	+	18.0238i	−0.451769	+	0.810111i
$496$		0			0
$497$	0.291341			0.0130684
$498$		0			0
$499$			15.9214i			0.712741i	0.934345	+	0.356370i	$0.115986\pi$
$499$			15.9214i			0.712741i	−0.934345	+	0.356370i	$0.884014\pi$
$500$		0			0
$501$	5.46894	+	9.31168i	0.244334	+	0.416015i
$502$		0			0
$503$	3.52133			0.157009			0.0785043	−	0.996914i	$-0.474986\pi$
$503$	3.52133			0.157009			0.0785043	+	0.996914i	$0.474986\pi$
$504$		0			0
$505$	8.17936			0.363977
$506$		0			0
$507$	−15.3903	+	9.03905i	−0.683508	+	0.401438i
$508$		0			0
$509$		−	17.6152i		−	0.780781i	−0.920649	−	0.390391i	$-0.872340\pi$
$509$		−	17.6152i		−	0.780781i	0.920649	−	0.390391i	$-0.127660\pi$
$510$		0			0
$511$			3.83806i			0.169786i
$512$		0			0
$513$	−0.459206	−	20.5545i	−0.0202744	−	0.907505i
$514$		0			0
$515$	6.22867			0.274468
$516$		0			0
$517$		−	27.5086i		−	1.20983i
$518$		0			0
$519$	−6.07142	−	10.3375i	−0.266506	−	0.453765i
$520$		0			0
$521$	−20.5991			−0.902464			−0.451232	−	0.892407i	$-0.649015\pi$
$521$	−20.5991			−0.902464			−0.451232	+	0.892407i	$0.649015\pi$
$522$		0			0
$523$	17.6925			0.773640			0.386820	−	0.922155i	$-0.373573\pi$
$523$	17.6925			0.773640			0.386820	+	0.922155i	$0.373573\pi$
$524$		0			0
$525$	1.27641	+	2.17328i	0.0557073	+	0.0948498i
$526$		0			0
$527$	−2.67700			−0.116612
$528$		0			0
$529$	13.6589			0.593866
$530$		0			0
$531$	−28.4850	−	15.8850i	−1.23614	−	0.689350i
$532$		0			0
$533$			32.4888i			1.40725i
$534$		0			0
$535$		−	15.3951i		−	0.665587i
$536$		0			0
$537$	−9.91342	+	5.82236i	−0.427796	+	0.251253i
$538$		0			0
$539$	39.6015			1.70576
$540$		0			0
$541$		−	5.24417i		−	0.225464i	−0.993625	−	0.112732i	$-0.964040\pi$
$541$		−	5.24417i		−	0.225464i	0.993625	−	0.112732i	$-0.0359602\pi$
$542$		0			0
$543$	33.6329	−	19.7533i	1.44333	−	0.847696i
$544$		0			0
$545$			4.52147i			0.193678i
$546$		0			0
$547$			23.8795i			1.02101i	0.859874	+	0.510507i	$0.170542\pi$
$547$			23.8795i			1.02101i	−0.859874	+	0.510507i	$0.829458\pi$
$548$		0			0
$549$	34.7268	+	19.3659i	1.48211	+	0.826515i
$550$		0			0
$551$		−	32.8692i		−	1.40028i
$552$		0			0
$553$	5.71808			0.243157
$554$		0			0
$555$	−1.69662	+	0.996463i	−0.0720177	+	0.0422975i
$556$		0			0
$557$		−	11.0878i		−	0.469805i	−0.972019	−	0.234903i	$-0.924523\pi$
$557$		−	11.0878i		−	0.469805i	0.972019	−	0.234903i	$-0.0754772\pi$
$558$		0			0
$559$	−36.9107			−1.56116
$560$		0			0
$561$	−2.49986	−	4.25639i	−0.105544	−	0.179705i
$562$		0			0
$563$	3.76127			0.158519			0.0792594	−	0.996854i	$-0.474744\pi$
$563$	3.76127			0.158519			0.0792594	+	0.996854i	$0.474744\pi$
$564$		0			0
$565$	−7.17680			−0.301930
$566$		0			0
$567$	−3.10348	+	1.91723i	−0.130334	+	0.0805161i
$568$		0			0
$569$			21.9066i			0.918374i	0.888340	+	0.459187i	$0.151859\pi$
$569$			21.9066i			0.918374i	−0.888340	+	0.459187i	$0.848141\pi$
$570$		0			0
$571$	20.4657			0.856464			0.428232	−	0.903669i	$-0.359137\pi$
$571$	20.4657			0.856464			0.428232	+	0.903669i	$0.359137\pi$
$572$		0			0
$573$	−25.5059	+	14.9801i	−1.06552	+	0.625804i
$574$		0			0
$575$			10.9724i			0.457582i
$576$		0			0
$577$		−	17.0870i		−	0.711343i	−0.934611	−	0.355672i	$-0.884252\pi$
$577$		−	17.0870i		−	0.711343i	0.934611	−	0.355672i	$-0.115748\pi$
$578$		0			0
$579$	−31.0260	+	18.2222i	−1.28940	+	0.757290i
$580$		0			0
$581$	6.43859			0.267118
$582$		0			0
$583$	−59.0912			−2.44731
$584$		0			0
$585$	−15.0190	−	8.37555i	−0.620960	−	0.346286i
$586$		0			0
$587$	−30.3897			−1.25432			−0.627158	−	0.778892i	$-0.715782\pi$
$587$	−30.3897			−1.25432			−0.627158	+	0.778892i	$0.715782\pi$
$588$		0			0
$589$		−	21.5317i		−	0.887197i
$590$		0			0
$591$	15.2336	−	8.94701i	0.626627	−	0.368031i
$592$		0			0
$593$	−29.8667			−1.22648			−0.613239	−	0.789897i	$-0.710134\pi$
$593$	−29.8667			−1.22648			−0.613239	+	0.789897i	$0.710134\pi$
$594$		0			0
$595$	0.236758			0.00970615
$596$		0			0
$597$	1.44713	−	0.849931i	0.0592271	−	0.0347853i
$598$		0			0
$599$	−47.8359			−1.95452			−0.977261	−	0.212039i	$-0.931990\pi$
$599$	−47.8359			−1.95452			−0.977261	+	0.212039i	$0.931990\pi$
$600$		0			0
$601$	19.0070			0.775312			0.387656	−	0.921804i	$-0.373285\pi$
$601$	19.0070			0.775312			0.387656	+	0.921804i	$0.373285\pi$
$602$		0			0
$603$	24.3032	+	3.51505i	0.989702	+	0.143144i
$604$		0			0
$605$	−26.7909			−1.08921
$606$		0			0
$607$	32.4975			1.31903			0.659516	−	0.751691i	$-0.270761\pi$
$607$	32.4975			1.31903			0.659516	+	0.751691i	$0.270761\pi$
$608$		0			0
$609$	−5.02884	+	2.95354i	−0.203779	+	0.119684i
$610$		0			0
$611$	22.9225			0.927346
$612$		0			0
$613$	5.95591			0.240557			0.120278	−	0.992740i	$-0.461621\pi$
$613$	5.95591			0.240557			0.120278	+	0.992740i	$0.461621\pi$
$614$		0			0
$615$	−11.9348	+	7.00958i	−0.481259	+	0.282653i
$616$		0			0
$617$			42.8539i			1.72523i	0.505858	+	0.862617i	$0.331176\pi$
$617$			42.8539i			1.72523i	−0.505858	+	0.862617i	$0.668824\pi$
$618$		0			0
$619$	3.52124			0.141531			0.0707654	−	0.997493i	$-0.477456\pi$
$619$	3.52124			0.141531			0.0707654	+	0.997493i	$0.477456\pi$
$620$		0			0
$621$	−15.8771	+	0.354709i	−0.637128	+	0.0142340i
$622$		0			0
$623$	4.27372			0.171223
$624$		0			0
$625$	5.83908			0.233563
$626$		0			0
$627$	34.2350	−	20.1069i	1.36722	−	0.802994i
$628$		0			0
$629$		−	0.470636i		−	0.0187655i
$630$		0			0
$631$			12.3122i			0.490140i	0.969505	+	0.245070i	$0.0788110\pi$
$631$			12.3122i			0.490140i	−0.969505	+	0.245070i	$0.921189\pi$
$632$		0			0
$633$	−21.2943	+	12.5066i	−0.846371	+	0.497092i
$634$		0			0
$635$	19.2166			0.762589
$636$		0			0
$637$			32.9994i			1.30748i
$638$		0			0
$639$	1.88330	+	1.05025i	0.0745023	+	0.0415472i
$640$		0			0
$641$	46.8182			1.84921			0.924604	−	0.380930i	$-0.124396\pi$
$641$	46.8182			1.84921			0.924604	+	0.380930i	$0.124396\pi$
$642$		0			0
$643$	0.472446			0.0186314			0.00931572	−	0.999957i	$-0.497035\pi$
$643$	0.472446			0.0186314			0.00931572	+	0.999957i	$0.497035\pi$
$644$		0			0
$645$	−7.96363	−	13.5592i	−0.313568	−	0.533895i
$646$		0			0
$647$	−33.8596			−1.33116			−0.665579	−	0.746327i	$-0.731815\pi$
$647$	−33.8596			−1.33116			−0.665579	+	0.746327i	$0.731815\pi$
$648$		0			0
$649$		−	62.9828i		−	2.47229i
$650$		0			0
$651$	−3.29424	+	1.93478i	−0.129112	+	0.0758299i
$652$		0			0
$653$	−32.6254			−1.27673			−0.638366	−	0.769733i	$-0.720389\pi$
$653$	−32.6254			−1.27673			−0.638366	+	0.769733i	$0.720389\pi$
$654$		0			0
$655$			2.48612i			0.0971408i
$656$		0			0
$657$	−13.8357	+	24.8102i	−0.539783	+	0.967937i
$658$		0			0
$659$		−	26.2371i		−	1.02205i	−0.859565	−	0.511026i	$-0.829266\pi$
$659$		−	26.2371i		−	1.02205i	0.859565	−	0.511026i	$-0.170734\pi$
$660$		0			0
$661$			16.7925i			0.653151i	0.945171	+	0.326575i	$0.105895\pi$
$661$			16.7925i			0.653151i	−0.945171	+	0.326575i	$0.894105\pi$
$662$		0			0
$663$	3.54679	−	2.08310i	0.137746	−	0.0809011i
$664$		0			0
$665$			1.90430i			0.0738456i
$666$		0			0
$667$	−25.3895			−0.983085
$668$		0			0
$669$	21.5501	−	12.6568i	0.833175	−	0.489341i
$670$		0			0
$671$			76.7842i			2.96422i
$672$		0			0
$673$			41.7731i			1.61023i	0.593116	+	0.805117i	$0.297898\pi$
$673$			41.7731i			1.61023i	−0.593116	+	0.805117i	$0.702102\pi$
$674$		0			0
$675$	0.416656	+	18.6499i	0.0160371	+	0.717837i
$676$		0			0
$677$	−27.8198			−1.06920			−0.534601	−	0.845105i	$-0.679538\pi$
$677$	−27.8198			−1.06920			−0.534601	+	0.845105i	$0.679538\pi$
$678$		0			0
$679$	−4.89204			−0.187739
$680$		0			0
$681$	−12.8905	−	21.9480i	−0.493966	−	0.841049i
$682$		0			0
$683$	7.23120			0.276694			0.138347	−	0.990384i	$-0.455821\pi$
$683$	7.23120			0.276694			0.138347	+	0.990384i	$0.455821\pi$
$684$		0			0
$685$	1.17316			0.0448242
$686$		0			0
$687$	−0.327911	−	0.558316i	−0.0125106	−	0.0213011i
$688$		0			0
$689$		−	49.2399i		−	1.87589i
$690$		0			0
$691$	−6.44690			−0.245252			−0.122626	−	0.992453i	$-0.539132\pi$
$691$	−6.44690			−0.245252			−0.122626	+	0.992453i	$0.539132\pi$
$692$		0			0
$693$	−6.15253	−	3.43104i	−0.233715	−	0.130334i
$694$		0			0
$695$			6.65890i			0.252586i
$696$		0			0
$697$		−	3.31067i		−	0.125400i
$698$		0			0
$699$	−10.7236	+	6.29819i	−0.405603	+	0.238219i
$700$		0			0
$701$	18.5822			0.701841			0.350921	−	0.936405i	$-0.385869\pi$
$701$	18.5822			0.701841			0.350921	+	0.936405i	$0.385869\pi$
$702$		0			0
$703$	3.78542			0.142770
$704$		0			0
$705$	4.94562	+	8.42064i	0.186263	+	0.317140i
$706$		0			0
$707$			2.79207i			0.105006i
$708$		0			0
$709$	−41.5629			−1.56093			−0.780464	−	0.625200i	$-0.785017\pi$
$709$	−41.5629			−1.56093			−0.780464	+	0.625200i	$0.785017\pi$
$710$		0			0
$711$	36.9631	+	20.6129i	1.38622	+	0.773046i
$712$		0			0
$713$	−16.6319			−0.622870
$714$		0			0
$715$		−	33.2084i		−	1.24192i
$716$		0			0
$717$	5.18238	−	3.04372i	0.193539	−	0.113670i
$718$		0			0
$719$			24.9842i			0.931755i	0.884849	+	0.465878i	$0.154261\pi$
$719$			24.9842i			0.931755i	−0.884849	+	0.465878i	$0.845739\pi$
$720$		0			0
$721$			2.12619i			0.0791834i
$722$		0			0
$723$	−15.3981	+	9.04365i	−0.572663	+	0.336337i
$724$		0			0
$725$			29.8236i			1.10762i
$726$		0			0
$727$		−	34.6762i		−	1.28607i	−0.765837	−	0.643035i	$-0.777675\pi$
$727$		−	34.6762i		−	1.28607i	0.765837	−	0.643035i	$-0.222325\pi$
$728$		0			0
$729$	−26.9731	+	1.20580i	−0.999002	+	0.0446594i
$730$		0			0
$731$	3.76127			0.139116
$732$		0			0
$733$		−	47.4391i		−	1.75220i	−0.482127	−	0.876102i	$-0.660135\pi$
$733$		−	47.4391i		−	1.75220i	0.482127	−	0.876102i	$-0.339865\pi$
$734$		0			0
$735$	−12.1224	+	7.11975i	−0.447142	+	0.262616i
$736$		0			0
$737$	16.9298	+	44.2954i	0.623617	+	1.63164i
$738$		0			0
$739$			7.73415i			0.284505i	0.989830	+	0.142253i	$0.0454346\pi$
$739$			7.73415i			0.284505i	−0.989830	+	0.142253i	$0.954565\pi$
$740$		0			0
$741$	16.7548	+	28.5276i	0.615505	+	1.04799i
$742$		0			0
$743$			11.1793i			0.410128i	0.978749	+	0.205064i	$0.0657402\pi$
$743$			11.1793i			0.410128i	−0.978749	+	0.205064i	$0.934260\pi$
$744$		0			0
$745$			1.30524i			0.0478203i
$746$		0			0
$747$	41.6207	+	23.2103i	1.52282	+	0.849221i
$748$		0			0
$749$	5.25519			0.192020
$750$		0			0
$751$	21.3384			0.778650			0.389325	−	0.921101i	$-0.372708\pi$
$751$	21.3384			0.778650			0.389325	+	0.921101i	$0.372708\pi$
$752$		0			0
$753$	−30.4672	+	17.8940i	−1.11029	+	0.652095i
$754$		0			0
$755$	17.4828			0.636264
$756$		0			0
$757$		−	32.8320i		−	1.19330i	−0.802502	−	0.596649i	$-0.796498\pi$
$757$		−	32.8320i		−	1.19330i	0.802502	−	0.596649i	$-0.203502\pi$
$758$		0			0
$759$	−15.5314	−	26.4445i	−0.563754	−	0.959874i
$760$		0			0
$761$		−	32.7326i		−	1.18655i	−0.804998	−	0.593277i	$-0.797834\pi$
$761$		−	32.7326i		−	1.18655i	0.804998	−	0.593277i	$-0.202166\pi$
$762$		0			0
$763$	−1.54343			−0.0558758
$764$		0			0
$765$	1.53047	+	0.853484i	0.0553341	+	0.0308578i
$766$		0			0
$767$	52.4827			1.89504
$768$		0			0
$769$		−	42.6531i		−	1.53811i	−0.639182	−	0.769055i	$-0.720727\pi$
$769$		−	42.6531i		−	1.53811i	0.639182	−	0.769055i	$-0.279273\pi$
$770$		0			0
$771$	0.645347	+	1.09880i	0.0232416	+	0.0395722i
$772$		0			0
$773$			25.1233i			0.903622i	0.892114	+	0.451811i	$0.149222\pi$
$773$			25.1233i			0.903622i	−0.892114	+	0.451811i	$0.850778\pi$
$774$		0			0
$775$			19.5365i			0.701773i
$776$		0			0
$777$	−0.340148	−	0.579151i	−0.0122027	−	0.0207769i
$778$		0			0
$779$	26.6284			0.954061
$780$		0			0
$781$			4.16415i			0.149005i
$782$		0			0
$783$	−43.1548	+	0.964115i	−1.54223	+	0.0344547i
$784$		0			0
$785$	0.752064			0.0268423
$786$		0			0
$787$			21.8415i			0.778566i	0.921118	+	0.389283i	$0.127277\pi$
$787$			21.8415i			0.778566i	−0.921118	+	0.389283i	$0.872723\pi$
$788$		0			0
$789$	13.9338	+	23.7244i	0.496058	+	0.844611i
$790$		0			0
$791$		−	2.44984i		−	0.0871062i
$792$		0			0
$793$	−63.9832			−2.27211
$794$		0			0
$795$	18.0884	−	10.6237i	0.641529	−	0.376783i
$796$		0			0
$797$		−	12.2307i		−	0.433235i	−0.976257	−	0.216617i	$-0.930498\pi$
$797$		−	12.2307i		−	0.433235i	0.976257	−	0.216617i	$-0.0695024\pi$
$798$		0			0
$799$	−2.33585			−0.0826363
$800$		0			0
$801$	27.6264	+	15.4062i	0.976131	+	0.544352i
$802$		0			0
$803$	−54.8575			−1.93588
$804$		0			0
$805$	1.47096			0.0518444
$806$		0			0
$807$	−14.0631	−	23.9444i	−0.495043	−	0.842884i
$808$		0			0
$809$	−7.41607			−0.260735			−0.130367	−	0.991466i	$-0.541616\pi$
$809$	−7.41607			−0.260735			−0.130367	+	0.991466i	$0.541616\pi$
$810$		0			0
$811$		−	4.66626i		−	0.163854i	−0.996638	−	0.0819272i	$-0.973892\pi$
$811$		−	4.66626i		−	0.163854i	0.996638	−	0.0819272i	$-0.0261075\pi$
$812$		0			0
$813$	19.2547	+	32.7840i	0.675292	+	1.14978i
$814$		0			0
$815$	18.9619			0.664205
$816$		0			0
$817$			30.2527i			1.05841i
$818$		0			0
$819$	2.85904	−	5.12682i	0.0999029	−	0.179146i
$820$		0			0
$821$		−	1.36978i		−	0.0478055i	−0.999714	−	0.0239028i	$-0.992391\pi$
$821$		−	1.36978i		−	0.0478055i	0.999714	−	0.0239028i	$-0.00760921\pi$
$822$		0			0
$823$	1.53428			0.0534817			0.0267409	−	0.999642i	$-0.491487\pi$
$823$	1.53428			0.0534817			0.0267409	+	0.999642i	$0.491487\pi$
$824$		0			0
$825$	−31.0628	+	18.2438i	−1.08147	+	0.635169i
$826$		0			0
$827$		−	53.8248i		−	1.87167i	−0.352437	−	0.935836i	$-0.614647\pi$
$827$		−	53.8248i		−	1.87167i	0.352437	−	0.935836i	$-0.385353\pi$
$828$		0			0
$829$	45.0098			1.56326			0.781628	−	0.623745i	$-0.214390\pi$
$829$	45.0098			1.56326			0.781628	+	0.623745i	$0.214390\pi$
$830$		0			0
$831$	42.2972	−	24.8420i	1.46727	−	0.861761i
$832$		0			0
$833$		−	3.36270i		−	0.116511i
$834$		0			0
$835$		−	7.40316i		−	0.256197i
$836$		0			0
$837$	−28.2694	+	0.631564i	−0.977135	+	0.0218300i
$838$		0			0
$839$			18.2706i			0.630771i	0.948964	+	0.315385i	$0.102134\pi$
$839$			18.2706i			0.630771i	−0.948964	+	0.315385i	$0.897866\pi$
$840$		0			0
$841$	−40.0098			−1.37965
$842$		0			0
$843$	−19.6673	+	11.5510i	−0.677379	+	0.397839i
$844$		0			0
$845$	12.2359			0.420928
$846$		0			0
$847$		−	9.14522i		−	0.314233i
$848$		0			0
$849$	−20.9975	+	12.3323i	−0.720632	+	0.423243i
$850$		0			0
$851$		−	2.92401i		−	0.100234i
$852$		0			0
$853$	−31.5541			−1.08039			−0.540195	−	0.841540i	$-0.681650\pi$
$853$	−31.5541			−1.08039			−0.540195	+	0.841540i	$0.681650\pi$
$854$		0			0
$855$	−6.86475	+	12.3099i	−0.234770	+	0.420989i
$856$		0			0
$857$	−35.4458			−1.21080			−0.605402	−	0.795920i	$-0.706988\pi$
$857$	−35.4458			−1.21080			−0.605402	+	0.795920i	$0.706988\pi$
$858$		0			0
$859$	22.6529			0.772906			0.386453	−	0.922309i	$-0.373700\pi$
$859$	22.6529			0.772906			0.386453	+	0.922309i	$0.373700\pi$
$860$		0			0
$861$	−2.39276	−	4.07402i	−0.0815449	−	0.138842i
$862$		0			0
$863$			10.8240i			0.368454i	0.982884	+	0.184227i	$0.0589781\pi$
$863$			10.8240i			0.368454i	−0.982884	+	0.184227i	$0.941022\pi$
$864$		0			0
$865$			8.21871i			0.279445i
$866$		0			0
$867$	25.0283	−	14.6996i	0.850004	−	0.499225i
$868$		0			0
$869$			81.7287i			2.77246i
$870$		0			0
$871$	−36.9107	+	14.1074i	−1.25067	+	0.478010i
$872$		0			0
$873$	−31.6234	−	17.6352i	−1.07029	−	0.596860i
$874$		0			0
$875$		−	4.13426i		−	0.139764i
$876$		0			0
$877$	27.0714			0.914136			0.457068	−	0.889432i	$-0.348900\pi$
$877$	27.0714			0.914136			0.457068	+	0.889432i	$0.348900\pi$
$878$		0			0
$879$	7.86170	+	13.3857i	0.265169	+	0.451488i
$880$		0			0
$881$			2.66085i			0.0896463i	0.998995	+	0.0448231i	$0.0142724\pi$
$881$			2.66085i			0.0896463i	−0.998995	+	0.0448231i	$0.985728\pi$
$882$		0			0
$883$			6.37322i			0.214476i	0.994233	+	0.107238i	$0.0342007\pi$
$883$			6.37322i			0.214476i	−0.994233	+	0.107238i	$0.965799\pi$
$884$		0			0
$885$	11.3233	+	19.2796i	0.380630	+	0.648078i
$886$		0			0
$887$		−	31.6270i		−	1.06193i	−0.847393	−	0.530966i	$-0.821829\pi$
$887$		−	31.6270i		−	1.06193i	0.847393	−	0.530966i	$-0.178171\pi$
$888$		0			0
$889$			6.55970i			0.220005i
$890$		0			0
$891$	−27.4031	−	44.3582i	−0.918037	−	1.48606i
$892$		0			0
$893$		−	18.7877i		−	0.628707i
$894$		0			0
$895$	7.88157			0.263452
$896$		0			0
$897$	22.0358	−	12.9421i	0.735755	−	0.432124i
$898$		0			0
$899$	−45.2063			−1.50772
$900$		0			0
$901$			5.01763i			0.167162i
$902$		0			0
$903$	4.62852	−	2.71843i	0.154028	−	0.0904636i
$904$		0			0
$905$	−26.7395			−0.888851
$906$		0			0
$907$	23.6229			0.784385			0.392193	−	0.919883i	$-0.371717\pi$
$907$	23.6229			0.784385			0.392193	+	0.919883i	$0.371717\pi$
$908$		0			0
$909$	−10.0650	+	18.0486i	−0.333836	+	0.598635i
$910$		0			0
$911$		−	39.8220i		−	1.31936i	−0.751546	−	0.659680i	$-0.770692\pi$
$911$		−	39.8220i		−	1.31936i	0.751546	−	0.659680i	$-0.229308\pi$
$912$		0			0
$913$			92.0270i			3.04565i
$914$		0			0
$915$	−13.8046	−	23.5044i	−0.456367	−	0.777031i
$916$		0			0
$917$	−0.848651			−0.0280249
$918$		0			0
$919$			43.8250i			1.44565i	0.691029	+	0.722827i	$0.257158\pi$
$919$			43.8250i			1.44565i	−0.691029	+	0.722827i	$0.742842\pi$
$920$		0			0
$921$	26.2631	−	15.4249i	0.865400	−	0.508268i
$922$		0			0
$923$	−3.46993			−0.114214
$924$		0			0
$925$	−3.43466			−0.112931
$926$		0			0
$927$	−7.66463	+	13.7442i	−0.251740	+	0.451419i
$928$		0			0
$929$	28.8638			0.946992			0.473496	−	0.880796i	$-0.342992\pi$
$929$	28.8638			0.946992			0.473496	+	0.880796i	$0.342992\pi$
$930$		0			0
$931$	27.0469			0.886427
$932$		0			0
$933$	14.4850	−	8.50732i	0.474216	−	0.278517i
$934$		0			0
$935$			3.38400i			0.110669i
$936$		0			0
$937$		−	16.8956i		−	0.551956i	−0.961164	−	0.275978i	$-0.910998\pi$
$937$		−	16.8956i		−	0.551956i	0.961164	−	0.275978i	$-0.0890016\pi$
$938$		0			0
$939$	10.2038	+	17.3734i	0.332987	+	0.566959i
$940$		0			0
$941$	41.1360			1.34099			0.670497	−	0.741912i	$-0.266081\pi$
$941$	41.1360			1.34099			0.670497	+	0.741912i	$0.266081\pi$
$942$		0			0
$943$		−	20.5688i		−	0.669813i
$944$		0			0
$945$	2.50020	−	0.0558566i	0.0813315	−	0.00181702i
$946$		0			0
$947$		−	8.19657i		−	0.266353i	−0.991092	−	0.133176i	$-0.957482\pi$
$947$		−	8.19657i		−	0.266353i	0.991092	−	0.133176i	$-0.0425177\pi$
$948$		0			0
$949$		−	45.7120i		−	1.48388i
$950$		0			0
$951$	18.8400	+	32.0778i	0.610928	+	1.04019i
$952$		0			0
$953$			21.0626i			0.682286i	0.940011	+	0.341143i	$0.110814\pi$
$953$			21.0626i			0.682286i	−0.940011	+	0.341143i	$0.889186\pi$
$954$		0			0
$955$	20.2782			0.656186
$956$		0			0
$957$	−42.2151	−	71.8773i	−1.36462	−	2.32346i
$958$		0			0
$959$			0.400465i			0.0129317i
$960$		0			0
$961$	1.38667			0.0447311
$962$		0			0
$963$	33.9708	+	18.9443i	1.09470	+	0.610471i
$964$		0			0
$965$	24.6669			0.794056
$966$		0			0
$967$	22.4461			0.721818			0.360909	−	0.932601i	$-0.382467\pi$
$967$	22.4461			0.721818			0.360909	+	0.932601i	$0.382467\pi$
$968$		0			0
$969$	−1.70735	−	2.90701i	−0.0548480	−	0.0933867i
$970$		0			0
$971$			4.91885i			0.157854i	0.996880	+	0.0789268i	$0.0251493\pi$
$971$			4.91885i			0.157854i	−0.996880	+	0.0789268i	$0.974851\pi$
$972$		0			0
$973$	−2.27305			−0.0728706
$974$		0			0
$975$	−15.2023	−	25.8842i	−0.486865	−	0.828958i
$976$		0			0
$977$		−	22.7232i		−	0.726978i	−0.931598	−	0.363489i	$-0.881585\pi$
$977$		−	22.7232i		−	0.726978i	0.931598	−	0.363489i	$-0.118415\pi$
$978$		0			0
$979$			61.0844i			1.95227i
$980$		0			0
$981$	−9.97709	−	5.56385i	−0.318544	−	0.177640i
$982$		0			0
$983$	0.732897			0.0233758			0.0116879	−	0.999932i	$-0.496280\pi$
$983$	0.732897			0.0233758			0.0116879	+	0.999932i	$0.496280\pi$
$984$		0			0
$985$	−12.1113			−0.385899
$986$		0			0
$987$	−2.87443	+	1.68821i	−0.0914941	+	0.0537364i
$988$		0			0
$989$	23.3684			0.743072
$990$		0			0
$991$		−	33.6930i		−	1.07029i	−0.844759	−	0.535146i	$-0.820257\pi$
$991$		−	33.6930i		−	1.07029i	0.844759	−	0.535146i	$-0.179743\pi$
$992$		0			0
$993$	10.5593	+	17.9788i	0.335089	+	0.570538i
$994$		0			0
$995$	−1.15053			−0.0364742
$996$		0			0
$997$	−24.0373			−0.761267			−0.380634	−	0.924726i	$-0.624294\pi$
$997$	−24.0373			−0.761267			−0.380634	+	0.924726i	$0.624294\pi$
$998$		0			0
$999$	−0.111033	−	4.96997i	−0.00351294	−	0.157243i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.g.c.401.13	yes	16
3.2	odd	2	inner	804.2.g.c.401.3	✓	16
67.66	odd	2	inner	804.2.g.c.401.4	yes	16
201.200	even	2	inner	804.2.g.c.401.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.g.c.401.3	✓	16	3.2	odd	2	inner
804.2.g.c.401.4	yes	16	67.66	odd	2	inner
804.2.g.c.401.13	yes	16	1.1	even	1	trivial
804.2.g.c.401.14	yes	16	201.200	even	2	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.49351 - 0.877170i) q^{3} -1.18740 q^{5} -0.405325i q^{7} +(1.46115 - 2.62012i) q^{9} +O(q^{10})\)	\(q+(1.49351 - 0.877170i) q^{3} -1.18740 q^{5} -0.405325i q^{7} +(1.46115 - 2.62012i) q^{9} +5.79333 q^{11} +4.82750i q^{13} +(-1.77339 + 1.04155i) q^{15} -0.491931i q^{17} +3.95671 q^{19} +(-0.355539 - 0.605358i) q^{21} -3.05632i q^{23} -3.59008 q^{25} +(-0.116058 - 5.19486i) q^{27} -8.30722i q^{29} -5.44181i q^{31} +(8.65240 - 5.08173i) q^{33} +0.481284i q^{35} +0.956710 q^{37} +(4.23454 + 7.20993i) q^{39} +6.72993 q^{41} +7.64593i q^{43} +(-1.73497 + 3.11114i) q^{45} -4.74832i q^{47} +6.83571 q^{49} +(-0.431507 - 0.734705i) q^{51} -10.1999 q^{53} -6.87900 q^{55} +(5.90939 - 3.47071i) q^{57} -10.8716i q^{59} +13.2539i q^{61} +(-1.06200 - 0.592240i) q^{63} -5.73218i q^{65} +(2.92229 + 7.64593i) q^{67} +(-2.68091 - 4.56464i) q^{69} +0.718784i q^{71} -9.46908 q^{73} +(-5.36182 + 3.14911i) q^{75} -2.34818i q^{77} +14.1074i q^{79} +(-4.73010 - 7.65677i) q^{81} +15.8850i q^{83} +0.584120i q^{85} +(-7.28684 - 12.4069i) q^{87} +10.5439i q^{89} +1.95671 q^{91} +(-4.77339 - 8.12740i) q^{93} -4.69820 q^{95} -12.0694i q^{97} +(8.46490 - 15.1792i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{9}+O(q^{10}) \) 16 * q + 2 * q^9	\( 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) \) 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

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