LMFDB - Embedded newform 810.3.b.c.809.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.b (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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.19
Character	$\chi$	$=$	810.809
Dual form			810.3.b.c.809.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(0.420464 - 4.98229i) q^{5} +1.96530i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(0.420464 - 4.98229i) q^{5} +1.96530i q^{7} +2.82843 q^{8} +(0.594626 - 7.04602i) q^{10} -3.81639i q^{11} -18.6194i q^{13} +2.77936i q^{14} +4.00000 q^{16} +17.6477 q^{17} -26.0605 q^{19} +(0.840928 - 9.96458i) q^{20} -5.39719i q^{22} -9.64111 q^{23} +(-24.6464 - 4.18975i) q^{25} -26.3318i q^{26} +3.93060i q^{28} +6.01270i q^{29} +25.3332 q^{31} +5.65685 q^{32} +24.9576 q^{34} +(9.79170 + 0.826339i) q^{35} -39.9637i q^{37} -36.8551 q^{38} +(1.18925 - 14.0920i) q^{40} -35.4706i q^{41} -61.0336i q^{43} -7.63278i q^{44} -13.6346 q^{46} -70.8869 q^{47} +45.1376 q^{49} +(-34.8553 - 5.92520i) q^{50} -37.2388i q^{52} +83.3187 q^{53} +(-19.0144 - 1.60465i) q^{55} +5.55871i q^{56} +8.50325i q^{58} +37.3045i q^{59} -4.75183 q^{61} +35.8265 q^{62} +8.00000 q^{64} +(-92.7672 - 7.82879i) q^{65} +9.27725i q^{67} +35.2954 q^{68} +(13.8476 + 1.16862i) q^{70} +34.1966i q^{71} -22.9490i q^{73} -56.5172i q^{74} -52.1210 q^{76} +7.50036 q^{77} -99.0887 q^{79} +(1.68186 - 19.9292i) q^{80} -50.1630i q^{82} -37.9042 q^{83} +(7.42023 - 87.9261i) q^{85} -86.3145i q^{86} -10.7944i q^{88} -43.0816i q^{89} +36.5927 q^{91} -19.2822 q^{92} -100.249 q^{94} +(-10.9575 + 129.841i) q^{95} -27.4575i q^{97} +63.8342 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4}+O(q^{10}) $ 24 * q + 48 * q^4	$ 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	0.420464	−	4.98229i	0.0840928	−	0.996458i
$5$	0.420464	−	4.98229i	0.0840928	−	0.996458i
$6$		0			0
$7$			1.96530i			0.280757i	0.990098	+	0.140379i	$0.0448320\pi$
$7$			1.96530i			0.280757i	−0.990098	+	0.140379i	$0.955168\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	0.594626	−	7.04602i	0.0594626	−	0.704602i
$11$		−	3.81639i		−	0.346945i	−0.984839	−	0.173472i	$-0.944501\pi$
$11$		−	3.81639i		−	0.346945i	0.984839	−	0.173472i	$-0.0554987\pi$
$12$		0			0
$13$		−	18.6194i		−	1.43226i	−0.697966	−	0.716131i	$-0.745912\pi$
$13$		−	18.6194i		−	1.43226i	0.697966	−	0.716131i	$-0.254088\pi$
$14$			2.77936i			0.198526i
$15$		0			0
$16$	4.00000			0.250000
$17$	17.6477			1.03810			0.519051	−	0.854743i	$-0.326286\pi$
$17$	17.6477			1.03810			0.519051	+	0.854743i	$0.326286\pi$
$18$		0			0
$19$	−26.0605			−1.37160			−0.685802	−	0.727788i	$-0.740549\pi$
$19$	−26.0605			−1.37160			−0.685802	+	0.727788i	$0.740549\pi$
$20$	0.840928	−	9.96458i	0.0420464	−	0.498229i
$21$		0			0
$22$		−	5.39719i		−	0.245327i
$23$	−9.64111			−0.419179			−0.209589	−	0.977790i	$-0.567213\pi$
$23$	−9.64111			−0.419179			−0.209589	+	0.977790i	$0.567213\pi$
$24$		0			0
$25$	−24.6464	−	4.18975i	−0.985857	−	0.167590i
$26$		−	26.3318i		−	1.01276i
$27$		0			0
$28$			3.93060i			0.140379i
$29$			6.01270i			0.207335i	0.994612	+	0.103667i	$0.0330577\pi$
$29$			6.01270i			0.207335i	−0.994612	+	0.103667i	$0.966942\pi$
$30$		0			0
$31$	25.3332			0.817199			0.408600	−	0.912714i	$-0.366017\pi$
$31$	25.3332			0.817199			0.408600	+	0.912714i	$0.366017\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	24.9576			0.734048
$35$	9.79170	+	0.826339i	0.279763	+	0.0236097i
$36$		0			0
$37$		−	39.9637i		−	1.08010i	−0.841633	−	0.540050i	$-0.818405\pi$
$37$		−	39.9637i		−	1.08010i	0.841633	−	0.540050i	$-0.181595\pi$
$38$	−36.8551			−0.969871
$39$		0			0
$40$	1.18925	−	14.0920i	0.0297313	−	0.352301i
$41$		−	35.4706i		−	0.865137i	−0.901601	−	0.432568i	$-0.857607\pi$
$41$		−	35.4706i		−	0.865137i	0.901601	−	0.432568i	$-0.142393\pi$
$42$		0			0
$43$		−	61.0336i		−	1.41939i	−0.704511	−	0.709693i	$-0.748834\pi$
$43$		−	61.0336i		−	1.41939i	0.704511	−	0.709693i	$-0.251166\pi$
$44$		−	7.63278i		−	0.173472i
$45$		0			0
$46$	−13.6346			−0.296404
$47$	−70.8869			−1.50823			−0.754116	−	0.656741i	$-0.771934\pi$
$47$	−70.8869			−1.50823			−0.754116	+	0.656741i	$0.771934\pi$
$48$		0			0
$49$	45.1376			0.921175
$50$	−34.8553	−	5.92520i	−0.697106	−	0.118504i
$51$		0			0
$52$		−	37.2388i		−	0.716131i
$53$	83.3187			1.57205			0.786025	−	0.618194i	$-0.212135\pi$
$53$	83.3187			1.57205			0.786025	+	0.618194i	$0.212135\pi$
$54$		0			0
$55$	−19.0144	−	1.60465i	−0.345716	−	0.0291755i
$56$			5.55871i			0.0992628i
$57$		0			0
$58$			8.50325i			0.146608i
$59$			37.3045i			0.632280i	0.948713	+	0.316140i	$0.102387\pi$
$59$			37.3045i			0.632280i	−0.948713	+	0.316140i	$0.897613\pi$
$60$		0			0
$61$	−4.75183			−0.0778989			−0.0389495	−	0.999241i	$-0.512401\pi$
$61$	−4.75183			−0.0778989			−0.0389495	+	0.999241i	$0.512401\pi$
$62$	35.8265			0.577847
$63$		0			0
$64$	8.00000			0.125000
$65$	−92.7672	−	7.82879i	−1.42719	−	0.120443i
$66$		0			0
$67$			9.27725i			0.138466i	0.997601	+	0.0692332i	$0.0220553\pi$
$67$			9.27725i			0.138466i	−0.997601	+	0.0692332i	$0.977945\pi$
$68$	35.2954			0.519051
$69$		0			0
$70$	13.8476	+	1.16862i	0.197822	+	0.0166946i
$71$			34.1966i			0.481643i	0.970569	+	0.240821i	$0.0774168\pi$
$71$			34.1966i			0.481643i	−0.970569	+	0.240821i	$0.922583\pi$
$72$		0			0
$73$		−	22.9490i		−	0.314370i	−0.987569	−	0.157185i	$-0.949758\pi$
$73$		−	22.9490i		−	0.314370i	0.987569	−	0.157185i	$-0.0502419\pi$
$74$		−	56.5172i		−	0.763747i
$75$		0			0
$76$	−52.1210			−0.685802
$77$	7.50036			0.0974073
$78$		0			0
$79$	−99.0887			−1.25429			−0.627144	−	0.778904i	$-0.715776\pi$
$79$	−99.0887			−1.25429			−0.627144	+	0.778904i	$0.715776\pi$
$80$	1.68186	−	19.9292i	0.0210232	−	0.249114i
$81$		0			0
$82$		−	50.1630i		−	0.611744i
$83$	−37.9042			−0.456677			−0.228339	−	0.973582i	$-0.573329\pi$
$83$	−37.9042			−0.456677			−0.228339	+	0.973582i	$0.573329\pi$
$84$		0			0
$85$	7.42023	−	87.9261i	0.0872969	−	1.03442i
$86$		−	86.3145i		−	1.00366i
$87$		0			0
$88$		−	10.7944i		−	0.122663i
$89$		−	43.0816i		−	0.484063i	−0.970268	−	0.242031i	$-0.922186\pi$
$89$		−	43.0816i		−	0.484063i	0.970268	−	0.242031i	$-0.0778137\pi$
$90$		0			0
$91$	36.5927			0.402118
$92$	−19.2822			−0.209589
$93$		0			0
$94$	−100.249			−1.06648
$95$	−10.9575	+	129.841i	−0.115342	+	1.36675i
$96$		0			0
$97$		−	27.4575i		−	0.283067i	−0.989933	−	0.141533i	$-0.954797\pi$
$97$		−	27.4575i		−	0.283067i	0.989933	−	0.141533i	$-0.0452033\pi$
$98$	63.8342			0.651369
$99$		0			0
$100$	−49.2928	−	8.37950i	−0.492928	−	0.0837950i
$101$			179.716i			1.77937i	0.456577	+	0.889684i	$0.349075\pi$
$101$			179.716i			1.77937i	−0.456577	+	0.889684i	$0.650925\pi$
$102$		0			0
$103$		−	131.077i		−	1.27259i	−0.771447	−	0.636294i	$-0.780466\pi$
$103$		−	131.077i		−	1.27259i	0.771447	−	0.636294i	$-0.219534\pi$
$104$		−	52.6636i		−	0.506381i
$105$		0			0
$106$	117.830			1.11161
$107$	208.077			1.94464			0.972320	−	0.233652i	$-0.0750675\pi$
$107$	208.077			1.94464			0.972320	+	0.233652i	$0.0750675\pi$
$108$		0			0
$109$	96.9352			0.889314			0.444657	−	0.895701i	$-0.353326\pi$
$109$	96.9352			0.889314			0.444657	+	0.895701i	$0.353326\pi$
$110$	−26.8904	−	2.26932i	−0.244458	−	0.0206302i
$111$		0			0
$112$			7.86121i			0.0701894i
$113$	11.9713			0.105941			0.0529703	−	0.998596i	$-0.483131\pi$
$113$	11.9713			0.105941			0.0529703	+	0.998596i	$0.483131\pi$
$114$		0			0
$115$	−4.05374	+	48.0348i	−0.0352499	+	0.417694i
$116$			12.0254i			0.103667i
$117$		0			0
$118$			52.7566i			0.447090i
$119$			34.6831i			0.291455i
$120$		0			0
$121$	106.435			0.879630
$122$	−6.72011			−0.0550828
$123$		0			0
$124$	50.6664			0.408600
$125$	−31.2375	+	121.034i	−0.249900	+	0.968272i
$126$		0			0
$127$		−	215.556i		−	1.69729i	−0.528964	−	0.848644i	$-0.677419\pi$
$127$		−	215.556i		−	1.69729i	0.528964	−	0.848644i	$-0.322581\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	−131.193	−	11.0716i	−1.00917	−	0.0851660i
$131$			79.3752i			0.605917i	0.953004	+	0.302959i	$0.0979744\pi$
$131$			79.3752i			0.605917i	−0.953004	+	0.302959i	$0.902026\pi$
$132$		0			0
$133$		−	51.2167i		−	0.385088i
$134$			13.1200i			0.0979105i
$135$		0			0
$136$	49.9153			0.367024
$137$	176.066			1.28515			0.642577	−	0.766221i	$-0.277865\pi$
$137$	176.066			1.28515			0.642577	+	0.766221i	$0.277865\pi$
$138$		0			0
$139$	−78.0544			−0.561543			−0.280771	−	0.959775i	$-0.590590\pi$
$139$	−78.0544			−0.561543			−0.280771	+	0.959775i	$0.590590\pi$
$140$	19.5834	+	1.65268i	0.139881	+	0.0118048i
$141$		0			0
$142$			48.3614i			0.340573i
$143$	−71.0589			−0.496915
$144$		0			0
$145$	29.9570	+	2.52813i	0.206600	+	0.0174353i
$146$		−	32.4548i		−	0.222293i
$147$		0			0
$148$		−	79.9275i		−	0.540050i
$149$			110.835i			0.743857i	0.928261	+	0.371928i	$0.121303\pi$
$149$			110.835i			0.743857i	−0.928261	+	0.371928i	$0.878697\pi$
$150$		0			0
$151$	−12.6330			−0.0836620			−0.0418310	−	0.999125i	$-0.513319\pi$
$151$	−12.6330			−0.0836620			−0.0418310	+	0.999125i	$0.513319\pi$
$152$	−73.7102			−0.484935
$153$		0			0
$154$	10.6071			0.0688773
$155$	10.6517	−	126.217i	0.0687206	−	0.814305i
$156$		0			0
$157$			206.144i			1.31302i	0.754318	+	0.656510i	$0.227968\pi$
$157$			206.144i			1.31302i	−0.754318	+	0.656510i	$0.772032\pi$
$158$	−140.133			−0.886915
$159$		0			0
$160$	2.37850	−	28.1841i	0.0148657	−	0.176151i
$161$		−	18.9477i		−	0.117688i
$162$		0			0
$163$			235.021i			1.44185i	0.693015	+	0.720923i	$0.256282\pi$
$163$			235.021i			1.44185i	−0.693015	+	0.720923i	$0.743718\pi$
$164$		−	70.9412i		−	0.432568i
$165$		0			0
$166$	−53.6046			−0.322919
$167$	126.179			0.755562			0.377781	−	0.925895i	$-0.376687\pi$
$167$	126.179			0.755562			0.377781	+	0.925895i	$0.376687\pi$
$168$		0			0
$169$	−177.682			−1.05137
$170$	10.4938	−	124.346i	0.0617282	−	0.731448i
$171$		0			0
$172$		−	122.067i		−	0.709693i
$173$	−96.9923			−0.560649			−0.280325	−	0.959905i	$-0.590442\pi$
$173$	−96.9923			−0.560649			−0.280325	+	0.959905i	$0.590442\pi$
$174$		0			0
$175$	8.23412	−	48.4377i	0.0470521	−	0.276787i
$176$		−	15.2656i		−	0.0867361i
$177$		0			0
$178$		−	60.9265i		−	0.342284i
$179$			239.017i			1.33529i	0.744479	+	0.667645i	$0.232698\pi$
$179$			239.017i			1.33529i	−0.744479	+	0.667645i	$0.767302\pi$
$180$		0			0
$181$	271.084			1.49770			0.748851	−	0.662738i	$-0.230606\pi$
$181$	271.084			1.49770			0.748851	+	0.662738i	$0.230606\pi$
$182$	51.7499			0.284340
$183$		0			0
$184$	−27.2692			−0.148202
$185$	−199.111	−	16.8033i	−1.07627	−	0.0908287i
$186$		0			0
$187$		−	67.3506i		−	0.360164i
$188$	−141.774			−0.754116
$189$		0			0
$190$	−15.4962	+	183.623i	−0.0815592	+	0.966435i
$191$			257.687i			1.34915i	0.738208	+	0.674574i	$0.235672\pi$
$191$			257.687i			1.34915i	−0.738208	+	0.674574i	$0.764328\pi$
$192$		0			0
$193$			117.010i			0.606269i	0.952948	+	0.303135i	$0.0980332\pi$
$193$			117.010i			0.606269i	−0.952948	+	0.303135i	$0.901967\pi$
$194$		−	38.8308i		−	0.200159i
$195$		0			0
$196$	90.2752			0.460588
$197$	−127.741			−0.648430			−0.324215	−	0.945983i	$-0.605100\pi$
$197$	−127.741			−0.648430			−0.324215	+	0.945983i	$0.605100\pi$
$198$		0			0
$199$	−352.563			−1.77167			−0.885836	−	0.463998i	$-0.846414\pi$
$199$	−352.563			−1.77167			−0.885836	+	0.463998i	$0.846414\pi$
$200$	−69.7106	−	11.8504i	−0.348553	−	0.0592520i
$201$		0			0
$202$			254.157i			1.25820i
$203$	−11.8168			−0.0582107
$204$		0			0
$205$	−176.725	−	14.9141i	−0.862073	−	0.0727518i
$206$		−	185.370i		−	0.899856i
$207$		0			0
$208$		−	74.4776i		−	0.358065i
$209$			99.4570i			0.475871i
$210$		0			0
$211$	278.402			1.31944			0.659720	−	0.751512i	$-0.270675\pi$
$211$	278.402			1.31944			0.659720	+	0.751512i	$0.270675\pi$
$212$	166.637			0.786025
$213$		0			0
$214$	294.265			1.37507
$215$	−304.087	−	25.6624i	−1.41436	−	0.119360i
$216$		0			0
$217$			49.7874i			0.229435i
$218$	137.087			0.628840
$219$		0			0
$220$	−38.0287	−	3.20931i	−0.172858	−	0.0145878i
$221$		−	328.590i		−	1.48683i
$222$		0			0
$223$			358.406i			1.60720i	0.595170	+	0.803600i	$0.297085\pi$
$223$			358.406i			1.60720i	−0.595170	+	0.803600i	$0.702915\pi$
$224$			11.1174i			0.0496314i
$225$		0			0
$226$	16.9300			0.0749113
$227$	319.102			1.40574			0.702868	−	0.711321i	$-0.251903\pi$
$227$	319.102			1.40574			0.702868	+	0.711321i	$0.251903\pi$
$228$		0			0
$229$	24.8580			0.108550			0.0542750	−	0.998526i	$-0.482715\pi$
$229$	24.8580			0.108550			0.0542750	+	0.998526i	$0.482715\pi$
$230$	−5.73285	+	67.9314i	−0.0249254	+	0.295354i
$231$		0			0
$232$			17.0065i			0.0733038i
$233$	−223.302			−0.958376			−0.479188	−	0.877712i	$-0.659069\pi$
$233$	−223.302			−0.958376			−0.479188	+	0.877712i	$0.659069\pi$
$234$		0			0
$235$	−29.8054	+	353.179i	−0.126832	+	1.50289i
$236$			74.6091i			0.316140i
$237$		0			0
$238$			49.0493i			0.206090i
$239$			89.7315i			0.375446i	0.982222	+	0.187723i	$0.0601107\pi$
$239$			89.7315i			0.375446i	−0.982222	+	0.187723i	$0.939889\pi$
$240$		0			0
$241$	−114.246			−0.474048			−0.237024	−	0.971504i	$-0.576172\pi$
$241$	−114.246			−0.474048			−0.237024	+	0.971504i	$0.576172\pi$
$242$	150.522			0.621992
$243$		0			0
$244$	−9.50367			−0.0389495
$245$	18.9787	−	224.889i	0.0774642	−	0.917912i
$246$		0			0
$247$			485.230i			1.96450i
$248$	71.6531			0.288924
$249$		0			0
$250$	−44.1765	+	171.168i	−0.176706	+	0.684671i
$251$			285.710i			1.13829i	0.822238	+	0.569144i	$0.192725\pi$
$251$			285.710i			1.13829i	−0.822238	+	0.569144i	$0.807275\pi$
$252$		0			0
$253$			36.7942i			0.145432i
$254$		−	304.842i		−	1.20016i
$255$		0			0
$256$	16.0000			0.0625000
$257$	257.018			1.00007			0.500035	−	0.866005i	$-0.333321\pi$
$257$	257.018			1.00007			0.500035	+	0.866005i	$0.333321\pi$
$258$		0			0
$259$	78.5408			0.303246
$260$	−185.534	−	15.6576i	−0.713594	−	0.0602214i
$261$		0			0
$262$			112.253i			0.428448i
$263$	182.038			0.692160			0.346080	−	0.938205i	$-0.387513\pi$
$263$	182.038			0.692160			0.346080	+	0.938205i	$0.387513\pi$
$264$		0			0
$265$	35.0325	−	415.118i	0.132198	−	1.56648i
$266$		−	72.4314i		−	0.272298i
$267$		0			0
$268$			18.5545i			0.0692332i
$269$		−	377.258i		−	1.40244i	−0.712943	−	0.701222i	$-0.752638\pi$
$269$		−	377.258i		−	1.40244i	0.712943	−	0.701222i	$-0.247362\pi$
$270$		0			0
$271$	−282.266			−1.04157			−0.520787	−	0.853687i	$-0.674361\pi$
$271$	−282.266			−1.04157			−0.520787	+	0.853687i	$0.674361\pi$
$272$	70.5909			0.259525
$273$		0			0
$274$	248.995			0.908741
$275$	−15.9897	+	94.0603i	−0.0581444	+	0.342038i
$276$		0			0
$277$			375.606i			1.35598i	0.735073	+	0.677988i	$0.237148\pi$
$277$			375.606i			1.35598i	−0.735073	+	0.677988i	$0.762852\pi$
$278$	−110.386			−0.397071
$279$		0			0
$280$	27.6951	+	2.33724i	0.0989112	+	0.00834728i
$281$		−	423.529i		−	1.50722i	−0.657322	−	0.753610i	$-0.728311\pi$
$281$		−	423.529i		−	1.50722i	0.657322	−	0.753610i	$-0.271689\pi$
$282$		0			0
$283$		−	12.6237i		−	0.0446069i	−0.999751	−	0.0223034i	$-0.992900\pi$
$283$		−	12.6237i		−	0.0446069i	0.999751	−	0.0223034i	$-0.00709999\pi$
$284$			68.3933i			0.240821i
$285$		0			0
$286$	−100.492			−0.351372
$287$	69.7105			0.242894
$288$		0			0
$289$	22.4421			0.0776543
$290$	42.3656	+	3.57531i	0.146088	+	0.0123287i
$291$		0			0
$292$		−	45.8980i		−	0.157185i
$293$	16.0317			0.0547158			0.0273579	−	0.999626i	$-0.491291\pi$
$293$	16.0317			0.0547158			0.0273579	+	0.999626i	$0.491291\pi$
$294$		0			0
$295$	185.862	+	15.6852i	0.630041	+	0.0531702i
$296$		−	113.034i		−	0.381873i
$297$		0			0
$298$			156.744i			0.525986i
$299$			179.512i			0.600373i
$300$		0			0
$301$	119.949			0.398503
$302$	−17.8657			−0.0591580
$303$		0			0
$304$	−104.242			−0.342901
$305$	−1.99798	+	23.6750i	−0.00655074	+	0.0776230i
$306$		0			0
$307$		−	460.226i		−	1.49911i	−0.661943	−	0.749554i	$-0.730268\pi$
$307$		−	460.226i		−	1.49911i	0.661943	−	0.749554i	$-0.269732\pi$
$308$	15.0007			0.0487036
$309$		0			0
$310$	15.0638	−	178.498i	0.0485928	−	0.575800i
$311$		−	147.557i		−	0.474461i	−0.971453	−	0.237231i	$-0.923760\pi$
$311$		−	147.557i		−	0.474461i	0.971453	−	0.237231i	$-0.0762397\pi$
$312$		0			0
$313$			208.899i			0.667408i	0.942678	+	0.333704i	$0.108299\pi$
$313$			208.899i			0.667408i	−0.942678	+	0.333704i	$0.891701\pi$
$314$			291.532i			0.928445i
$315$		0			0
$316$	−198.177			−0.627144
$317$	494.700			1.56057			0.780284	−	0.625425i	$-0.215074\pi$
$317$	494.700			1.56057			0.780284	+	0.625425i	$0.215074\pi$
$318$		0			0
$319$	22.9468			0.0719336
$320$	3.36371	−	39.8583i	0.0105116	−	0.124557i
$321$		0			0
$322$		−	26.7961i		−	0.0832176i
$323$	−459.908			−1.42386
$324$		0			0
$325$	−78.0106	+	458.901i	−0.240033	+	1.41200i
$326$			332.370i			1.01954i
$327$		0			0
$328$		−	100.326i		−	0.305872i
$329$		−	139.314i		−	0.423447i
$330$		0			0
$331$	505.012			1.52572			0.762859	−	0.646565i	$-0.223795\pi$
$331$	505.012			1.52572			0.762859	+	0.646565i	$0.223795\pi$
$332$	−75.8084			−0.228339
$333$		0			0
$334$	178.444			0.534263
$335$	46.2219	+	3.90075i	0.137976	+	0.0116440i
$336$		0			0
$337$			615.872i			1.82751i	0.406263	+	0.913756i	$0.366832\pi$
$337$			615.872i			1.82751i	−0.406263	+	0.913756i	$0.633168\pi$
$338$	−251.280			−0.743432
$339$		0			0
$340$	14.8405	−	175.852i	0.0436484	−	0.517212i
$341$		−	96.6813i		−	0.283523i
$342$		0			0
$343$			185.009i			0.539384i
$344$		−	172.629i		−	0.501829i
$345$		0			0
$346$	−137.168			−0.396439
$347$	258.712			0.745568			0.372784	−	0.927918i	$-0.378403\pi$
$347$	258.712			0.745568			0.372784	+	0.927918i	$0.378403\pi$
$348$		0			0
$349$	−466.958			−1.33799			−0.668994	−	0.743268i	$-0.733275\pi$
$349$	−466.958			−1.33799			−0.668994	+	0.743268i	$0.733275\pi$
$350$	11.6448	−	68.5012i	0.0332709	−	0.195718i
$351$		0			0
$352$		−	21.5888i		−	0.0613317i
$353$	−33.3361			−0.0944364			−0.0472182	−	0.998885i	$-0.515036\pi$
$353$	−33.3361			−0.0944364			−0.0472182	+	0.998885i	$0.515036\pi$
$354$		0			0
$355$	170.378	+	14.3785i	0.479937	+	0.0405027i
$356$		−	86.1631i		−	0.242031i
$357$		0			0
$358$			338.021i			0.944193i
$359$		−	618.698i		−	1.72339i	−0.507423	−	0.861697i	$-0.669402\pi$
$359$		−	618.698i		−	1.72339i	0.507423	−	0.861697i	$-0.330598\pi$
$360$		0			0
$361$	318.149			0.881299
$362$	383.371			1.05904
$363$		0			0
$364$	73.1855			0.201059
$365$	−114.339	−	9.64924i	−0.313257	−	0.0264363i
$366$		0			0
$367$		−	364.512i		−	0.993220i	−0.867974	−	0.496610i	$-0.834578\pi$
$367$		−	364.512i		−	0.993220i	0.867974	−	0.496610i	$-0.165422\pi$
$368$	−38.5644			−0.104795
$369$		0			0
$370$	−281.585	−	23.7635i	−0.761041	−	0.0642256i
$371$			163.746i			0.441365i
$372$		0			0
$373$		−	500.193i		−	1.34100i	−0.741910	−	0.670499i	$-0.766080\pi$
$373$		−	500.193i		−	1.34100i	0.741910	−	0.670499i	$-0.233920\pi$
$374$		−	95.2481i		−	0.254674i
$375$		0			0
$376$	−200.498			−0.533241
$377$	111.953			0.296957
$378$		0			0
$379$	153.088			0.403927			0.201963	−	0.979393i	$-0.435268\pi$
$379$	153.088			0.403927			0.201963	+	0.979393i	$0.435268\pi$
$380$	−21.9150	+	259.682i	−0.0576710	+	0.683373i
$381$		0			0
$382$			364.425i			0.953991i
$383$	−623.551			−1.62807			−0.814036	−	0.580815i	$-0.802734\pi$
$383$	−623.551			−1.62807			−0.814036	+	0.580815i	$0.802734\pi$
$384$		0			0
$385$	3.15363	−	37.3690i	0.00819125	−	0.0970622i
$386$			165.477i			0.428697i
$387$		0			0
$388$		−	54.9150i		−	0.141533i
$389$		−	278.937i		−	0.717061i	−0.933518	−	0.358531i	$-0.883278\pi$
$389$		−	278.937i		−	0.717061i	0.933518	−	0.358531i	$-0.116722\pi$
$390$		0			0
$391$	−170.144			−0.435150
$392$	127.668			0.325685
$393$		0			0
$394$	−180.653			−0.458509
$395$	−41.6632	+	493.689i	−0.105477	+	1.24984i
$396$		0			0
$397$		−	685.202i		−	1.72595i	−0.505246	−	0.862975i	$-0.668598\pi$
$397$		−	685.202i		−	1.72595i	0.505246	−	0.862975i	$-0.331402\pi$
$398$	−498.599			−1.25276
$399$		0			0
$400$	−98.5857	−	16.7590i	−0.246464	−	0.0418975i
$401$			393.384i			0.981009i	0.871439	+	0.490504i	$0.163187\pi$
$401$			393.384i			0.981009i	−0.871439	+	0.490504i	$0.836813\pi$
$402$		0			0
$403$		−	471.689i		−	1.17044i
$404$			359.432i			0.889684i
$405$		0			0
$406$	−16.7114			−0.0411612
$407$	−152.517			−0.374735
$408$		0			0
$409$	−581.610			−1.42203			−0.711014	−	0.703177i	$-0.751764\pi$
$409$	−581.610			−1.42203			−0.711014	+	0.703177i	$0.751764\pi$
$410$	−249.927	−	21.0918i	−0.609577	−	0.0514433i
$411$		0			0
$412$		−	262.153i		−	0.636294i
$413$	−73.3147			−0.177517
$414$		0			0
$415$	−15.9374	+	188.850i	−0.0384033	+	0.455060i
$416$		−	105.327i		−	0.253190i
$417$		0			0
$418$			140.653i			0.336491i
$419$			187.594i			0.447719i	0.974621	+	0.223860i	$0.0718657\pi$
$419$			187.594i			0.447719i	−0.974621	+	0.223860i	$0.928134\pi$
$420$		0			0
$421$	174.904			0.415450			0.207725	−	0.978187i	$-0.433394\pi$
$421$	174.904			0.415450			0.207725	+	0.978187i	$0.433394\pi$
$422$	393.720			0.932985
$423$		0			0
$424$	235.661			0.555804
$425$	−434.953	−	73.9395i	−1.02342	−	0.173975i
$426$		0			0
$427$		−	9.33879i		−	0.0218707i
$428$	416.153			0.972320
$429$		0			0
$430$	−430.044	−	36.2922i	−1.00010	−	0.0844004i
$431$		−	1.74212i		−	0.00404203i	−0.999998	−	0.00202102i	$-0.999357\pi$
$431$		−	1.74212i		−	0.00404203i	0.999998	−	0.00202102i	$-0.000643310\pi$
$432$		0			0
$433$			180.676i			0.417266i	0.977994	+	0.208633i	$0.0669014\pi$
$433$			180.676i			0.417266i	−0.977994	+	0.208633i	$0.933099\pi$
$434$			70.4100i			0.162235i
$435$		0			0
$436$	193.870			0.444657
$437$	251.252			0.574947
$438$		0			0
$439$	620.243			1.41285			0.706427	−	0.707786i	$-0.250306\pi$
$439$	620.243			1.41285			0.706427	+	0.707786i	$0.250306\pi$
$440$	−53.7807	−	4.53865i	−0.122229	−	0.0103151i
$441$		0			0
$442$		−	464.696i		−	1.05135i
$443$	−224.384			−0.506509			−0.253255	−	0.967400i	$-0.581501\pi$
$443$	−224.384			−0.506509			−0.253255	+	0.967400i	$0.581501\pi$
$444$		0			0
$445$	−214.645	−	18.1143i	−0.482348	−	0.0407062i
$446$			506.862i			1.13646i
$447$		0			0
$448$			15.7224i			0.0350947i
$449$		−	14.6582i		−	0.0326462i	−0.999867	−	0.0163231i	$-0.994804\pi$
$449$		−	14.6582i		−	0.0326462i	0.999867	−	0.0163231i	$-0.00519604\pi$
$450$		0			0
$451$	−135.370			−0.300154
$452$	23.9426			0.0529703
$453$		0			0
$454$	451.278			0.994005
$455$	15.3859	−	182.316i	0.0338152	−	0.400694i
$456$		0			0
$457$		−	442.741i		−	0.968799i	−0.874847	−	0.484400i	$-0.839038\pi$
$457$		−	442.741i		−	0.968799i	0.874847	−	0.484400i	$-0.160962\pi$
$458$	35.1545			0.0767565
$459$		0			0
$460$	−8.10748	+	96.0696i	−0.0176250	+	0.208847i
$461$		−	524.928i		−	1.13867i	−0.822104	−	0.569337i	$-0.807200\pi$
$461$		−	524.928i		−	1.13867i	0.822104	−	0.569337i	$-0.192800\pi$
$462$		0			0
$463$			2.86959i			0.00619782i	0.999995	+	0.00309891i	$0.000986415\pi$
$463$			2.86959i			0.00619782i	−0.999995	+	0.00309891i	$0.999014\pi$
$464$			24.0508i			0.0518336i
$465$		0			0
$466$	−315.796			−0.677674
$467$	692.793			1.48350			0.741748	−	0.670679i	$-0.233997\pi$
$467$	692.793			1.48350			0.741748	+	0.670679i	$0.233997\pi$
$468$		0			0
$469$	−18.2326			−0.0388755
$470$	−42.1512	+	499.471i	−0.0896834	+	1.06270i
$471$		0			0
$472$			105.513i			0.223545i
$473$	−232.928			−0.492448
$474$		0			0
$475$	642.298	+	109.187i	1.35221	+	0.229867i
$476$			69.3662i			0.145727i
$477$		0			0
$478$			126.899i			0.265480i
$479$			886.191i			1.85009i	0.379863	+	0.925043i	$0.375971\pi$
$479$			886.191i			1.85009i	−0.379863	+	0.925043i	$0.624029\pi$
$480$		0			0
$481$	−744.100			−1.54699
$482$	−161.568			−0.335203
$483$		0			0
$484$	212.870			0.439815
$485$	−136.801	−	11.5449i	−0.282064	−	0.0238039i
$486$		0			0
$487$			151.898i			0.311905i	0.987765	+	0.155953i	$0.0498447\pi$
$487$			151.898i			0.311905i	−0.987765	+	0.155953i	$0.950155\pi$
$488$	−13.4402			−0.0275414
$489$		0			0
$490$	26.8400	−	318.040i	0.0547755	−	0.649062i
$491$			9.86744i			0.0200966i	0.999950	+	0.0100483i	$0.00319853\pi$
$491$			9.86744i			0.0200966i	−0.999950	+	0.0100483i	$0.996801\pi$
$492$		0			0
$493$			106.111i			0.215234i
$494$			686.219i			1.38911i
$495$		0			0
$496$	101.333			0.204300
$497$	−67.2067			−0.135225
$498$		0			0
$499$	−214.377			−0.429612			−0.214806	−	0.976657i	$-0.568912\pi$
$499$	−214.377			−0.429612			−0.214806	+	0.976657i	$0.568912\pi$
$500$	−62.4749	+	242.068i	−0.124950	+	0.484136i
$501$		0			0
$502$			404.055i			0.804891i
$503$	778.384			1.54748			0.773741	−	0.633502i	$-0.218383\pi$
$503$	778.384			1.54748			0.773741	+	0.633502i	$0.218383\pi$
$504$		0			0
$505$	895.398	+	75.5642i	1.77306	+	0.149632i
$506$			52.0349i			0.102836i
$507$		0			0
$508$		−	431.111i		−	0.848644i
$509$		−	448.369i		−	0.880881i	−0.897782	−	0.440441i	$-0.854822\pi$
$509$		−	448.369i		−	0.880881i	0.897782	−	0.440441i	$-0.145178\pi$
$510$		0			0
$511$	45.1018			0.0882618
$512$	22.6274			0.0441942
$513$		0			0
$514$	363.478			0.707156
$515$	−653.062	−	55.1130i	−1.26808	−	0.107016i
$516$		0			0
$517$			270.532i			0.523273i
$518$	111.073			0.214428
$519$		0			0
$520$	−262.385	−	22.1432i	−0.504587	−	0.0425830i
$521$		−	603.440i		−	1.15823i	−0.815244	−	0.579117i	$-0.803397\pi$
$521$		−	603.440i		−	1.15823i	0.815244	−	0.579117i	$-0.196603\pi$
$522$		0			0
$523$		−	963.204i		−	1.84169i	−0.389928	−	0.920845i	$-0.627500\pi$
$523$		−	963.204i		−	1.84169i	0.389928	−	0.920845i	$-0.372500\pi$
$524$			158.750i			0.302959i
$525$		0			0
$526$	257.441			0.489431
$527$	447.073			0.848336
$528$		0			0
$529$	−436.049			−0.824289
$530$	49.5435	−	587.065i	0.0934782	−	1.10767i
$531$		0			0
$532$		−	102.433i		−	0.192544i
$533$	−660.441			−1.23910
$534$		0			0
$535$	87.4887	−	1036.70i	0.163530	−	1.93775i
$536$			26.2400i			0.0489553i
$537$		0			0
$538$		−	533.523i		−	0.991678i
$539$		−	172.263i		−	0.319597i
$540$		0			0
$541$	305.923			0.565477			0.282738	−	0.959197i	$-0.408757\pi$
$541$	305.923			0.565477			0.282738	+	0.959197i	$0.408757\pi$
$542$	−399.185			−0.736504
$543$		0			0
$544$	99.8306			0.183512
$545$	40.7578	−	482.959i	0.0747849	−	0.886164i
$546$		0			0
$547$			499.265i			0.912732i	0.889792	+	0.456366i	$0.150849\pi$
$547$			499.265i			0.912732i	−0.889792	+	0.456366i	$0.849151\pi$
$548$	352.132			0.642577
$549$		0			0
$550$	−22.6129	+	133.021i	−0.0411143	+	0.241857i
$551$		−	156.694i		−	0.284381i
$552$		0			0
$553$		−	194.739i		−	0.352150i
$554$			531.186i			0.958820i
$555$		0			0
$556$	−156.109			−0.280771
$557$	842.880			1.51325			0.756625	−	0.653849i	$-0.226847\pi$
$557$	842.880			1.51325			0.756625	+	0.653849i	$0.226847\pi$
$558$		0			0
$559$	−1136.41			−2.03293
$560$	39.1668	+	3.30536i	0.0699407	+	0.00590242i
$561$		0			0
$562$		−	598.960i		−	1.06577i
$563$	627.024			1.11372			0.556859	−	0.830607i	$-0.312006\pi$
$563$	627.024			1.11372			0.556859	+	0.830607i	$0.312006\pi$
$564$		0			0
$565$	5.03350	−	59.6444i	0.00890884	−	0.105565i
$566$		−	17.8527i		−	0.0315418i
$567$		0			0
$568$			96.7227i			0.170286i
$569$			164.726i			0.289501i	0.989468	+	0.144750i	$0.0462379\pi$
$569$			164.726i			0.289501i	−0.989468	+	0.144750i	$0.953762\pi$
$570$		0			0
$571$	−754.411			−1.32121			−0.660605	−	0.750733i	$-0.729700\pi$
$571$	−754.411			−1.32121			−0.660605	+	0.750733i	$0.729700\pi$
$572$	−142.118			−0.248458
$573$		0			0
$574$	98.5855			0.171752
$575$	237.619	+	40.3938i	0.413250	+	0.0702501i
$576$		0			0
$577$		−	924.607i		−	1.60244i	−0.598370	−	0.801220i	$-0.704185\pi$
$577$		−	924.607i		−	1.60244i	0.598370	−	0.801220i	$-0.295815\pi$
$578$	31.7379			0.0549099
$579$		0			0
$580$	59.9141	+	5.05625i	0.103300	+	0.00871767i
$581$		−	74.4932i		−	0.128216i
$582$		0			0
$583$		−	317.977i		−	0.545414i
$584$		−	64.9096i		−	0.111147i
$585$		0			0
$586$	22.6723			0.0386899
$587$	99.9008			0.170189			0.0850944	−	0.996373i	$-0.472881\pi$
$587$	99.9008			0.170189			0.0850944	+	0.996373i	$0.472881\pi$
$588$		0			0
$589$	−660.195			−1.12087
$590$	262.849	+	22.1822i	0.445506	+	0.0375970i
$591$		0			0
$592$		−	159.855i		−	0.270025i
$593$	519.869			0.876676			0.438338	−	0.898810i	$-0.355567\pi$
$593$	519.869			0.876676			0.438338	+	0.898810i	$0.355567\pi$
$594$		0			0
$595$	172.801	+	14.5830i	0.290422	+	0.0245092i
$596$			221.669i			0.371928i
$597$		0			0
$598$			253.868i			0.424528i
$599$			212.861i			0.355360i	0.984088	+	0.177680i	$0.0568593\pi$
$599$			212.861i			0.355360i	−0.984088	+	0.177680i	$0.943141\pi$
$600$		0			0
$601$	−586.694			−0.976196			−0.488098	−	0.872789i	$-0.662309\pi$
$601$	−586.694			−0.976196			−0.488098	+	0.872789i	$0.662309\pi$
$602$	169.634			0.281784
$603$		0			0
$604$	−25.2659			−0.0418310
$605$	44.7522	−	530.291i	0.0739705	−	0.876514i
$606$		0			0
$607$			348.743i			0.574536i	0.957850	+	0.287268i	$0.0927470\pi$
$607$			348.743i			0.574536i	−0.957850	+	0.287268i	$0.907253\pi$
$608$	−147.420			−0.242468
$609$		0			0
$610$	−2.82556	+	33.4815i	−0.00463207	+	0.0548877i
$611$			1319.87i			2.16018i
$612$		0			0
$613$			513.615i			0.837871i	0.908016	+	0.418935i	$0.137597\pi$
$613$			513.615i			0.837871i	−0.908016	+	0.418935i	$0.862403\pi$
$614$		−	650.858i		−	1.06003i
$615$		0			0
$616$	21.2142			0.0344387
$617$	−359.530			−0.582707			−0.291353	−	0.956616i	$-0.594106\pi$
$617$	−359.530			−0.582707			−0.291353	+	0.956616i	$0.594106\pi$
$618$		0			0
$619$	1032.38			1.66782			0.833912	−	0.551897i	$-0.186096\pi$
$619$	1032.38			1.66782			0.833912	+	0.551897i	$0.186096\pi$
$620$	21.3034	−	252.435i	0.0343603	−	0.407152i
$621$		0			0
$622$		−	208.678i		−	0.335495i
$623$	84.6683			0.135904
$624$		0			0
$625$	589.892	+	206.525i	0.943827	+	0.330439i
$626$			295.427i			0.471929i
$627$		0			0
$628$			412.288i			0.656510i
$629$		−	705.269i		−	1.12125i
$630$		0			0
$631$	−1042.04			−1.65141			−0.825705	−	0.564102i	$-0.809222\pi$
$631$	−1042.04			−1.65141			−0.825705	+	0.564102i	$0.809222\pi$
$632$	−280.265			−0.443457
$633$		0			0
$634$	699.612			1.10349
$635$	−1073.96	−	90.6334i	−1.69128	−	0.142730i
$636$		0			0
$637$		−	840.435i		−	1.31936i
$638$	32.4517			0.0508647
$639$		0			0
$640$	4.75701	−	56.3682i	0.00743283	−	0.0880753i
$641$		−	968.370i		−	1.51072i	−0.655312	−	0.755359i	$-0.727463\pi$
$641$		−	968.370i		−	1.51072i	0.655312	−	0.755359i	$-0.272537\pi$
$642$		0			0
$643$		−	866.324i		−	1.34732i	−0.739043	−	0.673658i	$-0.764722\pi$
$643$		−	866.324i		−	1.34732i	0.739043	−	0.673658i	$-0.235278\pi$
$644$		−	37.8954i		−	0.0588438i
$645$		0			0
$646$	−650.408			−1.00682
$647$	−211.789			−0.327340			−0.163670	−	0.986515i	$-0.552333\pi$
$647$	−211.789			−0.327340			−0.163670	+	0.986515i	$0.552333\pi$
$648$		0			0
$649$	142.369			0.219366
$650$	−110.324	+	648.985i	−0.169729	+	0.998438i
$651$		0			0
$652$			470.042i			0.720923i
$653$	−259.698			−0.397700			−0.198850	−	0.980030i	$-0.563721\pi$
$653$	−259.698			−0.397700			−0.198850	+	0.980030i	$0.563721\pi$
$654$		0			0
$655$	395.470	+	33.3744i	0.603771	+	0.0509533i
$656$		−	141.882i		−	0.216284i
$657$		0			0
$658$		−	197.020i		−	0.299423i
$659$			521.908i			0.791970i	0.918257	+	0.395985i	$0.129597\pi$
$659$			521.908i			0.791970i	−0.918257	+	0.395985i	$0.870403\pi$
$660$		0			0
$661$	605.685			0.916316			0.458158	−	0.888871i	$-0.348509\pi$
$661$	605.685			0.916316			0.458158	+	0.888871i	$0.348509\pi$
$662$	714.195			1.07884
$663$		0			0
$664$	−107.209			−0.161460
$665$	−255.177	−	21.5348i	−0.383724	−	0.0323832i
$666$		0			0
$667$		−	57.9691i		−	0.0869102i
$668$	252.358			0.377781
$669$		0			0
$670$	65.3677	+	5.51649i	0.0975637	+	0.00823357i
$671$			18.1348i			0.0270266i
$672$		0			0
$673$		−	485.275i		−	0.721063i	−0.932747	−	0.360531i	$-0.882595\pi$
$673$		−	485.275i		−	0.721063i	0.932747	−	0.360531i	$-0.117405\pi$
$674$			870.974i			1.29225i
$675$		0			0
$676$	−355.364			−0.525686
$677$	−908.353			−1.34173			−0.670867	−	0.741578i	$-0.734078\pi$
$677$	−908.353			−1.34173			−0.670867	+	0.741578i	$0.734078\pi$
$678$		0			0
$679$	53.9623			0.0794732
$680$	20.9876	−	248.692i	0.0308641	−	0.365724i
$681$		0			0
$682$		−	136.728i		−	0.200481i
$683$	−285.449			−0.417934			−0.208967	−	0.977923i	$-0.567010\pi$
$683$	−285.449			−0.417934			−0.208967	+	0.977923i	$0.567010\pi$
$684$		0			0
$685$	74.0295	−	877.213i	0.108072	−	1.28060i
$686$			261.642i			0.381402i
$687$		0			0
$688$		−	244.134i		−	0.354846i
$689$		−	1551.34i		−	2.25159i
$690$		0			0
$691$	388.446			0.562151			0.281075	−	0.959686i	$-0.409309\pi$
$691$	388.446			0.562151			0.281075	+	0.959686i	$0.409309\pi$
$692$	−193.985			−0.280325
$693$		0			0
$694$	365.874			0.527196
$695$	−32.8191	+	388.890i	−0.0472217	+	0.559554i
$696$		0			0
$697$		−	625.976i		−	0.898100i
$698$	−660.378			−0.946100
$699$		0			0
$700$	16.4682	−	96.8753i	0.0235261	−	0.138393i
$701$		−	519.499i		−	0.741083i	−0.928816	−	0.370542i	$-0.879172\pi$
$701$		−	519.499i		−	0.741083i	0.928816	−	0.370542i	$-0.120828\pi$
$702$		0			0
$703$			1041.47i			1.48147i
$704$		−	30.5311i		−	0.0433681i
$705$		0			0
$706$	−47.1443			−0.0667766
$707$	−353.196			−0.499571
$708$		0			0
$709$	−796.707			−1.12371			−0.561853	−	0.827237i	$-0.689911\pi$
$709$	−796.707			−1.12371			−0.561853	+	0.827237i	$0.689911\pi$
$710$	240.950	+	20.3342i	0.339367	+	0.0286397i
$711$		0			0
$712$		−	121.853i		−	0.171142i
$713$	−244.240			−0.342552
$714$		0			0
$715$	−29.8777	+	354.036i	−0.0417870	+	0.495155i
$716$			478.034i			0.667645i
$717$		0			0
$718$		−	874.972i		−	1.21862i
$719$		−	785.781i		−	1.09288i	−0.837498	−	0.546440i	$-0.815982\pi$
$719$		−	785.781i		−	1.09288i	0.837498	−	0.546440i	$-0.184018\pi$
$720$		0			0
$721$	257.605			0.357289
$722$	449.930			0.623172
$723$		0			0
$724$	542.168			0.748851
$725$	25.1917	−	148.192i	0.0347472	−	0.204402i
$726$		0			0
$727$		−	398.281i		−	0.547842i	−0.961752	−	0.273921i	$-0.911679\pi$
$727$		−	398.281i		−	0.547842i	0.961752	−	0.273921i	$-0.0883207\pi$
$728$	103.500			0.142170
$729$		0			0
$730$	−161.699	−	13.6461i	−0.221506	−	0.0186933i
$731$		−	1077.10i		−	1.47347i
$732$		0			0
$733$			514.270i			0.701596i	0.936451	+	0.350798i	$0.114090\pi$
$733$			514.270i			0.701596i	−0.936451	+	0.350798i	$0.885910\pi$
$734$		−	515.498i		−	0.702313i
$735$		0			0
$736$	−54.5383			−0.0741010
$737$	35.4056			0.0480402
$738$		0			0
$739$	−105.919			−0.143328			−0.0716639	−	0.997429i	$-0.522831\pi$
$739$	−105.919			−0.143328			−0.0716639	+	0.997429i	$0.522831\pi$
$740$	−398.222	−	33.6066i	−0.538137	−	0.0454144i
$741$		0			0
$742$			231.572i			0.312092i
$743$	124.265			0.167248			0.0836239	−	0.996497i	$-0.473351\pi$
$743$	124.265			0.167248			0.0836239	+	0.996497i	$0.473351\pi$
$744$		0			0
$745$	552.210	+	46.6020i	0.741222	+	0.0625530i
$746$		−	707.379i		−	0.948229i
$747$		0			0
$748$		−	134.701i		−	0.180082i
$749$			408.933i			0.545972i
$750$		0			0
$751$	585.674			0.779859			0.389930	−	0.920845i	$-0.372499\pi$
$751$	585.674			0.779859			0.389930	+	0.920845i	$0.372499\pi$
$752$	−283.548			−0.377058
$753$		0			0
$754$	158.325			0.209980
$755$	−5.31171	+	62.9411i	−0.00703538	+	0.0833657i
$756$		0			0
$757$			374.753i			0.495050i	0.968881	+	0.247525i	$0.0796173\pi$
$757$			374.753i			0.495050i	−0.968881	+	0.247525i	$0.920383\pi$
$758$	216.500			0.285619
$759$		0			0
$760$	−30.9925	+	367.245i	−0.0407796	+	0.483218i
$761$			965.839i			1.26917i	0.772853	+	0.634585i	$0.218829\pi$
$761$			965.839i			1.26917i	−0.772853	+	0.634585i	$0.781171\pi$
$762$		0			0
$763$			190.507i			0.249681i
$764$			515.374i			0.674574i
$765$		0			0
$766$	−881.835			−1.15122
$767$	694.588			0.905590
$768$		0			0
$769$	278.353			0.361968			0.180984	−	0.983486i	$-0.442072\pi$
$769$	278.353			0.361968			0.180984	+	0.983486i	$0.442072\pi$
$770$	4.45991	−	52.8477i	0.00579209	−	0.0686334i
$771$		0			0
$772$			234.020i			0.303135i
$773$	−621.766			−0.804354			−0.402177	−	0.915562i	$-0.631746\pi$
$773$	−621.766			−0.804354			−0.402177	+	0.915562i	$0.631746\pi$
$774$		0			0
$775$	−624.372	−	106.140i	−0.805642	−	0.136954i
$776$		−	77.6615i		−	0.100079i
$777$		0			0
$778$		−	394.476i		−	0.507039i
$779$			924.381i			1.18663i
$780$		0			0
$781$	130.508			0.167103
$782$	−240.619			−0.307697
$783$		0			0
$784$	180.550			0.230294
$785$	1027.07	+	86.6762i	1.30837	+	0.110415i
$786$		0			0
$787$			415.818i			0.528358i	0.964474	+	0.264179i	$0.0851009\pi$
$787$			415.818i			0.528358i	−0.964474	+	0.264179i	$0.914899\pi$
$788$	−255.481			−0.324215
$789$		0			0
$790$	−58.9207	+	698.181i	−0.0745832	+	0.883773i
$791$			23.5272i			0.0297436i
$792$		0			0
$793$			88.4763i			0.111572i
$794$		−	969.022i		−	1.22043i
$795$		0			0
$796$	−705.126			−0.885836
$797$	94.6817			0.118798			0.0593988	−	0.998234i	$-0.481082\pi$
$797$	94.6817			0.118798			0.0593988	+	0.998234i	$0.481082\pi$
$798$		0			0
$799$	−1250.99			−1.56570
$800$	−139.421	−	23.7008i	−0.174277	−	0.0296260i
$801$		0			0
$802$			556.330i			0.693678i
$803$	−87.5824			−0.109069
$804$		0			0
$805$	−94.4029	−	7.96682i	−0.117271	−	0.00989667i
$806$		−	667.068i		−	0.827628i
$807$		0			0
$808$			508.314i			0.629101i
$809$		−	1038.91i		−	1.28419i	−0.766624	−	0.642096i	$-0.778065\pi$
$809$		−	1038.91i		−	1.28419i	0.766624	−	0.642096i	$-0.221935\pi$
$810$		0			0
$811$	−414.046			−0.510537			−0.255269	−	0.966870i	$-0.582164\pi$
$811$	−414.046			−0.510537			−0.255269	+	0.966870i	$0.582164\pi$
$812$	−23.6336			−0.0291054
$813$		0			0
$814$	−215.692			−0.264978
$815$	1170.94	+	98.8178i	1.43674	+	0.121249i
$816$		0			0
$817$			1590.56i			1.94684i
$818$	−822.521			−1.00553
$819$		0			0
$820$	−353.450	−	29.8282i	−0.431036	−	0.0363759i
$821$			360.929i			0.439622i	0.975543	+	0.219811i	$0.0705440\pi$
$821$			360.929i			0.439622i	−0.975543	+	0.219811i	$0.929456\pi$
$822$		0			0
$823$			1120.87i			1.36194i	0.732312	+	0.680969i	$0.238441\pi$
$823$			1120.87i			1.36194i	−0.732312	+	0.680969i	$0.761559\pi$
$824$		−	370.741i		−	0.449928i
$825$		0			0
$826$	−103.683			−0.125524
$827$	−1158.15			−1.40042			−0.700212	−	0.713935i	$-0.746911\pi$
$827$	−1158.15			−1.40042			−0.700212	+	0.713935i	$0.746911\pi$
$828$		0			0
$829$	500.809			0.604112			0.302056	−	0.953290i	$-0.402327\pi$
$829$	500.809			0.604112			0.302056	+	0.953290i	$0.402327\pi$
$830$	−22.5388	+	267.074i	−0.0271552	+	0.321776i
$831$		0			0
$832$		−	148.955i		−	0.179033i
$833$	796.576			0.956273
$834$		0			0
$835$	53.0537	−	628.660i	0.0635374	−	0.752886i
$836$			198.914i			0.237935i
$837$		0			0
$838$			265.298i			0.316585i
$839$		−	438.216i		−	0.522308i	−0.965297	−	0.261154i	$-0.915897\pi$
$839$		−	438.216i		−	0.522308i	0.965297	−	0.261154i	$-0.0841030\pi$
$840$		0			0
$841$	804.847			0.957012
$842$	247.352			0.293767
$843$		0			0
$844$	556.804			0.659720
$845$	−74.7088	+	885.262i	−0.0884128	+	1.04765i
$846$		0			0
$847$			209.177i			0.246963i
$848$	333.275			0.393013
$849$		0			0
$850$	−615.117	−	104.566i	−0.723667	−	0.123019i
$851$			385.295i			0.452755i
$852$		0			0
$853$			541.187i			0.634451i	0.948350	+	0.317225i	$0.102751\pi$
$853$			541.187i			0.634451i	−0.948350	+	0.317225i	$0.897249\pi$
$854$		−	13.2070i		−	0.0154649i
$855$		0			0
$856$	588.529			0.687534
$857$	325.496			0.379809			0.189904	−	0.981803i	$-0.439182\pi$
$857$	325.496			0.379809			0.189904	+	0.981803i	$0.439182\pi$
$858$		0			0
$859$	−94.7565			−0.110310			−0.0551551	−	0.998478i	$-0.517565\pi$
$859$	−94.7565			−0.110310			−0.0551551	+	0.998478i	$0.517565\pi$
$860$	−608.174	−	51.3249i	−0.707179	−	0.0596801i
$861$		0			0
$862$		−	2.46373i		−	0.00285815i
$863$	598.282			0.693259			0.346629	−	0.938002i	$-0.387326\pi$
$863$	598.282			0.693259			0.346629	+	0.938002i	$0.387326\pi$
$864$		0			0
$865$	−40.7818	+	483.244i	−0.0471466	+	0.558663i
$866$			255.514i			0.295051i
$867$		0			0
$868$			99.5747i			0.114717i
$869$			378.161i			0.435168i
$870$		0			0
$871$	172.737			0.198320
$872$	274.174			0.314420
$873$		0			0
$874$	355.324			0.406549
$875$	−237.868	−	61.3911i	−0.271850	−	0.0701612i
$876$		0			0
$877$		−	528.196i		−	0.602276i	−0.953581	−	0.301138i	$-0.902634\pi$
$877$		−	528.196i		−	0.602276i	0.953581	−	0.301138i	$-0.0973665\pi$
$878$	877.156			0.999038
$879$		0			0
$880$	−76.0574	−	6.41862i	−0.0864289	−	0.00729389i
$881$			240.234i			0.272684i	0.990662	+	0.136342i	$0.0435345\pi$
$881$			240.234i			0.272684i	−0.990662	+	0.136342i	$0.956465\pi$
$882$		0			0
$883$		−	195.733i		−	0.221668i	−0.993839	−	0.110834i	$-0.964648\pi$
$883$		−	195.733i		−	0.221668i	0.993839	−	0.110834i	$-0.0353522\pi$
$884$		−	657.180i		−	0.743416i
$885$		0			0
$886$	−317.326			−0.358156
$887$	−776.601			−0.875537			−0.437768	−	0.899088i	$-0.644231\pi$
$887$	−776.601			−0.875537			−0.437768	+	0.899088i	$0.644231\pi$
$888$		0			0
$889$	423.632			0.476527
$890$	−303.554	−	25.6174i	−0.341072	−	0.0287836i
$891$		0			0
$892$			716.811i			0.803600i
$893$	1847.35			2.06870
$894$		0			0
$895$	1190.85	+	100.498i	1.33056	+	0.112288i
$896$			22.2349i			0.0248157i
$897$		0			0
$898$		−	20.7298i		−	0.0230844i
$899$			152.321i			0.169434i
$900$		0			0
$901$	1470.39			1.63195
$902$	−191.442			−0.212241
$903$		0			0
$904$	33.8599			0.0374556
$905$	113.981	−	1350.62i	0.125946	−	1.49240i
$906$		0			0
$907$			1116.26i			1.23071i	0.788249	+	0.615356i	$0.210988\pi$
$907$			1116.26i			1.23071i	−0.788249	+	0.615356i	$0.789012\pi$
$908$	638.204			0.702868
$909$		0			0
$910$	21.7590	−	257.833i	0.0239110	−	0.283333i
$911$		−	241.280i		−	0.264851i	−0.991193	−	0.132426i	$-0.957723\pi$
$911$		−	241.280i		−	0.264851i	0.991193	−	0.132426i	$-0.0422766\pi$
$912$		0			0
$913$			144.657i			0.158442i
$914$		−	626.131i		−	0.685045i
$915$		0			0
$916$	49.7159			0.0542750
$917$	−155.996			−0.170116
$918$		0			0
$919$	−80.4600			−0.0875517			−0.0437758	−	0.999041i	$-0.513939\pi$
$919$	−80.4600			−0.0875517			−0.0437758	+	0.999041i	$0.513939\pi$
$920$	−11.4657	+	135.863i	−0.0124627	+	0.147677i
$921$		0			0
$922$		−	742.361i		−	0.805164i
$923$	636.721			0.689838
$924$		0			0
$925$	−167.438	+	984.963i	−0.181014	+	1.06482i
$926$			4.05821i			0.00438252i
$927$		0			0
$928$			34.0130i			0.0366519i
$929$			1246.34i			1.34159i	0.741643	+	0.670795i	$0.234047\pi$
$929$			1246.34i			1.34159i	−0.741643	+	0.670795i	$0.765953\pi$
$930$		0			0
$931$	−1176.31			−1.26349
$932$	−446.603			−0.479188
$933$		0			0
$934$	979.757			1.04899
$935$	−335.560	−	28.3185i	−0.358888	−	0.0302872i
$936$		0			0
$937$			937.917i			1.00098i	0.865743	+	0.500489i	$0.166846\pi$
$937$			937.917i			1.00098i	−0.865743	+	0.500489i	$0.833154\pi$
$938$	−25.7848			−0.0274891
$939$		0			0
$940$	−59.6108	+	706.358i	−0.0634158	+	0.751445i
$941$		−	227.634i		−	0.241906i	−0.992658	−	0.120953i	$-0.961405\pi$
$941$		−	227.634i		−	0.241906i	0.992658	−	0.120953i	$-0.0385951\pi$
$942$		0			0
$943$			341.976i			0.362647i
$944$			149.218i			0.158070i
$945$		0			0
$946$	−329.410			−0.348213
$947$	271.423			0.286614			0.143307	−	0.989678i	$-0.454226\pi$
$947$	271.423			0.286614			0.143307	+	0.989678i	$0.454226\pi$
$948$		0			0
$949$	−427.297			−0.450260
$950$	908.346	+	154.414i	0.956154	+	0.162541i
$951$		0			0
$952$			98.0986i			0.103045i
$953$	−1026.91			−1.07756			−0.538780	−	0.842447i	$-0.681114\pi$
$953$	−1026.91			−1.07756			−0.538780	+	0.842447i	$0.681114\pi$
$954$		0			0
$955$	1283.87	+	108.348i	1.34437	+	0.113454i
$956$			179.463i			0.187723i
$957$		0			0
$958$			1253.26i			1.30821i
$959$			346.023i			0.360817i
$960$		0			0
$961$	−319.230			−0.332185
$962$	−1052.32			−1.09388
$963$		0			0
$964$	−228.491			−0.237024
$965$	582.978	+	49.1985i	0.604122	+	0.0509829i
$966$		0			0
$967$		−	659.712i		−	0.682225i	−0.940022	−	0.341113i	$-0.889196\pi$
$967$		−	659.712i		−	0.682225i	0.940022	−	0.341113i	$-0.110804\pi$
$968$	301.044			0.310996
$969$		0			0
$970$	−193.466	−	16.3269i	−0.199450	−	0.0168319i
$971$			1463.21i			1.50691i	0.657499	+	0.753455i	$0.271614\pi$
$971$			1463.21i			1.50691i	−0.657499	+	0.753455i	$0.728386\pi$
$972$		0			0
$973$		−	153.401i		−	0.157657i
$974$			214.816i			0.220550i
$975$		0			0
$976$	−19.0073			−0.0194747
$977$	−103.801			−0.106245			−0.0531225	−	0.998588i	$-0.516917\pi$
$977$	−103.801			−0.106245			−0.0531225	+	0.998588i	$0.516917\pi$
$978$		0			0
$979$	−164.416			−0.167943
$980$	37.9575	−	449.777i	0.0387321	−	0.458956i
$981$		0			0
$982$			13.9547i			0.0142104i
$983$	186.758			0.189988			0.0949941	−	0.995478i	$-0.469717\pi$
$983$	186.758			0.189988			0.0949941	+	0.995478i	$0.469717\pi$
$984$		0			0
$985$	−53.7103	+	636.441i	−0.0545283	+	0.646133i
$986$			150.063i			0.152194i
$987$		0			0
$988$			970.461i			0.982248i
$989$			588.431i			0.594976i
$990$		0			0
$991$	−1064.29			−1.07395			−0.536977	−	0.843597i	$-0.680434\pi$
$991$	−1064.29			−1.07395			−0.536977	+	0.843597i	$0.680434\pi$
$992$	143.306			0.144462
$993$		0			0
$994$	−95.0447			−0.0956184
$995$	−148.240	+	1756.57i	−0.148985	+	1.76540i
$996$		0			0
$997$		−	470.263i		−	0.471678i	−0.971792	−	0.235839i	$-0.924216\pi$
$997$		−	470.263i		−	0.471678i	0.971792	−	0.235839i	$-0.0757838\pi$
$998$	−303.174			−0.303782
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.c.809.19	yes	24
3.2	odd	2	inner	810.3.b.c.809.6	yes	24
5.4	even	2	inner	810.3.b.c.809.5	✓	24
9.2	odd	6		810.3.j.h.539.9		24
9.4	even	3		810.3.j.g.269.4		24
9.5	odd	6		810.3.j.g.269.8		24
9.7	even	3		810.3.j.h.539.4		24
15.14	odd	2	inner	810.3.b.c.809.20	yes	24
45.4	even	6		810.3.j.h.269.8		24
45.14	odd	6		810.3.j.h.269.4		24
45.29	odd	6		810.3.j.g.539.4		24
45.34	even	6		810.3.j.g.539.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.3.b.c.809.5	✓	24	5.4	even	2	inner
810.3.b.c.809.6	yes	24	3.2	odd	2	inner
810.3.b.c.809.19	yes	24	1.1	even	1	trivial
810.3.b.c.809.20	yes	24	15.14	odd	2	inner
810.3.j.g.269.4		24	9.4	even	3
810.3.j.g.269.8		24	9.5	odd	6
810.3.j.g.539.4		24	45.29	odd	6
810.3.j.g.539.9		24	45.34	even	6
810.3.j.h.269.4		24	45.14	odd	6
810.3.j.h.269.8		24	45.4	even	6
810.3.j.h.539.4		24	9.7	even	3
810.3.j.h.539.9		24	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(0.420464 - 4.98229i) q^{5} +1.96530i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(0.420464 - 4.98229i) q^{5} +1.96530i q^{7} +2.82843 q^{8} +(0.594626 - 7.04602i) q^{10} -3.81639i q^{11} -18.6194i q^{13} +2.77936i q^{14} +4.00000 q^{16} +17.6477 q^{17} -26.0605 q^{19} +(0.840928 - 9.96458i) q^{20} -5.39719i q^{22} -9.64111 q^{23} +(-24.6464 - 4.18975i) q^{25} -26.3318i q^{26} +3.93060i q^{28} +6.01270i q^{29} +25.3332 q^{31} +5.65685 q^{32} +24.9576 q^{34} +(9.79170 + 0.826339i) q^{35} -39.9637i q^{37} -36.8551 q^{38} +(1.18925 - 14.0920i) q^{40} -35.4706i q^{41} -61.0336i q^{43} -7.63278i q^{44} -13.6346 q^{46} -70.8869 q^{47} +45.1376 q^{49} +(-34.8553 - 5.92520i) q^{50} -37.2388i q^{52} +83.3187 q^{53} +(-19.0144 - 1.60465i) q^{55} +5.55871i q^{56} +8.50325i q^{58} +37.3045i q^{59} -4.75183 q^{61} +35.8265 q^{62} +8.00000 q^{64} +(-92.7672 - 7.82879i) q^{65} +9.27725i q^{67} +35.2954 q^{68} +(13.8476 + 1.16862i) q^{70} +34.1966i q^{71} -22.9490i q^{73} -56.5172i q^{74} -52.1210 q^{76} +7.50036 q^{77} -99.0887 q^{79} +(1.68186 - 19.9292i) q^{80} -50.1630i q^{82} -37.9042 q^{83} +(7.42023 - 87.9261i) q^{85} -86.3145i q^{86} -10.7944i q^{88} -43.0816i q^{89} +36.5927 q^{91} -19.2822 q^{92} -100.249 q^{94} +(-10.9575 + 129.841i) q^{95} -27.4575i q^{97} +63.8342 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4}+O(q^{10}) \) 24 * q + 48 * q^4	\( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

Introduction

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