LMFDB - Embedded newform 800.2.d.a.401.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(401,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.2
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.2.d.a.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575i q^{3} -4.00000 q^{7} -4.00000 q^{9} +O(q^{10})$	$q+2.64575i q^{3} -4.00000 q^{7} -4.00000 q^{9} -2.64575i q^{11} -3.00000 q^{17} -2.64575i q^{19} -10.5830i q^{21} -4.00000 q^{23} -2.64575i q^{27} -4.00000 q^{31} +7.00000 q^{33} -10.5830i q^{37} -5.00000 q^{41} +5.29150i q^{43} +8.00000 q^{47} +9.00000 q^{49} -7.93725i q^{51} +10.5830i q^{53} +7.00000 q^{57} -5.29150i q^{59} +10.5830i q^{61} +16.0000 q^{63} -7.93725i q^{67} -10.5830i q^{69} -8.00000 q^{71} -7.00000 q^{73} +10.5830i q^{77} -4.00000 q^{79} -5.00000 q^{81} -7.93725i q^{83} -1.00000 q^{89} -10.5830i q^{93} -2.00000 q^{97} +10.5830i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 - 8 * q^9	$ 2 q - 8 q^{7} - 8 q^{9} - 6 q^{17} - 8 q^{23} - 8 q^{31} + 14 q^{33} - 10 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{57} + 32 q^{63} - 16 q^{71} - 14 q^{73} - 8 q^{79} - 10 q^{81} - 2 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 8 * q^9 - 6 * q^17 - 8 * q^23 - 8 * q^31 + 14 * q^33 - 10 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^57 + 32 * q^63 - 16 * q^71 - 14 * q^73 - 8 * q^79 - 10 * q^81 - 2 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.64575i	1.52753i	0.645497	+	0.763763i	$0.276650\pi$
$3$	2.64575i	1.52753i	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−4.00000	−1.33333
$10$	0	0
$11$	− 2.64575i	− 0.797724i	−0.917011	−	0.398862i	$-0.869405\pi$
$11$	− 2.64575i	− 0.797724i	0.917011	−	0.398862i	$-0.130595\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 2.64575i	− 0.606977i	−0.952835	−	0.303488i	$-0.901849\pi$
$19$	− 2.64575i	− 0.606977i	0.952835	−	0.303488i	$-0.0981514\pi$
$20$	0	0
$21$	− 10.5830i	− 2.30940i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.64575i	− 0.509175i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	7.00000	1.21854
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.5830i	− 1.73984i	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	− 10.5830i	− 1.73984i	0.493197	−	0.869918i	$-0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	5.29150i	0.806947i	0.914991	+	0.403473i	$0.132197\pi$
$43$	5.29150i	0.806947i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	− 7.93725i	− 1.11144i
$52$	0	0
$53$	10.5830i	1.45369i	0.686803	+	0.726844i	$0.259014\pi$
$53$	10.5830i	1.45369i	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	− 5.29150i	− 0.688895i	−0.938806	−	0.344447i	$-0.888066\pi$
$59$	− 5.29150i	− 0.688895i	0.938806	−	0.344447i	$-0.111934\pi$
$60$	0	0
$61$	10.5830i	1.35501i	0.735516	+	0.677507i	$0.236940\pi$
$61$	10.5830i	1.35501i	−0.735516	+	0.677507i	$0.763060\pi$
$62$	0	0
$63$	16.0000	2.01581
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.93725i	− 0.969690i	−0.874600	−	0.484845i	$-0.838876\pi$
$67$	− 7.93725i	− 0.969690i	0.874600	−	0.484845i	$-0.161124\pi$
$68$	0	0
$69$	− 10.5830i	− 1.27404i
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5830i	1.20605i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 7.93725i	− 0.871227i	−0.900134	−	0.435613i	$-0.856531\pi$
$83$	− 7.93725i	− 0.871227i	0.900134	−	0.435613i	$-0.143469\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 10.5830i	− 1.09741i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	10.5830i	1.06363i
$100$	0	0
$101$	− 10.5830i	− 1.05305i	−0.850160	−	0.526524i	$-0.823495\pi$
$101$	− 10.5830i	− 1.05305i	0.850160	−	0.526524i	$-0.176505\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.64575i	0.255774i	0.991789	+	0.127887i	$0.0408196\pi$
$107$	2.64575i	0.255774i	−0.991789	+	0.127887i	$0.959180\pi$
$108$	0	0
$109$	10.5830i	1.01367i	0.862044	+	0.506834i	$0.169184\pi$
$109$	10.5830i	1.01367i	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	28.0000	2.65764
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	4.00000	0.363636
$122$	0	0
$123$	− 13.2288i	− 1.19280i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	−14.0000	−1.23263
$130$	0	0
$131$	15.8745i	1.38696i	0.720475	+	0.693481i	$0.243924\pi$
$131$	15.8745i	1.38696i	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	10.5830i	0.917663i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.0000	−1.62328	−0.811640	−	0.584158i	$-0.801425\pi$
$137$	−19.0000	−1.62328	−0.811640	+	0.584158i	$0.801425\pi$
$138$	0	0
$139$	18.5203i	1.57087i	0.618945	+	0.785434i	$0.287560\pi$
$139$	18.5203i	1.57087i	−0.618945	+	0.785434i	$0.712440\pi$
$140$	0	0
$141$	21.1660i	1.78250i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	23.8118i	1.96396i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.5830i	0.844616i	0.906452	+	0.422308i	$0.138780\pi$
$157$	10.5830i	0.844616i	−0.906452	+	0.422308i	$0.861220\pi$
$158$	0	0
$159$	−28.0000	−2.22054
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	13.2288i	1.03616i	0.855333	+	0.518078i	$0.173352\pi$
$163$	13.2288i	1.03616i	−0.855333	+	0.518078i	$0.826648\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	10.5830i	0.809303i
$172$	0	0
$173$	− 21.1660i	− 1.60922i	−0.593802	−	0.804611i	$-0.702374\pi$
$173$	− 21.1660i	− 1.60922i	0.593802	−	0.804611i	$-0.297626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.0000	1.05230
$178$	0	0
$179$	− 23.8118i	− 1.77977i	−0.456180	−	0.889887i	$-0.650783\pi$
$179$	− 23.8118i	− 1.77977i	0.456180	−	0.889887i	$-0.349217\pi$
$180$	0	0
$181$	10.5830i	0.786629i	0.919404	+	0.393314i	$0.128672\pi$
$181$	10.5830i	0.786629i	−0.919404	+	0.393314i	$0.871328\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.93725i	0.580429i
$188$	0	0
$189$	10.5830i	0.769800i
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.5830i	0.754008i	0.926212	+	0.377004i	$0.123046\pi$
$197$	10.5830i	0.754008i	−0.926212	+	0.377004i	$0.876954\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	21.0000	1.48123
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	16.0000	1.11208
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	7.93725i	0.546423i	0.961954	+	0.273212i	$0.0880859\pi$
$211$	7.93725i	0.546423i	−0.961954	+	0.273212i	$0.911914\pi$
$212$	0	0
$213$	− 21.1660i	− 1.45027i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	− 18.5203i	− 1.25148i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 15.8745i	− 1.05363i	−0.849981	−	0.526814i	$-0.823386\pi$
$227$	− 15.8745i	− 1.05363i	0.849981	−	0.526814i	$-0.176614\pi$
$228$	0	0
$229$	21.1660i	1.39869i	0.714785	+	0.699345i	$0.246525\pi$
$229$	21.1660i	1.39869i	−0.714785	+	0.699345i	$0.753475\pi$
$230$	0	0
$231$	−28.0000	−1.84226
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.5830i	− 0.687440i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	0	0
$243$	− 21.1660i	− 1.35780i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	21.0000	1.33082
$250$	0	0
$251$	7.93725i	0.500995i	0.968117	+	0.250498i	$0.0805942\pi$
$251$	7.93725i	0.500995i	−0.968117	+	0.250498i	$0.919406\pi$
$252$	0	0
$253$	10.5830i	0.665348i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	42.3320i	2.63038i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.64575i	− 0.161917i
$268$	0	0
$269$	− 21.1660i	− 1.29051i	−0.763965	−	0.645257i	$-0.776750\pi$
$269$	− 21.1660i	− 1.29051i	0.763965	−	0.645257i	$-0.223250\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 21.1660i	− 1.27174i	−0.771795	−	0.635871i	$-0.780641\pi$
$277$	− 21.1660i	− 1.27174i	0.771795	−	0.635871i	$-0.219359\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	13.2288i	0.786368i	0.919460	+	0.393184i	$0.128626\pi$
$283$	13.2288i	0.786368i	−0.919460	+	0.393184i	$0.871374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000	1.18056
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 5.29150i	− 0.310193i
$292$	0	0
$293$	− 10.5830i	− 0.618266i	−0.951019	−	0.309133i	$-0.899961\pi$
$293$	− 10.5830i	− 0.618266i	0.951019	−	0.309133i	$-0.100039\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.00000	−0.406181
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 21.1660i	− 1.21999i
$302$	0	0
$303$	28.0000	1.60856
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.64575i	0.151001i	0.997146	+	0.0755005i	$0.0240554\pi$
$307$	2.64575i	0.151001i	−0.997146	+	0.0755005i	$0.975945\pi$
$308$	0	0
$309$	− 21.1660i	− 1.20409i
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.00000	−0.390702
$322$	0	0
$323$	7.93725i	0.441641i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−28.0000	−1.54840
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	− 2.64575i	− 0.145424i	−0.997353	−	0.0727118i	$-0.976835\pi$
$331$	− 2.64575i	− 0.145424i	0.997353	−	0.0727118i	$-0.0231653\pi$
$332$	0	0
$333$	42.3320i	2.31978i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.0000	−0.817102	−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000	−0.817102	−0.408551	+	0.912735i	$0.633966\pi$
$338$	0	0
$339$	− 39.6863i	− 2.15546i
$340$	0	0
$341$	10.5830i	0.573102i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.64575i	0.142031i	0.997475	+	0.0710157i	$0.0226240\pi$
$347$	2.64575i	0.142031i	−0.997475	+	0.0710157i	$0.977376\pi$
$348$	0	0
$349$	− 10.5830i	− 0.566495i	−0.959047	−	0.283248i	$-0.908588\pi$
$349$	− 10.5830i	− 0.566495i	0.959047	−	0.283248i	$-0.0914118\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	31.7490i	1.68034i
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	12.0000	0.631579
$362$	0	0
$363$	10.5830i	0.555464i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	20.0000	1.04116
$370$	0	0
$371$	− 42.3320i	− 2.19777i
$372$	0	0
$373$	10.5830i	0.547967i	0.961734	+	0.273984i	$0.0883414\pi$
$373$	10.5830i	0.547967i	−0.961734	+	0.273984i	$0.911659\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	7.93725i	0.407709i	0.979001	+	0.203855i	$0.0653470\pi$
$379$	7.93725i	0.407709i	−0.979001	+	0.203855i	$0.934653\pi$
$380$	0	0
$381$	− 31.7490i	− 1.62655i
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 21.1660i	− 1.07593i
$388$	0	0
$389$	10.5830i	0.536580i	0.963338	+	0.268290i	$0.0864585\pi$
$389$	10.5830i	0.536580i	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−42.0000	−2.11862
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 21.1660i	− 1.06229i	−0.847280	−	0.531146i	$-0.821762\pi$
$397$	− 21.1660i	− 1.06229i	0.847280	−	0.531146i	$-0.178238\pi$
$398$	0	0
$399$	−28.0000	−1.40175
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.0000	−1.38791
$408$	0	0
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	0	0
$411$	− 50.2693i	− 2.47960i
$412$	0	0
$413$	21.1660i	1.04151i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−49.0000	−2.39954
$418$	0	0
$419$	18.5203i	0.904774i	0.891822	+	0.452387i	$0.149427\pi$
$419$	18.5203i	0.904774i	−0.891822	+	0.452387i	$0.850573\pi$
$420$	0	0
$421$	21.1660i	1.03157i	0.856719	+	0.515784i	$0.172499\pi$
$421$	21.1660i	1.03157i	−0.856719	+	0.515784i	$0.827501\pi$
$422$	0	0
$423$	−32.0000	−1.55589
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 42.3320i	− 2.04859i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	37.0000	1.77811	0.889053	−	0.457804i	$-0.151364\pi$
$433$	37.0000	1.77811	0.889053	+	0.457804i	$0.151364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5830i	0.506254i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	− 29.1033i	− 1.38274i	−0.722502	−	0.691369i	$-0.757008\pi$
$443$	− 29.1033i	− 1.38274i	0.722502	−	0.691369i	$-0.242992\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	13.2288i	0.622918i
$452$	0	0
$453$	− 10.5830i	− 0.497233i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.0000	−1.26301	−0.631503	−	0.775373i	$-0.717562\pi$
$457$	−27.0000	−1.26301	−0.631503	+	0.775373i	$0.717562\pi$
$458$	0	0
$459$	7.93725i	0.370479i
$460$	0	0
$461$	− 42.3320i	− 1.97160i	−0.167927	−	0.985799i	$-0.553707\pi$
$461$	− 42.3320i	− 1.97160i	0.167927	−	0.985799i	$-0.446293\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4575i	1.22431i	0.790739	+	0.612154i	$0.209697\pi$
$467$	26.4575i	1.22431i	−0.790739	+	0.612154i	$0.790303\pi$
$468$	0	0
$469$	31.7490i	1.46603i
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	14.0000	0.643721
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 42.3320i	− 1.93825i
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	42.3320i	1.92617i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−35.0000	−1.58275
$490$	0	0
$491$	− 5.29150i	− 0.238802i	−0.992846	−	0.119401i	$-0.961903\pi$
$491$	− 5.29150i	− 0.238802i	0.992846	−	0.119401i	$-0.0380974\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	− 26.4575i	− 1.18440i	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	− 26.4575i	− 1.18440i	0.805791	−	0.592200i	$-0.201741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	34.3948i	1.52753i
$508$	0	0
$509$	− 31.7490i	− 1.40725i	−0.710571	−	0.703625i	$-0.751563\pi$
$509$	− 31.7490i	− 1.40725i	0.710571	−	0.703625i	$-0.248437\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 21.1660i	− 0.930880i
$518$	0	0
$519$	56.0000	2.45813
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	2.64575i	0.115691i	0.998326	+	0.0578453i	$0.0184230\pi$
$523$	2.64575i	0.115691i	−0.998326	+	0.0578453i	$0.981577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	21.1660i	0.918527i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	63.0000	2.71865
$538$	0	0
$539$	− 23.8118i	− 1.02565i
$540$	0	0
$541$	− 21.1660i	− 0.909998i	−0.890492	−	0.454999i	$-0.849640\pi$
$541$	− 21.1660i	− 0.909998i	0.890492	−	0.454999i	$-0.150360\pi$
$542$	0	0
$543$	−28.0000	−1.20160
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.5203i	− 0.791869i	−0.918279	−	0.395935i	$-0.870421\pi$
$547$	− 18.5203i	− 0.791869i	0.918279	−	0.395935i	$-0.129579\pi$
$548$	0	0
$549$	− 42.3320i	− 1.80669i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−21.0000	−0.886621
$562$	0	0
$563$	− 15.8745i	− 0.669031i	−0.942390	−	0.334515i	$-0.891427\pi$
$563$	− 15.8745i	− 0.669031i	0.942390	−	0.334515i	$-0.108573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.0000	0.839921
$568$	0	0
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	0	0
$571$	37.0405i	1.55010i	0.631901	+	0.775049i	$0.282275\pi$
$571$	37.0405i	1.55010i	−0.631901	+	0.775049i	$0.717725\pi$
$572$	0	0
$573$	− 10.5830i	− 0.442111i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	13.2288i	0.549768i
$580$	0	0
$581$	31.7490i	1.31717i
$582$	0	0
$583$	28.0000	1.15964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.93725i	− 0.327606i	−0.986493	−	0.163803i	$-0.947624\pi$
$587$	− 7.93725i	− 0.327606i	0.986493	−	0.163803i	$-0.0523761\pi$
$588$	0	0
$589$	10.5830i	0.436065i
$590$	0	0
$591$	−28.0000	−1.15177
$592$	0	0
$593$	41.0000	1.68367	0.841834	−	0.539736i	$-0.181476\pi$
$593$	41.0000	1.68367	0.841834	+	0.539736i	$0.181476\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	63.4980i	2.59880i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	7.00000	0.285536	0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000	0.285536	0.142768	+	0.989756i	$0.454400\pi$
$602$	0	0
$603$	31.7490i	1.29292i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 10.5830i	− 0.427444i	−0.976895	−	0.213722i	$-0.931441\pi$
$613$	− 10.5830i	− 0.427444i	0.976895	−	0.213722i	$-0.0685586\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	− 5.29150i	− 0.212683i	−0.994330	−	0.106342i	$-0.966086\pi$
$619$	− 5.29150i	− 0.212683i	0.994330	−	0.106342i	$-0.0339137\pi$
$620$	0	0
$621$	10.5830i	0.424681i
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 18.5203i	− 0.739628i
$628$	0	0
$629$	31.7490i	1.26592i
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	−21.0000	−0.834675
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	32.0000	1.26590
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 15.8745i	− 0.626029i	−0.949748	−	0.313015i	$-0.898661\pi$
$643$	− 15.8745i	− 0.626029i	0.949748	−	0.313015i	$-0.101339\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−14.0000	−0.549548
$650$	0	0
$651$	42.3320i	1.65912i
$652$	0	0
$653$	− 31.7490i	− 1.24243i	−0.783638	−	0.621217i	$-0.786638\pi$
$653$	− 31.7490i	− 1.24243i	0.783638	−	0.621217i	$-0.213362\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	7.93725i	0.309192i	0.987978	+	0.154596i	$0.0494075\pi$
$659$	7.93725i	0.309192i	−0.987978	+	0.154596i	$0.950592\pi$
$660$	0	0
$661$	21.1660i	0.823262i	0.911351	+	0.411631i	$0.135041\pi$
$661$	21.1660i	0.823262i	−0.911351	+	0.411631i	$0.864959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	42.3320i	1.63665i
$670$	0	0
$671$	28.0000	1.08093
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 42.3320i	− 1.62695i	−0.581599	−	0.813476i	$-0.697573\pi$
$677$	− 42.3320i	− 1.62695i	0.581599	−	0.813476i	$-0.302427\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	42.0000	1.60944
$682$	0	0
$683$	23.8118i	0.911132i	0.890202	+	0.455566i	$0.150563\pi$
$683$	23.8118i	0.911132i	−0.890202	+	0.455566i	$0.849437\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−56.0000	−2.13653
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 23.8118i	− 0.905842i	−0.891551	−	0.452921i	$-0.850382\pi$
$691$	− 23.8118i	− 0.905842i	0.891551	−	0.452921i	$-0.149618\pi$
$692$	0	0
$693$	− 42.3320i	− 1.60806i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.0000	0.568166
$698$	0	0
$699$	15.8745i	0.600429i
$700$	0	0
$701$	21.1660i	0.799429i	0.916640	+	0.399715i	$0.130891\pi$
$701$	21.1660i	0.799429i	−0.916640	+	0.399715i	$0.869109\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	42.3320i	1.59206i
$708$	0	0
$709$	21.1660i	0.794906i	0.917622	+	0.397453i	$0.130106\pi$
$709$	21.1660i	0.794906i	−0.917622	+	0.397453i	$0.869894\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.1660i	0.790459i
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	− 55.5608i	− 2.06633i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	41.0000	1.51852
$730$	0	0
$731$	− 15.8745i	− 0.587140i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.0000	−0.773545
$738$	0	0
$739$	15.8745i	0.583953i	0.956425	+	0.291977i	$0.0943129\pi$
$739$	15.8745i	0.583953i	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	31.7490i	1.16164i
$748$	0	0
$749$	− 10.5830i	− 0.386695i
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.1660i	0.769292i	0.923064	+	0.384646i	$0.125676\pi$
$757$	21.1660i	0.769292i	−0.923064	+	0.384646i	$0.874324\pi$
$758$	0	0
$759$	−28.0000	−1.01634
$760$	0	0
$761$	−29.0000	−1.05125	−0.525625	−	0.850717i	$-0.676168\pi$
$761$	−29.0000	−1.05125	−0.525625	+	0.850717i	$0.676168\pi$
$762$	0	0
$763$	− 42.3320i	− 1.53252i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.0000	−0.757279	−0.378640	−	0.925544i	$-0.623608\pi$
$769$	−21.0000	−0.757279	−0.378640	+	0.925544i	$0.623608\pi$
$770$	0	0
$771$	37.0405i	1.33398i
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−112.000	−4.01798
$778$	0	0
$779$	13.2288i	0.473969i
$780$	0	0
$781$	21.1660i	0.757379i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.4575i	0.943108i	0.881837	+	0.471554i	$0.156307\pi$
$787$	26.4575i	0.943108i	−0.881837	+	0.471554i	$0.843693\pi$
$788$	0	0
$789$	− 31.7490i	− 1.13029i
$790$	0	0
$791$	60.0000	2.13335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.7490i	1.12461i	0.826931	+	0.562304i	$0.190085\pi$
$797$	31.7490i	1.12461i	−0.826931	+	0.562304i	$0.809915\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	18.5203i	0.653566i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	56.0000	1.97129
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	− 5.29150i	− 0.185810i	−0.995675	−	0.0929049i	$-0.970385\pi$
$811$	− 5.29150i	− 0.185810i	0.995675	−	0.0929049i	$-0.0296153\pi$
$812$	0	0
$813$	52.9150i	1.85581i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.0000	0.489798
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.5830i	− 0.369349i	−0.982800	−	0.184675i	$-0.940877\pi$
$821$	− 10.5830i	− 0.369349i	0.982800	−	0.184675i	$-0.0591232\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.2288i	0.460009i	0.973190	+	0.230004i	$0.0738741\pi$
$827$	13.2288i	0.460009i	−0.973190	+	0.230004i	$0.926126\pi$
$828$	0	0
$829$	− 21.1660i	− 0.735126i	−0.929999	−	0.367563i	$-0.880192\pi$
$829$	− 21.1660i	− 0.735126i	0.929999	−	0.367563i	$-0.119808\pi$
$830$	0	0
$831$	56.0000	1.94262
$832$	0	0
$833$	−27.0000	−0.935495
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.5830i	0.365802i
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	− 58.2065i	− 2.00474i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.0000	−0.549767
$848$	0	0
$849$	−35.0000	−1.20120
$850$	0	0
$851$	42.3320i	1.45112i
$852$	0	0
$853$	52.9150i	1.81178i	0.423517	+	0.905888i	$0.360795\pi$
$853$	52.9150i	1.81178i	−0.423517	+	0.905888i	$0.639205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	− 2.64575i	− 0.0902719i	−0.998981	−	0.0451359i	$-0.985628\pi$
$859$	− 2.64575i	− 0.0902719i	0.998981	−	0.0451359i	$-0.0143721\pi$
$860$	0	0
$861$	52.9150i	1.80334i
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 21.1660i	− 0.718835i
$868$	0	0
$869$	10.5830i	0.359004i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.3320i	1.42945i	0.699405	+	0.714725i	$0.253448\pi$
$877$	42.3320i	1.42945i	−0.699405	+	0.714725i	$0.746552\pi$
$878$	0	0
$879$	28.0000	0.944417
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	44.9778i	1.51362i	0.653633	+	0.756811i	$0.273244\pi$
$883$	44.9778i	1.51362i	−0.653633	+	0.756811i	$0.726756\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	13.2288i	0.443180i
$892$	0	0
$893$	− 21.1660i	− 0.708294i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 31.7490i	− 1.05771i
$902$	0	0
$903$	56.0000	1.86356
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.29150i	0.175701i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$	5.29150i	0.175701i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$	0	0
$909$	42.3320i	1.40406i
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−21.0000	−0.694999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 63.4980i	− 2.09689i
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.0000	1.05102
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	− 23.8118i	− 0.780399i
$932$	0	0
$933$	10.5830i	0.346472i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	15.8745i	0.518045i
$940$	0	0
$941$	42.3320i	1.37998i	0.723817	+	0.689992i	$0.242386\pi$
$941$	42.3320i	1.37998i	−0.723817	+	0.689992i	$0.757614\pi$
$942$	0	0
$943$	20.0000	0.651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.8745i	− 0.515852i	−0.966165	−	0.257926i	$-0.916961\pi$
$947$	− 15.8745i	− 0.515852i	0.966165	−	0.257926i	$-0.0830391\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.00000	0.161966	0.0809829	−	0.996715i	$-0.474194\pi$
$953$	5.00000	0.161966	0.0809829	+	0.996715i	$0.474194\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	76.0000	2.45417
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 10.5830i	− 0.341033i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−21.0000	−0.674617
$970$	0	0
$971$	− 23.8118i	− 0.764156i	−0.924130	−	0.382078i	$-0.875209\pi$
$971$	− 23.8118i	− 0.764156i	0.924130	−	0.382078i	$-0.124791\pi$
$972$	0	0
$973$	− 74.0810i	− 2.37493i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	0	0
$979$	2.64575i	0.0845586i
$980$	0	0
$981$	− 42.3320i	− 1.35156i
$982$	0	0
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 84.6640i	− 2.69489i
$988$	0	0
$989$	− 21.1660i	− 0.673040i
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.3320i	1.34067i	0.742059	+	0.670334i	$0.233849\pi$
$997$	42.3320i	1.34067i	−0.742059	+	0.670334i	$0.766151\pi$
$998$	0	0
$999$	−28.0000	−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.d.a.401.2		2
3.2	odd	2		7200.2.k.b.3601.2		2
4.3	odd	2		200.2.d.c.101.1	yes	2
5.2	odd	4		800.2.f.d.49.3		4
5.3	odd	4		800.2.f.d.49.2		4
5.4	even	2		800.2.d.d.401.1		2
8.3	odd	2		200.2.d.c.101.2	yes	2
8.5	even	2	inner	800.2.d.a.401.1		2
12.11	even	2		1800.2.k.d.901.2		2
15.2	even	4		7200.2.d.m.2449.2		4
15.8	even	4		7200.2.d.m.2449.4		4
15.14	odd	2		7200.2.k.i.3601.2		2
16.3	odd	4		6400.2.a.bg.1.2		2
16.5	even	4		6400.2.a.cb.1.2		2
16.11	odd	4		6400.2.a.bg.1.1		2
16.13	even	4		6400.2.a.cb.1.1		2
20.3	even	4		200.2.f.d.149.1		4
20.7	even	4		200.2.f.d.149.4		4
20.19	odd	2		200.2.d.b.101.2	yes	2
24.5	odd	2		7200.2.k.b.3601.1		2
24.11	even	2		1800.2.k.d.901.1		2
40.3	even	4		200.2.f.d.149.3		4
40.13	odd	4		800.2.f.d.49.4		4
40.19	odd	2		200.2.d.b.101.1	✓	2
40.27	even	4		200.2.f.d.149.2		4
40.29	even	2		800.2.d.d.401.2		2
40.37	odd	4		800.2.f.d.49.1		4
60.23	odd	4		1800.2.d.m.1549.4		4
60.47	odd	4		1800.2.d.m.1549.1		4
60.59	even	2		1800.2.k.f.901.1		2
80.19	odd	4		6400.2.a.cc.1.1		2
80.29	even	4		6400.2.a.bh.1.2		2
80.59	odd	4		6400.2.a.cc.1.2		2
80.69	even	4		6400.2.a.bh.1.1		2
120.29	odd	2		7200.2.k.i.3601.1		2
120.53	even	4		7200.2.d.m.2449.3		4
120.59	even	2		1800.2.k.f.901.2		2
120.77	even	4		7200.2.d.m.2449.1		4
120.83	odd	4		1800.2.d.m.1549.2		4
120.107	odd	4		1800.2.d.m.1549.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.d.b.101.1	✓	2	40.19	odd	2
200.2.d.b.101.2	yes	2	20.19	odd	2
200.2.d.c.101.1	yes	2	4.3	odd	2
200.2.d.c.101.2	yes	2	8.3	odd	2
200.2.f.d.149.1		4	20.3	even	4
200.2.f.d.149.2		4	40.27	even	4
200.2.f.d.149.3		4	40.3	even	4
200.2.f.d.149.4		4	20.7	even	4
800.2.d.a.401.1		2	8.5	even	2	inner
800.2.d.a.401.2		2	1.1	even	1	trivial
800.2.d.d.401.1		2	5.4	even	2
800.2.d.d.401.2		2	40.29	even	2
800.2.f.d.49.1		4	40.37	odd	4
800.2.f.d.49.2		4	5.3	odd	4
800.2.f.d.49.3		4	5.2	odd	4
800.2.f.d.49.4		4	40.13	odd	4
1800.2.d.m.1549.1		4	60.47	odd	4
1800.2.d.m.1549.2		4	120.83	odd	4
1800.2.d.m.1549.3		4	120.107	odd	4
1800.2.d.m.1549.4		4	60.23	odd	4
1800.2.k.d.901.1		2	24.11	even	2
1800.2.k.d.901.2		2	12.11	even	2
1800.2.k.f.901.1		2	60.59	even	2
1800.2.k.f.901.2		2	120.59	even	2
6400.2.a.bg.1.1		2	16.11	odd	4
6400.2.a.bg.1.2		2	16.3	odd	4
6400.2.a.bh.1.1		2	80.69	even	4
6400.2.a.bh.1.2		2	80.29	even	4
6400.2.a.cb.1.1		2	16.13	even	4
6400.2.a.cb.1.2		2	16.5	even	4
6400.2.a.cc.1.1		2	80.19	odd	4
6400.2.a.cc.1.2		2	80.59	odd	4
7200.2.d.m.2449.1		4	120.77	even	4
7200.2.d.m.2449.2		4	15.2	even	4
7200.2.d.m.2449.3		4	120.53	even	4
7200.2.d.m.2449.4		4	15.8	even	4
7200.2.k.b.3601.1		2	24.5	odd	2
7200.2.k.b.3601.2		2	3.2	odd	2
7200.2.k.i.3601.1		2	120.29	odd	2
7200.2.k.i.3601.2		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575i q^{3} -4.00000 q^{7} -4.00000 q^{9} +O(q^{10})\)	\(q+2.64575i q^{3} -4.00000 q^{7} -4.00000 q^{9} -2.64575i q^{11} -3.00000 q^{17} -2.64575i q^{19} -10.5830i q^{21} -4.00000 q^{23} -2.64575i q^{27} -4.00000 q^{31} +7.00000 q^{33} -10.5830i q^{37} -5.00000 q^{41} +5.29150i q^{43} +8.00000 q^{47} +9.00000 q^{49} -7.93725i q^{51} +10.5830i q^{53} +7.00000 q^{57} -5.29150i q^{59} +10.5830i q^{61} +16.0000 q^{63} -7.93725i q^{67} -10.5830i q^{69} -8.00000 q^{71} -7.00000 q^{73} +10.5830i q^{77} -4.00000 q^{79} -5.00000 q^{81} -7.93725i q^{83} -1.00000 q^{89} -10.5830i q^{93} -2.00000 q^{97} +10.5830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 - 8 * q^9	\( 2 q - 8 q^{7} - 8 q^{9} - 6 q^{17} - 8 q^{23} - 8 q^{31} + 14 q^{33} - 10 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{57} + 32 q^{63} - 16 q^{71} - 14 q^{73} - 8 q^{79} - 10 q^{81} - 2 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 8 * q^9 - 6 * q^17 - 8 * q^23 - 8 * q^31 + 14 * q^33 - 10 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^57 + 32 * q^63 - 16 * q^71 - 14 * q^73 - 8 * q^79 - 10 * q^81 - 2 * q^89 - 4 * q^97

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