LMFDB - Embedded newform 800.2.f.d.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(49,800)]

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, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.2.f.d.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{3} +4.00000i q^{7} +4.00000 q^{9} +2.64575i q^{11} +3.00000i q^{17} -2.64575i q^{19} +10.5830i q^{21} -4.00000i q^{23} +2.64575 q^{27} -4.00000 q^{31} +7.00000i q^{33} +10.5830 q^{37} -5.00000 q^{41} +5.29150 q^{43} -8.00000i q^{47} -9.00000 q^{49} +7.93725i q^{51} +10.5830 q^{53} -7.00000i q^{57} -5.29150i q^{59} -10.5830i q^{61} +16.0000i q^{63} +7.93725 q^{67} -10.5830i q^{69} -8.00000 q^{71} -7.00000i q^{73} -10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} +1.00000 q^{89} -10.5830 q^{93} +2.00000i q^{97} +10.5830i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{9} - 16 q^{31} - 20 q^{41} - 36 q^{49} - 32 q^{71} + 16 q^{79} - 20 q^{81} + 4 q^{89}+O(q^{100}) $ 4 * q + 16 * q^9 - 16 * q^31 - 20 * q^41 - 36 * q^49 - 32 * q^71 + 16 * q^79 - 20 * q^81 + 4 * q^89

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
$3$	2.64575	1.52753	0.763763	−	0.645497i	$-0.223350\pi$
$3$	2.64575	1.52753	0.763763	+	0.645497i	$0.223350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	2.64575i	0.797724i	0.917011	+	0.398862i	$0.130595\pi$
$11$	2.64575i	0.797724i	−0.917011	+	0.398862i	$0.869405\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	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
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.64575	0.509175
$28$	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
$33$	7.00000i	1.21854i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.5830	1.73984	0.869918	−	0.493197i	$-0.164172\pi$
$37$	10.5830	1.73984	0.869918	+	0.493197i	$0.164172\pi$
$38$	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
$43$	5.29150	0.806947	0.403473	−	0.914991i	$-0.367803\pi$
$43$	5.29150	0.806947	0.403473	+	0.914991i	$0.367803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	7.93725i	1.11144i
$52$	0	0
$53$	10.5830	1.45369	0.726844	−	0.686803i	$-0.240986\pi$
$53$	10.5830	1.45369	0.726844	+	0.686803i	$0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 7.00000i	− 0.927173i
$58$	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	$-0.763060\pi$
$61$	− 10.5830i	− 1.35501i	0.735516	−	0.677507i	$-0.236940\pi$
$62$	0	0
$63$	16.0000i	2.01581i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.93725	0.969690	0.484845	−	0.874600i	$-0.338876\pi$
$67$	7.93725	0.969690	0.484845	+	0.874600i	$0.338876\pi$
$68$	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
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.5830	−1.20605
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−7.93725	−0.871227	−0.435613	−	0.900134i	$-0.643469\pi$
$83$	−7.93725	−0.871227	−0.435613	+	0.900134i	$0.643469\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.5830	−1.09741
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	10.5830i	1.06363i
$100$	0	0
$101$	10.5830i	1.05305i	0.850160	+	0.526524i	$0.176505\pi$
$101$	10.5830i	1.05305i	−0.850160	+	0.526524i	$0.823495\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.64575	−0.255774	−0.127887	−	0.991789i	$-0.540820\pi$
$107$	−2.64575	−0.255774	−0.127887	+	0.991789i	$0.540820\pi$
$108$	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
$113$	− 15.0000i	− 1.41108i	−0.708669	−	0.705541i	$-0.750704\pi$
$113$	− 15.0000i	− 1.41108i	0.708669	−	0.705541i	$-0.249296\pi$
$114$	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
$123$	−13.2288	−1.19280
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	0	0
$129$	14.0000	1.23263
$130$	0	0
$131$	− 15.8745i	− 1.38696i	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	− 15.8745i	− 1.38696i	0.720475	−	0.693481i	$-0.243924\pi$
$132$	0	0
$133$	10.5830	0.917663
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.0000i	1.62328i	0.584158	+	0.811640i	$0.301425\pi$
$137$	19.0000i	1.62328i	−0.584158	+	0.811640i	$0.698575\pi$
$138$	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
$147$	−23.8118	−1.96396
$148$	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
$153$	12.0000i	0.970143i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.5830	−0.844616	−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830	−0.844616	−0.422308	+	0.906452i	$0.638780\pi$
$158$	0	0
$159$	28.0000	2.22054
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	13.2288	1.03616	0.518078	−	0.855333i	$-0.326648\pi$
$163$	13.2288	1.03616	0.518078	+	0.855333i	$0.326648\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	− 10.5830i	− 0.809303i
$172$	0	0
$173$	−21.1660	−1.60922	−0.804611	−	0.593802i	$-0.797626\pi$
$173$	−21.1660	−1.60922	−0.804611	+	0.593802i	$0.797626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 14.0000i	− 1.05230i
$178$	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	$-0.871328\pi$
$181$	− 10.5830i	− 0.786629i	0.919404	−	0.393314i	$-0.128672\pi$
$182$	0	0
$183$	− 28.0000i	− 2.06982i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.93725	−0.580429
$188$	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
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.5830	−0.754008	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	−10.5830	−0.754008	−0.377004	+	0.926212i	$0.623046\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	21.0000	1.48123
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 16.0000i	− 1.11208i
$208$	0	0
$209$	7.00000	0.484200
$210$	0	0
$211$	− 7.93725i	− 0.546423i	−0.961954	−	0.273212i	$-0.911914\pi$
$211$	− 7.93725i	− 0.546423i	0.961954	−	0.273212i	$-0.0880859\pi$
$212$	0	0
$213$	−21.1660	−1.45027
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 16.0000i	− 1.08615i
$218$	0	0
$219$	− 18.5203i	− 1.25148i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.8745	1.05363	0.526814	−	0.849981i	$-0.323386\pi$
$227$	15.8745	1.05363	0.526814	+	0.849981i	$0.323386\pi$
$228$	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
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.5830	0.687440
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	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
$243$	−21.1660	−1.35780
$244$	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	$-0.919406\pi$
$251$	− 7.93725i	− 0.500995i	0.968117	−	0.250498i	$-0.0805942\pi$
$252$	0	0
$253$	10.5830	0.665348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	42.3320i	2.63038i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.64575	0.161917
$268$	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
$277$	21.1660	1.27174	0.635871	−	0.771795i	$-0.280641\pi$
$277$	21.1660	1.27174	0.635871	+	0.771795i	$0.280641\pi$
$278$	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
$283$	13.2288	0.786368	0.393184	−	0.919460i	$-0.371374\pi$
$283$	13.2288	0.786368	0.393184	+	0.919460i	$0.371374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20.0000i	− 1.18056i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	5.29150i	0.310193i
$292$	0	0
$293$	−10.5830	−0.618266	−0.309133	−	0.951019i	$-0.600039\pi$
$293$	−10.5830	−0.618266	−0.309133	+	0.951019i	$0.600039\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.00000i	0.406181i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	21.1660i	1.21999i
$302$	0	0
$303$	28.0000i	1.60856i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.64575	−0.151001	−0.0755005	−	0.997146i	$-0.524055\pi$
$307$	−2.64575	−0.151001	−0.0755005	+	0.997146i	$0.524055\pi$
$308$	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
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.00000	−0.390702
$322$	0	0
$323$	7.93725	0.441641
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.0000i	1.54840i
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	2.64575i	0.145424i	0.997353	+	0.0727118i	$0.0231653\pi$
$331$	2.64575i	0.145424i	−0.997353	+	0.0727118i	$0.976835\pi$
$332$	0	0
$333$	42.3320	2.31978
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.0000i	0.817102i	0.912735	+	0.408551i	$0.133966\pi$
$337$	15.0000i	0.817102i	−0.912735	+	0.408551i	$0.866034\pi$
$338$	0	0
$339$	− 39.6863i	− 2.15546i
$340$	0	0
$341$	− 10.5830i	− 0.573102i
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.64575	−0.142031	−0.0710157	−	0.997475i	$-0.522624\pi$
$347$	−2.64575	−0.142031	−0.0710157	+	0.997475i	$0.522624\pi$
$348$	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
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−31.7490	−1.68034
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	12.0000	0.631579
$362$	0	0
$363$	10.5830	0.555464
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−20.0000	−1.04116
$370$	0	0
$371$	42.3320i	2.19777i
$372$	0	0
$373$	10.5830	0.547967	0.273984	−	0.961734i	$-0.411659\pi$
$373$	10.5830	0.547967	0.273984	+	0.961734i	$0.411659\pi$
$374$	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
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.1660	1.07593
$388$	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
$393$	− 42.0000i	− 2.11862i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.1660	1.06229	0.531146	−	0.847280i	$-0.321762\pi$
$397$	21.1660	1.06229	0.531146	+	0.847280i	$0.321762\pi$
$398$	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
$407$	28.0000i	1.38791i
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	0	0
$411$	50.2693i	2.47960i
$412$	0	0
$413$	21.1660	1.04151
$414$	0	0
$415$	0	0
$416$	0	0
$417$	49.0000i	2.39954i
$418$	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	$-0.827501\pi$
$421$	− 21.1660i	− 1.03157i	0.856719	−	0.515784i	$-0.172499\pi$
$422$	0	0
$423$	− 32.0000i	− 1.55589i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.3320	2.04859
$428$	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
$433$	37.0000i	1.77811i	0.457804	+	0.889053i	$0.348636\pi$
$433$	37.0000i	1.77811i	−0.457804	+	0.889053i	$0.651364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.5830	−0.506254
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	−29.1033	−1.38274	−0.691369	−	0.722502i	$-0.742992\pi$
$443$	−29.1033	−1.38274	−0.691369	+	0.722502i	$0.742992\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	− 13.2288i	− 0.622918i
$452$	0	0
$453$	−10.5830	−0.497233
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.0000i	1.26301i	0.775373	+	0.631503i	$0.217562\pi$
$457$	27.0000i	1.26301i	−0.775373	+	0.631503i	$0.782438\pi$
$458$	0	0
$459$	7.93725i	0.370479i
$460$	0	0
$461$	42.3320i	1.97160i	0.167927	+	0.985799i	$0.446293\pi$
$461$	42.3320i	1.97160i	−0.167927	+	0.985799i	$0.553707\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.4575	−1.22431	−0.612154	−	0.790739i	$-0.709697\pi$
$467$	−26.4575	−1.22431	−0.612154	+	0.790739i	$0.709697\pi$
$468$	0	0
$469$	31.7490i	1.46603i
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	14.0000i	0.643721i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	42.3320	1.93825
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	42.3320	1.92617
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	0	0
$489$	35.0000	1.58275
$490$	0	0
$491$	5.29150i	0.238802i	0.992846	+	0.119401i	$0.0380974\pi$
$491$	5.29150i	0.238802i	−0.992846	+	0.119401i	$0.961903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.0000i	− 1.43540i
$498$	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	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−34.3948	−1.52753
$508$	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
$513$	− 7.00000i	− 0.309058i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21.1660	0.930880
$518$	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
$523$	2.64575	0.115691	0.0578453	−	0.998326i	$-0.481577\pi$
$523$	2.64575	0.115691	0.0578453	+	0.998326i	$0.481577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	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
$537$	− 63.0000i	− 2.71865i
$538$	0	0
$539$	− 23.8118i	− 1.02565i
$540$	0	0
$541$	21.1660i	0.909998i	0.890492	+	0.454999i	$0.150360\pi$
$541$	21.1660i	0.909998i	−0.890492	+	0.454999i	$0.849640\pi$
$542$	0	0
$543$	− 28.0000i	− 1.20160i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.5203	0.791869	0.395935	−	0.918279i	$-0.370421\pi$
$547$	18.5203	0.791869	0.395935	+	0.918279i	$0.370421\pi$
$548$	0	0
$549$	− 42.3320i	− 1.80669i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000i	0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−21.0000	−0.886621
$562$	0	0
$563$	−15.8745	−0.669031	−0.334515	−	0.942390i	$-0.608573\pi$
$563$	−15.8745	−0.669031	−0.334515	+	0.942390i	$0.608573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 20.0000i	− 0.839921i
$568$	0	0
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	− 37.0405i	− 1.55010i	−0.631901	−	0.775049i	$-0.717725\pi$
$571$	− 37.0405i	− 1.55010i	0.631901	−	0.775049i	$-0.282275\pi$
$572$	0	0
$573$	−10.5830	−0.442111
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	0	0
$579$	13.2288i	0.549768i
$580$	0	0
$581$	− 31.7490i	− 1.31717i
$582$	0	0
$583$	28.0000i	1.15964i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.93725	0.327606	0.163803	−	0.986493i	$-0.447624\pi$
$587$	7.93725	0.327606	0.163803	+	0.986493i	$0.447624\pi$
$588$	0	0
$589$	10.5830i	0.436065i
$590$	0	0
$591$	−28.0000	−1.15177
$592$	0	0
$593$	41.0000i	1.68367i	0.539736	+	0.841834i	$0.318524\pi$
$593$	41.0000i	1.68367i	−0.539736	+	0.841834i	$0.681476\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−63.4980	−2.59880
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	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
$603$	31.7490	1.29292
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.5830	−0.427444	−0.213722	−	0.976895i	$-0.568559\pi$
$613$	−10.5830	−0.427444	−0.213722	+	0.976895i	$0.568559\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	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
$623$	4.00000i	0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.5203	0.739628
$628$	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
$633$	− 21.0000i	− 0.834675i
$634$	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
$643$	−15.8745	−0.626029	−0.313015	−	0.949748i	$-0.601339\pi$
$643$	−15.8745	−0.626029	−0.313015	+	0.949748i	$0.601339\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	− 42.3320i	− 1.65912i
$652$	0	0
$653$	−31.7490	−1.24243	−0.621217	−	0.783638i	$-0.713362\pi$
$653$	−31.7490	−1.24243	−0.621217	+	0.783638i	$0.713362\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 28.0000i	− 1.09238i
$658$	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	$-0.864959\pi$
$661$	− 21.1660i	− 0.823262i	0.911351	−	0.411631i	$-0.135041\pi$
$662$	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
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.3320	1.62695	0.813476	−	0.581599i	$-0.197573\pi$
$677$	42.3320	1.62695	0.813476	+	0.581599i	$0.197573\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	42.0000	1.60944
$682$	0	0
$683$	23.8118	0.911132	0.455566	−	0.890202i	$-0.349437\pi$
$683$	23.8118	0.911132	0.455566	+	0.890202i	$0.349437\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	56.0000i	2.13653i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	23.8118i	0.905842i	0.891551	+	0.452921i	$0.149618\pi$
$691$	23.8118i	0.905842i	−0.891551	+	0.452921i	$0.850382\pi$
$692$	0	0
$693$	−42.3320	−1.60806
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 15.0000i	− 0.568166i
$698$	0	0
$699$	15.8745i	0.600429i
$700$	0	0
$701$	− 21.1660i	− 0.799429i	−0.916640	−	0.399715i	$-0.869109\pi$
$701$	− 21.1660i	− 0.799429i	0.916640	−	0.399715i	$-0.130891\pi$
$702$	0	0
$703$	− 28.0000i	− 1.05604i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−42.3320	−1.59206
$708$	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
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.1660	−0.790459
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	−55.5608	−2.06633
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	15.8745i	0.587140i
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.0000i	0.773545i
$738$	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
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−31.7490	−1.16164
$748$	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
$753$	− 21.0000i	− 0.765283i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21.1660	−0.769292	−0.384646	−	0.923064i	$-0.625676\pi$
$757$	−21.1660	−0.769292	−0.384646	+	0.923064i	$0.625676\pi$
$758$	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
$763$	−42.3320	−1.53252
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	− 37.0405i	− 1.33398i
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	112.000i	4.01798i
$778$	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
$787$	−26.4575	−0.943108	−0.471554	−	0.881837i	$-0.656307\pi$
$787$	−26.4575	−0.943108	−0.471554	+	0.881837i	$0.656307\pi$
$788$	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
$797$	−31.7490	−1.12461	−0.562304	−	0.826931i	$-0.690085\pi$
$797$	−31.7490	−1.12461	−0.562304	+	0.826931i	$0.690085\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	18.5203	0.653566
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 56.0000i	− 1.97129i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	5.29150i	0.185810i	0.995675	+	0.0929049i	$0.0296153\pi$
$811$	5.29150i	0.185810i	−0.995675	+	0.0929049i	$0.970385\pi$
$812$	0	0
$813$	52.9150	1.85581
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 14.0000i	− 0.489798i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.5830i	0.369349i	0.982800	+	0.184675i	$0.0591232\pi$
$821$	10.5830i	0.369349i	−0.982800	+	0.184675i	$0.940877\pi$
$822$	0	0
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.2288	−0.460009	−0.230004	−	0.973190i	$-0.573874\pi$
$827$	−13.2288	−0.460009	−0.230004	+	0.973190i	$0.573874\pi$
$828$	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
$833$	− 27.0000i	− 0.935495i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.5830	−0.365802
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	−58.2065	−2.00474
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.0000i	0.549767i
$848$	0	0
$849$	35.0000	1.20120
$850$	0	0
$851$	− 42.3320i	− 1.45112i
$852$	0	0
$853$	52.9150	1.81178	0.905888	−	0.423517i	$-0.139205\pi$
$853$	52.9150	1.81178	0.905888	+	0.423517i	$0.139205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 21.0000i	− 0.717346i	−0.933463	−	0.358673i	$-0.883229\pi$
$857$	− 21.0000i	− 0.717346i	0.933463	−	0.358673i	$-0.116771\pi$
$858$	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
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.1660	0.718835
$868$	0	0
$869$	10.5830i	0.359004i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.00000i	0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.3320	−1.42945	−0.714725	−	0.699405i	$-0.753448\pi$
$877$	−42.3320	−1.42945	−0.714725	+	0.699405i	$0.753448\pi$
$878$	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
$883$	44.9778	1.51362	0.756811	−	0.653633i	$-0.226756\pi$
$883$	44.9778	1.51362	0.756811	+	0.653633i	$0.226756\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	56.0000i	1.88030i	0.340766	+	0.940148i	$0.389313\pi$
$887$	56.0000i	1.88030i	−0.340766	+	0.940148i	$0.610687\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	− 13.2288i	− 0.443180i
$892$	0	0
$893$	−21.1660	−0.708294
$894$	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
$903$	56.0000i	1.86356i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.29150	−0.175701	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	−5.29150	−0.175701	−0.0878507	+	0.996134i	$0.528000\pi$
$908$	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
$913$	− 21.0000i	− 0.694999i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.4980	2.09689
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 32.0000i	− 1.05102i
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	23.8118i	0.780399i
$932$	0	0
$933$	10.5830	0.346472
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000i	0.228680i	0.993442	+	0.114340i	$0.0364753\pi$
$937$	7.00000i	0.228680i	−0.993442	+	0.114340i	$0.963525\pi$
$938$	0	0
$939$	15.8745i	0.518045i
$940$	0	0
$941$	− 42.3320i	− 1.37998i	−0.723817	−	0.689992i	$-0.757614\pi$
$941$	− 42.3320i	− 1.37998i	0.723817	−	0.689992i	$-0.242386\pi$
$942$	0	0
$943$	20.0000i	0.651290i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.8745	0.515852	0.257926	−	0.966165i	$-0.416961\pi$
$947$	15.8745	0.515852	0.257926	+	0.966165i	$0.416961\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.00000i	0.161966i	0.996715	+	0.0809829i	$0.0258059\pi$
$953$	5.00000i	0.161966i	−0.996715	+	0.0809829i	$0.974194\pi$
$954$	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
$963$	−10.5830	−0.341033
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	23.8118i	0.764156i	0.924130	+	0.382078i	$0.124791\pi$
$971$	23.8118i	0.764156i	−0.924130	+	0.382078i	$0.875209\pi$
$972$	0	0
$973$	−74.0810	−2.37493
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.0000i	− 1.18373i	−0.806035	−	0.591867i	$-0.798391\pi$
$977$	− 37.0000i	− 1.18373i	0.806035	−	0.591867i	$-0.201609\pi$
$978$	0	0
$979$	2.64575i	0.0845586i
$980$	0	0
$981$	42.3320i	1.35156i
$982$	0	0
$983$	28.0000i	0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$	28.0000i	0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	84.6640	2.69489
$988$	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
$993$	7.00000i	0.222138i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42.3320	−1.34067	−0.670334	−	0.742059i	$-0.733849\pi$
$997$	−42.3320	−1.34067	−0.670334	+	0.742059i	$0.733849\pi$
$998$	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.f.d.49.4		4
3.2	odd	2		7200.2.d.m.2449.3		4
4.3	odd	2		200.2.f.d.149.3		4
5.2	odd	4		800.2.d.a.401.1		2
5.3	odd	4		800.2.d.d.401.2		2
5.4	even	2	inner	800.2.f.d.49.1		4
8.3	odd	2		200.2.f.d.149.1		4
8.5	even	2	inner	800.2.f.d.49.2		4
12.11	even	2		1800.2.d.m.1549.2		4
15.2	even	4		7200.2.k.b.3601.1		2
15.8	even	4		7200.2.k.i.3601.1		2
15.14	odd	2		7200.2.d.m.2449.1		4
20.3	even	4		200.2.d.b.101.1	✓	2
20.7	even	4		200.2.d.c.101.2	yes	2
20.19	odd	2		200.2.f.d.149.2		4
24.5	odd	2		7200.2.d.m.2449.4		4
24.11	even	2		1800.2.d.m.1549.4		4
40.3	even	4		200.2.d.b.101.2	yes	2
40.13	odd	4		800.2.d.d.401.1		2
40.19	odd	2		200.2.f.d.149.4		4
40.27	even	4		200.2.d.c.101.1	yes	2
40.29	even	2	inner	800.2.f.d.49.3		4
40.37	odd	4		800.2.d.a.401.2		2
60.23	odd	4		1800.2.k.f.901.2		2
60.47	odd	4		1800.2.k.d.901.1		2
60.59	even	2		1800.2.d.m.1549.3		4
80.3	even	4		6400.2.a.cc.1.2		2
80.13	odd	4		6400.2.a.bh.1.1		2
80.27	even	4		6400.2.a.bg.1.2		2
80.37	odd	4		6400.2.a.cb.1.1		2
80.43	even	4		6400.2.a.cc.1.1		2
80.53	odd	4		6400.2.a.bh.1.2		2
80.67	even	4		6400.2.a.bg.1.1		2
80.77	odd	4		6400.2.a.cb.1.2		2
120.29	odd	2		7200.2.d.m.2449.2		4
120.53	even	4		7200.2.k.i.3601.2		2
120.59	even	2		1800.2.d.m.1549.1		4
120.77	even	4		7200.2.k.b.3601.2		2
120.83	odd	4		1800.2.k.f.901.1		2
120.107	odd	4		1800.2.k.d.901.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.d.b.101.1	✓	2	20.3	even	4
200.2.d.b.101.2	yes	2	40.3	even	4
200.2.d.c.101.1	yes	2	40.27	even	4
200.2.d.c.101.2	yes	2	20.7	even	4
200.2.f.d.149.1		4	8.3	odd	2
200.2.f.d.149.2		4	20.19	odd	2
200.2.f.d.149.3		4	4.3	odd	2
200.2.f.d.149.4		4	40.19	odd	2
800.2.d.a.401.1		2	5.2	odd	4
800.2.d.a.401.2		2	40.37	odd	4
800.2.d.d.401.1		2	40.13	odd	4
800.2.d.d.401.2		2	5.3	odd	4
800.2.f.d.49.1		4	5.4	even	2	inner
800.2.f.d.49.2		4	8.5	even	2	inner
800.2.f.d.49.3		4	40.29	even	2	inner
800.2.f.d.49.4		4	1.1	even	1	trivial
1800.2.d.m.1549.1		4	120.59	even	2
1800.2.d.m.1549.2		4	12.11	even	2
1800.2.d.m.1549.3		4	60.59	even	2
1800.2.d.m.1549.4		4	24.11	even	2
1800.2.k.d.901.1		2	60.47	odd	4
1800.2.k.d.901.2		2	120.107	odd	4
1800.2.k.f.901.1		2	120.83	odd	4
1800.2.k.f.901.2		2	60.23	odd	4
6400.2.a.bg.1.1		2	80.67	even	4
6400.2.a.bg.1.2		2	80.27	even	4
6400.2.a.bh.1.1		2	80.13	odd	4
6400.2.a.bh.1.2		2	80.53	odd	4
6400.2.a.cb.1.1		2	80.37	odd	4
6400.2.a.cb.1.2		2	80.77	odd	4
6400.2.a.cc.1.1		2	80.43	even	4
6400.2.a.cc.1.2		2	80.3	even	4
7200.2.d.m.2449.1		4	15.14	odd	2
7200.2.d.m.2449.2		4	120.29	odd	2
7200.2.d.m.2449.3		4	3.2	odd	2
7200.2.d.m.2449.4		4	24.5	odd	2
7200.2.k.b.3601.1		2	15.2	even	4
7200.2.k.b.3601.2		2	120.77	even	4
7200.2.k.i.3601.1		2	15.8	even	4
7200.2.k.i.3601.2		2	120.53	even	4

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575 q^{3} +4.00000i q^{7} +4.00000 q^{9} +2.64575i q^{11} +3.00000i q^{17} -2.64575i q^{19} +10.5830i q^{21} -4.00000i q^{23} +2.64575 q^{27} -4.00000 q^{31} +7.00000i q^{33} +10.5830 q^{37} -5.00000 q^{41} +5.29150 q^{43} -8.00000i q^{47} -9.00000 q^{49} +7.93725i q^{51} +10.5830 q^{53} -7.00000i q^{57} -5.29150i q^{59} -10.5830i q^{61} +16.0000i q^{63} +7.93725 q^{67} -10.5830i q^{69} -8.00000 q^{71} -7.00000i q^{73} -10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} +1.00000 q^{89} -10.5830 q^{93} +2.00000i q^{97} +10.5830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{9} - 16 q^{31} - 20 q^{41} - 36 q^{49} - 32 q^{71} + 16 q^{79} - 20 q^{81} + 4 q^{89}+O(q^{100}) \) 4 * q + 16 * q^9 - 16 * q^31 - 20 * q^41 - 36 * q^49 - 32 * q^71 + 16 * q^79 - 20 * q^81 + 4 * q^89

Introduction

L-functions

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