LMFDB - Embedded newform 6400.2.a.cs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.a (trivial)

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:	yes
Analytic conductor:	$51.1042572936$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 3200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} +2.82843 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} +2.82843 q^{7} +2.00000 q^{9} +2.23607 q^{11} +6.32456 q^{13} +5.00000 q^{17} +2.23607 q^{19} +6.32456 q^{21} -5.65685 q^{23} -2.23607 q^{27} +6.32456 q^{29} +5.00000 q^{33} -6.32456 q^{37} +14.1421 q^{39} -3.00000 q^{41} +8.94427 q^{43} -2.82843 q^{47} +1.00000 q^{49} +11.1803 q^{51} -12.6491 q^{53} +5.00000 q^{57} +8.94427 q^{59} -6.32456 q^{61} +5.65685 q^{63} -11.1803 q^{67} -12.6491 q^{69} +14.1421 q^{71} +15.0000 q^{73} +6.32456 q^{77} -14.1421 q^{79} -11.0000 q^{81} +6.70820 q^{83} +14.1421 q^{87} -1.00000 q^{89} +17.8885 q^{91} -10.0000 q^{97} +4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^9	$ 4 q + 8 q^{9} + 20 q^{17} + 20 q^{33} - 12 q^{41} + 4 q^{49} + 20 q^{57} + 60 q^{73} - 44 q^{81} - 4 q^{89} - 40 q^{97}+O(q^{100}) $ 4 * q + 8 * q^9 + 20 * q^17 + 20 * q^33 - 12 * q^41 + 4 * q^49 + 20 * q^57 + 60 * q^73 - 44 * q^81 - 4 * q^89 - 40 * q^97

Display 100 coefficients 10 coefficients

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
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	0	0
$13$	6.32456	1.75412	0.877058	−	0.480384i	$-0.159503\pi$
$13$	6.32456	1.75412	0.877058	+	0.480384i	$0.159503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	6.32456	1.38013
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	6.32456	1.17444	0.587220	−	0.809427i	$-0.300222\pi$
$29$	6.32456	1.17444	0.587220	+	0.809427i	$0.300222\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.32456	−1.03975	−0.519875	−	0.854242i	$-0.674022\pi$
$37$	−6.32456	−1.03975	−0.519875	+	0.854242i	$0.674022\pi$
$38$	0	0
$39$	14.1421	2.26455
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.1803	1.56556
$52$	0	0
$53$	−12.6491	−1.73749	−0.868744	−	0.495261i	$-0.835073\pi$
$53$	−12.6491	−1.73749	−0.868744	+	0.495261i	$0.835073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.00000	0.662266
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−6.32456	−0.809776	−0.404888	−	0.914366i	$-0.632690\pi$
$61$	−6.32456	−0.809776	−0.404888	+	0.914366i	$0.632690\pi$
$62$	0	0
$63$	5.65685	0.712697
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.1803	−1.36590	−0.682948	−	0.730467i	$-0.739302\pi$
$67$	−11.1803	−1.36590	−0.682948	+	0.730467i	$0.739302\pi$
$68$	0	0
$69$	−12.6491	−1.52277
$70$	0	0
$71$	14.1421	1.67836	0.839181	−	0.543852i	$-0.183035\pi$
$71$	14.1421	1.67836	0.839181	+	0.543852i	$0.183035\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.32456	0.720750
$78$	0	0
$79$	−14.1421	−1.59111	−0.795557	−	0.605878i	$-0.792822\pi$
$79$	−14.1421	−1.59111	−0.795557	+	0.605878i	$0.792822\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.70820	0.736321	0.368161	−	0.929762i	$-0.379988\pi$
$83$	6.70820	0.736321	0.368161	+	0.929762i	$0.379988\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	14.1421	1.51620
$88$	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
$91$	17.8885	1.87523
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	4.47214	0.449467
$100$	0	0
$101$	−6.32456	−0.629317	−0.314658	−	0.949205i	$-0.601890\pi$
$101$	−6.32456	−0.629317	−0.314658	+	0.949205i	$0.601890\pi$
$102$	0	0
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.6525	−1.51318	−0.756591	−	0.653888i	$-0.773137\pi$
$107$	−15.6525	−1.51318	−0.756591	+	0.653888i	$0.773137\pi$
$108$	0	0
$109$	−6.32456	−0.605783	−0.302891	−	0.953025i	$-0.597952\pi$
$109$	−6.32456	−0.605783	−0.302891	+	0.953025i	$0.597952\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.6491	1.16941
$118$	0	0
$119$	14.1421	1.29641
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	−6.70820	−0.604858
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	0	0
$129$	20.0000	1.76090
$130$	0	0
$131$	8.94427	0.781465	0.390732	−	0.920504i	$-0.372222\pi$
$131$	8.94427	0.781465	0.390732	+	0.920504i	$0.372222\pi$
$132$	0	0
$133$	6.32456	0.548408
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	−15.6525	−1.32763	−0.663813	−	0.747899i	$-0.731063\pi$
$139$	−15.6525	−1.32763	−0.663813	+	0.747899i	$0.731063\pi$
$140$	0	0
$141$	−6.32456	−0.532624
$142$	0	0
$143$	14.1421	1.18262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.23607	0.184428
$148$	0	0
$149$	18.9737	1.55438	0.777192	−	0.629264i	$-0.216644\pi$
$149$	18.9737	1.55438	0.777192	+	0.629264i	$0.216644\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	0	0
$153$	10.0000	0.808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−28.2843	−2.24309
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	6.70820	0.525427	0.262714	−	0.964874i	$-0.415383\pi$
$163$	6.70820	0.525427	0.262714	+	0.964874i	$0.415383\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.82843	−0.218870	−0.109435	−	0.993994i	$-0.534904\pi$
$167$	−2.82843	−0.218870	−0.109435	+	0.993994i	$0.534904\pi$
$168$	0	0
$169$	27.0000	2.07692
$170$	0	0
$171$	4.47214	0.341993
$172$	0	0
$173$	−12.6491	−0.961694	−0.480847	−	0.876804i	$-0.659671\pi$
$173$	−12.6491	−0.961694	−0.480847	+	0.876804i	$0.659671\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.0000	1.50329
$178$	0	0
$179$	2.23607	0.167132	0.0835658	−	0.996502i	$-0.473369\pi$
$179$	2.23607	0.167132	0.0835658	+	0.996502i	$0.473369\pi$
$180$	0	0
$181$	−12.6491	−0.940201	−0.470100	−	0.882613i	$-0.655782\pi$
$181$	−12.6491	−0.940201	−0.470100	+	0.882613i	$0.655782\pi$
$182$	0	0
$183$	−14.1421	−1.04542
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.1803	0.817587
$188$	0	0
$189$	−6.32456	−0.460044
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	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
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−25.0000	−1.76336
$202$	0	0
$203$	17.8885	1.25553
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.3137	−0.786357
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	6.70820	0.461812	0.230906	−	0.972976i	$-0.425831\pi$
$211$	6.70820	0.461812	0.230906	+	0.972976i	$0.425831\pi$
$212$	0	0
$213$	31.6228	2.16676
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	33.5410	2.26649
$220$	0	0
$221$	31.6228	2.12718
$222$	0	0
$223$	22.6274	1.51524	0.757622	−	0.652694i	$-0.226361\pi$
$223$	22.6274	1.51524	0.757622	+	0.652694i	$0.226361\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.94427	−0.593652	−0.296826	−	0.954932i	$-0.595928\pi$
$227$	−8.94427	−0.593652	−0.296826	+	0.954932i	$0.595928\pi$
$228$	0	0
$229$	12.6491	0.835877	0.417938	−	0.908475i	$-0.362753\pi$
$229$	12.6491	0.835877	0.417938	+	0.908475i	$0.362753\pi$
$230$	0	0
$231$	14.1421	0.930484
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−31.6228	−2.05412
$238$	0	0
$239$	−14.1421	−0.914779	−0.457389	−	0.889267i	$-0.651215\pi$
$239$	−14.1421	−0.914779	−0.457389	+	0.889267i	$0.651215\pi$
$240$	0	0
$241$	13.0000	0.837404	0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000	0.837404	0.418702	+	0.908124i	$0.362485\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	14.1421	0.899843
$248$	0	0
$249$	15.0000	0.950586
$250$	0	0
$251$	−11.1803	−0.705697	−0.352848	−	0.935681i	$-0.614787\pi$
$251$	−11.1803	−0.705697	−0.352848	+	0.935681i	$0.614787\pi$
$252$	0	0
$253$	−12.6491	−0.795243
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	−17.8885	−1.11154
$260$	0	0
$261$	12.6491	0.782960
$262$	0	0
$263$	22.6274	1.39527	0.697633	−	0.716455i	$-0.254237\pi$
$263$	22.6274	1.39527	0.697633	+	0.716455i	$0.254237\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.23607	−0.136845
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	40.0000	2.42091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.9737	−1.14002	−0.570009	−	0.821639i	$-0.693060\pi$
$277$	−18.9737	−1.14002	−0.570009	+	0.821639i	$0.693060\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−11.1803	−0.664602	−0.332301	−	0.943173i	$-0.607825\pi$
$283$	−11.1803	−0.664602	−0.332301	+	0.943173i	$0.607825\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−22.3607	−1.31081
$292$	0	0
$293$	6.32456	0.369484	0.184742	−	0.982787i	$-0.440855\pi$
$293$	6.32456	0.369484	0.184742	+	0.982787i	$0.440855\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−35.7771	−2.06904
$300$	0	0
$301$	25.2982	1.45817
$302$	0	0
$303$	−14.1421	−0.812444
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.1246	1.14857	0.574286	−	0.818655i	$-0.305280\pi$
$307$	20.1246	1.14857	0.574286	+	0.818655i	$0.305280\pi$
$308$	0	0
$309$	−18.9737	−1.07937
$310$	0	0
$311$	14.1421	0.801927	0.400963	−	0.916094i	$-0.368675\pi$
$311$	14.1421	0.801927	0.400963	+	0.916094i	$0.368675\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6491	0.710445	0.355222	−	0.934782i	$-0.384405\pi$
$317$	12.6491	0.710445	0.355222	+	0.934782i	$0.384405\pi$
$318$	0	0
$319$	14.1421	0.791808
$320$	0	0
$321$	−35.0000	−1.95351
$322$	0	0
$323$	11.1803	0.622091
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.1421	−0.782062
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−33.5410	−1.84358	−0.921791	−	0.387688i	$-0.873274\pi$
$331$	−33.5410	−1.84358	−0.921791	+	0.387688i	$0.873274\pi$
$332$	0	0
$333$	−12.6491	−0.693167
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.0000	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$338$	0	0
$339$	33.5410	1.82170
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.1246	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$347$	20.1246	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$348$	0	0
$349$	−18.9737	−1.01564	−0.507819	−	0.861464i	$-0.669548\pi$
$349$	−18.9737	−1.01564	−0.507819	+	0.861464i	$0.669548\pi$
$350$	0	0
$351$	−14.1421	−0.754851
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	31.6228	1.67365
$358$	0	0
$359$	14.1421	0.746393	0.373197	−	0.927752i	$-0.378262\pi$
$359$	14.1421	0.746393	0.373197	+	0.927752i	$0.378262\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	−13.4164	−0.704179
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.9706	0.885856	0.442928	−	0.896557i	$-0.353940\pi$
$367$	16.9706	0.885856	0.442928	+	0.896557i	$0.353940\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−35.7771	−1.85745
$372$	0	0
$373$	−12.6491	−0.654946	−0.327473	−	0.944861i	$-0.606197\pi$
$373$	−12.6491	−0.654946	−0.327473	+	0.944861i	$0.606197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.0000	2.06010
$378$	0	0
$379$	6.70820	0.344577	0.172289	−	0.985047i	$-0.444884\pi$
$379$	6.70820	0.344577	0.172289	+	0.985047i	$0.444884\pi$
$380$	0	0
$381$	−37.9473	−1.94410
$382$	0	0
$383$	−19.7990	−1.01168	−0.505841	−	0.862627i	$-0.668818\pi$
$383$	−19.7990	−1.01168	−0.505841	+	0.862627i	$0.668818\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.8885	0.909326
$388$	0	0
$389$	25.2982	1.28267	0.641335	−	0.767261i	$-0.278381\pi$
$389$	25.2982	1.28267	0.641335	+	0.767261i	$0.278381\pi$
$390$	0	0
$391$	−28.2843	−1.43040
$392$	0	0
$393$	20.0000	1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.2982	1.26968	0.634841	−	0.772643i	$-0.281066\pi$
$397$	25.2982	1.26968	0.634841	+	0.772643i	$0.281066\pi$
$398$	0	0
$399$	14.1421	0.707992
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14.1421	−0.701000
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	11.1803	0.551485
$412$	0	0
$413$	25.2982	1.24484
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−35.0000	−1.71396
$418$	0	0
$419$	−33.5410	−1.63859	−0.819293	−	0.573375i	$-0.805634\pi$
$419$	−33.5410	−1.63859	−0.819293	+	0.573375i	$0.805634\pi$
$420$	0	0
$421$	−12.6491	−0.616480	−0.308240	−	0.951309i	$-0.599740\pi$
$421$	−12.6491	−0.616480	−0.308240	+	0.951309i	$0.599740\pi$
$422$	0	0
$423$	−5.65685	−0.275046
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17.8885	−0.865687
$428$	0	0
$429$	31.6228	1.52676
$430$	0	0
$431$	14.1421	0.681203	0.340601	−	0.940208i	$-0.389369\pi$
$431$	14.1421	0.681203	0.340601	+	0.940208i	$0.389369\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.6491	−0.605089
$438$	0	0
$439$	−14.1421	−0.674967	−0.337484	−	0.941331i	$-0.609576\pi$
$439$	−14.1421	−0.674967	−0.337484	+	0.941331i	$0.609576\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−29.0689	−1.38110	−0.690552	−	0.723283i	$-0.742632\pi$
$443$	−29.0689	−1.38110	−0.690552	+	0.723283i	$0.742632\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	42.4264	2.00670
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	−6.70820	−0.315877
$452$	0	0
$453$	−31.6228	−1.48577
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	0	0
$459$	−11.1803	−0.521854
$460$	0	0
$461$	−37.9473	−1.76738	−0.883692	−	0.468069i	$-0.844950\pi$
$461$	−37.9473	−1.76738	−0.883692	+	0.468069i	$0.844950\pi$
$462$	0	0
$463$	−5.65685	−0.262896	−0.131448	−	0.991323i	$-0.541963\pi$
$463$	−5.65685	−0.262896	−0.131448	+	0.991323i	$0.541963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−31.6228	−1.46020
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.2982	−1.15833
$478$	0	0
$479$	28.2843	1.29234	0.646171	−	0.763193i	$-0.276369\pi$
$479$	28.2843	1.29234	0.646171	+	0.763193i	$0.276369\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	−35.7771	−1.62791
$484$	0	0
$485$	0	0
$486$	0	0
$487$	39.5980	1.79436	0.897178	−	0.441669i	$-0.145614\pi$
$487$	39.5980	1.79436	0.897178	+	0.441669i	$0.145614\pi$
$488$	0	0
$489$	15.0000	0.678323
$490$	0	0
$491$	−26.8328	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$491$	−26.8328	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$492$	0	0
$493$	31.6228	1.42422
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	−26.8328	−1.20120	−0.600601	−	0.799549i	$-0.705072\pi$
$499$	−26.8328	−1.20120	−0.600601	+	0.799549i	$0.705072\pi$
$500$	0	0
$501$	−6.32456	−0.282560
$502$	0	0
$503$	33.9411	1.51336	0.756680	−	0.653785i	$-0.226820\pi$
$503$	33.9411	1.51336	0.756680	+	0.653785i	$0.226820\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	60.3738	2.68130
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	42.4264	1.87683
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.32456	−0.278154
$518$	0	0
$519$	−28.2843	−1.24154
$520$	0	0
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	20.1246	0.879988	0.439994	−	0.898001i	$-0.354981\pi$
$523$	20.1246	0.879988	0.439994	+	0.898001i	$0.354981\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	17.8885	0.776297
$532$	0	0
$533$	−18.9737	−0.821841
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.00000	0.215766
$538$	0	0
$539$	2.23607	0.0963143
$540$	0	0
$541$	−12.6491	−0.543828	−0.271914	−	0.962322i	$-0.587657\pi$
$541$	−12.6491	−0.543828	−0.271914	+	0.962322i	$0.587657\pi$
$542$	0	0
$543$	−28.2843	−1.21379
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.1246	0.860466	0.430233	−	0.902718i	$-0.358431\pi$
$547$	20.1246	0.860466	0.430233	+	0.902718i	$0.358431\pi$
$548$	0	0
$549$	−12.6491	−0.539851
$550$	0	0
$551$	14.1421	0.602475
$552$	0	0
$553$	−40.0000	−1.70097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.2719	1.87586	0.937930	−	0.346825i	$-0.112740\pi$
$557$	44.2719	1.87586	0.937930	+	0.346825i	$0.112740\pi$
$558$	0	0
$559$	56.5685	2.39259
$560$	0	0
$561$	25.0000	1.05550
$562$	0	0
$563$	−44.7214	−1.88478	−0.942390	−	0.334515i	$-0.891427\pi$
$563$	−44.7214	−1.88478	−0.942390	+	0.334515i	$0.891427\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−31.1127	−1.30661
$568$	0	0
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	−26.8328	−1.12292	−0.561459	−	0.827504i	$-0.689760\pi$
$571$	−26.8328	−1.12292	−0.561459	+	0.827504i	$0.689760\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	11.1803	0.464639
$580$	0	0
$581$	18.9737	0.787160
$582$	0	0
$583$	−28.2843	−1.17141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.1803	−0.461462	−0.230731	−	0.973018i	$-0.574112\pi$
$587$	−11.1803	−0.461462	−0.230731	+	0.973018i	$0.574112\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.1421	−0.577832	−0.288916	−	0.957354i	$-0.593295\pi$
$599$	−14.1421	−0.577832	−0.288916	+	0.957354i	$0.593295\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	−22.3607	−0.910597
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	40.0000	1.62088
$610$	0	0
$611$	−17.8885	−0.723693
$612$	0	0
$613$	−25.2982	−1.02179	−0.510893	−	0.859644i	$-0.670685\pi$
$613$	−25.2982	−1.02179	−0.510893	+	0.859644i	$0.670685\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−26.8328	−1.07850	−0.539251	−	0.842145i	$-0.681293\pi$
$619$	−26.8328	−1.07850	−0.539251	+	0.842145i	$0.681293\pi$
$620$	0	0
$621$	12.6491	0.507591
$622$	0	0
$623$	−2.82843	−0.113319
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.1803	0.446500
$628$	0	0
$629$	−31.6228	−1.26088
$630$	0	0
$631$	−28.2843	−1.12598	−0.562990	−	0.826464i	$-0.690349\pi$
$631$	−28.2843	−1.12598	−0.562990	+	0.826464i	$0.690349\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.32456	0.250588
$638$	0	0
$639$	28.2843	1.11891
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	26.8328	1.05818	0.529091	−	0.848565i	$-0.322533\pi$
$643$	26.8328	1.05818	0.529091	+	0.848565i	$0.322533\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.2548	−1.77915	−0.889576	−	0.456788i	$-0.849000\pi$
$647$	−45.2548	−1.77915	−0.889576	+	0.456788i	$0.849000\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.32456	0.247499	0.123749	−	0.992313i	$-0.460508\pi$
$653$	6.32456	0.247499	0.123749	+	0.992313i	$0.460508\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000	1.17041
$658$	0	0
$659$	6.70820	0.261315	0.130657	−	0.991428i	$-0.458291\pi$
$659$	6.70820	0.261315	0.130657	+	0.991428i	$0.458291\pi$
$660$	0	0
$661$	44.2719	1.72198	0.860988	−	0.508625i	$-0.169846\pi$
$661$	44.2719	1.72198	0.860988	+	0.508625i	$0.169846\pi$
$662$	0	0
$663$	70.7107	2.74618
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.7771	−1.38529
$668$	0	0
$669$	50.5964	1.95617
$670$	0	0
$671$	−14.1421	−0.545951
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.9473	−1.45843	−0.729217	−	0.684282i	$-0.760116\pi$
$677$	−37.9473	−1.45843	−0.729217	+	0.684282i	$0.760116\pi$
$678$	0	0
$679$	−28.2843	−1.08545
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	2.23607	0.0855608	0.0427804	−	0.999085i	$-0.486378\pi$
$683$	2.23607	0.0855608	0.0427804	+	0.999085i	$0.486378\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.2843	1.07911
$688$	0	0
$689$	−80.0000	−3.04776
$690$	0	0
$691$	−33.5410	−1.27596	−0.637980	−	0.770053i	$-0.720230\pi$
$691$	−33.5410	−1.27596	−0.637980	+	0.770053i	$0.720230\pi$
$692$	0	0
$693$	12.6491	0.480500
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.0000	−0.568166
$698$	0	0
$699$	22.3607	0.845759
$700$	0	0
$701$	31.6228	1.19438	0.597188	−	0.802101i	$-0.296285\pi$
$701$	31.6228	1.19438	0.597188	+	0.802101i	$0.296285\pi$
$702$	0	0
$703$	−14.1421	−0.533381
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.8885	−0.672768
$708$	0	0
$709$	12.6491	0.475047	0.237524	−	0.971382i	$-0.423664\pi$
$709$	12.6491	0.475047	0.237524	+	0.971382i	$0.423664\pi$
$710$	0	0
$711$	−28.2843	−1.06074
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−31.6228	−1.18097
$718$	0	0
$719$	−14.1421	−0.527413	−0.263706	−	0.964603i	$-0.584945\pi$
$719$	−14.1421	−0.527413	−0.263706	+	0.964603i	$0.584945\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	29.0689	1.08108
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.4558	−0.944105	−0.472052	−	0.881570i	$-0.656487\pi$
$727$	−25.4558	−0.944105	−0.472052	+	0.881570i	$0.656487\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	44.7214	1.65408
$732$	0	0
$733$	6.32456	0.233603	0.116801	−	0.993155i	$-0.462736\pi$
$733$	6.32456	0.233603	0.116801	+	0.993155i	$0.462736\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.0000	−0.920887
$738$	0	0
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	0	0
$741$	31.6228	1.16169
$742$	0	0
$743$	5.65685	0.207530	0.103765	−	0.994602i	$-0.466911\pi$
$743$	5.65685	0.207530	0.103765	+	0.994602i	$0.466911\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.4164	0.490881
$748$	0	0
$749$	−44.2719	−1.61766
$750$	0	0
$751$	42.4264	1.54816	0.774081	−	0.633087i	$-0.218212\pi$
$751$	42.4264	1.54816	0.774081	+	0.633087i	$0.218212\pi$
$752$	0	0
$753$	−25.0000	−0.911051
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.6491	0.459740	0.229870	−	0.973221i	$-0.426170\pi$
$757$	12.6491	0.459740	0.229870	+	0.973221i	$0.426170\pi$
$758$	0	0
$759$	−28.2843	−1.02665
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	−17.8885	−0.647609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	56.5685	2.04257
$768$	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
$771$	22.3607	0.805300
$772$	0	0
$773$	31.6228	1.13739	0.568696	−	0.822548i	$-0.307448\pi$
$773$	31.6228	1.13739	0.568696	+	0.822548i	$0.307448\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−40.0000	−1.43499
$778$	0	0
$779$	−6.70820	−0.240346
$780$	0	0
$781$	31.6228	1.13155
$782$	0	0
$783$	−14.1421	−0.505399
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.8328	−0.956487	−0.478243	−	0.878227i	$-0.658726\pi$
$787$	−26.8328	−0.956487	−0.478243	+	0.878227i	$0.658726\pi$
$788$	0	0
$789$	50.5964	1.80128
$790$	0	0
$791$	42.4264	1.50851
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.2982	−0.896109	−0.448054	−	0.894006i	$-0.647883\pi$
$797$	−25.2982	−0.896109	−0.448054	+	0.894006i	$0.647883\pi$
$798$	0	0
$799$	−14.1421	−0.500313
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	33.5410	1.18364
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	8.94427	0.314076	0.157038	−	0.987593i	$-0.449806\pi$
$811$	8.94427	0.314076	0.157038	+	0.987593i	$0.449806\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.0000	0.699711
$818$	0	0
$819$	35.7771	1.25015
$820$	0	0
$821$	−37.9473	−1.32437	−0.662186	−	0.749340i	$-0.730371\pi$
$821$	−37.9473	−1.32437	−0.662186	+	0.749340i	$0.730371\pi$
$822$	0	0
$823$	−5.65685	−0.197186	−0.0985928	−	0.995128i	$-0.531434\pi$
$823$	−5.65685	−0.197186	−0.0985928	+	0.995128i	$0.531434\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.1803	−0.388779	−0.194389	−	0.980924i	$-0.562272\pi$
$827$	−11.1803	−0.388779	−0.194389	+	0.980924i	$0.562272\pi$
$828$	0	0
$829$	6.32456	0.219661	0.109830	−	0.993950i	$-0.464969\pi$
$829$	6.32456	0.219661	0.109830	+	0.993950i	$0.464969\pi$
$830$	0	0
$831$	−42.4264	−1.47176
$832$	0	0
$833$	5.00000	0.173240
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−56.5685	−1.95296	−0.976481	−	0.215601i	$-0.930829\pi$
$839$	−56.5685	−1.95296	−0.976481	+	0.215601i	$0.930829\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	4.47214	0.154029
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.9706	−0.583115
$848$	0	0
$849$	−25.0000	−0.857998
$850$	0	0
$851$	35.7771	1.22642
$852$	0	0
$853$	12.6491	0.433097	0.216549	−	0.976272i	$-0.430520\pi$
$853$	12.6491	0.433097	0.216549	+	0.976272i	$0.430520\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	0	0
$859$	2.23607	0.0762937	0.0381468	−	0.999272i	$-0.487855\pi$
$859$	2.23607	0.0762937	0.0381468	+	0.999272i	$0.487855\pi$
$860$	0	0
$861$	−18.9737	−0.646621
$862$	0	0
$863$	−8.48528	−0.288842	−0.144421	−	0.989516i	$-0.546132\pi$
$863$	−8.48528	−0.288842	−0.144421	+	0.989516i	$0.546132\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.8885	0.607527
$868$	0	0
$869$	−31.6228	−1.07273
$870$	0	0
$871$	−70.7107	−2.39594
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.6491	−0.427130	−0.213565	−	0.976929i	$-0.568508\pi$
$877$	−12.6491	−0.427130	−0.213565	+	0.976929i	$0.568508\pi$
$878$	0	0
$879$	14.1421	0.477002
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−15.6525	−0.526748	−0.263374	−	0.964694i	$-0.584835\pi$
$883$	−15.6525	−0.526748	−0.263374	+	0.964694i	$0.584835\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.3137	−0.379877	−0.189939	−	0.981796i	$-0.560829\pi$
$887$	−11.3137	−0.379877	−0.189939	+	0.981796i	$0.560829\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−24.5967	−0.824022
$892$	0	0
$893$	−6.32456	−0.211643
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−80.0000	−2.67112
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−63.2456	−2.10701
$902$	0	0
$903$	56.5685	1.88248
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.94427	0.296990	0.148495	−	0.988913i	$-0.452557\pi$
$907$	8.94427	0.296990	0.148495	+	0.988913i	$0.452557\pi$
$908$	0	0
$909$	−12.6491	−0.419545
$910$	0	0
$911$	−28.2843	−0.937100	−0.468550	−	0.883437i	$-0.655223\pi$
$911$	−28.2843	−0.937100	−0.468550	+	0.883437i	$0.655223\pi$
$912$	0	0
$913$	15.0000	0.496428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25.2982	0.835421
$918$	0	0
$919$	−28.2843	−0.933012	−0.466506	−	0.884518i	$-0.654487\pi$
$919$	−28.2843	−0.933012	−0.466506	+	0.884518i	$0.654487\pi$
$920$	0	0
$921$	45.0000	1.48280
$922$	0	0
$923$	89.4427	2.94404
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.9706	−0.557386
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	2.23607	0.0732842
$932$	0	0
$933$	31.6228	1.03528
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	67.0820	2.18914
$940$	0	0
$941$	−31.6228	−1.03087	−0.515437	−	0.856928i	$-0.672370\pi$
$941$	−31.6228	−1.03087	−0.515437	+	0.856928i	$0.672370\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.8328	0.871949	0.435975	−	0.899959i	$-0.356404\pi$
$947$	26.8328	0.871949	0.435975	+	0.899959i	$0.356404\pi$
$948$	0	0
$949$	94.8683	3.07956
$950$	0	0
$951$	28.2843	0.917180
$952$	0	0
$953$	45.0000	1.45769	0.728846	−	0.684677i	$-0.240057\pi$
$953$	45.0000	1.45769	0.728846	+	0.684677i	$0.240057\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	31.6228	1.02222
$958$	0	0
$959$	14.1421	0.456673
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−31.3050	−1.00879
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.9706	0.545737	0.272868	−	0.962051i	$-0.412028\pi$
$967$	16.9706	0.545737	0.272868	+	0.962051i	$0.412028\pi$
$968$	0	0
$969$	25.0000	0.803116
$970$	0	0
$971$	2.23607	0.0717588	0.0358794	−	0.999356i	$-0.488577\pi$
$971$	2.23607	0.0717588	0.0358794	+	0.999356i	$0.488577\pi$
$972$	0	0
$973$	−44.2719	−1.41929
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	0	0
$979$	−2.23607	−0.0714650
$980$	0	0
$981$	−12.6491	−0.403855
$982$	0	0
$983$	−19.7990	−0.631490	−0.315745	−	0.948844i	$-0.602254\pi$
$983$	−19.7990	−0.631490	−0.315745	+	0.948844i	$0.602254\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−17.8885	−0.569399
$988$	0	0
$989$	−50.5964	−1.60887
$990$	0	0
$991$	42.4264	1.34772	0.673860	−	0.738859i	$-0.264635\pi$
$991$	42.4264	1.34772	0.673860	+	0.738859i	$0.264635\pi$
$992$	0	0
$993$	−75.0000	−2.38005
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.2982	0.801203	0.400601	−	0.916252i	$-0.368801\pi$
$997$	25.2982	0.801203	0.400601	+	0.916252i	$0.368801\pi$
$998$	0	0
$999$	14.1421	0.447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.cs.1.4		4
4.3	odd	2	inner	6400.2.a.cs.1.1		4
5.4	even	2		6400.2.a.cp.1.1		4
8.3	odd	2	inner	6400.2.a.cs.1.3		4
8.5	even	2	inner	6400.2.a.cs.1.2		4
16.3	odd	4		3200.2.d.r.1601.2	yes	4
16.5	even	4		3200.2.d.r.1601.1	yes	4
16.11	odd	4		3200.2.d.r.1601.4	yes	4
16.13	even	4		3200.2.d.r.1601.3	yes	4
20.19	odd	2		6400.2.a.cp.1.4		4
40.19	odd	2		6400.2.a.cp.1.2		4
40.29	even	2		6400.2.a.cp.1.3		4
80.3	even	4		3200.2.f.r.449.6		8
80.13	odd	4		3200.2.f.r.449.3		8
80.19	odd	4		3200.2.d.m.1601.3	yes	4
80.27	even	4		3200.2.f.r.449.7		8
80.29	even	4		3200.2.d.m.1601.2	yes	4
80.37	odd	4		3200.2.f.r.449.2		8
80.43	even	4		3200.2.f.r.449.1		8
80.53	odd	4		3200.2.f.r.449.8		8
80.59	odd	4		3200.2.d.m.1601.1	✓	4
80.67	even	4		3200.2.f.r.449.4		8
80.69	even	4		3200.2.d.m.1601.4	yes	4
80.77	odd	4		3200.2.f.r.449.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.m.1601.1	✓	4	80.59	odd	4
3200.2.d.m.1601.2	yes	4	80.29	even	4
3200.2.d.m.1601.3	yes	4	80.19	odd	4
3200.2.d.m.1601.4	yes	4	80.69	even	4
3200.2.d.r.1601.1	yes	4	16.5	even	4
3200.2.d.r.1601.2	yes	4	16.3	odd	4
3200.2.d.r.1601.3	yes	4	16.13	even	4
3200.2.d.r.1601.4	yes	4	16.11	odd	4
3200.2.f.r.449.1		8	80.43	even	4
3200.2.f.r.449.2		8	80.37	odd	4
3200.2.f.r.449.3		8	80.13	odd	4
3200.2.f.r.449.4		8	80.67	even	4
3200.2.f.r.449.5		8	80.77	odd	4
3200.2.f.r.449.6		8	80.3	even	4
3200.2.f.r.449.7		8	80.27	even	4
3200.2.f.r.449.8		8	80.53	odd	4
6400.2.a.cp.1.1		4	5.4	even	2
6400.2.a.cp.1.2		4	40.19	odd	2
6400.2.a.cp.1.3		4	40.29	even	2
6400.2.a.cp.1.4		4	20.19	odd	2
6400.2.a.cs.1.1		4	4.3	odd	2	inner
6400.2.a.cs.1.2		4	8.5	even	2	inner
6400.2.a.cs.1.3		4	8.3	odd	2	inner
6400.2.a.cs.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} +2.82843 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} +2.82843 q^{7} +2.00000 q^{9} +2.23607 q^{11} +6.32456 q^{13} +5.00000 q^{17} +2.23607 q^{19} +6.32456 q^{21} -5.65685 q^{23} -2.23607 q^{27} +6.32456 q^{29} +5.00000 q^{33} -6.32456 q^{37} +14.1421 q^{39} -3.00000 q^{41} +8.94427 q^{43} -2.82843 q^{47} +1.00000 q^{49} +11.1803 q^{51} -12.6491 q^{53} +5.00000 q^{57} +8.94427 q^{59} -6.32456 q^{61} +5.65685 q^{63} -11.1803 q^{67} -12.6491 q^{69} +14.1421 q^{71} +15.0000 q^{73} +6.32456 q^{77} -14.1421 q^{79} -11.0000 q^{81} +6.70820 q^{83} +14.1421 q^{87} -1.00000 q^{89} +17.8885 q^{91} -10.0000 q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^9	\( 4 q + 8 q^{9} + 20 q^{17} + 20 q^{33} - 12 q^{41} + 4 q^{49} + 20 q^{57} + 60 q^{73} - 44 q^{81} - 4 q^{89} - 40 q^{97}+O(q^{100}) \) 4 * q + 8 * q^9 + 20 * q^17 + 20 * q^33 - 12 * q^41 + 4 * q^49 + 20 * q^57 + 60 * q^73 - 44 * q^81 - 4 * q^89 - 40 * q^97

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