LMFDB - Embedded newform 3200.2.f.r.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(449,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, 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("3200.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.r.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} +2.23607i q^{11} -6.32456 q^{13} +5.00000i q^{17} +2.23607i q^{19} +6.32456i q^{21} -5.65685i q^{23} +2.23607 q^{27} +6.32456i q^{29} -5.00000i q^{33} +6.32456 q^{37} +14.1421 q^{39} +3.00000 q^{41} +8.94427 q^{43} -2.82843i q^{47} -1.00000 q^{49} -11.1803i q^{51} -12.6491 q^{53} -5.00000i q^{57} -8.94427i q^{59} +6.32456i q^{61} -5.65685i q^{63} -11.1803 q^{67} +12.6491i q^{69} -14.1421 q^{71} +15.0000i q^{73} +6.32456 q^{77} +14.1421 q^{79} -11.0000 q^{81} -6.70820 q^{83} -14.1421i q^{87} -1.00000 q^{89} +17.8885i q^{91} -10.0000i q^{97} +4.47214i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 24 q^{41} - 8 q^{49} - 88 q^{81} - 8 q^{89}+O(q^{100}) $ 8 * q + 16 * q^9 + 24 * q^41 - 8 * q^49 - 88 * q^81 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.23607i	0.674200i	0.941469	+	0.337100i	$0.109446\pi$
$11$	2.23607i	0.674200i	−0.941469	+	0.337100i	$0.890554\pi$
$12$	0	0
$13$	−6.32456	−1.75412	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	−6.32456	−1.75412	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	0	0
$19$	2.23607i	0.512989i	0.966546	+	0.256495i	$0.0825676\pi$
$19$	2.23607i	0.512989i	−0.966546	+	0.256495i	$0.917432\pi$
$20$	0	0
$21$	6.32456i	1.38013i
$22$	0	0
$23$	− 5.65685i	− 1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$	− 5.65685i	− 1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	6.32456i	1.17444i	0.809427	+	0.587220i	$0.199778\pi$
$29$	6.32456i	1.17444i	−0.809427	+	0.587220i	$0.800222\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 5.00000i	− 0.870388i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.32456	1.03975	0.519875	−	0.854242i	$-0.325978\pi$
$37$	6.32456	1.03975	0.519875	+	0.854242i	$0.325978\pi$
$38$	0	0
$39$	14.1421	2.26455
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	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
$47$	− 2.82843i	− 0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$	− 2.82843i	− 0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	− 11.1803i	− 1.56556i
$52$	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
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	− 8.94427i	− 1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$	− 8.94427i	− 1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$	0	0
$61$	6.32456i	0.809776i	0.914366	+	0.404888i	$0.132690\pi$
$61$	6.32456i	0.809776i	−0.914366	+	0.404888i	$0.867310\pi$
$62$	0	0
$63$	− 5.65685i	− 0.712697i
$64$	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
$69$	12.6491i	1.52277i
$70$	0	0
$71$	−14.1421	−1.67836	−0.839181	−	0.543852i	$-0.816965\pi$
$71$	−14.1421	−1.67836	−0.839181	+	0.543852i	$0.816965\pi$
$72$	0	0
$73$	15.0000i	1.75562i	0.479012	+	0.877809i	$0.340995\pi$
$73$	15.0000i	1.75562i	−0.479012	+	0.877809i	$0.659005\pi$
$74$	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	$-0.207178\pi$
$79$	14.1421	1.59111	0.795557	+	0.605878i	$0.207178\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−6.70820	−0.736321	−0.368161	−	0.929762i	$-0.620012\pi$
$83$	−6.70820	−0.736321	−0.368161	+	0.929762i	$0.620012\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 14.1421i	− 1.51620i
$88$	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
$91$	17.8885i	1.87523i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	4.47214i	0.449467i
$100$	0	0
$101$	− 6.32456i	− 0.629317i	−0.949205	−	0.314658i	$-0.898110\pi$
$101$	− 6.32456i	− 0.629317i	0.949205	−	0.314658i	$-0.101890\pi$
$102$	0	0
$103$	− 8.48528i	− 0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$	− 8.48528i	− 0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.6525	1.51318	0.756591	−	0.653888i	$-0.226863\pi$
$107$	15.6525	1.51318	0.756591	+	0.653888i	$0.226863\pi$
$108$	0	0
$109$	− 6.32456i	− 0.605783i	−0.953025	−	0.302891i	$-0.902048\pi$
$109$	− 6.32456i	− 0.605783i	0.953025	−	0.302891i	$-0.0979519\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	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
$117$	−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
$127$	− 16.9706i	− 1.50589i	−0.658081	−	0.752947i	$-0.728632\pi$
$127$	− 16.9706i	− 1.50589i	0.658081	−	0.752947i	$-0.271368\pi$
$128$	0	0
$129$	−20.0000	−1.76090
$130$	0	0
$131$	− 8.94427i	− 0.781465i	−0.920504	−	0.390732i	$-0.872222\pi$
$131$	− 8.94427i	− 0.781465i	0.920504	−	0.390732i	$-0.127778\pi$
$132$	0	0
$133$	6.32456	0.548408
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.00000i	− 0.427179i	−0.976924	−	0.213589i	$-0.931485\pi$
$137$	− 5.00000i	− 0.427179i	0.976924	−	0.213589i	$-0.0685155\pi$
$138$	0	0
$139$	15.6525i	1.32763i	0.747899	+	0.663813i	$0.231063\pi$
$139$	15.6525i	1.32763i	−0.747899	+	0.663813i	$0.768937\pi$
$140$	0	0
$141$	6.32456i	0.532624i
$142$	0	0
$143$	− 14.1421i	− 1.18262i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.23607	0.184428
$148$	0	0
$149$	− 18.9737i	− 1.55438i	−0.629264	−	0.777192i	$-0.716644\pi$
$149$	− 18.9737i	− 1.55438i	0.629264	−	0.777192i	$-0.283356\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	10.0000i	0.808452i
$154$	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	$-0.584617\pi$
$163$	−6.70820	−0.525427	−0.262714	+	0.964874i	$0.584617\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.82843i	0.218870i	0.993994	+	0.109435i	$0.0349042\pi$
$167$	2.82843i	0.218870i	−0.993994	+	0.109435i	$0.965096\pi$
$168$	0	0
$169$	27.0000	2.07692
$170$	0	0
$171$	4.47214i	0.341993i
$172$	0	0
$173$	12.6491	0.961694	0.480847	−	0.876804i	$-0.340329\pi$
$173$	12.6491	0.961694	0.480847	+	0.876804i	$0.340329\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.0000i	1.50329i
$178$	0	0
$179$	2.23607i	0.167132i	0.996502	+	0.0835658i	$0.0266309\pi$
$179$	2.23607i	0.167132i	−0.996502	+	0.0835658i	$0.973369\pi$
$180$	0	0
$181$	− 12.6491i	− 0.940201i	−0.882613	−	0.470100i	$-0.844218\pi$
$181$	− 12.6491i	− 0.940201i	0.882613	−	0.470100i	$-0.155782\pi$
$182$	0	0
$183$	− 14.1421i	− 1.04542i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.1803	−0.817587
$188$	0	0
$189$	− 6.32456i	− 0.460044i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	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
$207$	− 11.3137i	− 0.786357i
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	− 6.70820i	− 0.461812i	−0.972976	−	0.230906i	$-0.925831\pi$
$211$	− 6.70820i	− 0.461812i	0.972976	−	0.230906i	$-0.0741690\pi$
$212$	0	0
$213$	31.6228	2.16676
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 33.5410i	− 2.26649i
$220$	0	0
$221$	− 31.6228i	− 2.12718i
$222$	0	0
$223$	− 22.6274i	− 1.51524i	−0.652694	−	0.757622i	$-0.726361\pi$
$223$	− 22.6274i	− 1.51524i	0.652694	−	0.757622i	$-0.273639\pi$
$224$	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
$229$	− 12.6491i	− 0.835877i	−0.908475	−	0.417938i	$-0.862753\pi$
$229$	− 12.6491i	− 0.835877i	0.908475	−	0.417938i	$-0.137247\pi$
$230$	0	0
$231$	−14.1421	−0.930484
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	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	$-0.348785\pi$
$239$	14.1421	0.914779	0.457389	+	0.889267i	$0.348785\pi$
$240$	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
$247$	− 14.1421i	− 0.899843i
$248$	0	0
$249$	15.0000	0.950586
$250$	0	0
$251$	− 11.1803i	− 0.705697i	−0.935681	−	0.352848i	$-0.885213\pi$
$251$	− 11.1803i	− 0.705697i	0.935681	−	0.352848i	$-0.114787\pi$
$252$	0	0
$253$	12.6491	0.795243
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0000i	0.623783i	0.950118	+	0.311891i	$0.100963\pi$
$257$	10.0000i	0.623783i	−0.950118	+	0.311891i	$0.899037\pi$
$258$	0	0
$259$	− 17.8885i	− 1.11154i
$260$	0	0
$261$	12.6491i	0.782960i
$262$	0	0
$263$	22.6274i	1.39527i	0.716455	+	0.697633i	$0.245763\pi$
$263$	22.6274i	1.39527i	−0.716455	+	0.697633i	$0.754237\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.23607	0.136845
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	− 40.0000i	− 2.42091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.9737	1.14002	0.570009	−	0.821639i	$-0.306940\pi$
$277$	18.9737	1.14002	0.570009	+	0.821639i	$0.306940\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	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
$287$	− 8.48528i	− 0.500870i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	22.3607i	1.31081i
$292$	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
$297$	5.00000i	0.290129i
$298$	0	0
$299$	35.7771i	2.06904i
$300$	0	0
$301$	− 25.2982i	− 1.45817i
$302$	0	0
$303$	14.1421i	0.812444i
$304$	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
$309$	18.9737i	1.07937i
$310$	0	0
$311$	−14.1421	−0.801927	−0.400963	−	0.916094i	$-0.631325\pi$
$311$	−14.1421	−0.801927	−0.400963	+	0.916094i	$0.631325\pi$
$312$	0	0
$313$	30.0000i	1.69570i	0.530236	+	0.847850i	$0.322103\pi$
$313$	30.0000i	1.69570i	−0.530236	+	0.847850i	$0.677897\pi$
$314$	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
$327$	14.1421i	0.782062i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	− 33.5410i	− 1.84358i	−0.387688	−	0.921791i	$-0.626726\pi$
$331$	− 33.5410i	− 1.84358i	0.387688	−	0.921791i	$-0.373274\pi$
$332$	0	0
$333$	12.6491	0.693167
$334$	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
$339$	33.5410i	1.82170i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.1246	−1.08035	−0.540173	−	0.841554i	$-0.681641\pi$
$347$	−20.1246	−1.08035	−0.540173	+	0.841554i	$0.681641\pi$
$348$	0	0
$349$	− 18.9737i	− 1.01564i	−0.861464	−	0.507819i	$-0.830452\pi$
$349$	− 18.9737i	− 1.01564i	0.861464	−	0.507819i	$-0.169548\pi$
$350$	0	0
$351$	−14.1421	−0.754851
$352$	0	0
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	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
$367$	16.9706i	0.885856i	0.896557	+	0.442928i	$0.146060\pi$
$367$	16.9706i	0.885856i	−0.896557	+	0.442928i	$0.853940\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	35.7771i	1.85745i
$372$	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
$377$	− 40.0000i	− 2.06010i
$378$	0	0
$379$	− 6.70820i	− 0.344577i	−0.985047	−	0.172289i	$-0.944884\pi$
$379$	− 6.70820i	− 0.344577i	0.985047	−	0.172289i	$-0.0551162\pi$
$380$	0	0
$381$	37.9473i	1.94410i
$382$	0	0
$383$	19.7990i	1.01168i	0.862627	+	0.505841i	$0.168818\pi$
$383$	19.7990i	1.01168i	−0.862627	+	0.505841i	$0.831182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.8885	0.909326
$388$	0	0
$389$	− 25.2982i	− 1.28267i	−0.767261	−	0.641335i	$-0.778381\pi$
$389$	− 25.2982i	− 1.28267i	0.767261	−	0.641335i	$-0.221619\pi$
$390$	0	0
$391$	28.2843	1.43040
$392$	0	0
$393$	20.0000i	1.00887i
$394$	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
$407$	14.1421i	0.701000i
$408$	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
$411$	11.1803i	0.551485i
$412$	0	0
$413$	−25.2982	−1.24484
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 35.0000i	− 1.71396i
$418$	0	0
$419$	− 33.5410i	− 1.63859i	−0.573375	−	0.819293i	$-0.694366\pi$
$419$	− 33.5410i	− 1.63859i	0.573375	−	0.819293i	$-0.305634\pi$
$420$	0	0
$421$	− 12.6491i	− 0.616480i	−0.951309	−	0.308240i	$-0.900260\pi$
$421$	− 12.6491i	− 0.616480i	0.951309	−	0.308240i	$-0.0997400\pi$
$422$	0	0
$423$	− 5.65685i	− 0.275046i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.8885	0.865687
$428$	0	0
$429$	31.6228i	1.52676i
$430$	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
$433$	− 5.00000i	− 0.240285i	−0.992757	−	0.120142i	$-0.961665\pi$
$433$	− 5.00000i	− 0.240285i	0.992757	−	0.120142i	$-0.0383351\pi$
$434$	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
$447$	42.4264i	2.00670i
$448$	0	0
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	6.70820i	0.315877i
$452$	0	0
$453$	−31.6228	−1.48577
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.00000i	− 0.233890i	−0.993138	−	0.116945i	$-0.962690\pi$
$457$	− 5.00000i	− 0.233890i	0.993138	−	0.116945i	$-0.0373101\pi$
$458$	0	0
$459$	11.1803i	0.521854i
$460$	0	0
$461$	37.9473i	1.76738i	0.468069	+	0.883692i	$0.344950\pi$
$461$	37.9473i	1.76738i	−0.468069	+	0.883692i	$0.655050\pi$
$462$	0	0
$463$	5.65685i	0.262896i	0.991323	+	0.131448i	$0.0419627\pi$
$463$	5.65685i	0.262896i	−0.991323	+	0.131448i	$0.958037\pi$
$464$	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
$469$	31.6228i	1.46020i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.0000i	0.919601i
$474$	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	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	35.7771	1.62791
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 39.5980i	− 1.79436i	−0.441669	−	0.897178i	$-0.645614\pi$
$487$	− 39.5980i	− 1.79436i	0.441669	−	0.897178i	$-0.354386\pi$
$488$	0	0
$489$	15.0000	0.678323
$490$	0	0
$491$	− 26.8328i	− 1.21095i	−0.795865	−	0.605474i	$-0.792984\pi$
$491$	− 26.8328i	− 1.21095i	0.795865	−	0.605474i	$-0.207016\pi$
$492$	0	0
$493$	−31.6228	−1.42422
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000i	1.79425i
$498$	0	0
$499$	− 26.8328i	− 1.20120i	−0.799549	−	0.600601i	$-0.794928\pi$
$499$	− 26.8328i	− 1.20120i	0.799549	−	0.600601i	$-0.205072\pi$
$500$	0	0
$501$	− 6.32456i	− 0.282560i
$502$	0	0
$503$	33.9411i	1.51336i	0.653785	+	0.756680i	$0.273180\pi$
$503$	33.9411i	1.51336i	−0.653785	+	0.756680i	$0.726820\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−60.3738	−2.68130
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	42.4264	1.87683
$512$	0	0
$513$	5.00000i	0.220755i
$514$	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	$-0.591917\pi$
$521$	−13.0000	−0.569540	−0.284770	+	0.958596i	$0.591917\pi$
$522$	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
$531$	− 17.8885i	− 0.776297i
$532$	0	0
$533$	−18.9737	−0.821841
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.00000i	− 0.215766i
$538$	0	0
$539$	− 2.23607i	− 0.0963143i
$540$	0	0
$541$	12.6491i	0.543828i	0.962322	+	0.271914i	$0.0876566\pi$
$541$	12.6491i	0.543828i	−0.962322	+	0.271914i	$0.912343\pi$
$542$	0	0
$543$	28.2843i	1.21379i
$544$	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
$549$	12.6491i	0.539851i
$550$	0	0
$551$	−14.1421	−0.602475
$552$	0	0
$553$	− 40.0000i	− 1.70097i
$554$	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	$-0.108573\pi$
$563$	44.7214	1.88478	0.942390	+	0.334515i	$0.108573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	31.1127i	1.30661i
$568$	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
$571$	− 26.8328i	− 1.12292i	−0.827504	−	0.561459i	$-0.810240\pi$
$571$	− 26.8328i	− 1.12292i	0.827504	−	0.561459i	$-0.189760\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 25.0000i	− 1.04076i	−0.853934	−	0.520382i	$-0.825790\pi$
$577$	− 25.0000i	− 1.04076i	0.853934	−	0.520382i	$-0.174210\pi$
$578$	0	0
$579$	11.1803i	0.464639i
$580$	0	0
$581$	18.9737i	0.787160i
$582$	0	0
$583$	− 28.2843i	− 1.17141i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.1803	0.461462	0.230731	−	0.973018i	$-0.425888\pi$
$587$	11.1803	0.461462	0.230731	+	0.973018i	$0.425888\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.0000i	− 0.615976i	−0.951390	−	0.307988i	$-0.900344\pi$
$593$	− 15.0000i	− 0.615976i	0.951390	−	0.307988i	$-0.0996557\pi$
$594$	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	$-0.387295\pi$
$601$	17.0000	0.693444	0.346722	+	0.937968i	$0.387295\pi$
$602$	0	0
$603$	−22.3607	−0.910597
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.9706i	− 0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$	− 16.9706i	− 0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$	0	0
$609$	−40.0000	−1.62088
$610$	0	0
$611$	17.8885i	0.723693i
$612$	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
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	26.8328i	1.07850i	0.842145	+	0.539251i	$0.181293\pi$
$619$	26.8328i	1.07850i	−0.842145	+	0.539251i	$0.818707\pi$
$620$	0	0
$621$	− 12.6491i	− 0.507591i
$622$	0	0
$623$	2.82843i	0.113319i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.1803	0.446500
$628$	0	0
$629$	31.6228i	1.26088i
$630$	0	0
$631$	28.2843	1.12598	0.562990	−	0.826464i	$-0.309651\pi$
$631$	28.2843	1.12598	0.562990	+	0.826464i	$0.309651\pi$
$632$	0	0
$633$	15.0000i	0.596196i
$634$	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	$-0.677467\pi$
$643$	−26.8328	−1.05818	−0.529091	+	0.848565i	$0.677467\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.2548i	1.77915i	0.456788	+	0.889576i	$0.349000\pi$
$647$	45.2548i	1.77915i	−0.456788	+	0.889576i	$0.651000\pi$
$648$	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	$-0.539492\pi$
$653$	−6.32456	−0.247499	−0.123749	+	0.992313i	$0.539492\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000i	1.17041i
$658$	0	0
$659$	6.70820i	0.261315i	0.991428	+	0.130657i	$0.0417087\pi$
$659$	6.70820i	0.261315i	−0.991428	+	0.130657i	$0.958291\pi$
$660$	0	0
$661$	44.2719i	1.72198i	0.508625	+	0.860988i	$0.330154\pi$
$661$	44.2719i	1.72198i	−0.508625	+	0.860988i	$0.669846\pi$
$662$	0	0
$663$	70.7107i	2.74618i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.7771	1.38529
$668$	0	0
$669$	50.5964i	1.95617i
$670$	0	0
$671$	−14.1421	−0.545951
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.9473	1.45843	0.729217	−	0.684282i	$-0.239884\pi$
$677$	37.9473	1.45843	0.729217	+	0.684282i	$0.239884\pi$
$678$	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
$687$	28.2843i	1.07911i
$688$	0	0
$689$	80.0000	3.04776
$690$	0	0
$691$	33.5410i	1.27596i	0.770053	+	0.637980i	$0.220230\pi$
$691$	33.5410i	1.27596i	−0.770053	+	0.637980i	$0.779770\pi$
$692$	0	0
$693$	12.6491	0.480500
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.0000i	0.568166i
$698$	0	0
$699$	− 22.3607i	− 0.845759i
$700$	0	0
$701$	− 31.6228i	− 1.19438i	−0.802101	−	0.597188i	$-0.796285\pi$
$701$	− 31.6228i	− 1.19438i	0.802101	−	0.597188i	$-0.203715\pi$
$702$	0	0
$703$	14.1421i	0.533381i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.8885	−0.672768
$708$	0	0
$709$	− 12.6491i	− 0.475047i	−0.971382	−	0.237524i	$-0.923664\pi$
$709$	− 12.6491i	− 0.475047i	0.971382	−	0.237524i	$-0.0763357\pi$
$710$	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	$-0.415055\pi$
$719$	14.1421	0.527413	0.263706	+	0.964603i	$0.415055\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	−29.0689	−1.08108
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.4558i	0.944105i	0.881570	+	0.472052i	$0.156487\pi$
$727$	25.4558i	0.944105i	−0.881570	+	0.472052i	$0.843513\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	44.7214i	1.65408i
$732$	0	0
$733$	−6.32456	−0.233603	−0.116801	−	0.993155i	$-0.537264\pi$
$733$	−6.32456	−0.233603	−0.116801	+	0.993155i	$0.537264\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 25.0000i	− 0.920887i
$738$	0	0
$739$	− 8.94427i	− 0.329020i	−0.986375	−	0.164510i	$-0.947396\pi$
$739$	− 8.94427i	− 0.329020i	0.986375	−	0.164510i	$-0.0526043\pi$
$740$	0	0
$741$	31.6228i	1.16169i
$742$	0	0
$743$	5.65685i	0.207530i	0.994602	+	0.103765i	$0.0330890\pi$
$743$	5.65685i	0.207530i	−0.994602	+	0.103765i	$0.966911\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−13.4164	−0.490881
$748$	0	0
$749$	− 44.2719i	− 1.61766i
$750$	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
$753$	25.0000i	0.911051i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.6491	−0.459740	−0.229870	−	0.973221i	$-0.573830\pi$
$757$	−12.6491	−0.459740	−0.229870	+	0.973221i	$0.573830\pi$
$758$	0	0
$759$	−28.2843	−1.02665
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	−17.8885	−0.647609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	56.5685i	2.04257i
$768$	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
$771$	− 22.3607i	− 0.805300i
$772$	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
$777$	40.0000i	1.43499i
$778$	0	0
$779$	6.70820i	0.240346i
$780$	0	0
$781$	− 31.6228i	− 1.13155i
$782$	0	0
$783$	14.1421i	0.505399i
$784$	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
$789$	− 50.5964i	− 1.80128i
$790$	0	0
$791$	−42.4264	−1.50851
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	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
$811$	8.94427i	0.314076i	0.987593	+	0.157038i	$0.0501945\pi$
$811$	8.94427i	0.314076i	−0.987593	+	0.157038i	$0.949806\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.0000i	0.699711i
$818$	0	0
$819$	35.7771i	1.25015i
$820$	0	0
$821$	− 37.9473i	− 1.32437i	−0.749340	−	0.662186i	$-0.769629\pi$
$821$	− 37.9473i	− 1.32437i	0.749340	−	0.662186i	$-0.230371\pi$
$822$	0	0
$823$	− 5.65685i	− 0.197186i	−0.995128	−	0.0985928i	$-0.968566\pi$
$823$	− 5.65685i	− 0.197186i	0.995128	−	0.0985928i	$-0.0314341\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.1803	0.388779	0.194389	−	0.980924i	$-0.437728\pi$
$827$	11.1803	0.388779	0.194389	+	0.980924i	$0.437728\pi$
$828$	0	0
$829$	6.32456i	0.219661i	0.993950	+	0.109830i	$0.0350308\pi$
$829$	6.32456i	0.219661i	−0.993950	+	0.109830i	$0.964969\pi$
$830$	0	0
$831$	−42.4264	−1.47176
$832$	0	0
$833$	− 5.00000i	− 0.173240i
$834$	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
$847$	− 16.9706i	− 0.583115i
$848$	0	0
$849$	25.0000	0.857998
$850$	0	0
$851$	− 35.7771i	− 1.22642i
$852$	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
$857$	− 45.0000i	− 1.53717i	−0.639747	−	0.768585i	$-0.720961\pi$
$857$	− 45.0000i	− 1.53717i	0.639747	−	0.768585i	$-0.279039\pi$
$858$	0	0
$859$	− 2.23607i	− 0.0762937i	−0.999272	−	0.0381468i	$-0.987855\pi$
$859$	− 2.23607i	− 0.0762937i	0.999272	−	0.0381468i	$-0.0121455\pi$
$860$	0	0
$861$	18.9737i	0.646621i
$862$	0	0
$863$	8.48528i	0.288842i	0.989516	+	0.144421i	$0.0461320\pi$
$863$	8.48528i	0.288842i	−0.989516	+	0.144421i	$0.953868\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.8885	0.607527
$868$	0	0
$869$	31.6228i	1.07273i
$870$	0	0
$871$	70.7107	2.39594
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	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	$-0.415165\pi$
$883$	15.6525	0.526748	0.263374	+	0.964694i	$0.415165\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.3137i	0.379877i	0.981796	+	0.189939i	$0.0608289\pi$
$887$	11.3137i	0.379877i	−0.981796	+	0.189939i	$0.939171\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	− 24.5967i	− 0.824022i
$892$	0	0
$893$	6.32456	0.211643
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 80.0000i	− 2.67112i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 63.2456i	− 2.10701i
$902$	0	0
$903$	56.5685i	1.88248i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.94427	−0.296990	−0.148495	−	0.988913i	$-0.547443\pi$
$907$	−8.94427	−0.296990	−0.148495	+	0.988913i	$0.547443\pi$
$908$	0	0
$909$	− 12.6491i	− 0.419545i
$910$	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
$913$	− 15.0000i	− 0.496428i
$914$	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
$927$	− 16.9706i	− 0.557386i
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 2.23607i	− 0.0732842i
$932$	0	0
$933$	31.6228	1.03528
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.0000i	0.816714i	0.912822	+	0.408357i	$0.133898\pi$
$937$	25.0000i	0.816714i	−0.912822	+	0.408357i	$0.866102\pi$
$938$	0	0
$939$	− 67.0820i	− 2.18914i
$940$	0	0
$941$	31.6228i	1.03087i	0.856928	+	0.515437i	$0.172370\pi$
$941$	31.6228i	1.03087i	−0.856928	+	0.515437i	$0.827630\pi$
$942$	0	0
$943$	− 16.9706i	− 0.552638i
$944$	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
$949$	− 94.8683i	− 3.07956i
$950$	0	0
$951$	−28.2843	−0.917180
$952$	0	0
$953$	45.0000i	1.45769i	0.684677	+	0.728846i	$0.259943\pi$
$953$	45.0000i	1.45769i	−0.684677	+	0.728846i	$0.740057\pi$
$954$	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
$967$	− 16.9706i	− 0.545737i	−0.962051	−	0.272868i	$-0.912028\pi$
$967$	− 16.9706i	− 0.545737i	0.962051	−	0.272868i	$-0.0879723\pi$
$968$	0	0
$969$	25.0000	0.803116
$970$	0	0
$971$	2.23607i	0.0717588i	0.999356	+	0.0358794i	$0.0114232\pi$
$971$	2.23607i	0.0717588i	−0.999356	+	0.0358794i	$0.988577\pi$
$972$	0	0
$973$	44.2719	1.41929
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.0000i	1.43968i	0.694141	+	0.719839i	$0.255784\pi$
$977$	45.0000i	1.43968i	−0.694141	+	0.719839i	$0.744216\pi$
$978$	0	0
$979$	− 2.23607i	− 0.0714650i
$980$	0	0
$981$	− 12.6491i	− 0.403855i
$982$	0	0
$983$	− 19.7990i	− 0.631490i	−0.948844	−	0.315745i	$-0.897746\pi$
$983$	− 19.7990i	− 0.631490i	0.948844	−	0.315745i	$-0.102254\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.8885	0.569399
$988$	0	0
$989$	− 50.5964i	− 1.60887i
$990$	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
$993$	75.0000i	2.38005i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25.2982	−0.801203	−0.400601	−	0.916252i	$-0.631199\pi$
$997$	−25.2982	−0.801203	−0.400601	+	0.916252i	$0.631199\pi$
$998$	0	0
$999$	14.1421	0.447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.f.r.449.2		8
4.3	odd	2	inner	3200.2.f.r.449.7		8
5.2	odd	4		3200.2.d.m.1601.4	yes	4
5.3	odd	4		3200.2.d.r.1601.1	yes	4
5.4	even	2	inner	3200.2.f.r.449.8		8
8.3	odd	2	inner	3200.2.f.r.449.4		8
8.5	even	2	inner	3200.2.f.r.449.5		8
20.3	even	4		3200.2.d.r.1601.4	yes	4
20.7	even	4		3200.2.d.m.1601.1	✓	4
20.19	odd	2	inner	3200.2.f.r.449.1		8
40.3	even	4		3200.2.d.r.1601.2	yes	4
40.13	odd	4		3200.2.d.r.1601.3	yes	4
40.19	odd	2	inner	3200.2.f.r.449.6		8
40.27	even	4		3200.2.d.m.1601.3	yes	4
40.29	even	2	inner	3200.2.f.r.449.3		8
40.37	odd	4		3200.2.d.m.1601.2	yes	4
80.3	even	4		6400.2.a.cs.1.1		4
80.13	odd	4		6400.2.a.cs.1.4		4
80.27	even	4		6400.2.a.cp.1.2		4
80.37	odd	4		6400.2.a.cp.1.3		4
80.43	even	4		6400.2.a.cs.1.3		4
80.53	odd	4		6400.2.a.cs.1.2		4
80.67	even	4		6400.2.a.cp.1.4		4
80.77	odd	4		6400.2.a.cp.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.m.1601.1	✓	4	20.7	even	4
3200.2.d.m.1601.2	yes	4	40.37	odd	4
3200.2.d.m.1601.3	yes	4	40.27	even	4
3200.2.d.m.1601.4	yes	4	5.2	odd	4
3200.2.d.r.1601.1	yes	4	5.3	odd	4
3200.2.d.r.1601.2	yes	4	40.3	even	4
3200.2.d.r.1601.3	yes	4	40.13	odd	4
3200.2.d.r.1601.4	yes	4	20.3	even	4
3200.2.f.r.449.1		8	20.19	odd	2	inner
3200.2.f.r.449.2		8	1.1	even	1	trivial
3200.2.f.r.449.3		8	40.29	even	2	inner
3200.2.f.r.449.4		8	8.3	odd	2	inner
3200.2.f.r.449.5		8	8.5	even	2	inner
3200.2.f.r.449.6		8	40.19	odd	2	inner
3200.2.f.r.449.7		8	4.3	odd	2	inner
3200.2.f.r.449.8		8	5.4	even	2	inner
6400.2.a.cp.1.1		4	80.77	odd	4
6400.2.a.cp.1.2		4	80.27	even	4
6400.2.a.cp.1.3		4	80.37	odd	4
6400.2.a.cp.1.4		4	80.67	even	4
6400.2.a.cs.1.1		4	80.3	even	4
6400.2.a.cs.1.2		4	80.53	odd	4
6400.2.a.cs.1.3		4	80.43	even	4
6400.2.a.cs.1.4		4	80.13	odd	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} +2.23607i q^{11} -6.32456 q^{13} +5.00000i q^{17} +2.23607i q^{19} +6.32456i q^{21} -5.65685i q^{23} +2.23607 q^{27} +6.32456i q^{29} -5.00000i q^{33} +6.32456 q^{37} +14.1421 q^{39} +3.00000 q^{41} +8.94427 q^{43} -2.82843i q^{47} -1.00000 q^{49} -11.1803i q^{51} -12.6491 q^{53} -5.00000i q^{57} -8.94427i q^{59} +6.32456i q^{61} -5.65685i q^{63} -11.1803 q^{67} +12.6491i q^{69} -14.1421 q^{71} +15.0000i q^{73} +6.32456 q^{77} +14.1421 q^{79} -11.0000 q^{81} -6.70820 q^{83} -14.1421i q^{87} -1.00000 q^{89} +17.8885i q^{91} -10.0000i q^{97} +4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 24 q^{41} - 8 q^{49} - 88 q^{81} - 8 q^{89}+O(q^{100}) \) 8 * q + 16 * q^9 + 24 * q^41 - 8 * q^49 - 88 * q^81 - 8 * q^89

Introduction

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Modular forms

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