LMFDB - Embedded newform 3200.2.d.r.1601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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, 0]))

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

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

chi := DirichletCharacter("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.5521286468$
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} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.1
Root			$-0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.r.1601.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	− 2.23607i	− 1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	− 2.23607i	− 1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	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
$13$	− 6.32456i	− 1.75412i	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	− 6.32456i	− 1.75412i	0.480384	−	0.877058i	$-0.340497\pi$
$14$	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
$19$	− 2.23607i	− 0.512989i	−0.966546	−	0.256495i	$-0.917432\pi$
$19$	− 2.23607i	− 0.512989i	0.966546	−	0.256495i	$-0.0825676\pi$
$20$	0	0
$21$	6.32456i	1.38013i
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.23607i	− 0.430331i
$28$	0	0
$29$	− 6.32456i	− 1.17444i	−0.809427	−	0.587220i	$-0.800222\pi$
$29$	− 6.32456i	− 1.17444i	0.809427	−	0.587220i	$-0.199778\pi$
$30$	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
$37$	− 6.32456i	− 1.03975i	−0.854242	−	0.519875i	$-0.825978\pi$
$37$	− 6.32456i	− 1.03975i	0.854242	−	0.519875i	$-0.174022\pi$
$38$	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
$43$	8.94427i	1.36399i	0.731357	+	0.681994i	$0.238887\pi$
$43$	8.94427i	1.36399i	−0.731357	+	0.681994i	$0.761113\pi$
$44$	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
$51$	− 11.1803i	− 1.56556i
$52$	0	0
$53$	− 12.6491i	− 1.73749i	−0.495261	−	0.868744i	$-0.664927\pi$
$53$	− 12.6491i	− 1.73749i	0.495261	−	0.868744i	$-0.335073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	8.94427i	1.16445i	0.813029	+	0.582223i	$0.197817\pi$
$59$	8.94427i	1.16445i	−0.813029	+	0.582223i	$0.802183\pi$
$60$	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
$63$	5.65685	0.712697
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.1803i	1.36590i	0.730467	+	0.682948i	$0.239302\pi$
$67$	11.1803i	1.36590i	−0.730467	+	0.682948i	$0.760698\pi$
$68$	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
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.32456i	− 0.720750i
$78$	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
$83$	− 6.70820i	− 0.736321i	−0.929762	−	0.368161i	$-0.879988\pi$
$83$	− 6.70820i	− 0.736321i	0.929762	−	0.368161i	$-0.120012\pi$
$84$	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	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	17.8885i	1.87523i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	− 4.47214i	− 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
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.6525i	− 1.51318i	−0.653888	−	0.756591i	$-0.726863\pi$
$107$	− 15.6525i	− 1.51318i	0.653888	−	0.756591i	$-0.273137\pi$
$108$	0	0
$109$	6.32456i	0.605783i	0.953025	+	0.302891i	$0.0979519\pi$
$109$	6.32456i	0.605783i	−0.953025	+	0.302891i	$0.902048\pi$
$110$	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
$117$	12.6491i	1.16941i
$118$	0	0
$119$	−14.1421	−1.29641
$120$	0	0
$121$	6.00000	0.545455
$122$	0	0
$123$	− 6.70820i	− 0.604858i
$124$	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
$131$	− 8.94427i	− 0.781465i	−0.920504	−	0.390732i	$-0.872222\pi$
$131$	− 8.94427i	− 0.781465i	0.920504	−	0.390732i	$-0.127778\pi$
$132$	0	0
$133$	6.32456i	0.548408i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	− 15.6525i	− 1.32763i	−0.747899	−	0.663813i	$-0.768937\pi$
$139$	− 15.6525i	− 1.32763i	0.747899	−	0.663813i	$-0.231063\pi$
$140$	0	0
$141$	6.32456i	0.532624i
$142$	0	0
$143$	14.1421	1.18262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.23607i	− 0.184428i
$148$	0	0
$149$	18.9737i	1.55438i	0.629264	+	0.777192i	$0.283356\pi$
$149$	18.9737i	1.55438i	−0.629264	+	0.777192i	$0.716644\pi$
$150$	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
$153$	−10.0000	−0.808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−28.2843	−2.24309
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	− 6.70820i	− 0.525427i	−0.964874	−	0.262714i	$-0.915383\pi$
$163$	− 6.70820i	− 0.525427i	0.964874	−	0.262714i	$-0.0846174\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.82843	0.218870	0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843	0.218870	0.109435	+	0.993994i	$0.465096\pi$
$168$	0	0
$169$	−27.0000	−2.07692
$170$	0	0
$171$	4.47214i	0.341993i
$172$	0	0
$173$	12.6491i	0.961694i	0.876804	+	0.480847i	$0.159671\pi$
$173$	12.6491i	0.961694i	−0.876804	+	0.480847i	$0.840329\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.0000	1.50329
$178$	0	0
$179$	− 2.23607i	− 0.167132i	−0.996502	−	0.0835658i	$-0.973369\pi$
$179$	− 2.23607i	− 0.167132i	0.996502	−	0.0835658i	$-0.0266309\pi$
$180$	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
$183$	14.1421	1.04542
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.1803i	0.817587i
$188$	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
$193$	5.00000	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	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	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
$203$	17.8885i	1.25553i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.3137	−0.786357
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	− 6.70820i	− 0.461812i	−0.972976	−	0.230906i	$-0.925831\pi$
$211$	− 6.70820i	− 0.461812i	0.972976	−	0.230906i	$-0.0741690\pi$
$212$	0	0
$213$	31.6228i	2.16676i
$214$	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
$223$	22.6274	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
$227$	8.94427i	0.593652i	0.954932	+	0.296826i	$0.0959282\pi$
$227$	8.94427i	0.593652i	−0.954932	+	0.296826i	$0.904072\pi$
$228$	0	0
$229$	12.6491i	0.835877i	0.908475	+	0.417938i	$0.137247\pi$
$229$	12.6491i	0.835877i	−0.908475	+	0.417938i	$0.862753\pi$
$230$	0	0
$231$	−14.1421	−0.930484
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	31.6228i	2.05412i
$238$	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
$243$	17.8885i	1.14755i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.1421	−0.899843
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	− 11.1803i	− 0.705697i	−0.935681	−	0.352848i	$-0.885213\pi$
$251$	− 11.1803i	− 0.705697i	0.935681	−	0.352848i	$-0.114787\pi$
$252$	0	0
$253$	12.6491i	0.795243i
$254$	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
$259$	17.8885i	1.11154i
$260$	0	0
$261$	12.6491i	0.782960i
$262$	0	0
$263$	−22.6274	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.23607i	− 0.136845i
$268$	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
$273$	40.0000	2.42091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 18.9737i	− 1.14002i	−0.821639	−	0.570009i	$-0.806940\pi$
$277$	− 18.9737i	− 1.14002i	0.821639	−	0.570009i	$-0.193060\pi$
$278$	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
$283$	− 11.1803i	− 0.664602i	−0.943173	−	0.332301i	$-0.892175\pi$
$283$	− 11.1803i	− 0.664602i	0.943173	−	0.332301i	$-0.107825\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	22.3607i	1.31081i
$292$	0	0
$293$	6.32456i	0.369484i	0.982787	+	0.184742i	$0.0591450\pi$
$293$	6.32456i	0.369484i	−0.982787	+	0.184742i	$0.940855\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	− 35.7771i	− 2.06904i
$300$	0	0
$301$	− 25.2982i	− 1.45817i
$302$	0	0
$303$	−14.1421	−0.812444
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 20.1246i	− 1.14857i	−0.818655	−	0.574286i	$-0.805280\pi$
$307$	− 20.1246i	− 1.14857i	0.818655	−	0.574286i	$-0.194720\pi$
$308$	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
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.6491i	− 0.710445i	−0.934782	−	0.355222i	$-0.884405\pi$
$317$	− 12.6491i	− 0.710445i	0.934782	−	0.355222i	$-0.115595\pi$
$318$	0	0
$319$	14.1421	0.791808
$320$	0	0
$321$	−35.0000	−1.95351
$322$	0	0
$323$	− 11.1803i	− 0.622091i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.1421	0.782062
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	− 33.5410i	− 1.84358i	−0.387688	−	0.921791i	$-0.626726\pi$
$331$	− 33.5410i	− 1.84358i	0.387688	−	0.921791i	$-0.373274\pi$
$332$	0	0
$333$	12.6491i	0.693167i
$334$	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
$339$	− 33.5410i	− 1.82170i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.1246i	1.08035i	0.841554	+	0.540173i	$0.181641\pi$
$347$	20.1246i	1.08035i	−0.841554	+	0.540173i	$0.818359\pi$
$348$	0	0
$349$	18.9737i	1.01564i	0.861464	+	0.507819i	$0.169548\pi$
$349$	18.9737i	1.01564i	−0.861464	+	0.507819i	$0.830452\pi$
$350$	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
$357$	31.6228i	1.67365i
$358$	0	0
$359$	−14.1421	−0.746393	−0.373197	−	0.927752i	$-0.621738\pi$
$359$	−14.1421	−0.746393	−0.373197	+	0.927752i	$0.621738\pi$
$360$	0	0
$361$	14.0000	0.736842
$362$	0	0
$363$	− 13.4164i	− 0.704179i
$364$	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
$371$	35.7771i	1.85745i
$372$	0	0
$373$	− 12.6491i	− 0.654946i	−0.944861	−	0.327473i	$-0.893803\pi$
$373$	− 12.6491i	− 0.654946i	0.944861	−	0.327473i	$-0.106197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−40.0000	−2.06010
$378$	0	0
$379$	6.70820i	0.344577i	0.985047	+	0.172289i	$0.0551162\pi$
$379$	6.70820i	0.344577i	−0.985047	+	0.172289i	$0.944884\pi$
$380$	0	0
$381$	37.9473i	1.94410i
$382$	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
$387$	− 17.8885i	− 0.909326i
$388$	0	0
$389$	25.2982i	1.28267i	0.767261	+	0.641335i	$0.221619\pi$
$389$	25.2982i	1.28267i	−0.767261	+	0.641335i	$0.778381\pi$
$390$	0	0
$391$	28.2843	1.43040
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 25.2982i	− 1.26968i	−0.772643	−	0.634841i	$-0.781066\pi$
$397$	− 25.2982i	− 1.26968i	0.772643	−	0.634841i	$-0.218934\pi$
$398$	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	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	11.1803i	0.551485i
$412$	0	0
$413$	− 25.2982i	− 1.24484i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−35.0000	−1.71396
$418$	0	0
$419$	33.5410i	1.63859i	0.573375	+	0.819293i	$0.305634\pi$
$419$	33.5410i	1.63859i	−0.573375	+	0.819293i	$0.694366\pi$
$420$	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
$423$	5.65685	0.275046
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 17.8885i	− 0.865687i
$428$	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
$433$	5.00000	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
$437$	− 12.6491i	− 0.605089i
$438$	0	0
$439$	14.1421	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$439$	14.1421	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	− 29.0689i	− 1.38110i	−0.723283	−	0.690552i	$-0.757368\pi$
$443$	− 29.0689i	− 1.38110i	0.723283	−	0.690552i	$-0.242632\pi$
$444$	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
$451$	6.70820i	0.315877i
$452$	0	0
$453$	− 31.6228i	− 1.48577i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	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
$463$	−5.65685	−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
$467$	8.94427i	0.413892i	0.978352	+	0.206946i	$0.0663524\pi$
$467$	8.94427i	0.413892i	−0.978352	+	0.206946i	$0.933648\pi$
$468$	0	0
$469$	− 31.6228i	− 1.46020i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.2982i	1.15833i
$478$	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
$483$	35.7771i	1.62791i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−39.5980	−1.79436	−0.897178	−	0.441669i	$-0.854386\pi$
$487$	−39.5980	−1.79436	−0.897178	+	0.441669i	$0.854386\pi$
$488$	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
$493$	− 31.6228i	− 1.42422i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	26.8328i	1.20120i	0.799549	+	0.600601i	$0.205072\pi$
$499$	26.8328i	1.20120i	−0.799549	+	0.600601i	$0.794928\pi$
$500$	0	0
$501$	− 6.32456i	− 0.282560i
$502$	0	0
$503$	−33.9411	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$503$	−33.9411	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	60.3738i	2.68130i
$508$	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
$513$	−5.00000	−0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.32456i	− 0.278154i
$518$	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
$523$	20.1246i	0.879988i	0.898001	+	0.439994i	$0.145019\pi$
$523$	20.1246i	0.879988i	−0.898001	+	0.439994i	$0.854981\pi$
$524$	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
$533$	− 18.9737i	− 0.821841i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.00000	−0.215766
$538$	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
$543$	−28.2843	−1.21379
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.1246i	− 0.860466i	−0.902718	−	0.430233i	$-0.858431\pi$
$547$	− 20.1246i	− 0.860466i	0.902718	−	0.430233i	$-0.141569\pi$
$548$	0	0
$549$	− 12.6491i	− 0.539851i
$550$	0	0
$551$	−14.1421	−0.602475
$552$	0	0
$553$	40.0000	1.70097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 44.2719i	− 1.87586i	−0.346825	−	0.937930i	$-0.612740\pi$
$557$	− 44.2719i	− 1.87586i	0.346825	−	0.937930i	$-0.387260\pi$
$558$	0	0
$559$	56.5685	2.39259
$560$	0	0
$561$	25.0000	1.05550
$562$	0	0
$563$	44.7214i	1.88478i	0.334515	+	0.942390i	$0.391427\pi$
$563$	44.7214i	1.88478i	−0.334515	+	0.942390i	$0.608573\pi$
$564$	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	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	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
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	− 11.1803i	− 0.464639i
$580$	0	0
$581$	18.9737i	0.787160i
$582$	0	0
$583$	28.2843	1.17141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11.1803i	− 0.461462i	−0.973018	−	0.230731i	$-0.925888\pi$
$587$	− 11.1803i	− 0.461462i	0.973018	−	0.230731i	$-0.0741117\pi$
$588$	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	$-0.406705\pi$
$599$	14.1421	0.577832	0.288916	+	0.957354i	$0.406705\pi$
$600$	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
$603$	− 22.3607i	− 0.910597i
$604$	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
$611$	17.8885i	0.723693i
$612$	0	0
$613$	− 25.2982i	− 1.02179i	−0.859644	−	0.510893i	$-0.829315\pi$
$613$	− 25.2982i	− 1.02179i	0.859644	−	0.510893i	$-0.170685\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	− 26.8328i	− 1.07850i	−0.842145	−	0.539251i	$-0.818707\pi$
$619$	− 26.8328i	− 1.07850i	0.842145	−	0.539251i	$-0.181293\pi$
$620$	0	0
$621$	− 12.6491i	− 0.507591i
$622$	0	0
$623$	−2.82843	−0.113319
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11.1803i	− 0.446500i
$628$	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
$633$	−15.0000	−0.596196
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.32456i	− 0.250588i
$638$	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
$643$	− 26.8328i	− 1.05818i	−0.848565	−	0.529091i	$-0.822533\pi$
$643$	− 26.8328i	− 1.05818i	0.848565	−	0.529091i	$-0.177467\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.2548	1.77915	0.889576	−	0.456788i	$-0.151000\pi$
$647$	45.2548	1.77915	0.889576	+	0.456788i	$0.151000\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.32456i	− 0.247499i	−0.992313	−	0.123749i	$-0.960508\pi$
$653$	− 6.32456i	− 0.247499i	0.992313	−	0.123749i	$-0.0394919\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000	1.17041
$658$	0	0
$659$	− 6.70820i	− 0.261315i	−0.991428	−	0.130657i	$-0.958291\pi$
$659$	− 6.70820i	− 0.261315i	0.991428	−	0.130657i	$-0.0417087\pi$
$660$	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
$663$	−70.7107	−2.74618
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 35.7771i	− 1.38529i
$668$	0	0
$669$	− 50.5964i	− 1.95617i
$670$	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
$677$	− 37.9473i	− 1.45843i	−0.684282	−	0.729217i	$-0.739884\pi$
$677$	− 37.9473i	− 1.45843i	0.684282	−	0.729217i	$-0.260116\pi$
$678$	0	0
$679$	28.2843	1.08545
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	2.23607i	0.0855608i	0.999085	+	0.0427804i	$0.0136216\pi$
$683$	2.23607i	0.0855608i	−0.999085	+	0.0427804i	$0.986378\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.2843	1.07911
$688$	0	0
$689$	−80.0000	−3.04776
$690$	0	0
$691$	33.5410i	1.27596i	0.770053	+	0.637980i	$0.220230\pi$
$691$	33.5410i	1.27596i	−0.770053	+	0.637980i	$0.779770\pi$
$692$	0	0
$693$	12.6491i	0.480500i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.0000	0.568166
$698$	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
$703$	−14.1421	−0.533381
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.8885i	0.672768i
$708$	0	0
$709$	12.6491i	0.475047i	0.971382	+	0.237524i	$0.0763357\pi$
$709$	12.6491i	0.475047i	−0.971382	+	0.237524i	$0.923664\pi$
$710$	0	0
$711$	28.2843	1.06074
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.6228i	1.18097i
$718$	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
$723$	− 29.0689i	− 1.08108i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.4558	0.944105	0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558	0.944105	0.472052	+	0.881570i	$0.343513\pi$
$728$	0	0
$729$	7.00000	0.259259
$730$	0	0
$731$	44.7214i	1.65408i
$732$	0	0
$733$	− 6.32456i	− 0.233603i	−0.993155	−	0.116801i	$-0.962736\pi$
$733$	− 6.32456i	− 0.233603i	0.993155	−	0.116801i	$-0.0372641\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.0000	−0.920887
$738$	0	0
$739$	8.94427i	0.329020i	0.986375	+	0.164510i	$0.0526043\pi$
$739$	8.94427i	0.329020i	−0.986375	+	0.164510i	$0.947396\pi$
$740$	0	0
$741$	31.6228i	1.16169i
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.4164i	0.490881i
$748$	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
$753$	−25.0000	−0.911051
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.6491i	0.459740i	0.973221	+	0.229870i	$0.0738301\pi$
$757$	12.6491i	0.459740i	−0.973221	+	0.229870i	$0.926170\pi$
$758$	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
$763$	− 17.8885i	− 0.647609i
$764$	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
$771$	− 22.3607i	− 0.805300i
$772$	0	0
$773$	31.6228i	1.13739i	0.822548	+	0.568696i	$0.192552\pi$
$773$	31.6228i	1.13739i	−0.822548	+	0.568696i	$0.807448\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	40.0000	1.43499
$778$	0	0
$779$	− 6.70820i	− 0.240346i
$780$	0	0
$781$	− 31.6228i	− 1.13155i
$782$	0	0
$783$	−14.1421	−0.505399
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.8328i	0.956487i	0.878227	+	0.478243i	$0.158726\pi$
$787$	26.8328i	0.956487i	−0.878227	+	0.478243i	$0.841274\pi$
$788$	0	0
$789$	50.5964i	1.80128i
$790$	0	0
$791$	−42.4264	−1.50851
$792$	0	0
$793$	40.0000	1.42044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.2982i	0.896109i	0.894006	+	0.448054i	$0.147883\pi$
$797$	25.2982i	0.896109i	−0.894006	+	0.448054i	$0.852117\pi$
$798$	0	0
$799$	−14.1421	−0.500313
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	− 33.5410i	− 1.18364i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	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
$817$	20.0000	0.699711
$818$	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
$823$	5.65685	0.197186	0.0985928	−	0.995128i	$-0.468566\pi$
$823$	5.65685	0.197186	0.0985928	+	0.995128i	$0.468566\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11.1803i	− 0.388779i	−0.980924	−	0.194389i	$-0.937728\pi$
$827$	− 11.1803i	− 0.388779i	0.980924	−	0.194389i	$-0.0622725\pi$
$828$	0	0
$829$	− 6.32456i	− 0.219661i	−0.993950	−	0.109830i	$-0.964969\pi$
$829$	− 6.32456i	− 0.219661i	0.993950	−	0.109830i	$-0.0350308\pi$
$830$	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	$-0.0691711\pi$
$839$	56.5685	1.95296	0.976481	+	0.215601i	$0.0691711\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	4.47214i	0.154029i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.9706	−0.583115
$848$	0	0
$849$	−25.0000	−0.857998
$850$	0	0
$851$	− 35.7771i	− 1.22642i
$852$	0	0
$853$	12.6491i	0.433097i	0.976272	+	0.216549i	$0.0694800\pi$
$853$	12.6491i	0.433097i	−0.976272	+	0.216549i	$0.930520\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	0	0
$859$	2.23607i	0.0762937i	0.999272	+	0.0381468i	$0.0121455\pi$
$859$	2.23607i	0.0762937i	−0.999272	+	0.0381468i	$0.987855\pi$
$860$	0	0
$861$	18.9737i	0.646621i
$862$	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
$867$	− 17.8885i	− 0.607527i
$868$	0	0
$869$	− 31.6228i	− 1.07273i
$870$	0	0
$871$	70.7107	2.39594
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.6491i	0.427130i	0.976929	+	0.213565i	$0.0685075\pi$
$877$	12.6491i	0.427130i	−0.976929	+	0.213565i	$0.931492\pi$
$878$	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
$883$	15.6525i	0.526748i	0.964694	+	0.263374i	$0.0848353\pi$
$883$	15.6525i	0.526748i	−0.964694	+	0.263374i	$0.915165\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.3137	0.379877	0.189939	−	0.981796i	$-0.439171\pi$
$887$	11.3137	0.379877	0.189939	+	0.981796i	$0.439171\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	− 24.5967i	− 0.824022i
$892$	0	0
$893$	6.32456i	0.211643i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−80.0000	−2.67112
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 63.2456i	− 2.10701i
$902$	0	0
$903$	−56.5685	−1.88248
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.94427i	0.296990i	0.988913	+	0.148495i	$0.0474428\pi$
$907$	8.94427i	0.296990i	−0.988913	+	0.148495i	$0.952557\pi$
$908$	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
$913$	15.0000	0.496428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25.2982i	0.835421i
$918$	0	0
$919$	28.2843	0.933012	0.466506	−	0.884518i	$-0.345513\pi$
$919$	28.2843	0.933012	0.466506	+	0.884518i	$0.345513\pi$
$920$	0	0
$921$	−45.0000	−1.48280
$922$	0	0
$923$	89.4427i	2.94404i
$924$	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
$931$	− 2.23607i	− 0.0732842i
$932$	0	0
$933$	31.6228i	1.03528i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	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
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26.8328i	− 0.871949i	−0.899959	−	0.435975i	$-0.856404\pi$
$947$	− 26.8328i	− 0.871949i	0.899959	−	0.435975i	$-0.143596\pi$
$948$	0	0
$949$	94.8683i	3.07956i
$950$	0	0
$951$	−28.2843	−0.917180
$952$	0	0
$953$	−45.0000	−1.45769	−0.728846	−	0.684677i	$-0.759943\pi$
$953$	−45.0000	−1.45769	−0.728846	+	0.684677i	$0.759943\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 31.6228i	− 1.02222i
$958$	0	0
$959$	14.1421	0.456673
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	31.3050i	1.00879i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.9706	−0.545737	−0.272868	−	0.962051i	$-0.587972\pi$
$967$	−16.9706	−0.545737	−0.272868	+	0.962051i	$0.587972\pi$
$968$	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
$973$	44.2719i	1.41929i
$974$	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
$979$	2.23607i	0.0714650i
$980$	0	0
$981$	− 12.6491i	− 0.403855i
$982$	0	0
$983$	19.7990	0.631490	0.315745	−	0.948844i	$-0.397746\pi$
$983$	19.7990	0.631490	0.315745	+	0.948844i	$0.397746\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 17.8885i	− 0.569399i
$988$	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
$993$	−75.0000	−2.38005
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.2982i	0.801203i	0.916252	+	0.400601i	$0.131199\pi$
$997$	25.2982i	0.801203i	−0.916252	+	0.400601i	$0.868801\pi$
$998$	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.d.r.1601.1	yes	4
4.3	odd	2	inner	3200.2.d.r.1601.4	yes	4
5.2	odd	4		3200.2.f.r.449.2		8
5.3	odd	4		3200.2.f.r.449.8		8
5.4	even	2		3200.2.d.m.1601.4	yes	4
8.3	odd	2	inner	3200.2.d.r.1601.2	yes	4
8.5	even	2	inner	3200.2.d.r.1601.3	yes	4
16.3	odd	4		6400.2.a.cs.1.1		4
16.5	even	4		6400.2.a.cs.1.2		4
16.11	odd	4		6400.2.a.cs.1.3		4
16.13	even	4		6400.2.a.cs.1.4		4
20.3	even	4		3200.2.f.r.449.1		8
20.7	even	4		3200.2.f.r.449.7		8
20.19	odd	2		3200.2.d.m.1601.1	✓	4
40.3	even	4		3200.2.f.r.449.6		8
40.13	odd	4		3200.2.f.r.449.3		8
40.19	odd	2		3200.2.d.m.1601.3	yes	4
40.27	even	4		3200.2.f.r.449.4		8
40.29	even	2		3200.2.d.m.1601.2	yes	4
40.37	odd	4		3200.2.f.r.449.5		8
80.19	odd	4		6400.2.a.cp.1.4		4
80.29	even	4		6400.2.a.cp.1.1		4
80.59	odd	4		6400.2.a.cp.1.2		4
80.69	even	4		6400.2.a.cp.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.m.1601.1	✓	4	20.19	odd	2
3200.2.d.m.1601.2	yes	4	40.29	even	2
3200.2.d.m.1601.3	yes	4	40.19	odd	2
3200.2.d.m.1601.4	yes	4	5.4	even	2
3200.2.d.r.1601.1	yes	4	1.1	even	1	trivial
3200.2.d.r.1601.2	yes	4	8.3	odd	2	inner
3200.2.d.r.1601.3	yes	4	8.5	even	2	inner
3200.2.d.r.1601.4	yes	4	4.3	odd	2	inner
3200.2.f.r.449.1		8	20.3	even	4
3200.2.f.r.449.2		8	5.2	odd	4
3200.2.f.r.449.3		8	40.13	odd	4
3200.2.f.r.449.4		8	40.27	even	4
3200.2.f.r.449.5		8	40.37	odd	4
3200.2.f.r.449.6		8	40.3	even	4
3200.2.f.r.449.7		8	20.7	even	4
3200.2.f.r.449.8		8	5.3	odd	4
6400.2.a.cp.1.1		4	80.29	even	4
6400.2.a.cp.1.2		4	80.59	odd	4
6400.2.a.cp.1.3		4	80.69	even	4
6400.2.a.cp.1.4		4	80.19	odd	4
6400.2.a.cs.1.1		4	16.3	odd	4
6400.2.a.cs.1.2		4	16.5	even	4
6400.2.a.cs.1.3		4	16.11	odd	4
6400.2.a.cs.1.4		4	16.13	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

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