LMFDB - Embedded newform 3600.3.c.k.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.c (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:	$98.0928951697$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.k.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.48683i q^{7} +O(q^{10})$	$q-5.48683i q^{7} +9.17377i q^{11} -11.4868i q^{13} +16.9706 q^{17} +26.9737 q^{19} +4.93113 q^{23} -20.5247i q^{29} -20.9737 q^{31} -62.4605i q^{37} +40.9377i q^{41} +1.02633i q^{43} -86.2298 q^{47} +18.8947 q^{49} +96.0920 q^{53} +112.374i q^{59} -66.9210 q^{61} +76.0000i q^{67} -24.0789i q^{71} +18.9210i q^{73} +50.3349 q^{77} +106.921 q^{79} +45.1804 q^{83} -115.928i q^{89} -63.0263 q^{91} -87.0263i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 64 q^{19} - 16 q^{31} - 456 q^{49} - 80 q^{61} + 400 q^{79} - 656 q^{91}+O(q^{100}) $ 8 * q + 64 * q^19 - 16 * q^31 - 456 * q^49 - 80 * q^61 + 400 * q^79 - 656 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.48683i	− 0.783833i	−0.920001	−	0.391917i	$-0.871812\pi$
$7$	− 5.48683i	− 0.783833i	0.920001	−	0.391917i	$-0.128188\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.17377i	0.833979i	0.908911	+	0.416989i	$0.136915\pi$
$11$	9.17377i	0.833979i	−0.908911	+	0.416989i	$0.863085\pi$
$12$	0	0
$13$	− 11.4868i	− 0.883603i	−0.897113	−	0.441801i	$-0.854340\pi$
$13$	− 11.4868i	− 0.883603i	0.897113	−	0.441801i	$-0.145660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16.9706	0.998268	0.499134	−	0.866525i	$-0.333651\pi$
$17$	16.9706	0.998268	0.499134	+	0.866525i	$0.333651\pi$
$18$	0	0
$19$	26.9737	1.41967	0.709833	−	0.704370i	$-0.248770\pi$
$19$	26.9737	1.41967	0.709833	+	0.704370i	$0.248770\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.93113	0.214397	0.107198	−	0.994238i	$-0.465812\pi$
$23$	4.93113	0.214397	0.107198	+	0.994238i	$0.465812\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 20.5247i	− 0.707749i	−0.935293	−	0.353874i	$-0.884864\pi$
$29$	− 20.5247i	− 0.707749i	0.935293	−	0.353874i	$-0.115136\pi$
$30$	0	0
$31$	−20.9737	−0.676570	−0.338285	−	0.941044i	$-0.609847\pi$
$31$	−20.9737	−0.676570	−0.338285	+	0.941044i	$0.609847\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 62.4605i	− 1.68812i	−0.536247	−	0.844061i	$-0.680159\pi$
$37$	− 62.4605i	− 1.68812i	0.536247	−	0.844061i	$-0.319841\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	40.9377i	0.998481i	0.866464	+	0.499240i	$0.166388\pi$
$41$	40.9377i	0.998481i	−0.866464	+	0.499240i	$0.833612\pi$
$42$	0	0
$43$	1.02633i	0.0238682i	0.999929	+	0.0119341i	$0.00379884\pi$
$43$	1.02633i	0.0238682i	−0.999929	+	0.0119341i	$0.996201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−86.2298	−1.83468	−0.917338	−	0.398109i	$-0.869667\pi$
$47$	−86.2298	−1.83468	−0.917338	+	0.398109i	$0.869667\pi$
$48$	0	0
$49$	18.8947	0.385605
$50$	0	0
$51$	0	0
$52$	0	0
$53$	96.0920	1.81306	0.906529	−	0.422144i	$-0.138722\pi$
$53$	96.0920	1.81306	0.906529	+	0.422144i	$0.138722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	112.374i	1.90465i	0.305092	+	0.952323i	$0.401313\pi$
$59$	112.374i	1.90465i	−0.305092	+	0.952323i	$0.598687\pi$
$60$	0	0
$61$	−66.9210	−1.09707	−0.548533	−	0.836129i	$-0.684813\pi$
$61$	−66.9210	−1.09707	−0.548533	+	0.836129i	$0.684813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	76.0000i	1.13433i	0.823605	+	0.567164i	$0.191960\pi$
$67$	76.0000i	1.13433i	−0.823605	+	0.567164i	$0.808040\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.0789i	− 0.339139i	−0.985518	−	0.169570i	$-0.945762\pi$
$71$	− 24.0789i	− 0.339139i	0.985518	−	0.169570i	$-0.0542377\pi$
$72$	0	0
$73$	18.9210i	0.259192i	0.991567	+	0.129596i	$0.0413680\pi$
$73$	18.9210i	0.259192i	−0.991567	+	0.129596i	$0.958632\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	50.3349	0.653700
$78$	0	0
$79$	106.921	1.35343	0.676715	−	0.736245i	$-0.263403\pi$
$79$	106.921	1.35343	0.676715	+	0.736245i	$0.263403\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	45.1804	0.544342	0.272171	−	0.962249i	$-0.412258\pi$
$83$	45.1804	0.544342	0.272171	+	0.962249i	$0.412258\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 115.928i	− 1.30256i	−0.758835	−	0.651282i	$-0.774231\pi$
$89$	− 115.928i	− 1.30256i	0.758835	−	0.651282i	$-0.225769\pi$
$90$	0	0
$91$	−63.0263	−0.692597
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 87.0263i	− 0.897179i	−0.893738	−	0.448589i	$-0.851927\pi$
$97$	− 87.0263i	− 0.897179i	0.893738	−	0.448589i	$-0.148073\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 129.233i	− 1.27953i	−0.768569	−	0.639767i	$-0.779031\pi$
$101$	− 129.233i	− 1.27953i	0.768569	−	0.639767i	$-0.220969\pi$
$102$	0	0
$103$	114.302i	1.10973i	0.831939	+	0.554866i	$0.187231\pi$
$103$	114.302i	1.10973i	−0.831939	+	0.554866i	$0.812769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−93.3381	−0.872319	−0.436159	−	0.899869i	$-0.643662\pi$
$107$	−93.3381	−0.872319	−0.436159	+	0.899869i	$0.643662\pi$
$108$	0	0
$109$	−120.868	−1.10888	−0.554442	−	0.832222i	$-0.687068\pi$
$109$	−120.868	−1.10888	−0.554442	+	0.832222i	$0.687068\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	110.309	0.976183	0.488091	−	0.872793i	$-0.337693\pi$
$113$	110.309	0.976183	0.488091	+	0.872793i	$0.337693\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 93.1146i	− 0.782476i
$120$	0	0
$121$	36.8420	0.304479
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 105.381i	− 0.829776i	−0.909873	−	0.414888i	$-0.863821\pi$
$127$	− 105.381i	− 0.829776i	0.909873	−	0.414888i	$-0.136179\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 140.584i	− 1.07316i	−0.843850	−	0.536580i	$-0.819716\pi$
$131$	− 140.584i	− 1.07316i	0.843850	−	0.536580i	$-0.180284\pi$
$132$	0	0
$133$	− 148.000i	− 1.11278i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	266.951	1.94855	0.974274	−	0.225365i	$-0.0723575\pi$
$137$	266.951	1.94855	0.974274	+	0.225365i	$0.0723575\pi$
$138$	0	0
$139$	−15.8420	−0.113971	−0.0569856	−	0.998375i	$-0.518149\pi$
$139$	−15.8420	−0.113971	−0.0569856	+	0.998375i	$0.518149\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	105.378	0.736906
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 41.6262i	− 0.279370i	−0.990196	−	0.139685i	$-0.955391\pi$
$149$	− 41.6262i	− 0.279370i	0.990196	−	0.139685i	$-0.0446091\pi$
$150$	0	0
$151$	−103.842	−0.687695	−0.343848	−	0.939025i	$-0.611730\pi$
$151$	−103.842	−0.687695	−0.343848	+	0.939025i	$0.611730\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.5132i	− 0.0797017i	−0.999206	−	0.0398509i	$-0.987312\pi$
$157$	− 12.5132i	− 0.0797017i	0.999206	−	0.0398509i	$-0.0126883\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 27.0563i	− 0.168051i
$162$	0	0
$163$	− 290.868i	− 1.78447i	−0.451573	−	0.892234i	$-0.649137\pi$
$163$	− 290.868i	− 1.78447i	0.451573	−	0.892234i	$-0.350863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	174.637	1.04573	0.522865	−	0.852416i	$-0.324863\pi$
$167$	174.637	1.04573	0.522865	+	0.852416i	$0.324863\pi$
$168$	0	0
$169$	37.0527	0.219247
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−43.5799	−0.251907	−0.125954	−	0.992036i	$-0.540199\pi$
$173$	−43.5799	−0.251907	−0.125954	+	0.992036i	$0.540199\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	88.2952i	0.493270i	0.969109	+	0.246635i	$0.0793248\pi$
$179$	88.2952i	0.493270i	−0.969109	+	0.246635i	$0.920675\pi$
$180$	0	0
$181$	−56.8683	−0.314190	−0.157095	−	0.987584i	$-0.550213\pi$
$181$	−56.8683	−0.314190	−0.157095	+	0.987584i	$0.550213\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	155.684i	0.832535i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 56.6430i	− 0.296560i	−0.988945	−	0.148280i	$-0.952626\pi$
$191$	− 56.6430i	− 0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$192$	0	0
$193$	− 110.000i	− 0.569948i	−0.958535	−	0.284974i	$-0.908015\pi$
$193$	− 110.000i	− 0.569948i	0.958535	−	0.284974i	$-0.0919850\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−245.049	−1.24391	−0.621953	−	0.783055i	$-0.713661\pi$
$197$	−245.049	−1.24391	−0.621953	+	0.783055i	$0.713661\pi$
$198$	0	0
$199$	−169.895	−0.853742	−0.426871	−	0.904313i	$-0.640384\pi$
$199$	−169.895	−0.853742	−0.426871	+	0.904313i	$0.640384\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−112.616	−0.554757
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	247.450i	1.18397i
$210$	0	0
$211$	265.579	1.25867	0.629333	−	0.777136i	$-0.283328\pi$
$211$	265.579	1.25867	0.629333	+	0.777136i	$0.283328\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	115.079i	0.530318i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 194.938i	− 0.882072i
$222$	0	0
$223$	− 187.329i	− 0.840040i	−0.907515	−	0.420020i	$-0.862023\pi$
$223$	− 187.329i	− 0.840040i	0.907515	−	0.420020i	$-0.137977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	421.063	1.85490	0.927452	−	0.373942i	$-0.121994\pi$
$227$	421.063	1.85490	0.927452	+	0.373942i	$0.121994\pi$
$228$	0	0
$229$	102.105	0.445875	0.222937	−	0.974833i	$-0.428435\pi$
$229$	102.105	0.445875	0.222937	+	0.974833i	$0.428435\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	99.0694	0.425191	0.212595	−	0.977140i	$-0.431808\pi$
$233$	99.0694	0.425191	0.212595	+	0.977140i	$0.431808\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 77.9680i	− 0.326226i	−0.986607	−	0.163113i	$-0.947847\pi$
$239$	− 77.9680i	− 0.326226i	0.986607	−	0.163113i	$-0.0521535\pi$
$240$	0	0
$241$	−257.947	−1.07032	−0.535160	−	0.844750i	$-0.679749\pi$
$241$	−257.947	−1.07032	−0.535160	+	0.844750i	$0.679749\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 309.842i	− 1.25442i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 274.971i	− 1.09550i	−0.836641	−	0.547752i	$-0.815484\pi$
$251$	− 274.971i	− 1.09550i	0.836641	−	0.547752i	$-0.184516\pi$
$252$	0	0
$253$	45.2370i	0.178802i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−131.634	−0.512193	−0.256096	−	0.966651i	$-0.582436\pi$
$257$	−131.634	−0.512193	−0.256096	+	0.966651i	$0.582436\pi$
$258$	0	0
$259$	−342.710	−1.32321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	458.782	1.74442	0.872209	−	0.489133i	$-0.162687\pi$
$263$	458.782	1.74442	0.872209	+	0.489133i	$0.162687\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	301.916i	1.12236i	0.827692	+	0.561182i	$0.189653\pi$
$269$	301.916i	1.12236i	−0.827692	+	0.561182i	$0.810347\pi$
$270$	0	0
$271$	−475.842	−1.75587	−0.877937	−	0.478776i	$-0.841081\pi$
$271$	−475.842	−1.75587	−0.877937	+	0.478776i	$0.841081\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	322.039i	1.16260i	0.813691	+	0.581298i	$0.197455\pi$
$277$	322.039i	1.16260i	−0.813691	+	0.581298i	$0.802545\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 352.139i	− 1.25316i	−0.779355	−	0.626582i	$-0.784453\pi$
$281$	− 352.139i	− 1.25316i	0.779355	−	0.626582i	$-0.215547\pi$
$282$	0	0
$283$	− 281.631i	− 0.995164i	−0.867417	−	0.497582i	$-0.834221\pi$
$283$	− 281.631i	− 0.995164i	0.867417	−	0.497582i	$-0.165779\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	224.618	0.782642
$288$	0	0
$289$	−1.00000	−0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	64.3281	0.219550	0.109775	−	0.993956i	$-0.464987\pi$
$293$	64.3281	0.219550	0.109775	+	0.993956i	$0.464987\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 56.6430i	− 0.189442i
$300$	0	0
$301$	5.63132	0.0187087
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 230.158i	− 0.749700i	−0.927085	−	0.374850i	$-0.877694\pi$
$307$	− 230.158i	− 0.749700i	0.927085	−	0.374850i	$-0.122306\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 8.48528i	− 0.0272839i	−0.999907	−	0.0136419i	$-0.995658\pi$
$311$	− 8.48528i	− 0.0272839i	0.999907	−	0.0136419i	$-0.00434250\pi$
$312$	0	0
$313$	− 605.579i	− 1.93476i	−0.253337	−	0.967378i	$-0.581528\pi$
$313$	− 605.579i	− 1.93476i	0.253337	−	0.967378i	$-0.418472\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	87.9601	0.277477	0.138738	−	0.990329i	$-0.455695\pi$
$317$	87.9601	0.277477	0.138738	+	0.990329i	$0.455695\pi$
$318$	0	0
$319$	188.289	0.590248
$320$	0	0
$321$	0	0
$322$	0	0
$323$	457.758	1.41721
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	473.128i	1.43808i
$330$	0	0
$331$	49.2370	0.148752	0.0743761	−	0.997230i	$-0.476303\pi$
$331$	49.2370	0.148752	0.0743761	+	0.997230i	$0.476303\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.71033i	− 0.00804251i	−0.999992	−	0.00402125i	$-0.998720\pi$
$337$	− 2.71033i	− 0.00804251i	0.999992	−	0.00402125i	$-0.00128001\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 192.408i	− 0.564245i
$342$	0	0
$343$	− 372.527i	− 1.08608i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	74.9906	0.216111	0.108056	−	0.994145i	$-0.465538\pi$
$347$	74.9906	0.216111	0.108056	+	0.994145i	$0.465538\pi$
$348$	0	0
$349$	150.921	0.432438	0.216219	−	0.976345i	$-0.430627\pi$
$349$	150.921	0.432438	0.216219	+	0.976345i	$0.430627\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	292.407	0.828349	0.414174	−	0.910198i	$-0.364070\pi$
$353$	292.407	0.828349	0.414174	+	0.910198i	$0.364070\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 576.776i	− 1.60662i	−0.595563	−	0.803309i	$-0.703071\pi$
$359$	− 576.776i	− 1.60662i	0.595563	−	0.803309i	$-0.296929\pi$
$360$	0	0
$361$	366.579	1.01545
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 243.540i	− 0.663595i	−0.943351	−	0.331798i	$-0.892345\pi$
$367$	− 243.540i	− 0.663595i	0.943351	−	0.331798i	$-0.107655\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 527.241i	− 1.42113i
$372$	0	0
$373$	− 115.908i	− 0.310746i	−0.987856	−	0.155373i	$-0.950342\pi$
$373$	− 115.908i	− 0.310746i	0.987856	−	0.155373i	$-0.0496579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−235.764	−0.625369
$378$	0	0
$379$	30.2107	0.0797115	0.0398558	−	0.999205i	$-0.487310\pi$
$379$	30.2107	0.0797115	0.0398558	+	0.999205i	$0.487310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	651.319	1.70057	0.850286	−	0.526320i	$-0.176429\pi$
$383$	651.319	1.70057	0.850286	+	0.526320i	$0.176429\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 205.600i	− 0.528536i	−0.964449	−	0.264268i	$-0.914870\pi$
$389$	− 205.600i	− 0.528536i	0.964449	−	0.264268i	$-0.0851303\pi$
$390$	0	0
$391$	83.6840	0.214026
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	118.355i	0.298124i	0.988828	+	0.149062i	$0.0476254\pi$
$397$	118.355i	0.298124i	−0.988828	+	0.149062i	$0.952375\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 170.971i	− 0.426361i	−0.977013	−	0.213181i	$-0.931618\pi$
$401$	− 170.971i	− 0.426361i	0.977013	−	0.213181i	$-0.0683823\pi$
$402$	0	0
$403$	240.921i	0.597819i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	572.998	1.40786
$408$	0	0
$409$	335.947	0.821387	0.410694	−	0.911773i	$-0.365287\pi$
$409$	335.947	0.821387	0.410694	+	0.911773i	$0.365287\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	616.578	1.49292
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	407.535i	0.972637i	0.873781	+	0.486319i	$0.161661\pi$
$419$	407.535i	0.972637i	−0.873781	+	0.486319i	$0.838339\pi$
$420$	0	0
$421$	−771.210	−1.83185	−0.915926	−	0.401346i	$-0.868542\pi$
$421$	−771.210	−1.83185	−0.915926	+	0.401346i	$0.868542\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	367.184i	0.859916i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	436.210i	1.01209i	0.862508	+	0.506044i	$0.168893\pi$
$431$	436.210i	1.01209i	−0.862508	+	0.506044i	$0.831107\pi$
$432$	0	0
$433$	838.500i	1.93649i	0.250006	+	0.968244i	$0.419568\pi$
$433$	838.500i	1.93649i	−0.250006	+	0.968244i	$0.580432\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	133.011	0.304372
$438$	0	0
$439$	50.0000	0.113895	0.0569476	−	0.998377i	$-0.481863\pi$
$439$	50.0000	0.113895	0.0569476	+	0.998377i	$0.481863\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−492.853	−1.11253	−0.556267	−	0.831003i	$-0.687767\pi$
$443$	−492.853	−1.11253	−0.556267	+	0.831003i	$0.687767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 483.102i	− 1.07595i	−0.842960	−	0.537976i	$-0.819189\pi$
$449$	− 483.102i	− 1.07595i	0.842960	−	0.537976i	$-0.180811\pi$
$450$	0	0
$451$	−375.553	−0.832712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 546.921i	− 1.19676i	−0.801211	−	0.598382i	$-0.795811\pi$
$457$	− 546.921i	− 1.19676i	0.801211	−	0.598382i	$-0.204189\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	130.386i	0.282834i	0.989950	+	0.141417i	$0.0451658\pi$
$461$	130.386i	0.282834i	−0.989950	+	0.141417i	$0.954834\pi$
$462$	0	0
$463$	24.7765i	0.0535130i	0.999642	+	0.0267565i	$0.00851787\pi$
$463$	24.7765i	0.0535130i	−0.999642	+	0.0267565i	$0.991482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−671.491	−1.43788	−0.718941	−	0.695071i	$-0.755373\pi$
$467$	−671.491	−1.43788	−0.718941	+	0.695071i	$0.755373\pi$
$468$	0	0
$469$	416.999	0.889124
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.41535	−0.0199056
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	538.257i	1.12371i	0.827236	+	0.561855i	$0.189912\pi$
$479$	538.257i	1.12371i	−0.827236	+	0.561855i	$0.810088\pi$
$480$	0	0
$481$	−717.473	−1.49163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 608.250i	− 1.24897i	−0.781036	−	0.624486i	$-0.785308\pi$
$487$	− 608.250i	− 1.24897i	0.781036	−	0.624486i	$-0.214692\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 143.561i	− 0.292386i	−0.989256	−	0.146193i	$-0.953298\pi$
$491$	− 143.561i	− 0.292386i	0.989256	−	0.146193i	$-0.0467020\pi$
$492$	0	0
$493$	− 348.316i	− 0.706523i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−132.117	−0.265828
$498$	0	0
$499$	616.605	1.23568	0.617841	−	0.786303i	$-0.288008\pi$
$499$	616.605	1.23568	0.617841	+	0.786303i	$0.288008\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	374.729	0.744989	0.372494	−	0.928034i	$-0.378503\pi$
$503$	374.729	0.744989	0.372494	+	0.928034i	$0.378503\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	173.036i	0.339953i	0.985448	+	0.169977i	$0.0543693\pi$
$509$	173.036i	0.339953i	−0.985448	+	0.169977i	$0.945631\pi$
$510$	0	0
$511$	103.816	0.203163
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 791.052i	− 1.53008i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	455.116i	0.873543i	0.899572	+	0.436772i	$0.143878\pi$
$521$	455.116i	0.873543i	−0.899572	+	0.436772i	$0.856122\pi$
$522$	0	0
$523$	− 295.395i	− 0.564809i	−0.959295	−	0.282404i	$-0.908868\pi$
$523$	− 295.395i	− 0.564809i	0.959295	−	0.282404i	$-0.0911320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−355.935	−0.675398
$528$	0	0
$529$	−504.684	−0.954034
$530$	0	0
$531$	0	0
$532$	0	0
$533$	470.245	0.882260
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	173.335i	0.321587i
$540$	0	0
$541$	906.394	1.67541	0.837703	−	0.546127i	$-0.183898\pi$
$541$	906.394	1.67541	0.837703	+	0.546127i	$0.183898\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	672.921i	1.23020i	0.788448	+	0.615101i	$0.210885\pi$
$547$	672.921i	1.23020i	−0.788448	+	0.615101i	$0.789115\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 553.627i	− 1.00477i
$552$	0	0
$553$	− 586.658i	− 1.06086i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	870.336	1.56254	0.781271	−	0.624192i	$-0.214572\pi$
$557$	870.336	1.56254	0.781271	+	0.624192i	$0.214572\pi$
$558$	0	0
$559$	11.7893	0.0210900
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.6320	0.0419751	0.0209875	−	0.999780i	$-0.493319\pi$
$563$	23.6320	0.0419751	0.0209875	+	0.999780i	$0.493319\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 295.273i	− 0.518933i	−0.965752	−	0.259466i	$-0.916453\pi$
$569$	− 295.273i	− 0.518933i	0.965752	−	0.259466i	$-0.0835467\pi$
$570$	0	0
$571$	−893.920	−1.56553	−0.782767	−	0.622314i	$-0.786193\pi$
$571$	−893.920	−1.56553	−0.782767	+	0.622314i	$0.786193\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 264.763i	− 0.458861i	−0.973325	−	0.229431i	$-0.926314\pi$
$577$	− 264.763i	− 0.458861i	0.973325	−	0.229431i	$-0.0736864\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 247.897i	− 0.426673i
$582$	0	0
$583$	881.526i	1.51205i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−791.885	−1.34904	−0.674519	−	0.738258i	$-0.735649\pi$
$587$	−791.885	−1.34904	−0.674519	+	0.738258i	$0.735649\pi$
$588$	0	0
$589$	−565.737	−0.960504
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.82388	0.00307567	0.00153784	−	0.999999i	$-0.499510\pi$
$593$	1.82388	0.00307567	0.00153784	+	0.999999i	$0.499510\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 758.204i	− 1.26578i	−0.774241	−	0.632891i	$-0.781868\pi$
$599$	− 758.204i	− 1.26578i	0.774241	−	0.632891i	$-0.218132\pi$
$600$	0	0
$601$	−283.579	−0.471845	−0.235922	−	0.971772i	$-0.575811\pi$
$601$	−283.579	−0.471845	−0.235922	+	0.971772i	$0.575811\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 167.828i	− 0.276488i	−0.990398	−	0.138244i	$-0.955854\pi$
$607$	− 167.828i	− 0.276488i	0.990398	−	0.138244i	$-0.0441459\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	990.507i	1.62112i
$612$	0	0
$613$	395.698i	0.645510i	0.946483	+	0.322755i	$0.104609\pi$
$613$	395.698i	0.645510i	−0.946483	+	0.322755i	$0.895391\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	855.190	1.38604	0.693022	−	0.720916i	$-0.256279\pi$
$617$	855.190	1.38604	0.693022	+	0.720916i	$0.256279\pi$
$618$	0	0
$619$	308.158	0.497832	0.248916	−	0.968525i	$-0.419926\pi$
$619$	308.158	0.497832	0.248916	+	0.968525i	$0.419926\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−636.079	−1.02099
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1059.99i	− 1.68520i
$630$	0	0
$631$	−344.974	−0.546709	−0.273355	−	0.961913i	$-0.588133\pi$
$631$	−344.974	−0.546709	−0.273355	+	0.961913i	$0.588133\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 217.040i	− 0.340722i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 202.641i	− 0.316133i	−0.987428	−	0.158066i	$-0.949474\pi$
$641$	− 202.641i	− 0.316133i	0.987428	−	0.158066i	$-0.0505260\pi$
$642$	0	0
$643$	724.605i	1.12691i	0.826146	+	0.563456i	$0.190529\pi$
$643$	724.605i	1.12691i	−0.826146	+	0.563456i	$0.809471\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−125.679	−0.194249	−0.0971243	−	0.995272i	$-0.530964\pi$
$647$	−125.679	−0.194249	−0.0971243	+	0.995272i	$0.530964\pi$
$648$	0	0
$649$	−1030.89	−1.58843
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−551.226	−0.844144	−0.422072	−	0.906562i	$-0.638697\pi$
$653$	−551.226	−0.844144	−0.422072	+	0.906562i	$0.638697\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	801.765i	1.21664i	0.793692	+	0.608320i	$0.208156\pi$
$659$	801.765i	1.21664i	−0.793692	+	0.608320i	$0.791844\pi$
$660$	0	0
$661$	−529.079	−0.800422	−0.400211	−	0.916423i	$-0.631063\pi$
$661$	−529.079	−0.800422	−0.400211	+	0.916423i	$0.631063\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 101.210i	− 0.151739i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 613.918i	− 0.914929i
$672$	0	0
$673$	− 413.395i	− 0.614257i	−0.951668	−	0.307129i	$-0.900632\pi$
$673$	− 413.395i	− 0.614257i	0.951668	−	0.307129i	$-0.0993682\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	359.936	0.531663	0.265832	−	0.964019i	$-0.414353\pi$
$677$	359.936	0.531663	0.265832	+	0.964019i	$0.414353\pi$
$678$	0	0
$679$	−477.499	−0.703239
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.5484	−0.0315496	−0.0157748	−	0.999876i	$-0.505021\pi$
$683$	−21.5484	−0.0315496	−0.0157748	+	0.999876i	$0.505021\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1103.79i	− 1.60202i
$690$	0	0
$691$	384.921	0.557049	0.278525	−	0.960429i	$-0.410155\pi$
$691$	384.921	0.557049	0.278525	+	0.960429i	$0.410155\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	694.736i	0.996752i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	439.057i	0.626330i	0.949699	+	0.313165i	$0.101389\pi$
$701$	439.057i	0.626330i	−0.949699	+	0.313165i	$0.898611\pi$
$702$	0	0
$703$	− 1684.79i	− 2.39657i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−709.080	−1.00294
$708$	0	0
$709$	126.421	0.178309	0.0891547	−	0.996018i	$-0.471583\pi$
$709$	126.421	0.178309	0.0891547	+	0.996018i	$0.471583\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−103.424	−0.145054
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 571.715i	− 0.795153i	−0.917569	−	0.397576i	$-0.869851\pi$
$719$	− 571.715i	− 0.795153i	0.917569	−	0.397576i	$-0.130149\pi$
$720$	0	0
$721$	627.159	0.869846
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	841.038i	1.15686i	0.815731	+	0.578431i	$0.196335\pi$
$727$	841.038i	1.15686i	−0.815731	+	0.578431i	$0.803665\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.4175i	0.0238269i
$732$	0	0
$733$	− 494.749i	− 0.674965i	−0.941332	−	0.337483i	$-0.890425\pi$
$733$	− 494.749i	− 0.674965i	0.941332	−	0.337483i	$-0.109575\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−697.206	−0.946006
$738$	0	0
$739$	1263.97	1.71038	0.855191	−	0.518312i	$-0.173439\pi$
$739$	1263.97	1.71038	0.855191	+	0.518312i	$0.173439\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−956.343	−1.28714	−0.643568	−	0.765389i	$-0.722547\pi$
$743$	−956.343	−1.28714	−0.643568	+	0.765389i	$0.722547\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	512.131i	0.683752i
$750$	0	0
$751$	1121.66	1.49355	0.746776	−	0.665076i	$-0.231601\pi$
$751$	1121.66	1.49355	0.746776	+	0.665076i	$0.231601\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 466.433i	− 0.616160i	−0.951360	−	0.308080i	$-0.900313\pi$
$757$	− 466.433i	− 0.616160i	0.951360	−	0.308080i	$-0.0996865\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1223.41i	− 1.60763i	−0.594880	−	0.803814i	$-0.702801\pi$
$761$	− 1223.41i	− 1.60763i	0.594880	−	0.803814i	$-0.297199\pi$
$762$	0	0
$763$	663.184i	0.869180i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1290.82	1.68295
$768$	0	0
$769$	1290.63	1.67832	0.839162	−	0.543882i	$-0.183046\pi$
$769$	1290.63	1.67832	0.839162	+	0.543882i	$0.183046\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	329.455	0.426204	0.213102	−	0.977030i	$-0.431643\pi$
$773$	329.455	0.426204	0.213102	+	0.977030i	$0.431643\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1104.24i	1.41751i
$780$	0	0
$781$	220.894	0.282835
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1506.13i	1.91376i	0.290480	+	0.956881i	$0.406185\pi$
$787$	1506.13i	1.91376i	−0.290480	+	0.956881i	$0.593815\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 605.245i	− 0.765165i
$792$	0	0
$793$	768.710i	0.969370i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−250.874	−0.314773	−0.157387	−	0.987537i	$-0.550307\pi$
$797$	−250.874	−0.314773	−0.157387	+	0.987537i	$0.550307\pi$
$798$	0	0
$799$	−1463.37	−1.83150
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−173.577	−0.216160
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1097.02i	1.35602i	0.735053	+	0.678010i	$0.237157\pi$
$809$	1097.02i	1.35602i	−0.735053	+	0.678010i	$0.762843\pi$
$810$	0	0
$811$	221.473	0.273087	0.136543	−	0.990634i	$-0.456401\pi$
$811$	221.473	0.273087	0.136543	+	0.990634i	$0.456401\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.6840i	0.0338849i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1003.92i	1.22281i	0.791319	+	0.611403i	$0.209395\pi$
$821$	1003.92i	1.22281i	−0.791319	+	0.611403i	$0.790605\pi$
$822$	0	0
$823$	335.013i	0.407063i	0.979068	+	0.203531i	$0.0652419\pi$
$823$	335.013i	0.407063i	−0.979068	+	0.203531i	$0.934758\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1004.72	−1.21490	−0.607451	−	0.794357i	$-0.707808\pi$
$827$	−1004.72	−1.21490	−0.607451	+	0.794357i	$0.707808\pi$
$828$	0	0
$829$	−676.763	−0.816361	−0.408180	−	0.912901i	$-0.633837\pi$
$829$	−676.763	−0.816361	−0.408180	+	0.912901i	$0.633837\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	320.653	0.384938
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	621.286i	0.740507i	0.928931	+	0.370254i	$0.120729\pi$
$839$	621.286i	0.740507i	−0.928931	+	0.370254i	$0.879271\pi$
$840$	0	0
$841$	419.736	0.499092
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 202.146i	− 0.238661i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 308.001i	− 0.361928i
$852$	0	0
$853$	544.591i	0.638443i	0.947680	+	0.319221i	$0.103421\pi$
$853$	544.591i	0.638443i	−0.947680	+	0.319221i	$0.896579\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	362.149	0.422578	0.211289	−	0.977424i	$-0.432234\pi$
$857$	362.149	0.422578	0.211289	+	0.977424i	$0.432234\pi$
$858$	0	0
$859$	−21.6840	−0.0252433	−0.0126216	−	0.999920i	$-0.504018\pi$
$859$	−21.6840	−0.0252433	−0.0126216	+	0.999920i	$0.504018\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	220.300	0.255273	0.127636	−	0.991821i	$-0.459261\pi$
$863$	220.300	0.255273	0.127636	+	0.991821i	$0.459261\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	980.868i	1.12873i
$870$	0	0
$871$	872.999	1.00230
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1136.41i	1.29579i	0.761730	+	0.647895i	$0.224350\pi$
$877$	1136.41i	1.29579i	−0.761730	+	0.647895i	$0.775650\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	711.758i	0.807898i	0.914782	+	0.403949i	$0.132363\pi$
$881$	711.758i	0.807898i	−0.914782	+	0.403949i	$0.867637\pi$
$882$	0	0
$883$	− 536.394i	− 0.607468i	−0.952757	−	0.303734i	$-0.901767\pi$
$883$	− 536.394i	− 0.607468i	0.952757	−	0.303734i	$-0.0982334\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1463.95	1.65045	0.825227	−	0.564801i	$-0.191047\pi$
$887$	1463.95	1.65045	0.825227	+	0.564801i	$0.191047\pi$
$888$	0	0
$889$	−578.211	−0.650406
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2325.93	−2.60463
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	430.479i	0.478842i
$900$	0	0
$901$	1630.74	1.80992
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 855.657i	− 0.943392i	−0.881761	−	0.471696i	$-0.843642\pi$
$907$	− 855.657i	− 0.943392i	0.881761	−	0.471696i	$-0.156358\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1581.02i	1.73547i	0.497024	+	0.867737i	$0.334426\pi$
$911$	1581.02i	1.73547i	−0.497024	+	0.867737i	$0.665574\pi$
$912$	0	0
$913$	414.474i	0.453969i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−771.360	−0.841178
$918$	0	0
$919$	−1489.08	−1.62033	−0.810163	−	0.586205i	$-0.800621\pi$
$919$	−1489.08	−1.62033	−0.810163	+	0.586205i	$0.800621\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−276.590	−0.299664
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	736.990i	0.793316i	0.917966	+	0.396658i	$0.129830\pi$
$929$	736.990i	0.793316i	−0.917966	+	0.396658i	$0.870170\pi$
$930$	0	0
$931$	509.658	0.547431
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1550.05i	1.65427i	0.562003	+	0.827135i	$0.310031\pi$
$937$	1550.05i	1.65427i	−0.562003	+	0.827135i	$0.689969\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	628.711i	0.668131i	0.942550	+	0.334065i	$0.108421\pi$
$941$	628.711i	0.668131i	−0.942550	+	0.334065i	$0.891579\pi$
$942$	0	0
$943$	201.869i	0.214071i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1498.73	−1.58261	−0.791304	−	0.611423i	$-0.790598\pi$
$947$	−1498.73	−1.58261	−0.791304	+	0.611423i	$0.790598\pi$
$948$	0	0
$949$	217.342	0.229022
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−224.748	−0.235832	−0.117916	−	0.993024i	$-0.537621\pi$
$953$	−224.748	−0.235832	−0.117916	+	0.993024i	$0.537621\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1464.72i	− 1.52734i
$960$	0	0
$961$	−521.105	−0.542253
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 233.986i	− 0.241972i	−0.992654	−	0.120986i	$-0.961394\pi$
$967$	− 233.986i	− 0.241972i	0.992654	−	0.120986i	$-0.0386055\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	691.940i	0.712606i	0.934371	+	0.356303i	$0.115963\pi$
$971$	691.940i	0.712606i	−0.934371	+	0.356303i	$0.884037\pi$
$972$	0	0
$973$	86.9224i	0.0893344i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−395.571	−0.404883	−0.202442	−	0.979294i	$-0.564888\pi$
$977$	−395.571	−0.404883	−0.202442	+	0.979294i	$0.564888\pi$
$978$	0	0
$979$	1063.50	1.08631
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.82388	0.00185542	0.000927709	−	1.00000i	$-0.499705\pi$
$983$	1.82388	0.00185542	0.000927709		1.00000i	$0.499705\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.06098i	0.00511727i
$990$	0	0
$991$	−506.316	−0.510914	−0.255457	−	0.966820i	$-0.582226\pi$
$991$	−506.316	−0.510914	−0.255457	+	0.966820i	$0.582226\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 295.670i	− 0.296560i	−0.988945	−	0.148280i	$-0.952626\pi$
$997$	− 295.670i	− 0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.k.449.4		8
3.2	odd	2	inner	3600.3.c.k.449.3		8
4.3	odd	2		900.3.b.b.449.5		8
5.2	odd	4		3600.3.l.n.1601.4		4
5.3	odd	4		720.3.l.c.161.1		4
5.4	even	2	inner	3600.3.c.k.449.6		8
12.11	even	2		900.3.b.b.449.6		8
15.2	even	4		3600.3.l.n.1601.3		4
15.8	even	4		720.3.l.c.161.3		4
15.14	odd	2	inner	3600.3.c.k.449.5		8
20.3	even	4		180.3.g.a.161.2	✓	4
20.7	even	4		900.3.g.d.701.1		4
20.19	odd	2		900.3.b.b.449.3		8
40.3	even	4		2880.3.l.b.1601.4		4
40.13	odd	4		2880.3.l.f.1601.3		4
60.23	odd	4		180.3.g.a.161.4	yes	4
60.47	odd	4		900.3.g.d.701.2		4
60.59	even	2		900.3.b.b.449.4		8
120.53	even	4		2880.3.l.f.1601.1		4
120.83	odd	4		2880.3.l.b.1601.2		4
180.23	odd	12		1620.3.o.f.1241.3		8
180.43	even	12		1620.3.o.f.701.3		8
180.83	odd	12		1620.3.o.f.701.1		8
180.103	even	12		1620.3.o.f.1241.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.g.a.161.2	✓	4	20.3	even	4
180.3.g.a.161.4	yes	4	60.23	odd	4
720.3.l.c.161.1		4	5.3	odd	4
720.3.l.c.161.3		4	15.8	even	4
900.3.b.b.449.3		8	20.19	odd	2
900.3.b.b.449.4		8	60.59	even	2
900.3.b.b.449.5		8	4.3	odd	2
900.3.b.b.449.6		8	12.11	even	2
900.3.g.d.701.1		4	20.7	even	4
900.3.g.d.701.2		4	60.47	odd	4
1620.3.o.f.701.1		8	180.83	odd	12
1620.3.o.f.701.3		8	180.43	even	12
1620.3.o.f.1241.1		8	180.103	even	12
1620.3.o.f.1241.3		8	180.23	odd	12
2880.3.l.b.1601.2		4	120.83	odd	4
2880.3.l.b.1601.4		4	40.3	even	4
2880.3.l.f.1601.1		4	120.53	even	4
2880.3.l.f.1601.3		4	40.13	odd	4
3600.3.c.k.449.3		8	3.2	odd	2	inner
3600.3.c.k.449.4		8	1.1	even	1	trivial
3600.3.c.k.449.5		8	15.14	odd	2	inner
3600.3.c.k.449.6		8	5.4	even	2	inner
3600.3.l.n.1601.3		4	15.2	even	4
3600.3.l.n.1601.4		4	5.2	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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.48683i q^{7} +O(q^{10})\)	\(q-5.48683i q^{7} +9.17377i q^{11} -11.4868i q^{13} +16.9706 q^{17} +26.9737 q^{19} +4.93113 q^{23} -20.5247i q^{29} -20.9737 q^{31} -62.4605i q^{37} +40.9377i q^{41} +1.02633i q^{43} -86.2298 q^{47} +18.8947 q^{49} +96.0920 q^{53} +112.374i q^{59} -66.9210 q^{61} +76.0000i q^{67} -24.0789i q^{71} +18.9210i q^{73} +50.3349 q^{77} +106.921 q^{79} +45.1804 q^{83} -115.928i q^{89} -63.0263 q^{91} -87.0263i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 64 q^{19} - 16 q^{31} - 456 q^{49} - 80 q^{61} + 400 q^{79} - 656 q^{91}+O(q^{100}) \) 8 * q + 64 * q^19 - 16 * q^31 - 456 * q^49 - 80 * q^61 + 400 * q^79 - 656 * q^91

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