LMFDB - Embedded newform 3600.3.c.g.449.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3600,3,Mod(449,3600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-88,0,0,0,0,0,0,0,0,0,0, 0,152] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 1800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.g.449.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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
$7$	− 9.32456i	− 1.33208i	−0.745916	−	0.666040i	$-0.767988\pi$
$7$	− 9.32456i	− 1.33208i	0.745916	−	0.666040i	$-0.232012\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.3028i	1.75480i	0.479763	+	0.877398i	$0.340723\pi$
$11$	19.3028i	1.75480i	−0.479763	+	0.877398i	$0.659277\pi$
$12$	0	0
$13$	19.6491i	1.51147i	0.654878	+	0.755735i	$0.272720\pi$
$13$	19.6491i	1.51147i	−0.654878	+	0.755735i	$0.727280\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.70163	−0.276567	−0.138283	−	0.990393i	$-0.544158\pi$
$17$	−4.70163	−0.276567	−0.138283	+	0.990393i	$0.544158\pi$
$18$	0	0
$19$	−29.9737	−1.57756	−0.788781	−	0.614675i	$-0.789287\pi$
$19$	−29.9737	−1.57756	−0.788781	+	0.614675i	$0.789287\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.89949	−0.430413	−0.215206	−	0.976569i	$-0.569042\pi$
$23$	−9.89949	−0.430413	−0.215206	+	0.976569i	$0.569042\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 24.5006i	− 0.844849i	−0.906398	−	0.422425i	$-0.861179\pi$
$29$	− 24.5006i	− 0.844849i	0.906398	−	0.422425i	$-0.138821\pi$
$30$	0	0
$31$	12.6754	0.408885	0.204443	−	0.978879i	$-0.434462\pi$
$31$	12.6754	0.408885	0.204443	+	0.978879i	$0.434462\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 37.9473i	− 1.02560i	−0.858507	−	0.512802i	$-0.828608\pi$
$37$	− 37.9473i	− 1.02560i	0.858507	−	0.512802i	$-0.171392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	73.0056i	1.78063i	0.455350	+	0.890313i	$0.349514\pi$
$41$	73.0056i	1.78063i	−0.455350	+	0.890313i	$0.650486\pi$
$42$	0	0
$43$	− 33.3246i	− 0.774990i	−0.921872	−	0.387495i	$-0.873341\pi$
$43$	− 33.3246i	− 0.774990i	0.921872	−	0.387495i	$-0.126659\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.70163	−0.100035	−0.0500174	−	0.998748i	$-0.515928\pi$
$47$	−4.70163	−0.100035	−0.0500174	+	0.998748i	$0.515928\pi$
$48$	0	0
$49$	−37.9473	−0.774435
$50$	0	0
$51$	0	0
$52$	0	0
$53$	73.0056	1.37746	0.688732	−	0.725016i	$-0.258168\pi$
$53$	73.0056	1.37746	0.688732	+	0.725016i	$0.258168\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 24.5006i	− 0.415265i	−0.978207	−	0.207632i	$-0.933424\pi$
$59$	− 24.5006i	− 0.415265i	0.978207	−	0.207632i	$-0.0665758\pi$
$60$	0	0
$61$	−53.7018	−0.880357	−0.440179	−	0.897910i	$-0.645085\pi$
$61$	−53.7018	−0.880357	−0.440179	+	0.897910i	$0.645085\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 101.974i	− 1.52200i	−0.648755	−	0.760998i	$-0.724710\pi$
$67$	− 101.974i	− 1.52200i	0.648755	−	0.760998i	$-0.275290\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 111.686i	− 1.57304i	−0.617567	−	0.786519i	$-0.711881\pi$
$71$	− 111.686i	− 1.57304i	0.617567	−	0.786519i	$-0.288119\pi$
$72$	0	0
$73$	− 48.5438i	− 0.664983i	−0.943106	−	0.332492i	$-0.892111\pi$
$73$	− 48.5438i	− 0.664983i	0.943106	−	0.332492i	$-0.107889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	179.990	2.33753
$78$	0	0
$79$	10.5964	0.134132	0.0670661	−	0.997749i	$-0.478636\pi$
$79$	10.5964	0.134132	0.0670661	+	0.997749i	$0.478636\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	44.2251	0.532833	0.266416	−	0.963858i	$-0.414160\pi$
$83$	44.2251	0.532833	0.266416	+	0.963858i	$0.414160\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 38.6800i	− 0.434607i	−0.976104	−	0.217303i	$-0.930274\pi$
$89$	− 38.6800i	− 0.434607i	0.976104	−	0.217303i	$-0.0697261\pi$
$90$	0	0
$91$	183.219	2.01340
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	183.491i	1.89166i	0.324661	+	0.945830i	$0.394750\pi$
$97$	183.491i	1.89166i	−0.324661	+	0.945830i	$0.605250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 87.9913i	− 0.871201i	−0.900140	−	0.435600i	$-0.856536\pi$
$101$	− 87.9913i	− 0.871201i	0.900140	−	0.435600i	$-0.143464\pi$
$102$	0	0
$103$	− 26.0000i	− 0.252427i	−0.992003	−	0.126214i	$-0.959718\pi$
$103$	− 26.0000i	− 0.252427i	0.992003	−	0.126214i	$-0.0402825\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	126.671	1.18384	0.591922	−	0.805995i	$-0.298369\pi$
$107$	126.671	1.18384	0.591922	+	0.805995i	$0.298369\pi$
$108$	0	0
$109$	200.789	1.84210	0.921052	−	0.389440i	$-0.127331\pi$
$109$	200.789	1.84210	0.921052	+	0.389440i	$0.127331\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−160.997	−1.42475	−0.712376	−	0.701798i	$-0.752381\pi$
$113$	−160.997	−1.42475	−0.712376	+	0.701798i	$0.752381\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	43.8406i	0.368409i
$120$	0	0
$121$	−251.596	−2.07931
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 71.4562i	− 0.562647i	−0.959613	−	0.281324i	$-0.909227\pi$
$127$	− 71.4562i	− 0.562647i	0.959613	−	0.281324i	$-0.0907735\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 189.046i	− 1.44310i	−0.692364	−	0.721548i	$-0.743431\pi$
$131$	− 189.046i	− 1.44310i	0.692364	−	0.721548i	$-0.256569\pi$
$132$	0	0
$133$	279.491i	2.10144i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	101.402	0.740158	0.370079	−	0.929000i	$-0.379331\pi$
$137$	101.402	0.740158	0.370079	+	0.929000i	$0.379331\pi$
$138$	0	0
$139$	−82.0527	−0.590307	−0.295153	−	0.955450i	$-0.595371\pi$
$139$	−82.0527	−0.590307	−0.295153	+	0.955450i	$0.595371\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−379.282	−2.65232
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 33.5194i	− 0.224962i	−0.993654	−	0.112481i	$-0.964120\pi$
$149$	− 33.5194i	− 0.224962i	0.993654	−	0.112481i	$-0.0358798\pi$
$150$	0	0
$151$	−6.51744	−0.0431619	−0.0215809	−	0.999767i	$-0.506870\pi$
$151$	−6.51744	−0.0431619	−0.0215809	+	0.999767i	$0.506870\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 98.8947i	− 0.629902i	−0.949108	−	0.314951i	$-0.898012\pi$
$157$	− 98.8947i	− 0.629902i	0.949108	−	0.314951i	$-0.101988\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	92.3084i	0.573344i
$162$	0	0
$163$	− 153.325i	− 0.940641i	−0.882496	−	0.470321i	$-0.844138\pi$
$163$	− 153.325i	− 0.940641i	0.882496	−	0.470321i	$-0.155862\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−87.9913	−0.526894	−0.263447	−	0.964674i	$-0.584859\pi$
$167$	−87.9913	−0.526894	−0.263447	+	0.964674i	$0.584859\pi$
$168$	0	0
$169$	−217.088	−1.28454
$170$	0	0
$171$	0	0
$172$	0	0
$173$	224.103	1.29539	0.647697	−	0.761898i	$-0.275732\pi$
$173$	224.103	1.29539	0.647697	+	0.761898i	$0.275732\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 101.861i	− 0.569054i	−0.958668	−	0.284527i	$-0.908164\pi$
$179$	− 101.861i	− 0.569054i	0.958668	−	0.284527i	$-0.0918365\pi$
$180$	0	0
$181$	−203.053	−1.12184	−0.560919	−	0.827871i	$-0.689552\pi$
$181$	−203.053	−1.12184	−0.560919	+	0.827871i	$0.689552\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 90.7544i	− 0.485318i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 335.020i	− 1.75403i	−0.480463	−	0.877015i	$-0.659532\pi$
$191$	− 335.020i	− 1.75403i	0.480463	−	0.877015i	$-0.340468\pi$
$192$	0	0
$193$	335.544i	1.73857i	0.494312	+	0.869284i	$0.335420\pi$
$193$	335.544i	1.73857i	−0.494312	+	0.869284i	$0.664580\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	32.7504	0.166246	0.0831228	−	0.996539i	$-0.473511\pi$
$197$	32.7504	0.166246	0.0831228	+	0.996539i	$0.473511\pi$
$198$	0	0
$199$	−307.114	−1.54329	−0.771643	−	0.636056i	$-0.780565\pi$
$199$	−307.114	−1.54329	−0.771643	+	0.636056i	$0.780565\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−228.457	−1.12541
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 578.574i	− 2.76830i
$210$	0	0
$211$	−240.359	−1.13914	−0.569572	−	0.821941i	$-0.692891\pi$
$211$	−240.359	−1.13914	−0.569572	+	0.821941i	$0.692891\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 118.193i	− 0.544668i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 92.3829i	− 0.418022i
$222$	0	0
$223$	174.517i	0.782589i	0.920265	+	0.391295i	$0.127973\pi$
$223$	174.517i	0.782589i	−0.920265	+	0.391295i	$0.872027\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−210.308	−0.926468	−0.463234	−	0.886236i	$-0.653311\pi$
$227$	−210.308	−0.926468	−0.463234	+	0.886236i	$0.653311\pi$
$228$	0	0
$229$	81.5964	0.356316	0.178158	−	0.984002i	$-0.442986\pi$
$229$	81.5964	0.356316	0.178158	+	0.984002i	$0.442986\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−126.671	−0.543653	−0.271827	−	0.962346i	$-0.587628\pi$
$233$	−126.671	−0.543653	−0.271827	+	0.962346i	$0.587628\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4.35437i	− 0.0182191i	−0.999959	−	0.00910955i	$-0.997100\pi$
$239$	− 4.35437i	− 0.0182191i	0.999959	−	0.00910955i	$-0.00289970\pi$
$240$	0	0
$241$	−316.895	−1.31492	−0.657458	−	0.753491i	$-0.728368\pi$
$241$	−316.895	−1.31492	−0.657458	+	0.753491i	$0.728368\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 588.956i	− 2.38444i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 247.066i	− 0.984325i	−0.870503	−	0.492163i	$-0.836207\pi$
$251$	− 247.066i	− 0.984325i	0.870503	−	0.492163i	$-0.163793\pi$
$252$	0	0
$253$	− 191.088i	− 0.755287i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	285.746	1.11185	0.555925	−	0.831232i	$-0.312364\pi$
$257$	285.746	1.11185	0.555925	+	0.831232i	$0.312364\pi$
$258$	0	0
$259$	−353.842	−1.36619
$260$	0	0
$261$	0	0
$262$	0	0
$263$	317.639	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$263$	317.639	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	400.160i	1.48758i	0.668411	+	0.743792i	$0.266975\pi$
$269$	400.160i	1.48758i	−0.668411	+	0.743792i	$0.733025\pi$
$270$	0	0
$271$	320.333	1.18204	0.591020	−	0.806657i	$-0.298725\pi$
$271$	320.333	1.18204	0.591020	+	0.806657i	$0.298725\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.24555i	0.0225471i	0.999936	+	0.0112736i	$0.00358856\pi$
$277$	6.24555i	0.0225471i	−0.999936	+	0.0112736i	$0.996411\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 145.974i	− 0.519480i	−0.965679	−	0.259740i	$-0.916363\pi$
$281$	− 145.974i	− 0.519480i	0.965679	−	0.259740i	$-0.0836369\pi$
$282$	0	0
$283$	− 4.62278i	− 0.0163349i	−0.999967	−	0.00816745i	$-0.997400\pi$
$283$	− 4.62278i	− 0.0163349i	0.999967	−	0.00816745i	$-0.00259981\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	680.745	2.37193
$288$	0	0
$289$	−266.895	−0.923511
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−151.097	−0.515691	−0.257845	−	0.966186i	$-0.583013\pi$
$293$	−151.097	−0.515691	−0.257845	+	0.966186i	$0.583013\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 194.516i	− 0.650556i
$300$	0	0
$301$	−310.737	−1.03235
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 365.710i	− 1.19124i	−0.803267	−	0.595619i	$-0.796907\pi$
$307$	− 365.710i	− 1.19124i	0.803267	−	0.595619i	$-0.203093\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	506.722i	1.62933i	0.579930	+	0.814666i	$0.303080\pi$
$311$	506.722i	1.62933i	−0.579930	+	0.814666i	$0.696920\pi$
$312$	0	0
$313$	121.544i	0.388319i	0.980970	+	0.194159i	$0.0621979\pi$
$313$	121.544i	0.388319i	−0.980970	+	0.194159i	$0.937802\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	569.059	1.79514	0.897570	−	0.440872i	$-0.145331\pi$
$317$	569.059	1.79514	0.897570	+	0.440872i	$0.145331\pi$
$318$	0	0
$319$	472.930	1.48254
$320$	0	0
$321$	0	0
$322$	0	0
$323$	140.925	0.436301
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	43.8406i	0.133254i
$330$	0	0
$331$	90.9295	0.274712	0.137356	−	0.990522i	$-0.456140\pi$
$331$	90.9295	0.274712	0.137356	+	0.990522i	$0.456140\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 420.947i	− 1.24910i	−0.780984	−	0.624551i	$-0.785282\pi$
$337$	− 420.947i	− 1.24910i	0.780984	−	0.624551i	$-0.214718\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	244.671i	0.717510i
$342$	0	0
$343$	− 103.061i	− 0.300470i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	588.747	1.69668	0.848338	−	0.529455i	$-0.177603\pi$
$347$	588.747	1.69668	0.848338	+	0.529455i	$0.177603\pi$
$348$	0	0
$349$	−250.596	−0.718041	−0.359021	−	0.933330i	$-0.616889\pi$
$349$	−250.596	−0.718041	−0.359021	+	0.933330i	$0.616889\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.97696	−0.0225976	−0.0112988	−	0.999936i	$-0.503597\pi$
$353$	−7.97696	−0.0225976	−0.0112988	+	0.999936i	$0.503597\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 296.377i	− 0.825562i	−0.910830	−	0.412781i	$-0.864557\pi$
$359$	− 296.377i	− 0.825562i	0.910830	−	0.412781i	$-0.135443\pi$
$360$	0	0
$361$	537.421	1.48870
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	363.991i	0.991800i	0.868380	+	0.495900i	$0.165162\pi$
$367$	363.991i	0.991800i	−0.868380	+	0.495900i	$0.834838\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 680.745i	− 1.83489i
$372$	0	0
$373$	− 501.017i	− 1.34321i	−0.740910	−	0.671605i	$-0.765605\pi$
$373$	− 501.017i	− 1.34321i	0.740910	−	0.671605i	$-0.234395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	481.415	1.27696
$378$	0	0
$379$	305.394	0.805790	0.402895	−	0.915246i	$-0.368004\pi$
$379$	305.394	0.805790	0.402895	+	0.915246i	$0.368004\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−646.419	−1.68778	−0.843890	−	0.536517i	$-0.819740\pi$
$383$	−646.419	−1.68778	−0.843890	+	0.536517i	$0.819740\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 719.425i	− 1.84942i	−0.380670	−	0.924711i	$-0.624307\pi$
$389$	− 719.425i	− 1.84942i	0.380670	−	0.924711i	$-0.375693\pi$
$390$	0	0
$391$	46.5438	0.119038
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 258.562i	− 0.651289i	−0.945492	−	0.325644i	$-0.894419\pi$
$397$	− 258.562i	− 0.651289i	0.945492	−	0.325644i	$-0.105581\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 343.766i	− 0.857271i	−0.903478	−	0.428635i	$-0.858994\pi$
$401$	− 343.766i	− 0.857271i	0.903478	−	0.428635i	$-0.141006\pi$
$402$	0	0
$403$	249.061i	0.618018i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	732.488	1.79973
$408$	0	0
$409$	716.105	1.75087	0.875434	−	0.483338i	$-0.160576\pi$
$409$	716.105	1.75087	0.875434	+	0.483338i	$0.160576\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−228.457	−0.553166
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 313.794i	− 0.748913i	−0.927245	−	0.374456i	$-0.877829\pi$
$419$	− 313.794i	− 0.748913i	0.927245	−	0.374456i	$-0.122171\pi$
$420$	0	0
$421$	429.737	1.02075	0.510376	−	0.859951i	$-0.329506\pi$
$421$	429.737	1.02075	0.510376	+	0.859951i	$0.329506\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	500.745i	1.17271i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	305.123i	0.707942i	0.935256	+	0.353971i	$0.115169\pi$
$431$	305.123i	0.707942i	−0.935256	+	0.353971i	$0.884831\pi$
$432$	0	0
$433$	185.122i	0.427534i	0.976885	+	0.213767i	$0.0685734\pi$
$433$	185.122i	0.427534i	−0.976885	+	0.213767i	$0.931427\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	296.724	0.679003
$438$	0	0
$439$	−757.131	−1.72467	−0.862336	−	0.506337i	$-0.830999\pi$
$439$	−757.131	−1.72467	−0.862336	+	0.506337i	$0.830999\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	455.451	1.02811	0.514053	−	0.857758i	$-0.328143\pi$
$443$	455.451	1.02811	0.514053	+	0.857758i	$0.328143\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 414.339i	− 0.922805i	−0.887191	−	0.461402i	$-0.847346\pi$
$449$	− 414.339i	− 0.922805i	0.887191	−	0.461402i	$-0.152654\pi$
$450$	0	0
$451$	−1409.21	−3.12463
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 194.000i	− 0.424508i	−0.977215	−	0.212254i	$-0.931920\pi$
$457$	− 194.000i	− 0.424508i	0.977215	−	0.212254i	$-0.0680804\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 466.083i	− 1.01102i	−0.862819	−	0.505512i	$-0.831303\pi$
$461$	− 466.083i	− 1.01102i	0.862819	−	0.505512i	$-0.168697\pi$
$462$	0	0
$463$	269.070i	0.581146i	0.956853	+	0.290573i	$0.0938459\pi$
$463$	269.070i	0.581146i	−0.956853	+	0.290573i	$0.906154\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−589.975	−1.26333	−0.631665	−	0.775242i	$-0.717628\pi$
$467$	−589.975	−1.26333	−0.631665	+	0.775242i	$0.717628\pi$
$468$	0	0
$469$	−950.859	−2.02742
$470$	0	0
$471$	0	0
$472$	0	0
$473$	643.256	1.35995
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	41.9181i	0.0875117i	0.999042	+	0.0437558i	$0.0139324\pi$
$479$	41.9181i	0.0875117i	−0.999042	+	0.0437558i	$0.986068\pi$
$480$	0	0
$481$	745.631	1.55017
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 532.535i	− 1.09350i	−0.837296	−	0.546750i	$-0.815865\pi$
$487$	− 532.535i	− 1.09350i	0.837296	−	0.546750i	$-0.184135\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	238.357i	0.485452i	0.970095	+	0.242726i	$0.0780416\pi$
$491$	238.357i	0.485452i	−0.970095	+	0.242726i	$0.921958\pi$
$492$	0	0
$493$	115.193i	0.233657i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1041.42	−2.09541
$498$	0	0
$499$	−387.447	−0.776447	−0.388223	−	0.921565i	$-0.626911\pi$
$499$	−387.447	−0.776447	−0.388223	+	0.921565i	$0.626911\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−174.060	−0.346044	−0.173022	−	0.984918i	$-0.555353\pi$
$503$	−174.060	−0.346044	−0.173022	+	0.984918i	$0.555353\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.1463i	− 0.0395801i	−0.999804	−	0.0197900i	$-0.993700\pi$
$509$	− 20.1463i	− 0.0395801i	0.999804	−	0.0197900i	$-0.00629977\pi$
$510$	0	0
$511$	−452.649	−0.885810
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 90.7544i	− 0.175541i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 992.070i	− 1.90417i	−0.305840	−	0.952083i	$-0.598937\pi$
$521$	− 992.070i	− 1.90417i	0.305840	−	0.952083i	$-0.401063\pi$
$522$	0	0
$523$	− 446.307i	− 0.853359i	−0.904403	−	0.426680i	$-0.859683\pi$
$523$	− 446.307i	− 0.853359i	0.904403	−	0.426680i	$-0.140317\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−59.5953	−0.113084
$528$	0	0
$529$	−431.000	−0.814745
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1434.50	−2.69136
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 732.488i	− 1.35898i
$540$	0	0
$541$	−665.298	−1.22976	−0.614878	−	0.788623i	$-0.710795\pi$
$541$	−665.298	−1.22976	−0.614878	+	0.788623i	$0.710795\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	353.018i	0.645371i	0.946506	+	0.322685i	$0.104586\pi$
$547$	353.018i	0.645371i	−0.946506	+	0.322685i	$0.895414\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	734.373i	1.33280i
$552$	0	0
$553$	− 98.8071i	− 0.178675i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−376.007	−0.675057	−0.337528	−	0.941315i	$-0.609591\pi$
$557$	−376.007	−0.675057	−0.337528	+	0.941315i	$0.609591\pi$
$558$	0	0
$559$	654.798	1.17137
$560$	0	0
$561$	0	0
$562$	0	0
$563$	290.832	0.516575	0.258288	−	0.966068i	$-0.416842\pi$
$563$	290.832	0.516575	0.258288	+	0.966068i	$0.416842\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1029.16i	1.80872i	0.426770	+	0.904360i	$0.359651\pi$
$569$	1029.16i	1.80872i	−0.426770	+	0.904360i	$0.640349\pi$
$570$	0	0
$571$	129.236	0.226333	0.113167	−	0.993576i	$-0.463901\pi$
$571$	129.236	0.226333	0.113167	+	0.993576i	$0.463901\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 575.280i	− 0.997020i	−0.866884	−	0.498510i	$-0.833881\pi$
$577$	− 575.280i	− 0.997020i	0.866884	−	0.498510i	$-0.166119\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 412.380i	− 0.709776i
$582$	0	0
$583$	1409.21i	2.41717i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	996.462	1.69755	0.848775	−	0.528754i	$-0.177341\pi$
$587$	996.462	1.69755	0.848775	+	0.528754i	$0.177341\pi$
$588$	0	0
$589$	−379.930	−0.645042
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−311.363	−0.525063	−0.262532	−	0.964923i	$-0.584557\pi$
$593$	−311.363	−0.525063	−0.262532	+	0.964923i	$0.584557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	190.968i	0.318812i	0.987213	+	0.159406i	$0.0509578\pi$
$599$	190.968i	0.318812i	−0.987213	+	0.159406i	$0.949042\pi$
$600$	0	0
$601$	−393.508	−0.654756	−0.327378	−	0.944894i	$-0.606165\pi$
$601$	−393.508	−0.654756	−0.327378	+	0.944894i	$0.606165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	195.895i	0.322726i	0.986895	+	0.161363i	$0.0515890\pi$
$607$	195.895i	0.322726i	−0.986895	+	0.161363i	$0.948411\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 92.3829i	− 0.151199i
$612$	0	0
$613$	74.1751i	0.121003i	0.998168	+	0.0605017i	$0.0192701\pi$
$613$	74.1751i	0.121003i	−0.998168	+	0.0605017i	$0.980730\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−180.337	−0.292280	−0.146140	−	0.989264i	$-0.546685\pi$
$617$	−180.337	−0.292280	−0.146140	+	0.989264i	$0.546685\pi$
$618$	0	0
$619$	−197.710	−0.319403	−0.159701	−	0.987165i	$-0.551053\pi$
$619$	−197.710	−0.319403	−0.159701	+	0.987165i	$0.551053\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−360.674	−0.578931
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	178.414i	0.283648i
$630$	0	0
$631$	−775.342	−1.22875	−0.614375	−	0.789014i	$-0.710592\pi$
$631$	−775.342	−1.22875	−0.614375	+	0.789014i	$0.710592\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 745.631i	− 1.17054i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	429.325i	0.669774i	0.942258	+	0.334887i	$0.108698\pi$
$641$	429.325i	0.669774i	−0.942258	+	0.334887i	$0.891302\pi$
$642$	0	0
$643$	689.210i	1.07187i	0.844260	+	0.535933i	$0.180040\pi$
$643$	689.210i	1.07187i	−0.844260	+	0.535933i	$0.819960\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−420.616	−0.650103	−0.325051	−	0.945696i	$-0.605382\pi$
$647$	−420.616	−0.650103	−0.325051	+	0.945696i	$0.605382\pi$
$648$	0	0
$649$	472.930	0.728705
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−325.616	−0.498647	−0.249323	−	0.968420i	$-0.580208\pi$
$653$	−325.616	−0.498647	−0.249323	+	0.968420i	$0.580208\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	749.396i	1.13717i	0.822624	+	0.568586i	$0.192509\pi$
$659$	749.396i	1.13717i	−0.822624	+	0.568586i	$0.807491\pi$
$660$	0	0
$661$	730.596	1.10529	0.552645	−	0.833417i	$-0.313619\pi$
$661$	730.596	1.10529	0.552645	+	0.833417i	$0.313619\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	242.544i	0.363634i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1036.59i	− 1.54485i
$672$	0	0
$673$	− 1114.51i	− 1.65603i	−0.560706	−	0.828015i	$-0.689470\pi$
$673$	− 1114.51i	− 1.65603i	0.560706	−	0.828015i	$-0.310530\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−974.852	−1.43996	−0.719979	−	0.693995i	$-0.755849\pi$
$677$	−974.852	−1.43996	−0.719979	+	0.693995i	$0.755849\pi$
$678$	0	0
$679$	1710.97	2.51984
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1170.52	1.71379	0.856897	−	0.515487i	$-0.172389\pi$
$683$	1170.52	1.71379	0.856897	+	0.515487i	$0.172389\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1434.50i	2.08200i
$690$	0	0
$691$	−654.982	−0.947876	−0.473938	−	0.880558i	$-0.657168\pi$
$691$	−654.982	−0.947876	−0.473938	+	0.880558i	$0.657168\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 343.246i	− 0.492461i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 309.477i	− 0.441480i	−0.975333	−	0.220740i	$-0.929153\pi$
$701$	− 309.477i	− 0.441480i	0.975333	−	0.220740i	$-0.0708472\pi$
$702$	0	0
$703$	1137.42i	1.61795i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−820.479	−1.16051
$708$	0	0
$709$	−180.404	−0.254448	−0.127224	−	0.991874i	$-0.540607\pi$
$709$	−180.404	−0.254448	−0.127224	+	0.991874i	$0.540607\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−125.481	−0.175989
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1020.96i	1.41998i	0.704214	+	0.709988i	$0.251300\pi$
$719$	1020.96i	1.41998i	−0.704214	+	0.709988i	$0.748700\pi$
$720$	0	0
$721$	−242.438	−0.336253
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 215.816i	− 0.296858i	−0.988923	−	0.148429i	$-0.952578\pi$
$727$	− 215.816i	− 0.296858i	0.988923	−	0.148429i	$-0.0474216\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	156.680i	0.214336i
$732$	0	0
$733$	86.4911i	0.117996i	0.998258	+	0.0589980i	$0.0187906\pi$
$733$	86.4911i	0.117996i	−0.998258	+	0.0589980i	$0.981209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1968.37	2.67079
$738$	0	0
$739$	−33.5089	−0.0453436	−0.0226718	−	0.999743i	$-0.507217\pi$
$739$	−33.5089	−0.0453436	−0.0226718	+	0.999743i	$0.507217\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−526.534	−0.708660	−0.354330	−	0.935120i	$-0.615291\pi$
$743$	−526.534	−0.708660	−0.354330	+	0.935120i	$0.615291\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1181.15i	− 1.57697i
$750$	0	0
$751$	−329.210	−0.438362	−0.219181	−	0.975684i	$-0.570339\pi$
$751$	−329.210	−0.438362	−0.219181	+	0.975684i	$0.570339\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1164.86i	1.53878i	0.638777	+	0.769392i	$0.279441\pi$
$757$	1164.86i	1.53878i	−0.638777	+	0.769392i	$0.720559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 184.691i	− 0.242695i	−0.992610	−	0.121348i	$-0.961278\pi$
$761$	− 184.691i	− 0.242695i	0.992610	−	0.121348i	$-0.0387216\pi$
$762$	0	0
$763$	− 1872.27i	− 2.45383i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	481.415	0.627660
$768$	0	0
$769$	327.754	0.426209	0.213104	−	0.977029i	$-0.431643\pi$
$769$	327.754	0.426209	0.213104	+	0.977029i	$0.431643\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1333.44	1.72502	0.862511	−	0.506039i	$-0.168891\pi$
$773$	1333.44	1.72502	0.862511	+	0.506039i	$0.168891\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2188.25i	− 2.80905i
$780$	0	0
$781$	2155.84	2.76036
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	209.483i	0.266179i	0.991104	+	0.133089i	$0.0424897\pi$
$787$	209.483i	0.266179i	−0.991104	+	0.133089i	$0.957510\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1501.22i	1.89788i
$792$	0	0
$793$	− 1055.19i	− 1.33063i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−292.023	−0.366402	−0.183201	−	0.983075i	$-0.558646\pi$
$797$	−292.023	−0.366402	−0.183201	+	0.983075i	$0.558646\pi$
$798$	0	0
$799$	22.1053	0.0276663
$800$	0	0
$801$	0	0
$802$	0	0
$803$	937.029	1.16691
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 353.888i	− 0.437438i	−0.975788	−	0.218719i	$-0.929812\pi$
$809$	− 353.888i	− 0.437438i	0.975788	−	0.218719i	$-0.0701879\pi$
$810$	0	0
$811$	196.535	0.242336	0.121168	−	0.992632i	$-0.461336\pi$
$811$	196.535	0.242336	0.121168	+	0.992632i	$0.461336\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	998.859i	1.22259i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	814.240i	0.991766i	0.868389	+	0.495883i	$0.165155\pi$
$821$	814.240i	0.991766i	−0.868389	+	0.495883i	$0.834845\pi$
$822$	0	0
$823$	− 659.955i	− 0.801890i	−0.916102	−	0.400945i	$-0.868682\pi$
$823$	− 659.955i	− 0.801890i	0.916102	−	0.400945i	$-0.131318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−131.535	−0.159051	−0.0795254	−	0.996833i	$-0.525340\pi$
$827$	−131.535	−0.159051	−0.0795254	+	0.996833i	$0.525340\pi$
$828$	0	0
$829$	909.929	1.09762	0.548811	−	0.835946i	$-0.315081\pi$
$829$	909.929	1.09762	0.548811	+	0.835946i	$0.315081\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	178.414	0.214183
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1429.84i	1.70422i	0.523359	+	0.852112i	$0.324679\pi$
$839$	1429.84i	1.70422i	−0.523359	+	0.852112i	$0.675321\pi$
$840$	0	0
$841$	240.720	0.286230
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2346.02i	2.76981i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	375.659i	0.441433i
$852$	0	0
$853$	362.543i	0.425021i	0.977159	+	0.212511i	$0.0681640\pi$
$853$	362.543i	0.425021i	−0.977159	+	0.212511i	$0.931836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−949.583	−1.10803	−0.554015	−	0.832506i	$-0.686905\pi$
$857$	−949.583	−1.10803	−0.554015	+	0.832506i	$0.686905\pi$
$858$	0	0
$859$	−794.982	−0.925474	−0.462737	−	0.886496i	$-0.653133\pi$
$859$	−794.982	−0.925474	−0.462737	+	0.886496i	$0.653133\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1363.92	−1.58044	−0.790221	−	0.612822i	$-0.790034\pi$
$863$	−1363.92	−1.58044	−0.790221	+	0.612822i	$0.790034\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	204.541i	0.235375i
$870$	0	0
$871$	2003.69	2.30045
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	292.895i	0.333973i	0.985959	+	0.166987i	$0.0534037\pi$
$877$	292.895i	0.333973i	−0.985959	+	0.166987i	$0.946596\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 980.967i	− 1.11347i	−0.830690	−	0.556735i	$-0.812054\pi$
$881$	− 980.967i	− 1.11347i	0.830690	−	0.556735i	$-0.187946\pi$
$882$	0	0
$883$	− 882.570i	− 0.999513i	−0.866166	−	0.499757i	$-0.833423\pi$
$883$	− 882.570i	− 0.999513i	0.866166	−	0.499757i	$-0.166577\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	815.268	0.919130	0.459565	−	0.888144i	$-0.348005\pi$
$887$	815.268	0.919130	0.459565	+	0.888144i	$0.348005\pi$
$888$	0	0
$889$	−666.298	−0.749491
$890$	0	0
$891$	0	0
$892$	0	0
$893$	140.925	0.157811
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 310.556i	− 0.345446i
$900$	0	0
$901$	−343.246	−0.380961
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	197.614i	0.217877i	0.994049	+	0.108938i	$0.0347451\pi$
$907$	197.614i	0.217877i	−0.994049	+	0.108938i	$0.965255\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	160.960i	0.176685i	0.996090	+	0.0883423i	$0.0281569\pi$
$911$	160.960i	0.176685i	−0.996090	+	0.0883423i	$0.971843\pi$
$912$	0	0
$913$	853.667i	0.935013i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1762.77	−1.92232
$918$	0	0
$919$	−954.937	−1.03910	−0.519552	−	0.854439i	$-0.673901\pi$
$919$	−954.937	−1.03910	−0.519552	+	0.854439i	$0.673901\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2194.52	2.37760
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 729.449i	− 0.785198i	−0.919710	−	0.392599i	$-0.871576\pi$
$929$	− 729.449i	− 0.785198i	0.919710	−	0.392599i	$-0.128424\pi$
$930$	0	0
$931$	1137.42	1.22172
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 25.7189i	− 0.0274481i	−0.999906	−	0.0137241i	$-0.995631\pi$
$937$	− 25.7189i	− 0.0274481i	0.999906	−	0.0137241i	$-0.00436864\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	513.508i	0.545705i	0.962056	+	0.272853i	$0.0879671\pi$
$941$	513.508i	0.545705i	−0.962056	+	0.272853i	$0.912033\pi$
$942$	0	0
$943$	− 722.719i	− 0.766404i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	136.955	0.144620	0.0723101	−	0.997382i	$-0.476963\pi$
$947$	136.955	0.144620	0.0723101	+	0.997382i	$0.476963\pi$
$948$	0	0
$949$	953.842	1.00510
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−358.242	−0.375910	−0.187955	−	0.982178i	$-0.560186\pi$
$953$	−358.242	−0.375910	−0.187955	+	0.982178i	$0.560186\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 945.525i	− 0.985949i
$960$	0	0
$961$	−800.333	−0.832813
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1316.56i	− 1.36149i	−0.732520	−	0.680745i	$-0.761656\pi$
$967$	− 1316.56i	− 1.36149i	0.732520	−	0.680745i	$-0.238344\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	551.679i	0.568156i	0.958801	+	0.284078i	$0.0916874\pi$
$971$	551.679i	0.568156i	−0.958801	+	0.284078i	$0.908313\pi$
$972$	0	0
$973$	765.105i	0.786336i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	294.107	0.301031	0.150515	−	0.988608i	$-0.451907\pi$
$977$	294.107	0.301031	0.150515	+	0.988608i	$0.451907\pi$
$978$	0	0
$979$	746.631	0.762646
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1528.76	−1.55520	−0.777601	−	0.628758i	$-0.783564\pi$
$983$	−1528.76	−1.55520	−0.777601	+	0.628758i	$0.783564\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	329.896i	0.333565i
$990$	0	0
$991$	768.184	0.775161	0.387580	−	0.921836i	$-0.373311\pi$
$991$	768.184	0.775161	0.387580	+	0.921836i	$0.373311\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 234.841i	− 0.235548i	−0.993040	−	0.117774i	$-0.962424\pi$
$997$	− 234.841i	− 0.235548i	0.993040	−	0.117774i	$-0.0375758\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.g.449.2		8
3.2	odd	2	inner	3600.3.c.g.449.1		8
4.3	odd	2		1800.3.c.e.449.7		8
5.2	odd	4		3600.3.l.u.1601.4		4
5.3	odd	4		3600.3.l.o.1601.2		4
5.4	even	2	inner	3600.3.c.g.449.8		8
12.11	even	2		1800.3.c.e.449.8		8
15.2	even	4		3600.3.l.u.1601.3		4
15.8	even	4		3600.3.l.o.1601.1		4
15.14	odd	2	inner	3600.3.c.g.449.7		8
20.3	even	4		1800.3.l.f.1601.3	yes	4
20.7	even	4		1800.3.l.b.1601.1	✓	4
20.19	odd	2		1800.3.c.e.449.1		8
60.23	odd	4		1800.3.l.f.1601.4	yes	4
60.47	odd	4		1800.3.l.b.1601.2	yes	4
60.59	even	2		1800.3.c.e.449.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.3.c.e.449.1		8	20.19	odd	2
1800.3.c.e.449.2		8	60.59	even	2
1800.3.c.e.449.7		8	4.3	odd	2
1800.3.c.e.449.8		8	12.11	even	2
1800.3.l.b.1601.1	✓	4	20.7	even	4
1800.3.l.b.1601.2	yes	4	60.47	odd	4
1800.3.l.f.1601.3	yes	4	20.3	even	4
1800.3.l.f.1601.4	yes	4	60.23	odd	4
3600.3.c.g.449.1		8	3.2	odd	2	inner
3600.3.c.g.449.2		8	1.1	even	1	trivial
3600.3.c.g.449.7		8	15.14	odd	2	inner
3600.3.c.g.449.8		8	5.4	even	2	inner
3600.3.l.o.1601.1		4	15.8	even	4
3600.3.l.o.1601.2		4	5.3	odd	4
3600.3.l.u.1601.3		4	15.2	even	4
3600.3.l.u.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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-9.32456i q^{7} +19.3028i q^{11} +19.6491i q^{13} -4.70163 q^{17} -29.9737 q^{19} -9.89949 q^{23} -24.5006i q^{29} +12.6754 q^{31} -37.9473i q^{37} +73.0056i q^{41} -33.3246i q^{43} -4.70163 q^{47} -37.9473 q^{49} +73.0056 q^{53} -24.5006i q^{59} -53.7018 q^{61} -101.974i q^{67} -111.686i q^{71} -48.5438i q^{73} +179.990 q^{77} +10.5964 q^{79} +44.2251 q^{83} -38.6800i q^{89} +183.219 q^{91} +183.491i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 88 q^{19} + 152 q^{31} - 632 q^{61} - 320 q^{79} + 808 q^{91}+O(q^{100}) \) 8 * q - 88 * q^19 + 152 * q^31 - 632 * q^61 - 320 * q^79 + 808 * q^91

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