LMFDB - Embedded newform 3600.3.e.bd.3151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(3151,3600)]

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([1, 0, 0, 0]))

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

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

chi := DirichletCharacter("3600.3151");

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.e (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:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3151.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.3151
Dual form			3600.3.e.bd.3151.1

$q$-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
$7$	8.48528i	1.21218i	0.795395	+	0.606092i	$0.207263\pi$
$7$	8.48528i	1.21218i	−0.795395	+	0.606092i	$0.792737\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.8564i	1.25967i	0.776728	+	0.629837i	$0.216878\pi$
$11$	13.8564i	1.25967i	−0.776728	+	0.629837i	$0.783122\pi$
$12$	0	0
$13$	9.79796	0.753689	0.376845	−	0.926277i	$-0.377009\pi$
$13$	9.79796	0.753689	0.376845	+	0.926277i	$0.377009\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−19.5959	−1.15270	−0.576351	−	0.817203i	$-0.695524\pi$
$17$	−19.5959	−1.15270	−0.576351	+	0.817203i	$0.695524\pi$
$18$	0	0
$19$	− 13.8564i	− 0.729285i	−0.931148	−	0.364642i	$-0.881191\pi$
$19$	− 13.8564i	− 0.729285i	0.931148	−	0.364642i	$-0.118809\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 25.4558i	− 1.10678i	−0.832924	−	0.553388i	$-0.813335\pi$
$23$	− 25.4558i	− 1.10678i	0.832924	−	0.553388i	$-0.186665\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−22.0000	−0.758621	−0.379310	−	0.925270i	$-0.623839\pi$
$29$	−22.0000	−0.758621	−0.379310	+	0.925270i	$0.623839\pi$
$30$	0	0
$31$	− 55.4256i	− 1.78792i	−0.448143	−	0.893962i	$-0.647915\pi$
$31$	− 55.4256i	− 1.78792i	0.448143	−	0.893962i	$-0.352085\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−48.9898	−1.32405	−0.662024	−	0.749482i	$-0.730302\pi$
$37$	−48.9898	−1.32405	−0.662024	+	0.749482i	$0.730302\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−22.0000	−0.536585	−0.268293	−	0.963337i	$-0.586459\pi$
$41$	−22.0000	−0.536585	−0.268293	+	0.963337i	$0.586459\pi$
$42$	0	0
$43$	− 59.3970i	− 1.38132i	−0.723177	−	0.690662i	$-0.757319\pi$
$43$	− 59.3970i	− 1.38132i	0.723177	−	0.690662i	$-0.242681\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.48528i	− 0.180538i	−0.995917	−	0.0902690i	$-0.971227\pi$
$47$	− 8.48528i	− 0.180538i	0.995917	−	0.0902690i	$-0.0287727\pi$
$48$	0	0
$49$	−23.0000	−0.469388
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−29.3939	−0.554601	−0.277301	−	0.960783i	$-0.589440\pi$
$53$	−29.3939	−0.554601	−0.277301	+	0.960783i	$0.589440\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.8564i	− 0.234854i	−0.993081	−	0.117427i	$-0.962535\pi$
$59$	− 13.8564i	− 0.234854i	0.993081	−	0.117427i	$-0.0374647\pi$
$60$	0	0
$61$	46.0000	0.754098	0.377049	−	0.926193i	$-0.376939\pi$
$61$	46.0000	0.754098	0.377049	+	0.926193i	$0.376939\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	59.3970i	0.886522i	0.896393	+	0.443261i	$0.146178\pi$
$67$	59.3970i	0.886522i	−0.896393	+	0.443261i	$0.853822\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 27.7128i	− 0.390321i	−0.980771	−	0.195161i	$-0.937477\pi$
$71$	− 27.7128i	− 0.390321i	0.980771	−	0.195161i	$-0.0625228\pi$
$72$	0	0
$73$	78.3837	1.07375	0.536874	−	0.843662i	$-0.319605\pi$
$73$	78.3837	1.07375	0.536874	+	0.843662i	$0.319605\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−117.576	−1.52695
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 76.3675i	− 0.920091i	−0.887895	−	0.460045i	$-0.847833\pi$
$83$	− 76.3675i	− 0.920091i	0.887895	−	0.460045i	$-0.152167\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	146.000	1.64045	0.820225	−	0.572041i	$-0.193848\pi$
$89$	146.000	1.64045	0.820225	+	0.572041i	$0.193848\pi$
$90$	0	0
$91$	83.1384i	0.913609i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	58.7878	0.606059	0.303030	−	0.952981i	$-0.402002\pi$
$97$	58.7878	0.606059	0.303030	+	0.952981i	$0.402002\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	70.0000	0.693069	0.346535	−	0.938037i	$-0.387358\pi$
$101$	70.0000	0.693069	0.346535	+	0.938037i	$0.387358\pi$
$102$	0	0
$103$	25.4558i	0.247144i	0.992336	+	0.123572i	$0.0394350\pi$
$103$	25.4558i	0.247144i	−0.992336	+	0.123572i	$0.960565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	42.4264i	0.396508i	0.980151	+	0.198254i	$0.0635272\pi$
$107$	42.4264i	0.396508i	−0.980151	+	0.198254i	$0.936473\pi$
$108$	0	0
$109$	146.000	1.33945	0.669725	−	0.742609i	$-0.266412\pi$
$109$	146.000	1.33945	0.669725	+	0.742609i	$0.266412\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−39.1918	−0.346830	−0.173415	−	0.984849i	$-0.555480\pi$
$113$	−39.1918	−0.346830	−0.173415	+	0.984849i	$0.555480\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 166.277i	− 1.39728i
$120$	0	0
$121$	−71.0000	−0.586777
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	110.309i	0.868572i	0.900775	+	0.434286i	$0.142999\pi$
$127$	110.309i	0.868572i	−0.900775	+	0.434286i	$0.857001\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 180.133i	− 1.37506i	−0.726154	−	0.687532i	$-0.758694\pi$
$131$	− 180.133i	− 1.37506i	0.726154	−	0.687532i	$-0.241306\pi$
$132$	0	0
$133$	117.576	0.884026
$134$	0	0
$135$	0	0
$136$	0	0
$137$	215.555	1.57339	0.786697	−	0.617339i	$-0.211789\pi$
$137$	215.555	1.57339	0.786697	+	0.617339i	$0.211789\pi$
$138$	0	0
$139$	13.8564i	0.0996864i	0.998757	+	0.0498432i	$0.0158722\pi$
$139$	13.8564i	0.0996864i	−0.998757	+	0.0498432i	$0.984128\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	135.765i	0.949402i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.0134228	−0.00671141	−	0.999977i	$-0.502136\pi$
$149$	−2.00000	−0.0134228	−0.00671141	+	0.999977i	$0.502136\pi$
$150$	0	0
$151$	27.7128i	0.183529i	0.995781	+	0.0917643i	$0.0292506\pi$
$151$	27.7128i	0.183529i	−0.995781	+	0.0917643i	$0.970749\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−68.5857	−0.436852	−0.218426	−	0.975854i	$-0.570092\pi$
$157$	−68.5857	−0.436852	−0.218426	+	0.975854i	$0.570092\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	216.000	1.34161
$162$	0	0
$163$	110.309i	0.676740i	0.941013	+	0.338370i	$0.109876\pi$
$163$	110.309i	0.676740i	−0.941013	+	0.338370i	$0.890124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	93.3381i	0.558911i	0.960159	+	0.279455i	$0.0901539\pi$
$167$	93.3381i	0.558911i	−0.960159	+	0.279455i	$0.909846\pi$
$168$	0	0
$169$	−73.0000	−0.431953
$170$	0	0
$171$	0	0
$172$	0	0
$173$	48.9898	0.283178	0.141589	−	0.989926i	$-0.454779\pi$
$173$	48.9898	0.283178	0.141589	+	0.989926i	$0.454779\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 263.272i	− 1.47079i	−0.677638	−	0.735396i	$-0.736996\pi$
$179$	− 263.272i	− 1.47079i	0.677638	−	0.735396i	$-0.263004\pi$
$180$	0	0
$181$	26.0000	0.143646	0.0718232	−	0.997417i	$-0.477118\pi$
$181$	26.0000	0.143646	0.0718232	+	0.997417i	$0.477118\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 271.529i	− 1.45203i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 110.851i	− 0.580373i	−0.956970	−	0.290187i	$-0.906283\pi$
$191$	− 110.851i	− 0.580373i	0.956970	−	0.290187i	$-0.0937174\pi$
$192$	0	0
$193$	−333.131	−1.72607	−0.863033	−	0.505148i	$-0.831438\pi$
$193$	−333.131	−1.72607	−0.863033	+	0.505148i	$0.831438\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	107.778	0.547094	0.273547	−	0.961859i	$-0.411803\pi$
$197$	107.778	0.547094	0.273547	+	0.961859i	$0.411803\pi$
$198$	0	0
$199$	− 249.415i	− 1.25334i	−0.779283	−	0.626672i	$-0.784417\pi$
$199$	− 249.415i	− 1.25334i	0.779283	−	0.626672i	$-0.215583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 186.676i	− 0.919587i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	192.000	0.918660
$210$	0	0
$211$	− 96.9948i	− 0.459691i	−0.973227	−	0.229846i	$-0.926178\pi$
$211$	− 96.9948i	− 0.459691i	0.973227	−	0.229846i	$-0.0738221\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	470.302	2.16729
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−192.000	−0.868778
$222$	0	0
$223$	229.103i	1.02737i	0.857980	+	0.513683i	$0.171719\pi$
$223$	229.103i	1.02737i	−0.857980	+	0.513683i	$0.828281\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 296.985i	− 1.30830i	−0.756363	−	0.654152i	$-0.773026\pi$
$227$	− 296.985i	− 1.30830i	0.756363	−	0.654152i	$-0.226974\pi$
$228$	0	0
$229$	70.0000	0.305677	0.152838	−	0.988251i	$-0.451159\pi$
$229$	70.0000	0.305677	0.152838	+	0.988251i	$0.451159\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−156.767	−0.672821	−0.336411	−	0.941715i	$-0.609213\pi$
$233$	−156.767	−0.672821	−0.336411	+	0.941715i	$0.609213\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 332.554i	− 1.39144i	−0.718314	−	0.695719i	$-0.755086\pi$
$239$	− 332.554i	− 1.39144i	0.718314	−	0.695719i	$-0.244914\pi$
$240$	0	0
$241$	−122.000	−0.506224	−0.253112	−	0.967437i	$-0.581454\pi$
$241$	−122.000	−0.506224	−0.253112	+	0.967437i	$0.581454\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 135.765i	− 0.549654i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	290.985i	1.15930i	0.814865	+	0.579650i	$0.196811\pi$
$251$	290.985i	1.15930i	−0.814865	+	0.579650i	$0.803189\pi$
$252$	0	0
$253$	352.727	1.39418
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−78.3837	−0.304995	−0.152497	−	0.988304i	$-0.548732\pi$
$257$	−78.3837	−0.304995	−0.152497	+	0.988304i	$0.548732\pi$
$258$	0	0
$259$	− 415.692i	− 1.60499i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	347.897i	1.32280i	0.750033	+	0.661400i	$0.230038\pi$
$263$	347.897i	1.32280i	−0.750033	+	0.661400i	$0.769962\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	142.000	0.527881	0.263941	−	0.964539i	$-0.414978\pi$
$269$	142.000	0.527881	0.263941	+	0.964539i	$0.414978\pi$
$270$	0	0
$271$	− 332.554i	− 1.22714i	−0.789642	−	0.613568i	$-0.789734\pi$
$271$	− 332.554i	− 1.22714i	0.789642	−	0.613568i	$-0.210266\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−88.1816	−0.318345	−0.159173	−	0.987251i	$-0.550883\pi$
$277$	−88.1816	−0.318345	−0.159173	+	0.987251i	$0.550883\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−406.000	−1.44484	−0.722420	−	0.691455i	$-0.756970\pi$
$281$	−406.000	−1.44484	−0.722420	+	0.691455i	$0.756970\pi$
$282$	0	0
$283$	− 93.3381i	− 0.329817i	−0.986309	−	0.164908i	$-0.947267\pi$
$283$	− 93.3381i	− 0.329817i	0.986309	−	0.164908i	$-0.0527328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 186.676i	− 0.650440i
$288$	0	0
$289$	95.0000	0.328720
$290$	0	0
$291$	0	0
$292$	0	0
$293$	303.737	1.03664	0.518322	−	0.855186i	$-0.326557\pi$
$293$	303.737	1.03664	0.518322	+	0.855186i	$0.326557\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 249.415i	− 0.834165i
$300$	0	0
$301$	504.000	1.67442
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 280.014i	− 0.912099i	−0.889955	−	0.456049i	$-0.849264\pi$
$307$	− 280.014i	− 0.912099i	0.889955	−	0.456049i	$-0.150736\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 138.564i	− 0.445544i	−0.974871	−	0.222772i	$-0.928490\pi$
$311$	− 138.564i	− 0.445544i	0.974871	−	0.222772i	$-0.0715105\pi$
$312$	0	0
$313$	−607.473	−1.94081	−0.970405	−	0.241484i	$-0.922366\pi$
$313$	−607.473	−1.94081	−0.970405	+	0.241484i	$0.922366\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−68.5857	−0.216359	−0.108179	−	0.994131i	$-0.534502\pi$
$317$	−68.5857	−0.216359	−0.108179	+	0.994131i	$0.534502\pi$
$318$	0	0
$319$	− 304.841i	− 0.955614i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	271.529i	0.840647i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	72.0000	0.218845
$330$	0	0
$331$	− 180.133i	− 0.544209i	−0.962268	−	0.272105i	$-0.912280\pi$
$331$	− 180.133i	− 0.544209i	0.962268	−	0.272105i	$-0.0877197\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−470.302	−1.39555	−0.697777	−	0.716315i	$-0.745828\pi$
$337$	−470.302	−1.39555	−0.697777	+	0.716315i	$0.745828\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	768.000	2.25220
$342$	0	0
$343$	220.617i	0.643199i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	619.426i	1.78509i	0.450960	+	0.892544i	$0.351082\pi$
$347$	619.426i	1.78509i	−0.450960	+	0.892544i	$0.648918\pi$
$348$	0	0
$349$	214.000	0.613181	0.306590	−	0.951842i	$-0.400812\pi$
$349$	214.000	0.613181	0.306590	+	0.951842i	$0.400812\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−352.727	−0.999225	−0.499613	−	0.866249i	$-0.666524\pi$
$353$	−352.727	−0.999225	−0.499613	+	0.866249i	$0.666524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 27.7128i	− 0.0771945i	−0.999255	−	0.0385972i	$-0.987711\pi$
$359$	− 27.7128i	− 0.0771945i	0.999255	−	0.0385972i	$-0.0122889\pi$
$360$	0	0
$361$	169.000	0.468144
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	246.073i	0.670499i	0.942129	+	0.335250i	$0.108821\pi$
$367$	246.073i	0.670499i	−0.942129	+	0.335250i	$0.891179\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 249.415i	− 0.672278i
$372$	0	0
$373$	−382.120	−1.02445	−0.512226	−	0.858851i	$-0.671179\pi$
$373$	−382.120	−1.02445	−0.512226	+	0.858851i	$0.671179\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−215.555	−0.571764
$378$	0	0
$379$	290.985i	0.767769i	0.923381	+	0.383885i	$0.125414\pi$
$379$	290.985i	0.767769i	−0.923381	+	0.383885i	$0.874586\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 568.514i	− 1.48437i	−0.670195	−	0.742185i	$-0.733789\pi$
$383$	− 568.514i	− 1.48437i	0.670195	−	0.742185i	$-0.266211\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	670.000	1.72237	0.861183	−	0.508296i	$-0.169724\pi$
$389$	670.000	1.72237	0.861183	+	0.508296i	$0.169724\pi$
$390$	0	0
$391$	498.831i	1.27578i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−362.524	−0.913160	−0.456580	−	0.889682i	$-0.650926\pi$
$397$	−362.524	−0.913160	−0.456580	+	0.889682i	$0.650926\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.00498753	−0.00249377	−	0.999997i	$-0.500794\pi$
$401$	−2.00000	−0.00498753	−0.00249377	+	0.999997i	$0.500794\pi$
$402$	0	0
$403$	− 543.058i	− 1.34754i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 678.823i	− 1.66787i
$408$	0	0
$409$	650.000	1.58924	0.794621	−	0.607106i	$-0.207670\pi$
$409$	650.000	1.58924	0.794621	+	0.607106i	$0.207670\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	117.576	0.284686
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 540.400i	− 1.28974i	−0.764293	−	0.644869i	$-0.776912\pi$
$419$	− 540.400i	− 1.28974i	0.764293	−	0.644869i	$-0.223088\pi$
$420$	0	0
$421$	−482.000	−1.14489	−0.572447	−	0.819942i	$-0.694006\pi$
$421$	−482.000	−1.14489	−0.572447	+	0.819942i	$0.694006\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	390.323i	0.914105i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 221.703i	− 0.514391i	−0.966359	−	0.257195i	$-0.917202\pi$
$431$	− 221.703i	− 0.514391i	0.966359	−	0.257195i	$-0.0827984\pi$
$432$	0	0
$433$	568.282	1.31243	0.656214	−	0.754575i	$-0.272157\pi$
$433$	568.282	1.31243	0.656214	+	0.754575i	$0.272157\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−352.727	−0.807155
$438$	0	0
$439$	526.543i	1.19942i	0.800219	+	0.599708i	$0.204717\pi$
$439$	526.543i	1.19942i	−0.800219	+	0.599708i	$0.795283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	432.749i	0.976861i	0.872603	+	0.488430i	$0.162430\pi$
$443$	432.749i	0.976861i	−0.872603	+	0.488430i	$0.837570\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.0000	−0.0579065	−0.0289532	−	0.999581i	$-0.509217\pi$
$449$	−26.0000	−0.0579065	−0.0289532	+	0.999581i	$0.509217\pi$
$450$	0	0
$451$	− 304.841i	− 0.675922i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	137.171	0.300156	0.150078	−	0.988674i	$-0.452047\pi$
$457$	137.171	0.300156	0.150078	+	0.988674i	$0.452047\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	502.000	1.08894	0.544469	−	0.838781i	$-0.316731\pi$
$461$	502.000	1.08894	0.544469	+	0.838781i	$0.316731\pi$
$462$	0	0
$463$	− 823.072i	− 1.77769i	−0.458204	−	0.888847i	$-0.651507\pi$
$463$	− 823.072i	− 1.77769i	0.458204	−	0.888847i	$-0.348493\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	178.191i	0.381565i	0.981632	+	0.190783i	$0.0611025\pi$
$467$	178.191i	0.381565i	−0.981632	+	0.190783i	$0.938897\pi$
$468$	0	0
$469$	−504.000	−1.07463
$470$	0	0
$471$	0	0
$472$	0	0
$473$	823.029	1.74002
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 775.959i	− 1.61996i	−0.586460	−	0.809978i	$-0.699479\pi$
$479$	− 775.959i	− 1.61996i	0.586460	−	0.809978i	$-0.300521\pi$
$480$	0	0
$481$	−480.000	−0.997921
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	347.897i	0.714367i	0.934034	+	0.357183i	$0.116263\pi$
$487$	347.897i	0.714367i	−0.934034	+	0.357183i	$0.883737\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 928.379i	− 1.89079i	−0.325923	−	0.945396i	$-0.605675\pi$
$491$	− 928.379i	− 1.89079i	0.325923	−	0.945396i	$-0.394325\pi$
$492$	0	0
$493$	431.110	0.874463
$494$	0	0
$495$	0	0
$496$	0	0
$497$	235.151	0.473141
$498$	0	0
$499$	− 512.687i	− 1.02743i	−0.857961	−	0.513714i	$-0.828269\pi$
$499$	− 512.687i	− 1.02743i	0.857961	−	0.513714i	$-0.171731\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 704.278i	− 1.40016i	−0.714066	−	0.700078i	$-0.753149\pi$
$503$	− 704.278i	− 1.40016i	0.714066	−	0.700078i	$-0.246851\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	170.000	0.333988	0.166994	−	0.985958i	$-0.446594\pi$
$509$	170.000	0.333988	0.166994	+	0.985958i	$0.446594\pi$
$510$	0	0
$511$	665.108i	1.30158i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	117.576	0.227419
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−722.000	−1.38580	−0.692898	−	0.721035i	$-0.743667\pi$
$521$	−722.000	−1.38580	−0.692898	+	0.721035i	$0.743667\pi$
$522$	0	0
$523$	246.073i	0.470503i	0.971935	+	0.235252i	$0.0755914\pi$
$523$	246.073i	0.470503i	−0.971935	+	0.235252i	$0.924409\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1086.12i	2.06094i
$528$	0	0
$529$	−119.000	−0.224953
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−215.555	−0.404419
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 318.697i	− 0.591275i
$540$	0	0
$541$	362.000	0.669131	0.334566	−	0.942372i	$-0.391410\pi$
$541$	362.000	0.669131	0.334566	+	0.942372i	$0.391410\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 42.4264i	− 0.0775620i	−0.999248	−	0.0387810i	$-0.987653\pi$
$547$	− 42.4264i	− 0.0775620i	0.999248	−	0.0387810i	$-0.0123475\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	304.841i	0.553250i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	656.463	1.17857	0.589285	−	0.807925i	$-0.299410\pi$
$557$	656.463	1.17857	0.589285	+	0.807925i	$0.299410\pi$
$558$	0	0
$559$	− 581.969i	− 1.04109i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 517.602i	− 0.919364i	−0.888083	−	0.459682i	$-0.847963\pi$
$563$	− 517.602i	− 0.919364i	0.888083	−	0.459682i	$-0.152037\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−842.000	−1.47979	−0.739895	−	0.672723i	$-0.765125\pi$
$569$	−842.000	−1.47979	−0.739895	+	0.672723i	$0.765125\pi$
$570$	0	0
$571$	− 401.836i	− 0.703740i	−0.936049	−	0.351870i	$-0.885546\pi$
$571$	− 401.836i	− 0.703740i	0.936049	−	0.351870i	$-0.114454\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.1918	−0.0679235	−0.0339617	−	0.999423i	$-0.510812\pi$
$577$	−39.1918	−0.0679235	−0.0339617	+	0.999423i	$0.510812\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	648.000	1.11532
$582$	0	0
$583$	− 407.294i	− 0.698617i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 59.3970i	− 0.101187i	−0.998719	−	0.0505937i	$-0.983889\pi$
$587$	− 59.3970i	− 0.101187i	0.998719	−	0.0505937i	$-0.0161113\pi$
$588$	0	0
$589$	−768.000	−1.30390
$590$	0	0
$591$	0	0
$592$	0	0
$593$	509.494	0.859180	0.429590	−	0.903024i	$-0.358658\pi$
$593$	509.494	0.859180	0.429590	+	0.903024i	$0.358658\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	859.097i	1.43422i	0.696961	+	0.717110i	$0.254535\pi$
$599$	859.097i	1.43422i	−0.696961	+	0.717110i	$0.745465\pi$
$600$	0	0
$601$	598.000	0.995008	0.497504	−	0.867462i	$-0.334250\pi$
$601$	598.000	0.995008	0.497504	+	0.867462i	$0.334250\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 127.279i	− 0.209686i	−0.994489	−	0.104843i	$-0.966566\pi$
$607$	− 127.279i	− 0.209686i	0.994489	−	0.104843i	$-0.0334340\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 83.1384i	− 0.136069i
$612$	0	0
$613$	−205.757	−0.335656	−0.167828	−	0.985816i	$-0.553675\pi$
$613$	−205.757	−0.335656	−0.167828	+	0.985816i	$0.553675\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	431.110	0.698720	0.349360	−	0.936989i	$-0.386399\pi$
$617$	431.110	0.698720	0.349360	+	0.936989i	$0.386399\pi$
$618$	0	0
$619$	− 41.5692i	− 0.0671554i	−0.999436	−	0.0335777i	$-0.989310\pi$
$619$	− 41.5692i	− 0.0671554i	0.999436	−	0.0335777i	$-0.0106901\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1238.85i	1.98853i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	960.000	1.52623
$630$	0	0
$631$	− 969.948i	− 1.53716i	−0.639753	−	0.768580i	$-0.720963\pi$
$631$	− 969.948i	− 1.53716i	0.639753	−	0.768580i	$-0.279037\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−225.353	−0.353772
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−166.000	−0.258970	−0.129485	−	0.991581i	$-0.541332\pi$
$641$	−166.000	−0.258970	−0.129485	+	0.991581i	$0.541332\pi$
$642$	0	0
$643$	144.250i	0.224339i	0.993689	+	0.112169i	$0.0357799\pi$
$643$	144.250i	0.224339i	−0.993689	+	0.112169i	$0.964220\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 687.308i	− 1.06230i	−0.847278	−	0.531150i	$-0.821760\pi$
$647$	− 687.308i	− 1.06230i	0.847278	−	0.531150i	$-0.178240\pi$
$648$	0	0
$649$	192.000	0.295840
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.3939	0.0450136	0.0225068	−	0.999747i	$-0.492835\pi$
$653$	29.3939	0.0450136	0.0225068	+	0.999747i	$0.492835\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	623.538i	0.946189i	0.881012	+	0.473094i	$0.156863\pi$
$659$	623.538i	0.946189i	−0.881012	+	0.473094i	$0.843137\pi$
$660$	0	0
$661$	−98.0000	−0.148260	−0.0741301	−	0.997249i	$-0.523618\pi$
$661$	−98.0000	−0.148260	−0.0741301	+	0.997249i	$0.523618\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	560.029i	0.839623i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	637.395i	0.949918i
$672$	0	0
$673$	489.898	0.727932	0.363966	−	0.931412i	$-0.381423\pi$
$673$	489.898	0.727932	0.363966	+	0.931412i	$0.381423\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	146.969	0.217089	0.108545	−	0.994092i	$-0.465381\pi$
$677$	146.969	0.217089	0.108545	+	0.994092i	$0.465381\pi$
$678$	0	0
$679$	498.831i	0.734655i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	398.808i	0.583907i	0.956433	+	0.291953i	$0.0943052\pi$
$683$	398.808i	0.583907i	−0.956433	+	0.291953i	$0.905695\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−288.000	−0.417997
$690$	0	0
$691$	789.815i	1.14300i	0.820601	+	0.571502i	$0.193639\pi$
$691$	789.815i	1.14300i	−0.820601	+	0.571502i	$0.806361\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	431.110	0.618523
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−718.000	−1.02425	−0.512126	−	0.858911i	$-0.671142\pi$
$701$	−718.000	−1.02425	−0.512126	+	0.858911i	$0.671142\pi$
$702$	0	0
$703$	678.823i	0.965608i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	593.970i	0.840127i
$708$	0	0
$709$	838.000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$709$	838.000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1410.91	−1.97883
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 221.703i	− 0.308348i	−0.988044	−	0.154174i	$-0.950728\pi$
$719$	− 221.703i	− 0.308348i	0.988044	−	0.154174i	$-0.0492717\pi$
$720$	0	0
$721$	−216.000	−0.299584
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	347.897i	0.478537i	0.970953	+	0.239269i	$0.0769076\pi$
$727$	347.897i	0.478537i	−0.970953	+	0.239269i	$0.923092\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1163.94i	1.59225i
$732$	0	0
$733$	−676.059	−0.922318	−0.461159	−	0.887317i	$-0.652566\pi$
$733$	−676.059	−0.922318	−0.461159	+	0.887317i	$0.652566\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−823.029	−1.11673
$738$	0	0
$739$	− 235.559i	− 0.318754i	−0.987218	−	0.159377i	$-0.949052\pi$
$739$	− 235.559i	− 0.318754i	0.987218	−	0.159377i	$-0.0509485\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1349.16i	− 1.81583i	−0.419157	−	0.907914i	$-0.637674\pi$
$743$	− 1349.16i	− 1.81583i	0.419157	−	0.907914i	$-0.362326\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−360.000	−0.480641
$750$	0	0
$751$	− 1163.94i	− 1.54985i	−0.632053	−	0.774926i	$-0.717787\pi$
$751$	− 1163.94i	− 1.54985i	0.632053	−	0.774926i	$-0.282213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1205.15	1.59201	0.796003	−	0.605292i	$-0.206944\pi$
$757$	1205.15	1.59201	0.796003	+	0.605292i	$0.206944\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−530.000	−0.696452	−0.348226	−	0.937411i	$-0.613216\pi$
$761$	−530.000	−0.696452	−0.348226	+	0.937411i	$0.613216\pi$
$762$	0	0
$763$	1238.85i	1.62366i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 135.765i	− 0.177007i
$768$	0	0
$769$	−386.000	−0.501951	−0.250975	−	0.967993i	$-0.580751\pi$
$769$	−386.000	−0.501951	−0.250975	+	0.967993i	$0.580751\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−695.655	−0.899942	−0.449971	−	0.893043i	$-0.648566\pi$
$773$	−695.655	−0.899942	−0.449971	+	0.893043i	$0.648566\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	304.841i	0.391323i
$780$	0	0
$781$	384.000	0.491677
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1298.25i	− 1.64962i	−0.565413	−	0.824808i	$-0.691283\pi$
$787$	− 1298.25i	− 1.64962i	0.565413	−	0.824808i	$-0.308717\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 332.554i	− 0.420422i
$792$	0	0
$793$	450.706	0.568356
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1361.92	−1.70880	−0.854402	−	0.519613i	$-0.826076\pi$
$797$	−1361.92	−1.70880	−0.854402	+	0.519613i	$0.826076\pi$
$798$	0	0
$799$	166.277i	0.208106i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1086.12i	1.35257i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1006.00	−1.24351	−0.621755	−	0.783212i	$-0.713580\pi$
$809$	−1006.00	−1.24351	−0.621755	+	0.783212i	$0.713580\pi$
$810$	0	0
$811$	651.251i	0.803022i	0.915854	+	0.401511i	$0.131515\pi$
$811$	651.251i	0.803022i	−0.915854	+	0.401511i	$0.868485\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−823.029	−1.00738
$818$	0	0
$819$	0	0
$820$	0	0
$821$	482.000	0.587089	0.293544	−	0.955945i	$-0.405165\pi$
$821$	482.000	0.587089	0.293544	+	0.955945i	$0.405165\pi$
$822$	0	0
$823$	636.396i	0.773264i	0.922234	+	0.386632i	$0.126362\pi$
$823$	636.396i	0.773264i	−0.922234	+	0.386632i	$0.873638\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 398.808i	− 0.482235i	−0.970496	−	0.241117i	$-0.922486\pi$
$827$	− 398.808i	− 0.482235i	0.970496	−	0.241117i	$-0.0775139\pi$
$828$	0	0
$829$	1106.00	1.33414	0.667069	−	0.744996i	$-0.267549\pi$
$829$	1106.00	1.33414	0.667069	+	0.744996i	$0.267549\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	450.706	0.541064
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 360.267i	− 0.429400i	−0.976680	−	0.214700i	$-0.931123\pi$
$839$	− 360.267i	− 0.429400i	0.976680	−	0.214700i	$-0.0688774\pi$
$840$	0	0
$841$	−357.000	−0.424495
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 602.455i	− 0.711281i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1247.08i	1.46542i
$852$	0	0
$853$	−676.059	−0.792566	−0.396283	−	0.918128i	$-0.629700\pi$
$853$	−676.059	−0.792566	−0.396283	+	0.918128i	$0.629700\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1077.78	1.25761	0.628807	−	0.777561i	$-0.283543\pi$
$857$	1077.78	1.25761	0.628807	+	0.777561i	$0.283543\pi$
$858$	0	0
$859$	568.113i	0.661365i	0.943742	+	0.330683i	$0.107279\pi$
$859$	568.113i	0.661365i	−0.943742	+	0.330683i	$0.892721\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 195.161i	− 0.226143i	−0.993587	−	0.113072i	$-0.963931\pi$
$863$	− 195.161i	− 0.226143i	0.993587	−	0.113072i	$-0.0360689\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	581.969i	0.668162i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	244.949	0.279303	0.139652	−	0.990201i	$-0.455402\pi$
$877$	244.949	0.279303	0.139652	+	0.990201i	$0.455402\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−550.000	−0.624291	−0.312145	−	0.950034i	$-0.601048\pi$
$881$	−550.000	−0.624291	−0.312145	+	0.950034i	$0.601048\pi$
$882$	0	0
$883$	− 636.396i	− 0.720720i	−0.932813	−	0.360360i	$-0.882654\pi$
$883$	− 636.396i	− 0.720720i	0.932813	−	0.360360i	$-0.117346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1230.37i	− 1.38711i	−0.720404	−	0.693555i	$-0.756044\pi$
$887$	− 1230.37i	− 1.38711i	0.720404	−	0.693555i	$-0.243956\pi$
$888$	0	0
$889$	−936.000	−1.05287
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−117.576	−0.131664
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1219.36i	1.35636i
$900$	0	0
$901$	576.000	0.639290
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 585.484i	− 0.645518i	−0.946481	−	0.322759i	$-0.895390\pi$
$907$	− 585.484i	− 0.645518i	0.946481	−	0.322759i	$-0.104610\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 886.810i	− 0.973447i	−0.873556	−	0.486723i	$-0.838192\pi$
$911$	− 886.810i	− 0.973447i	0.873556	−	0.486723i	$-0.161808\pi$
$912$	0	0
$913$	1058.18	1.15901
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1528.48	1.66683
$918$	0	0
$919$	− 193.990i	− 0.211088i	−0.994415	−	0.105544i	$-0.966342\pi$
$919$	− 193.990i	− 0.211088i	0.994415	−	0.105544i	$-0.0336584\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 271.529i	− 0.294181i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1318.00	1.41873	0.709365	−	0.704841i	$-0.248982\pi$
$929$	1318.00	1.41873	0.709365	+	0.704841i	$0.248982\pi$
$930$	0	0
$931$	318.697i	0.342317i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−548.686	−0.585577	−0.292789	−	0.956177i	$-0.594583\pi$
$937$	−548.686	−0.585577	−0.292789	+	0.956177i	$0.594583\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1654.00	1.75770	0.878852	−	0.477094i	$-0.158310\pi$
$941$	1654.00	1.75770	0.878852	+	0.477094i	$0.158310\pi$
$942$	0	0
$943$	560.029i	0.593880i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1417.04i	− 1.49635i	−0.663502	−	0.748174i	$-0.730931\pi$
$947$	− 1417.04i	− 1.49635i	0.663502	−	0.748174i	$-0.269069\pi$
$948$	0	0
$949$	768.000	0.809273
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−137.171	−0.143936	−0.0719682	−	0.997407i	$-0.522928\pi$
$953$	−137.171	−0.143936	−0.0719682	+	0.997407i	$0.522928\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1829.05i	1.90724i
$960$	0	0
$961$	−2111.00	−2.19667
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1450.98i	− 1.50050i	−0.661154	−	0.750250i	$-0.729933\pi$
$967$	− 1450.98i	− 1.50050i	0.661154	−	0.750250i	$-0.270067\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 817.528i	− 0.841944i	−0.907074	−	0.420972i	$-0.861689\pi$
$971$	− 817.528i	− 0.841944i	0.907074	−	0.420972i	$-0.138311\pi$
$972$	0	0
$973$	−117.576	−0.120838
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1704.84	−1.74498	−0.872490	−	0.488633i	$-0.837496\pi$
$977$	−1704.84	−1.74498	−0.872490	+	0.488633i	$0.837496\pi$
$978$	0	0
$979$	2023.04i	2.06643i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 738.219i	− 0.750986i	−0.926825	−	0.375493i	$-0.877473\pi$
$983$	− 738.219i	− 0.750986i	0.926825	−	0.375493i	$-0.122527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1512.00	−1.52882
$990$	0	0
$991$	775.959i	0.783006i	0.920177	+	0.391503i	$0.128045\pi$
$991$	775.959i	0.783006i	−0.920177	+	0.391503i	$0.871955\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1420.70	1.42498	0.712489	−	0.701683i	$-0.247568\pi$
$997$	1420.70	1.42498	0.712489	+	0.701683i	$0.247568\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.e.bd.3151.4		4
3.2	odd	2		400.3.b.h.351.1		4
4.3	odd	2	inner	3600.3.e.bd.3151.1		4
5.2	odd	4		720.3.j.e.559.3		4
5.3	odd	4		720.3.j.e.559.2		4
5.4	even	2	inner	3600.3.e.bd.3151.2		4
12.11	even	2		400.3.b.h.351.4		4
15.2	even	4		80.3.h.b.79.1	✓	4
15.8	even	4		80.3.h.b.79.4	yes	4
15.14	odd	2		400.3.b.h.351.3		4
20.3	even	4		720.3.j.e.559.1		4
20.7	even	4		720.3.j.e.559.4		4
20.19	odd	2	inner	3600.3.e.bd.3151.3		4
24.5	odd	2		1600.3.b.t.1151.4		4
24.11	even	2		1600.3.b.t.1151.1		4
60.23	odd	4		80.3.h.b.79.2	yes	4
60.47	odd	4		80.3.h.b.79.3	yes	4
60.59	even	2		400.3.b.h.351.2		4
120.29	odd	2		1600.3.b.t.1151.2		4
120.53	even	4		320.3.h.e.319.1		4
120.59	even	2		1600.3.b.t.1151.3		4
120.77	even	4		320.3.h.e.319.4		4
120.83	odd	4		320.3.h.e.319.3		4
120.107	odd	4		320.3.h.e.319.2		4
240.53	even	4		1280.3.e.j.639.1		8
240.77	even	4		1280.3.e.j.639.2		8
240.83	odd	4		1280.3.e.j.639.4		8
240.107	odd	4		1280.3.e.j.639.3		8
240.173	even	4		1280.3.e.j.639.8		8
240.197	even	4		1280.3.e.j.639.7		8
240.203	odd	4		1280.3.e.j.639.5		8
240.227	odd	4		1280.3.e.j.639.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.h.b.79.1	✓	4	15.2	even	4
80.3.h.b.79.2	yes	4	60.23	odd	4
80.3.h.b.79.3	yes	4	60.47	odd	4
80.3.h.b.79.4	yes	4	15.8	even	4
320.3.h.e.319.1		4	120.53	even	4
320.3.h.e.319.2		4	120.107	odd	4
320.3.h.e.319.3		4	120.83	odd	4
320.3.h.e.319.4		4	120.77	even	4
400.3.b.h.351.1		4	3.2	odd	2
400.3.b.h.351.2		4	60.59	even	2
400.3.b.h.351.3		4	15.14	odd	2
400.3.b.h.351.4		4	12.11	even	2
720.3.j.e.559.1		4	20.3	even	4
720.3.j.e.559.2		4	5.3	odd	4
720.3.j.e.559.3		4	5.2	odd	4
720.3.j.e.559.4		4	20.7	even	4
1280.3.e.j.639.1		8	240.53	even	4
1280.3.e.j.639.2		8	240.77	even	4
1280.3.e.j.639.3		8	240.107	odd	4
1280.3.e.j.639.4		8	240.83	odd	4
1280.3.e.j.639.5		8	240.203	odd	4
1280.3.e.j.639.6		8	240.227	odd	4
1280.3.e.j.639.7		8	240.197	even	4
1280.3.e.j.639.8		8	240.173	even	4
1600.3.b.t.1151.1		4	24.11	even	2
1600.3.b.t.1151.2		4	120.29	odd	2
1600.3.b.t.1151.3		4	120.59	even	2
1600.3.b.t.1151.4		4	24.5	odd	2
3600.3.e.bd.3151.1		4	4.3	odd	2	inner
3600.3.e.bd.3151.2		4	5.4	even	2	inner
3600.3.e.bd.3151.3		4	20.19	odd	2	inner
3600.3.e.bd.3151.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.48528i q^{7} +13.8564i q^{11} +9.79796 q^{13} -19.5959 q^{17} -13.8564i q^{19} -25.4558i q^{23} -22.0000 q^{29} -55.4256i q^{31} -48.9898 q^{37} -22.0000 q^{41} -59.3970i q^{43} -8.48528i q^{47} -23.0000 q^{49} -29.3939 q^{53} -13.8564i q^{59} +46.0000 q^{61} +59.3970i q^{67} -27.7128i q^{71} +78.3837 q^{73} -117.576 q^{77} -76.3675i q^{83} +146.000 q^{89} +83.1384i q^{91} +58.7878 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 88 q^{29} - 88 q^{41} - 92 q^{49} + 184 q^{61} + 584 q^{89}+O(q^{100}) \) 4 * q - 88 * q^29 - 88 * q^41 - 92 * q^49 + 184 * q^61 + 584 * q^89

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