LMFDB - Embedded newform 900.3.l.f.757.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(757,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("900.757");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.l (of order $4$, degree $2$, 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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			757.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.757
Dual form			900.3.l.f.793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.79796 - 9.79796i) q^{7} +O(q^{10})$	$q+(-9.79796 - 9.79796i) q^{7} +(-9.79796 + 9.79796i) q^{13} +26.0000i q^{19} +46.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(58.7878 - 58.7878i) q^{43} +143.000i q^{49} +74.0000 q^{61} +(-39.1918 - 39.1918i) q^{67} +(-97.9796 + 97.9796i) q^{73} +142.000i q^{79} +192.000 q^{91} +(-137.171 - 137.171i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 184 q^{31} + 296 q^{61} + 768 q^{91}+O(q^{100}) $ 4 * q + 184 * q^31 + 296 * q^61 + 768 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$7$	−9.79796	−	9.79796i	−1.39971	−	1.39971i	−0.800869	−	0.598839i	$-0.795629\pi$
$7$	−9.79796	−	9.79796i	−1.39971	−	1.39971i	−0.598839	−	0.800869i	$-0.704371\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−9.79796	+	9.79796i	−0.753689	+	0.753689i	−0.975166	−	0.221477i	$-0.928912\pi$
$13$	−9.79796	+	9.79796i	−0.753689	+	0.753689i	0.221477	+	0.975166i	$0.428912\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			26.0000i			1.36842i	0.729285	+	0.684211i	$0.239853\pi$
$19$			26.0000i			1.36842i	−0.729285	+	0.684211i	$0.760147\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	46.0000			1.48387			0.741935	−	0.670471i	$-0.233908\pi$
$31$	46.0000			1.48387			0.741935	+	0.670471i	$0.233908\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	48.9898	+	48.9898i	1.32405	+	1.32405i	0.910467	+	0.413581i	$0.135722\pi$
$37$	48.9898	+	48.9898i	1.32405	+	1.32405i	0.413581	+	0.910467i	$0.364278\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	58.7878	−	58.7878i	1.36716	−	1.36716i	0.502691	−	0.864466i	$-0.332343\pi$
$43$	58.7878	−	58.7878i	1.36716	−	1.36716i	0.864466	−	0.502691i	$-0.167657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$			143.000i			2.91837i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	74.0000			1.21311			0.606557	−	0.795040i	$-0.292550\pi$
$61$	74.0000			1.21311			0.606557	+	0.795040i	$0.292550\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−39.1918	−	39.1918i	−0.584953	−	0.584953i	0.351307	−	0.936260i	$-0.385737\pi$
$67$	−39.1918	−	39.1918i	−0.584953	−	0.584953i	−0.936260	+	0.351307i	$0.885737\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−97.9796	+	97.9796i	−1.34219	+	1.34219i	−0.448306	+	0.893880i	$0.647973\pi$
$73$	−97.9796	+	97.9796i	−1.34219	+	1.34219i	−0.893880	+	0.448306i	$0.852027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			142.000i			1.79747i	0.438494	+	0.898734i	$0.355512\pi$
$79$			142.000i			1.79747i	−0.438494	+	0.898734i	$0.644488\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	192.000			2.10989
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−137.171	−	137.171i	−1.41414	−	1.41414i	−0.714359	−	0.699779i	$-0.753282\pi$
$97$	−137.171	−	137.171i	−1.41414	−	1.41414i	−0.699779	−	0.714359i	$-0.746718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−48.9898	+	48.9898i	−0.475629	+	0.475629i	−0.903731	−	0.428102i	$-0.859183\pi$
$103$	−48.9898	+	48.9898i	−0.475629	+	0.475629i	0.428102	+	0.903731i	$0.359183\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$			214.000i			1.96330i	0.190684	+	0.981651i	$0.438929\pi$
$109$			214.000i			1.96330i	−0.190684	+	0.981651i	$0.561071\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	146.969	+	146.969i	1.15724	+	1.15724i	0.985067	+	0.172172i	$0.0550786\pi$
$127$	146.969	+	146.969i	1.15724	+	1.15724i	0.172172	+	0.985067i	$0.444921\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	254.747	−	254.747i	1.91539	−	1.91539i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	22.0000i		−	0.158273i	−0.996864	−	0.0791367i	$-0.974784\pi$
$139$		−	22.0000i		−	0.158273i	0.996864	−	0.0791367i	$-0.0252164\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	286.000			1.89404			0.947020	−	0.321175i	$-0.104078\pi$
$151$	286.000			1.89404			0.947020	+	0.321175i	$0.104078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	205.757	+	205.757i	1.31056	+	1.31056i	0.921006	+	0.389549i	$0.127369\pi$
$157$	205.757	+	205.757i	1.31056	+	1.31056i	0.389549	+	0.921006i	$0.372631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−137.171	+	137.171i	−0.841542	+	0.841542i	−0.989060	−	0.147517i	$-0.952872\pi$
$163$	−137.171	+	137.171i	−0.841542	+	0.841542i	0.147517	+	0.989060i	$0.452872\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		−	23.0000i		−	0.136095i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−314.000			−1.73481			−0.867403	−	0.497606i	$-0.834213\pi$
$181$	−314.000			−1.73481			−0.867403	+	0.497606i	$0.834213\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	39.1918	−	39.1918i	0.203067	−	0.203067i	−0.598246	−	0.801313i	$-0.704135\pi$
$193$	39.1918	−	39.1918i	0.203067	−	0.203067i	0.801313	+	0.598246i	$0.204135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	386.000i		−	1.93970i	−0.243706	−	0.969849i	$-0.578363\pi$
$199$		−	386.000i		−	1.93970i	0.243706	−	0.969849i	$-0.421637\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−166.000			−0.786730			−0.393365	−	0.919382i	$-0.628689\pi$
$211$	−166.000			−0.786730			−0.393365	+	0.919382i	$0.628689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−450.706	−	450.706i	−2.07699	−	2.07699i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−205.757	+	205.757i	−0.922678	+	0.922678i	−0.997218	−	0.0745403i	$-0.976251\pi$
$223$	−205.757	+	205.757i	−0.922678	+	0.922678i	0.0745403	+	0.997218i	$0.476251\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			26.0000i			0.113537i	0.998387	+	0.0567686i	$0.0180797\pi$
$229$			26.0000i			0.113537i	−0.998387	+	0.0567686i	$0.981920\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−286.000			−1.18672			−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−286.000			−1.18672			−0.593361	+	0.804936i	$0.702199\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−254.747	−	254.747i	−1.03136	−	1.03136i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		−	960.000i		−	3.70656i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	242.000			0.892989			0.446494	−	0.894786i	$-0.352672\pi$
$271$	242.000			0.892989			0.446494	+	0.894786i	$0.352672\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	382.120	+	382.120i	1.37950	+	1.37950i	0.845465	+	0.534031i	$0.179324\pi$
$277$	382.120	+	382.120i	1.37950	+	1.37950i	0.534031	+	0.845465i	$0.320676\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	235.151	−	235.151i	0.830922	−	0.830922i	−0.156721	−	0.987643i	$-0.550092\pi$
$283$	235.151	−	235.151i	0.830922	−	0.830922i	0.987643	+	0.156721i	$0.0500922\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	289.000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1152.00			−3.82724
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	352.727	+	352.727i	1.14895	+	1.14895i	0.986760	+	0.162186i	$0.0518545\pi$
$307$	352.727	+	352.727i	1.14895	+	1.14895i	0.162186	+	0.986760i	$0.448145\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−431.110	+	431.110i	−1.37735	+	1.37735i	−0.528276	+	0.849072i	$0.677162\pi$
$313$	−431.110	+	431.110i	−1.37735	+	1.37735i	−0.849072	+	0.528276i	$0.822838\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	362.000			1.09366			0.546828	−	0.837245i	$-0.315835\pi$
$331$	362.000			1.09366			0.546828	+	0.837245i	$0.315835\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	333.131	+	333.131i	0.988518	+	0.988518i	0.999935	−	0.0114167i	$-0.00363413\pi$
$337$	333.131	+	333.131i	0.988518	+	0.988518i	−0.0114167	+	0.999935i	$0.503634\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	921.008	−	921.008i	2.68515	−	2.68515i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	502.000i		−	1.43840i	−0.694805	−	0.719198i	$-0.744510\pi$
$349$		−	502.000i		−	1.43840i	0.694805	−	0.719198i	$-0.255490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−315.000			−0.872576
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−107.778	−	107.778i	−0.293672	−	0.293672i	0.544857	−	0.838529i	$-0.316584\pi$
$367$	−107.778	−	107.778i	−0.293672	−	0.293672i	−0.838529	+	0.544857i	$0.816584\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−186.161	+	186.161i	−0.499092	+	0.499092i	−0.911155	−	0.412063i	$-0.864808\pi$
$373$	−186.161	+	186.161i	−0.499092	+	0.499092i	0.412063	+	0.911155i	$0.364808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	694.000i		−	1.83113i	−0.402165	−	0.915567i	$-0.631742\pi$
$379$		−	694.000i		−	1.83113i	0.402165	−	0.915567i	$-0.368258\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−499.696	−	499.696i	−1.25868	−	1.25868i	−0.951724	−	0.306956i	$-0.900689\pi$
$397$	−499.696	−	499.696i	−1.25868	−	1.25868i	−0.306956	−	0.951724i	$-0.599311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−450.706	+	450.706i	−1.11838	+	1.11838i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			626.000i			1.53056i	0.643696	+	0.765281i	$0.277400\pi$
$409$			626.000i			1.53056i	−0.643696	+	0.765281i	$0.722600\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	358.000			0.850356			0.425178	−	0.905110i	$-0.360211\pi$
$421$	358.000			0.850356			0.425178	+	0.905110i	$0.360211\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−725.049	−	725.049i	−1.69801	−	1.69801i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−58.7878	+	58.7878i	−0.135768	+	0.135768i	−0.771725	−	0.635956i	$-0.780606\pi$
$433$	−58.7878	+	58.7878i	−0.135768	+	0.135768i	0.635956	+	0.771725i	$0.280606\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			94.0000i			0.214123i	0.994252	+	0.107062i	$0.0341442\pi$
$439$			94.0000i			0.214123i	−0.994252	+	0.107062i	$0.965856\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	293.939	+	293.939i	0.643192	+	0.643192i	0.951339	−	0.308147i	$-0.0997088\pi$
$457$	293.939	+	293.939i	0.643192	+	0.643192i	−0.308147	+	0.951339i	$0.599709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−538.888	+	538.888i	−1.16390	+	1.16390i	−0.180291	+	0.983613i	$0.557704\pi$
$463$	−538.888	+	538.888i	−1.16390	+	1.16390i	−0.983613	+	0.180291i	$0.942296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$			768.000i			1.63753i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−960.000			−1.99584
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	107.778	+	107.778i	0.221309	+	0.221309i	0.809050	−	0.587740i	$-0.199982\pi$
$487$	107.778	+	107.778i	0.221309	+	0.221309i	−0.587740	+	0.809050i	$0.699982\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	26.0000i		−	0.0521042i	−0.999661	−	0.0260521i	$-0.991706\pi$
$499$		−	26.0000i		−	0.0521042i	0.999661	−	0.0260521i	$-0.00829358\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	1920.00			3.75734
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	254.747	−	254.747i	0.487088	−	0.487088i	−0.420298	−	0.907386i	$-0.638075\pi$
$523$	254.747	−	254.747i	0.487088	−	0.487088i	0.907386	+	0.420298i	$0.138075\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			529.000i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1034.00			1.91128			0.955638	−	0.294545i	$-0.0951680\pi$
$541$	1034.00			1.91128			0.955638	+	0.294545i	$0.0951680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	685.857	+	685.857i	1.25385	+	1.25385i	0.953979	+	0.299873i	$0.0969443\pi$
$547$	685.857	+	685.857i	1.25385	+	1.25385i	0.299873	+	0.953979i	$0.403056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1391.31	−	1391.31i	2.51593	−	2.51593i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$			1152.00i			2.06082i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−886.000			−1.55166			−0.775832	−	0.630940i	$-0.782670\pi$
$571$	−886.000			−1.55166			−0.775832	+	0.630940i	$0.782670\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	450.706	+	450.706i	0.781120	+	0.781120i	0.980020	−	0.198900i	$-0.0637369\pi$
$577$	450.706	+	450.706i	0.781120	+	0.781120i	−0.198900	+	0.980020i	$0.563737\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$			1196.00i			2.03056i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	526.000			0.875208			0.437604	−	0.899168i	$-0.355827\pi$
$601$	526.000			0.875208			0.437604	+	0.899168i	$0.355827\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	636.867	+	636.867i	1.04920	+	1.04920i	0.998725	+	0.0504797i	$0.0160750\pi$
$607$	636.867	+	636.867i	1.04920	+	1.04920i	0.0504797	+	0.998725i	$0.483925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−342.929	+	342.929i	−0.559427	+	0.559427i	−0.929144	−	0.369718i	$-0.879454\pi$
$613$	−342.929	+	342.929i	−0.559427	+	0.559427i	0.369718	+	0.929144i	$0.379454\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$			214.000i			0.345719i	0.984947	+	0.172859i	$0.0553006\pi$
$619$			214.000i			0.345719i	−0.984947	+	0.172859i	$0.944699\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	674.000			1.06815			0.534073	−	0.845438i	$-0.320661\pi$
$631$	674.000			1.06815			0.534073	+	0.845438i	$0.320661\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1401.11	−	1401.11i	−2.19954	−	2.19954i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−881.816	+	881.816i	−1.37141	+	1.37141i	−0.513052	+	0.858358i	$0.671485\pi$
$643$	−881.816	+	881.816i	−1.37141	+	1.37141i	−0.858358	+	0.513052i	$0.828515\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−122.000			−0.184569			−0.0922844	−	0.995733i	$-0.529417\pi$
$661$	−122.000			−0.184569			−0.0922844	+	0.995733i	$0.529417\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	489.898	−	489.898i	0.727932	−	0.727932i	−0.242276	−	0.970207i	$-0.577894\pi$
$673$	489.898	−	489.898i	0.727932	−	0.727932i	0.970207	+	0.242276i	$0.0778939\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$			2688.00i			3.95876i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1318.00			1.90738			0.953690	−	0.300790i	$-0.0972504\pi$
$691$	1318.00			1.90738			0.953690	+	0.300790i	$0.0972504\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−1273.73	+	1273.73i	−1.81186	+	1.81186i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	934.000i		−	1.31735i	−0.752428	−	0.658674i	$-0.771118\pi$
$709$		−	934.000i		−	1.31735i	0.752428	−	0.658674i	$-0.228882\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	960.000			1.33148
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−969.998	−	969.998i	−1.33425	−	1.33425i	−0.901529	−	0.432718i	$-0.857554\pi$
$727$	−969.998	−	969.998i	−1.33425	−	1.33425i	−0.432718	−	0.901529i	$-0.642446\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−734.847	+	734.847i	−1.00252	+	1.00252i	−0.00252286	+	0.999997i	$0.500803\pi$
$733$	−734.847	+	734.847i	−1.00252	+	1.00252i	−0.999997	+	0.00252286i	$0.999197\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	1222.00i		−	1.65359i	−0.562506	−	0.826793i	$-0.690163\pi$
$739$		−	1222.00i		−	1.65359i	0.562506	−	0.826793i	$-0.309837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1202.00			−1.60053			−0.800266	−	0.599645i	$-0.795309\pi$
$751$	−1202.00			−1.60053			−0.800266	+	0.599645i	$0.795309\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−891.614	−	891.614i	−1.17783	−	1.17783i	−0.980297	−	0.197529i	$-0.936708\pi$
$757$	−891.614	−	891.614i	−1.17783	−	1.17783i	−0.197529	−	0.980297i	$-0.563292\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	2096.76	−	2096.76i	2.74805	−	2.74805i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			1534.00i			1.99480i	0.0720749	+	0.997399i	$0.477038\pi$
$769$			1534.00i			1.99480i	−0.0720749	+	0.997399i	$0.522962\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	137.171	+	137.171i	0.174297	+	0.174297i	0.788864	−	0.614568i	$-0.210669\pi$
$787$	137.171	+	137.171i	0.174297	+	0.174297i	−0.614568	+	0.788864i	$0.710669\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−725.049	+	725.049i	−0.914311	+	0.914311i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−1514.00			−1.86683			−0.933416	−	0.358797i	$-0.883187\pi$
$811$	−1514.00			−1.86683			−0.933416	+	0.358797i	$0.883187\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1528.48	+	1528.48i	1.87085	+	1.87085i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	891.614	−	891.614i	1.08337	−	1.08337i	0.0871782	−	0.996193i	$-0.472215\pi$
$823$	891.614	−	891.614i	1.08337	−	1.08337i	0.996193	−	0.0871782i	$-0.0277849\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			458.000i			0.552473i	0.961090	+	0.276236i	$0.0890873\pi$
$829$			458.000i			0.552473i	−0.961090	+	0.276236i	$0.910913\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1185.55	+	1185.55i	1.39971	+	1.39971i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	284.141	−	284.141i	0.333108	−	0.333108i	−0.520658	−	0.853765i	$-0.674313\pi$
$853$	284.141	−	284.141i	0.333108	−	0.333108i	0.853765	+	0.520658i	$0.174313\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	1418.00i		−	1.65076i	−0.564580	−	0.825378i	$-0.690962\pi$
$859$		−	1418.00i		−	1.65076i	0.564580	−	0.825378i	$-0.309038\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	768.000			0.881745
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1165.96	+	1165.96i	1.32948	+	1.32948i	0.905819	+	0.423664i	$0.139256\pi$
$877$	1165.96	+	1165.96i	1.32948	+	1.32948i	0.423664	+	0.905819i	$0.360744\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−333.131	+	333.131i	−0.377271	+	0.377271i	−0.870117	−	0.492845i	$-0.835957\pi$
$883$	−333.131	+	333.131i	−0.377271	+	0.377271i	0.492845	+	0.870117i	$0.335957\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		−	2880.00i		−	3.23960i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1273.73	−	1273.73i	−1.40434	−	1.40434i	−0.785587	−	0.618751i	$-0.787639\pi$
$907$	−1273.73	−	1273.73i	−1.40434	−	1.40434i	−0.618751	−	0.785587i	$-0.712361\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			866.000i			0.942329i	0.882045	+	0.471164i	$0.156166\pi$
$919$			866.000i			0.942329i	−0.882045	+	0.471164i	$0.843834\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−3718.00			−3.99356
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1018.99	+	1018.99i	1.08750	+	1.08750i	0.995785	+	0.0917149i	$0.0292349\pi$
$937$	1018.99	+	1018.99i	1.08750	+	1.08750i	0.0917149	+	0.995785i	$0.470765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		−	1920.00i		−	2.02318i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1155.00			1.20187
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−832.827	−	832.827i	−0.861248	−	0.861248i	0.130235	−	0.991483i	$-0.458427\pi$
$967$	−832.827	−	832.827i	−0.861248	−	0.861248i	−0.991483	+	0.130235i	$0.958427\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	−215.555	+	215.555i	−0.221537	+	0.221537i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−46.0000			−0.0464178			−0.0232089	−	0.999731i	$-0.507388\pi$
$991$	−46.0000			−0.0464178			−0.0232089	+	0.999731i	$0.507388\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−440.908	−	440.908i	−0.442235	−	0.442235i	0.450528	−	0.892762i	$-0.351236\pi$
$997$	−440.908	−	440.908i	−0.442235	−	0.442235i	−0.892762	+	0.450528i	$0.851236\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.f.757.1	✓	4
3.2	odd	2	CM	900.3.l.f.757.1	✓	4
5.2	odd	4	inner	900.3.l.f.793.2	yes	4
5.3	odd	4	inner	900.3.l.f.793.1	yes	4
5.4	even	2	inner	900.3.l.f.757.2	yes	4
15.2	even	4	inner	900.3.l.f.793.2	yes	4
15.8	even	4	inner	900.3.l.f.793.1	yes	4
15.14	odd	2	inner	900.3.l.f.757.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.l.f.757.1	✓	4	1.1	even	1	trivial
900.3.l.f.757.1	✓	4	3.2	odd	2	CM
900.3.l.f.757.2	yes	4	5.4	even	2	inner
900.3.l.f.757.2	yes	4	15.14	odd	2	inner
900.3.l.f.793.1	yes	4	5.3	odd	4	inner
900.3.l.f.793.1	yes	4	15.8	even	4	inner
900.3.l.f.793.2	yes	4	5.2	odd	4	inner
900.3.l.f.793.2	yes	4	15.2	even	4	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.79796 - 9.79796i) q^{7} +O(q^{10})\)	\(q+(-9.79796 - 9.79796i) q^{7} +(-9.79796 + 9.79796i) q^{13} +26.0000i q^{19} +46.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(58.7878 - 58.7878i) q^{43} +143.000i q^{49} +74.0000 q^{61} +(-39.1918 - 39.1918i) q^{67} +(-97.9796 + 97.9796i) q^{73} +142.000i q^{79} +192.000 q^{91} +(-137.171 - 137.171i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 184 q^{31} + 296 q^{61} + 768 q^{91}+O(q^{100}) \) 4 * q + 184 * q^31 + 296 * q^61 + 768 * q^91

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