LMFDB - Embedded newform 225.3.g.d.82.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([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("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.g (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:	$6.13080594811$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			82.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	225.82
Dual form			225.3.g.d.118.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{4} +(6.12372 + 6.12372i) q^{7} +(18.3712 - 18.3712i) q^{13} -16.0000 q^{16} -37.0000i q^{19} +(24.4949 - 24.4949i) q^{28} +13.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(-42.8661 + 42.8661i) q^{43} +26.0000i q^{49} +(-73.4847 - 73.4847i) q^{52} +47.0000 q^{61} +64.0000i q^{64} +(-55.1135 - 55.1135i) q^{67} +(-97.9796 + 97.9796i) q^{73} -148.000 q^{76} +142.000i q^{79} +225.000 q^{91} +(67.3610 + 67.3610i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{16} + 52 q^{31} + 188 q^{61} - 592 q^{76} + 900 q^{91}+O(q^{100}) $ 4 * q - 64 * q^16 + 52 * q^31 + 188 * q^61 - 592 * q^76 + 900 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$		0			0
$3$		0			0
$4$		−	4.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.12372	+	6.12372i	0.874818	+	0.874818i	0.992993	−	0.118175i	$-0.0377044\pi$
$7$	6.12372	+	6.12372i	0.874818	+	0.874818i	−0.118175	+	0.992993i	$0.537704\pi$
$8$		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
$13$	18.3712	−	18.3712i	1.41317	−	1.41317i	0.679387	−	0.733780i	$-0.262246\pi$
$13$	18.3712	−	18.3712i	1.41317	−	1.41317i	0.733780	−	0.679387i	$-0.237754\pi$
$14$		0			0
$15$		0			0
$16$	−16.0000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		−	37.0000i		−	1.94737i	−0.227901	−	0.973684i	$-0.573186\pi$
$19$		−	37.0000i		−	1.94737i	0.227901	−	0.973684i	$-0.426814\pi$
$20$		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
$28$	24.4949	−	24.4949i	0.874818	−	0.874818i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	13.0000			0.419355			0.209677	−	0.977771i	$-0.432759\pi$
$31$	13.0000			0.419355			0.209677	+	0.977771i	$0.432759\pi$
$32$		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
$43$	−42.8661	+	42.8661i	−0.996885	+	0.996885i	−0.999995	−	0.00310980i	$-0.999010\pi$
$43$	−42.8661	+	42.8661i	−0.996885	+	0.996885i	0.00310980	+	0.999995i	$0.499010\pi$
$44$		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
$49$			26.0000i			0.530612i
$50$		0			0
$51$		0			0
$52$	−73.4847	−	73.4847i	−1.41317	−	1.41317i
$53$		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
$61$	47.0000			0.770492			0.385246	−	0.922814i	$-0.374117\pi$
$61$	47.0000			0.770492			0.385246	+	0.922814i	$0.374117\pi$
$62$		0			0
$63$		0			0
$64$			64.0000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−55.1135	−	55.1135i	−0.822590	−	0.822590i	0.163889	−	0.986479i	$-0.447596\pi$
$67$	−55.1135	−	55.1135i	−0.822590	−	0.822590i	−0.986479	+	0.163889i	$0.947596\pi$
$68$		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
$76$	−148.000			−1.94737
$77$		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
$91$	225.000			2.47253
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	67.3610	+	67.3610i	0.694443	+	0.694443i	0.963206	−	0.268763i	$-0.0866149\pi$
$97$	67.3610	+	67.3610i	0.694443	+	0.694443i	−0.268763	+	0.963206i	$0.586615\pi$
$98$		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
$109$		−	143.000i		−	1.31193i	−0.754793	−	0.655963i	$-0.772263\pi$
$109$		−	143.000i		−	1.31193i	0.754793	−	0.655963i	$-0.227737\pi$
$110$		0			0
$111$		0			0
$112$	−97.9796	−	97.9796i	−0.874818	−	0.874818i
$113$		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
$124$		−	52.0000i		−	0.419355i
$125$		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
$133$	226.578	−	226.578i	1.70359	−	1.70359i
$134$		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
$148$	195.959	−	195.959i	1.32405	−	1.32405i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−227.000			−1.50331			−0.751656	−	0.659556i	$-0.770744\pi$
$151$	−227.000			−1.50331			−0.751656	+	0.659556i	$0.770744\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−30.6186	−	30.6186i	−0.195023	−	0.195023i	0.602839	−	0.797863i	$-0.294036\pi$
$157$	−30.6186	−	30.6186i	−0.195023	−	0.195023i	−0.797863	+	0.602839i	$0.794036\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−91.8559	+	91.8559i	−0.563533	+	0.563533i	−0.930309	−	0.366776i	$-0.880461\pi$
$163$	−91.8559	+	91.8559i	−0.563533	+	0.563533i	0.366776	+	0.930309i	$0.380461\pi$
$164$		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
$169$		−	506.000i		−	2.99408i
$170$		0			0
$171$		0			0
$172$	171.464	+	171.464i	0.996885	+	0.996885i
$173$		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
$181$	313.000			1.72928			0.864641	−	0.502390i	$-0.167546\pi$
$181$	313.000			1.72928			0.864641	+	0.502390i	$0.167546\pi$
$182$		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
$193$	214.330	−	214.330i	1.11052	−	1.11052i	0.117440	−	0.993080i	$-0.462531\pi$
$193$	214.330	−	214.330i	1.11052	−	1.11052i	0.993080	−	0.117440i	$-0.0374688\pi$
$194$		0			0
$195$		0			0
$196$	104.000			0.530612
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			277.000i			1.39196i	0.718061	+	0.695980i	$0.245030\pi$
$199$			277.000i			1.39196i	−0.718061	+	0.695980i	$0.754970\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−293.939	+	293.939i	−1.41317	+	1.41317i
$209$		0			0
$210$		0			0
$211$	−253.000			−1.19905			−0.599526	−	0.800355i	$-0.704644\pi$
$211$	−253.000			−1.19905			−0.599526	+	0.800355i	$0.704644\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	79.6084	+	79.6084i	0.366859	+	0.366859i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−104.103	+	104.103i	−0.466831	+	0.466831i	−0.900886	−	0.434055i	$-0.857082\pi$
$223$	−104.103	+	104.103i	−0.466831	+	0.466831i	0.434055	+	0.900886i	$0.357082\pi$
$224$		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
$229$			383.000i			1.67249i	0.548357	+	0.836245i	$0.315254\pi$
$229$			383.000i			1.67249i	−0.548357	+	0.836245i	$0.684746\pi$
$230$		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
$241$	−193.000			−0.800830			−0.400415	−	0.916334i	$-0.631134\pi$
$241$	−193.000			−0.800830			−0.400415	+	0.916334i	$0.631134\pi$
$242$		0			0
$243$		0			0
$244$		−	188.000i		−	0.770492i
$245$		0			0
$246$		0			0
$247$	−679.733	−	679.733i	−2.75196	−	2.75196i
$248$		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
$256$	256.000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			600.000i			2.31660i
$260$		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
$268$	−220.454	+	220.454i	−0.822590	+	0.822590i
$269$		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
$277$	−116.351	−	116.351i	−0.420039	−	0.420039i	0.465178	−	0.885217i	$-0.345990\pi$
$277$	−116.351	−	116.351i	−0.420039	−	0.420039i	−0.885217	+	0.465178i	$0.845990\pi$
$278$		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
$283$	−398.042	+	398.042i	−1.40651	+	1.40651i	−0.629546	+	0.776964i	$0.716759\pi$
$283$	−398.042	+	398.042i	−1.40651	+	1.40651i	−0.776964	+	0.629546i	$0.783241\pi$
$284$		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
$292$	391.918	+	391.918i	1.34219	+	1.34219i
$293$		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
$301$	−525.000			−1.74419
$302$		0			0
$303$		0			0
$304$			592.000i			1.94737i
$305$		0			0
$306$		0			0
$307$	42.8661	+	42.8661i	0.139629	+	0.139629i	0.773466	−	0.633837i	$-0.218521\pi$
$307$	42.8661	+	42.8661i	0.139629	+	0.139629i	−0.633837	+	0.773466i	$0.718521\pi$
$308$		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
$313$	128.598	−	128.598i	0.410857	−	0.410857i	−0.471180	−	0.882037i	$-0.656172\pi$
$313$	128.598	−	128.598i	0.410857	−	0.410857i	0.882037	+	0.471180i	$0.156172\pi$
$314$		0			0
$315$		0			0
$316$	568.000			1.79747
$317$		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
$337$	128.598	+	128.598i	0.381597	+	0.381597i	0.871677	−	0.490080i	$-0.163033\pi$
$337$	128.598	+	128.598i	0.381597	+	0.381597i	−0.490080	+	0.871677i	$0.663033\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	140.846	−	140.846i	0.410629	−	0.410629i
$344$		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
$361$	−1008.00			−2.79224
$362$		0			0
$363$		0			0
$364$		−	900.000i		−	2.47253i
$365$		0			0
$366$		0			0
$367$	−385.795	−	385.795i	−1.05121	−	1.05121i	−0.998616	−	0.0525956i	$-0.983251\pi$
$367$	−385.795	−	385.795i	−1.05121	−	1.05121i	−0.0525956	−	0.998616i	$-0.516749\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	520.517	−	520.517i	1.39549	−	1.39549i	0.583052	−	0.812435i	$-0.301858\pi$
$373$	520.517	−	520.517i	1.39549	−	1.39549i	0.812435	−	0.583052i	$-0.198142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			83.0000i			0.218997i	0.993987	+	0.109499i	$0.0349245\pi$
$379$			83.0000i			0.218997i	−0.993987	+	0.109499i	$0.965075\pi$
$380$		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
$388$	269.444	−	269.444i	0.694443	−	0.694443i
$389$		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
$397$	471.527	+	471.527i	1.18772	+	1.18772i	0.977695	+	0.210030i	$0.0673561\pi$
$397$	471.527	+	471.527i	1.18772	+	1.18772i	0.210030	+	0.977695i	$0.432644\pi$
$398$		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
$403$	238.825	−	238.825i	0.592618	−	0.592618i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			143.000i			0.349633i	0.984601	+	0.174817i	$0.0559333\pi$
$409$			143.000i			0.349633i	−0.984601	+	0.174817i	$0.944067\pi$
$410$		0			0
$411$		0			0
$412$	195.959	+	195.959i	0.475629	+	0.475629i
$413$		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
$427$	287.815	+	287.815i	0.674040	+	0.674040i
$428$		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
$433$	557.259	−	557.259i	1.28697	−	1.28697i	0.350355	−	0.936617i	$-0.386061\pi$
$433$	557.259	−	557.259i	1.28697	−	1.28697i	0.936617	−	0.350355i	$-0.113939\pi$
$434$		0			0
$435$		0			0
$436$	−572.000			−1.31193
$437$		0			0
$438$		0			0
$439$		−	803.000i		−	1.82916i	−0.404408	−	0.914579i	$-0.632522\pi$
$439$		−	803.000i		−	1.82916i	0.404408	−	0.914579i	$-0.367478\pi$
$440$		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
$448$	−391.918	+	391.918i	−0.874818	+	0.874818i
$449$		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
$469$		−	675.000i		−	1.43923i
$470$		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
$481$	1800.00			3.74220
$482$		0			0
$483$		0			0
$484$			484.000i			1.00000i
$485$		0			0
$486$		0			0
$487$	−642.991	−	642.991i	−1.32031	−	1.32031i	−0.913523	−	0.406787i	$-0.866649\pi$
$487$	−642.991	−	642.991i	−1.32031	−	1.32031i	−0.406787	−	0.913523i	$-0.633351\pi$
$488$		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
$496$	−208.000			−0.419355
$497$		0			0
$498$		0			0
$499$			877.000i			1.75752i	0.477269	+	0.878758i	$0.341627\pi$
$499$			877.000i			1.75752i	−0.477269	+	0.878758i	$0.658373\pi$
$500$		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
$508$	587.878	−	587.878i	1.15724	−	1.15724i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−1200.00			−2.34834
$512$		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
$523$	−728.723	+	728.723i	−1.39335	+	1.39335i	−0.575670	+	0.817682i	$0.695259\pi$
$523$	−728.723	+	728.723i	−1.39335	+	1.39335i	−0.817682	+	0.575670i	$0.804741\pi$
$524$		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
$532$	−906.311	−	906.311i	−1.70359	−	1.70359i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−793.000			−1.46580			−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−793.000			−1.46580			−0.732902	+	0.680334i	$0.761835\pi$
$542$		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
$553$	−869.569	+	869.569i	−1.57246	+	1.57246i
$554$		0			0
$555$		0			0
$556$	−88.0000			−0.158273
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$			1575.00i			2.81753i
$560$		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
$571$	1067.00			1.86865			0.934326	−	0.356420i	$-0.116003\pi$
$571$	1067.00			1.86865			0.934326	+	0.356420i	$0.116003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−814.455	−	814.455i	−1.41153	−	1.41153i	−0.749272	−	0.662262i	$-0.769596\pi$
$577$	−814.455	−	814.455i	−1.41153	−	1.41153i	−0.662262	−	0.749272i	$-0.730404\pi$
$578$		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
$589$		−	481.000i		−	0.816638i
$590$		0			0
$591$		0			0
$592$	−783.837	−	783.837i	−1.32405	−	1.32405i
$593$		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
$601$	673.000			1.11980			0.559900	−	0.828560i	$-0.310839\pi$
$601$	673.000			1.11980			0.559900	+	0.828560i	$0.310839\pi$
$602$		0			0
$603$		0			0
$604$			908.000i			1.50331i
$605$		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
$619$		−	1163.00i		−	1.87884i	−0.342773	−	0.939418i	$-0.611366\pi$
$619$		−	1163.00i		−	1.87884i	0.342773	−	0.939418i	$-0.388634\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−122.474	+	122.474i	−0.195023	+	0.195023i
$629$		0			0
$630$		0			0
$631$	587.000			0.930269			0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000			0.930269			0.465135	+	0.885240i	$0.346006\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	477.650	+	477.650i	0.749844	+	0.749844i
$638$		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
$652$	367.423	+	367.423i	0.563533	+	0.563533i
$653$		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
$676$	−2024.00			−2.99408
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$			825.000i			1.21502i
$680$		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
$688$	685.857	−	685.857i	0.996885	−	0.996885i
$689$		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
$703$	1812.62	−	1812.62i	2.57841	−	2.57841i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	457.000i		−	0.644570i	−0.946643	−	0.322285i	$-0.895549\pi$
$709$		−	457.000i		−	0.644570i	0.946643	−	0.322285i	$-0.104451\pi$
$710$		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
$721$	−600.000			−0.832178
$722$		0			0
$723$		0			0
$724$		−	1252.00i		−	1.72928i
$725$		0			0
$726$		0			0
$727$	189.835	+	189.835i	0.261122	+	0.261122i	0.825510	−	0.564388i	$-0.190888\pi$
$727$	189.835	+	189.835i	0.261122	+	0.261122i	−0.564388	+	0.825510i	$0.690888\pi$
$728$		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
$757$	−67.3610	−	67.3610i	−0.0889841	−	0.0889841i	0.661214	−	0.750198i	$-0.270042\pi$
$757$	−67.3610	−	67.3610i	−0.0889841	−	0.0889841i	−0.750198	+	0.661214i	$0.770042\pi$
$758$		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
$763$	875.693	−	875.693i	1.14770	−	1.14770i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	863.000i		−	1.12224i	−0.827736	−	0.561118i	$-0.810371\pi$
$769$		−	863.000i		−	1.12224i	0.827736	−	0.561118i	$-0.189629\pi$
$770$		0			0
$771$		0			0
$772$	−857.321	−	857.321i	−1.11052	−	1.11052i
$773$		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
$784$		−	416.000i		−	0.530612i
$785$		0			0
$786$		0			0
$787$	887.940	+	887.940i	1.12826	+	1.12826i	0.990460	+	0.137799i	$0.0440028\pi$
$787$	887.940	+	887.940i	1.12826	+	1.12826i	0.137799	+	0.990460i	$0.455997\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	863.445	−	863.445i	1.08883	−	1.08883i
$794$		0			0
$795$		0			0
$796$	1108.00			1.39196
$797$		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
$811$	253.000			0.311961			0.155980	−	0.987760i	$-0.450146\pi$
$811$	253.000			0.311961			0.155980	+	0.987760i	$0.450146\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1586.04	+	1586.04i	1.94130	+	1.94130i
$818$		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
$823$	202.083	−	202.083i	0.245544	−	0.245544i	−0.573595	−	0.819139i	$-0.694452\pi$
$823$	202.083	−	202.083i	0.245544	−	0.245544i	0.819139	+	0.573595i	$0.194452\pi$
$824$		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
$832$	1175.76	+	1175.76i	1.41317	+	1.41317i
$833$		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
$844$			1012.00i			1.19905i
$845$		0			0
$846$		0			0
$847$	−740.971	−	740.971i	−0.874818	−	0.874818i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1157.38	+	1157.38i	−1.35684	+	1.35684i	−0.479054	+	0.877786i	$0.659020\pi$
$853$	−1157.38	+	1157.38i	−1.35684	+	1.35684i	−0.877786	+	0.479054i	$0.840980\pi$
$854$		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
$868$	318.434	−	318.434i	0.366859	−	0.366859i
$869$		0			0
$870$		0			0
$871$	−2025.00			−2.32491
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−949.177	−	949.177i	−1.08230	−	1.08230i	−0.996295	−	0.0860055i	$-0.972590\pi$
$877$	−949.177	−	949.177i	−1.08230	−	1.08230i	−0.0860055	−	0.996295i	$-0.527410\pi$
$878$		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
$883$	−875.693	+	875.693i	−0.991724	+	0.991724i	−0.999966	−	0.00824171i	$-0.997377\pi$
$883$	−875.693	+	875.693i	−0.991724	+	0.991724i	0.00824171	+	0.999966i	$0.497377\pi$
$884$		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
$889$			1800.00i			2.02475i
$890$		0			0
$891$		0			0
$892$	416.413	+	416.413i	0.466831	+	0.466831i
$893$		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
$916$	1532.00			1.67249
$917$		0			0
$918$		0			0
$919$		−	1837.00i		−	1.99891i	−0.0329825	−	0.999456i	$-0.510501\pi$
$919$		−	1837.00i		−	1.99891i	0.0329825	−	0.999456i	$-0.489499\pi$
$920$		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
$931$	962.000			1.03330
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1243.12	−	1243.12i	−1.32670	−	1.32670i	−0.908233	−	0.418465i	$-0.862568\pi$
$937$	−1243.12	−	1243.12i	−1.32670	−	1.32670i	−0.418465	−	0.908233i	$-0.637432\pi$
$938$		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
$949$			3600.00i			3.79347i
$950$		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
$961$	−792.000			−0.824142
$962$		0			0
$963$		0			0
$964$			772.000i			0.800830i
$965$		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
$973$	134.722	−	134.722i	0.138460	−	0.138460i
$974$		0			0
$975$		0			0
$976$	−752.000			−0.770492
$977$		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
$988$	−2718.93	+	2718.93i	−2.75196	+	2.75196i
$989$		0			0
$990$		0			0
$991$	−1693.00			−1.70838			−0.854188	−	0.519965i	$-0.825945\pi$
$991$	−1693.00			−1.70838			−0.854188	+	0.519965i	$0.825945\pi$
$992$		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	225.3.g.d.82.2	yes	4
3.2	odd	2	CM	225.3.g.d.82.2	yes	4
5.2	odd	4	inner	225.3.g.d.118.1	yes	4
5.3	odd	4	inner	225.3.g.d.118.2	yes	4
5.4	even	2	inner	225.3.g.d.82.1	✓	4
15.2	even	4	inner	225.3.g.d.118.1	yes	4
15.8	even	4	inner	225.3.g.d.118.2	yes	4
15.14	odd	2	inner	225.3.g.d.82.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.g.d.82.1	✓	4	5.4	even	2	inner
225.3.g.d.82.1	✓	4	15.14	odd	2	inner
225.3.g.d.82.2	yes	4	1.1	even	1	trivial
225.3.g.d.82.2	yes	4	3.2	odd	2	CM
225.3.g.d.118.1	yes	4	5.2	odd	4	inner
225.3.g.d.118.1	yes	4	15.2	even	4	inner
225.3.g.d.118.2	yes	4	5.3	odd	4	inner
225.3.g.d.118.2	yes	4	15.8	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{4} +(6.12372 + 6.12372i) q^{7} +(18.3712 - 18.3712i) q^{13} -16.0000 q^{16} -37.0000i q^{19} +(24.4949 - 24.4949i) q^{28} +13.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(-42.8661 + 42.8661i) q^{43} +26.0000i q^{49} +(-73.4847 - 73.4847i) q^{52} +47.0000 q^{61} +64.0000i q^{64} +(-55.1135 - 55.1135i) q^{67} +(-97.9796 + 97.9796i) q^{73} -148.000 q^{76} +142.000i q^{79} +225.000 q^{91} +(67.3610 + 67.3610i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{16} + 52 q^{31} + 188 q^{61} - 592 q^{76} + 900 q^{91}+O(q^{100}) \) 4 * q - 64 * q^16 + 52 * q^31 + 188 * q^61 - 592 * q^76 + 900 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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