LMFDB - Embedded newform 495.2.k.a.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(208,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("495.208");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.k (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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			307.1
Root			$1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	495.307
Dual form			495.2.k.a.208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -3.31662 q^{11} -4.00000 q^{16} +(-3.00000 - 3.31662i) q^{20} +(-2.84169 + 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} -9.94987 q^{31} +(1.47494 + 1.47494i) q^{37} -6.63325i q^{44} +(9.31662 + 9.31662i) q^{47} +7.00000i q^{49} +(-3.63325 + 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} +3.31662i q^{59} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +3.00000 q^{71} +(6.63325 - 6.00000i) q^{80} -9.00000i q^{89} +(-5.68338 - 5.68338i) q^{92} +(-3.52506 - 3.52506i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{16} - 12 q^{20} - 18 q^{23} + 2 q^{25} - 14 q^{37} + 24 q^{47} + 12 q^{53} + 22 q^{55} + 26 q^{67} + 12 q^{71} - 36 q^{92} - 34 q^{97}+O(q^{100}) $ 4 * q - 16 * q^16 - 12 * q^20 - 18 * q^23 + 2 * q^25 - 14 * q^37 + 24 * q^47 + 12 * q^53 + 22 * q^55 + 26 * q^67 + 12 * q^71 - 36 * q^92 - 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$		0			0
$3$		0			0
$4$			2.00000i			1.00000i
$5$	−1.65831	+	1.50000i	−0.741620	+	0.670820i
$5$	−1.65831	+	1.50000i	−0.741620	+	0.670820i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−3.00000	−	3.31662i	−0.670820	−	0.741620i
$21$		0			0
$22$		0			0
$23$	−2.84169	+	2.84169i	−0.592533	+	0.592533i	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−2.84169	+	2.84169i	−0.592533	+	0.592533i	0.345782	+	0.938315i	$0.387614\pi$
$24$		0			0
$25$	0.500000	−	4.97494i	0.100000	−	0.994987i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−9.94987			−1.78705			−0.893525	−	0.449013i	$-0.851776\pi$
$31$	−9.94987			−1.78705			−0.893525	+	0.449013i	$0.851776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.47494	+	1.47494i	0.242478	+	0.242478i	0.817875	−	0.575396i	$-0.195152\pi$
$37$	1.47494	+	1.47494i	0.242478	+	0.242478i	−0.575396	+	0.817875i	$0.695152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		−	6.63325i		−	1.00000i
$45$		0			0
$46$		0			0
$47$	9.31662	+	9.31662i	1.35897	+	1.35897i	0.875190	+	0.483779i	$0.160736\pi$
$47$	9.31662	+	9.31662i	1.35897	+	1.35897i	0.483779	+	0.875190i	$0.339264\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.63325	+	3.63325i	−0.499065	+	0.499065i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	−3.63325	+	3.63325i	−0.499065	+	0.499065i	0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	5.50000	−	4.97494i	0.741620	−	0.670820i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			3.31662i			0.431788i	0.976417	+	0.215894i	$0.0692665\pi$
$59$			3.31662i			0.431788i	−0.976417	+	0.215894i	$0.930733\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.794101	+	0.607785i	$0.207942\pi$
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.607785	+	0.794101i	$0.292058\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	6.63325	−	6.00000i	0.741620	−	0.670820i
$81$		0			0
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.00000i		−	0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$		−	9.00000i		−	0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$		0			0
$91$		0			0
$92$	−5.68338	−	5.68338i	−0.592533	−	0.592533i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.52506	−	3.52506i	−0.357916	−	0.357916i	0.505128	−	0.863044i	$-0.331445\pi$
$97$	−3.52506	−	3.52506i	−0.357916	−	0.357916i	−0.863044	+	0.505128i	$0.831445\pi$
$98$		0			0
$99$		0			0
$100$	9.94987	+	1.00000i	0.994987	+	0.100000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−7.94987	+	7.94987i	−0.783324	+	0.783324i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	−7.94987	+	7.94987i	−0.783324	+	0.783324i	0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.1583	−	12.1583i	1.14376	−	1.14376i	0.156001	−	0.987757i	$-0.450140\pi$
$113$	12.1583	−	12.1583i	1.14376	−	1.14376i	0.987757	−	0.156001i	$-0.0498603\pi$
$114$		0			0
$115$	0.449874	−	8.97494i	0.0419510	−	0.836917i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		−	19.8997i		−	1.78705i
$125$	6.63325	+	9.00000i	0.593296	+	0.804984i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.1082	+	10.1082i	0.863601	+	0.863601i	0.991754	−	0.128154i	$-0.0409051\pi$
$137$	10.1082	+	10.1082i	0.863601	+	0.863601i	−0.128154	+	0.991754i	$0.540905\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2.94987	+	2.94987i	−0.242478	+	0.242478i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.5000	−	14.9248i	1.32531	−	1.19879i
$156$		0			0
$157$	16.4749	+	16.4749i	1.31484	+	1.31484i	0.917800	+	0.397043i	$0.129964\pi$
$157$	16.4749	+	16.4749i	1.31484	+	1.31484i	0.397043	+	0.917800i	$0.370036\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−17.9499	+	17.9499i	−1.40594	+	1.40594i	−0.626608	+	0.779334i	$0.715557\pi$
$163$	−17.9499	+	17.9499i	−1.40594	+	1.40594i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$	13.2665			1.00000
$177$		0			0
$178$		0			0
$179$			21.0000i			1.56961i	0.619740	+	0.784807i	$0.287238\pi$
$179$			21.0000i			1.56961i	−0.619740	+	0.784807i	$0.712762\pi$
$180$		0			0
$181$	−9.94987			−0.739568			−0.369784	−	0.929118i	$-0.620568\pi$
$181$	−9.94987			−0.739568			−0.369784	+	0.929118i	$0.620568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.65831	−	0.233501i	−0.342486	−	0.0171673i
$186$		0			0
$187$		0			0
$188$	−18.6332	+	18.6332i	−1.35897	+	1.35897i
$189$		0			0
$190$		0			0
$191$	−23.2164			−1.67988			−0.839939	−	0.542681i	$-0.817409\pi$
$191$	−23.2164			−1.67988			−0.839939	+	0.542681i	$0.817409\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		−	19.8997i		−	1.41066i	−0.708881	−	0.705328i	$-0.750800\pi$
$199$		−	19.8997i		−	1.41066i	0.708881	−	0.705328i	$-0.249200\pi$
$200$		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
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−7.26650	−	7.26650i	−0.499065	−	0.499065i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	9.94987	+	11.0000i	0.670820	+	0.741620i
$221$		0			0
$222$		0			0
$223$	14.4248	−	14.4248i	0.965957	−	0.965957i	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	14.4248	−	14.4248i	0.965957	−	0.965957i	0.999439	+	0.0334825i	$0.0106598\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			29.8496i			1.97252i	0.165205	+	0.986259i	$0.447172\pi$
$229$			29.8496i			1.97252i	−0.165205	+	0.986259i	$0.552828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$	−29.4248	−	1.47494i	−1.91946	−	0.0962143i
$236$	−6.63325			−0.431788
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.5000	−	11.6082i	−0.670820	−	0.741620i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.0000			−1.70422			−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000			−1.70422			−0.852112	+	0.523359i	$0.824679\pi$
$252$		0			0
$253$	9.42481	−	9.42481i	0.592533	−	0.592533i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−22.2665	−	22.2665i	−1.38895	−	1.38895i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	−22.2665	−	22.2665i	−1.38895	−	1.38895i	−0.561405	−	0.827541i	$-0.689739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	0.575188	−	11.4749i	0.0353335	−	0.704900i
$266$		0			0
$267$		0			0
$268$	−22.9499	+	22.9499i	−1.40189	+	1.40189i
$269$		−	13.2665i		−	0.808873i	−0.914566	−	0.404436i	$-0.867468\pi$
$269$		−	13.2665i		−	0.808873i	0.914566	−	0.404436i	$-0.132532\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.65831	+	16.5000i	−0.100000	+	0.994987i
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$			6.00000i			0.356034i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$	−4.97494	−	5.50000i	−0.289652	−	0.320222i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	24.4248	−	24.4248i	1.38057	−	1.38057i	0.536972	−	0.843600i	$-0.319568\pi$
$313$	24.4248	−	24.4248i	1.38057	−	1.38057i	0.843600	−	0.536972i	$-0.180432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	25.1082	+	25.1082i	1.41022	+	1.41022i	0.758236	+	0.651981i	$0.226062\pi$
$317$	25.1082	+	25.1082i	1.41022	+	1.41022i	0.651981	+	0.758236i	$0.273938\pi$
$318$		0			0
$319$		0			0
$320$	12.0000	+	13.2665i	0.670820	+	0.741620i
$321$		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
$331$	−9.94987			−0.546895			−0.273447	−	0.961887i	$-0.588164\pi$
$331$	−9.94987			−0.546895			−0.273447	+	0.961887i	$0.588164\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−36.2414	−	1.81662i	−1.98008	−	0.0992528i
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.0000			1.78705
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.7414	−	13.7414i	0.731383	−	0.731383i	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	13.7414	−	13.7414i	0.731383	−	0.731383i	0.970894	+	0.239511i	$0.0769871\pi$
$354$		0			0
$355$	−4.97494	+	4.50000i	−0.264042	+	0.238835i
$356$	18.0000			0.953998
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.5251	−	13.5251i	−0.706003	−	0.706003i	0.259690	−	0.965692i	$-0.416380\pi$
$367$	−13.5251	−	13.5251i	−0.706003	−	0.706003i	−0.965692	+	0.259690i	$0.916380\pi$
$368$	11.3668	−	11.3668i	0.592533	−	0.592533i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			29.8496i			1.53327i	0.642082	+	0.766636i	$0.278071\pi$
$379$			29.8496i			1.53327i	−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.8417	+	17.8417i	−0.911668	+	0.911668i	−0.996403	−	0.0847358i	$-0.972995\pi$
$383$	−17.8417	+	17.8417i	−0.911668	+	0.911668i	0.0847358	+	0.996403i	$0.472995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	7.05013	−	7.05013i	0.357916	−	0.357916i
$389$			36.4829i			1.84976i	0.380265	+	0.924878i	$0.375833\pi$
$389$			36.4829i			1.84976i	−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.0501886	−	0.998740i	$-0.515982\pi$
$398$		0			0
$399$		0			0
$400$	−2.00000	+	19.8997i	−0.100000	+	0.994987i
$401$	26.5330			1.32499			0.662497	−	0.749064i	$-0.269497\pi$
$401$	26.5330			1.32499			0.662497	+	0.749064i	$0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.89181	−	4.89181i	−0.242478	−	0.242478i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	−15.8997	−	15.8997i	−0.783324	−	0.783324i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.0000i		−	1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$		−	24.0000i		−	1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$		0			0
$421$	39.7995			1.93971			0.969854	−	0.243685i	$-0.0783563\pi$
$421$	39.7995			1.93971			0.969854	+	0.243685i	$0.0783563\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	29.4248	−	29.4248i	1.41407	−	1.41407i	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.4248	−	29.4248i	1.41407	−	1.41407i	0.717241	−	0.696826i	$-0.245405\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	28.7414	−	28.7414i	1.36555	−	1.36555i	0.498870	−	0.866677i	$-0.333748\pi$
$443$	28.7414	−	28.7414i	1.36555	−	1.36555i	0.866677	−	0.498870i	$-0.166252\pi$
$444$		0			0
$445$	13.5000	+	14.9248i	0.639961	+	0.707504i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	39.0000i		−	1.84052i	−0.391303	−	0.920262i	$-0.627976\pi$
$449$		−	39.0000i		−	1.84052i	0.391303	−	0.920262i	$-0.372024\pi$
$450$		0			0
$451$		0			0
$452$	24.3166	+	24.3166i	1.14376	+	1.14376i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$	17.9499	+	0.899749i	0.836917	+	0.0419510i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	0.693615	+	0.720346i	$0.256017\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.0581	−	23.0581i	−1.06700	−	1.06700i	−0.997588	−	0.0694117i	$-0.977888\pi$
$467$	−23.0581	−	23.0581i	−1.06700	−	1.06700i	−0.0694117	−	0.997588i	$-0.522112\pi$
$468$		0			0
$469$		0			0
$470$		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			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$			22.0000i			1.00000i
$485$	11.1332	+	0.558061i	0.505535	+	0.0253403i
$486$		0			0
$487$	26.4749	+	26.4749i	1.19969	+	1.19969i	0.974258	+	0.225436i	$0.0723806\pi$
$487$	26.4749	+	26.4749i	1.19969	+	1.19969i	0.225436	+	0.974258i	$0.427619\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	39.7995			1.78705
$497$		0			0
$498$		0			0
$499$		−	19.8997i		−	0.890835i	−0.895323	−	0.445418i	$-0.853055\pi$
$499$		−	19.8997i		−	0.890835i	0.895323	−	0.445418i	$-0.146945\pi$
$500$	−18.0000	+	13.2665i	−0.804984	+	0.593296i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			3.31662i			0.147007i	0.997295	+	0.0735034i	$0.0234180\pi$
$509$			3.31662i			0.147007i	−0.997295	+	0.0735034i	$0.976582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.25856	−	25.1082i	0.0554589	−	1.10640i
$516$		0			0
$517$	−30.8997	−	30.8997i	−1.35897	−	1.35897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	43.1161			1.88895			0.944476	−	0.328581i	$-0.106570\pi$
$521$	43.1161			1.88895			0.944476	+	0.328581i	$0.106570\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			6.84962i			0.297810i
$530$		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
$539$		−	23.2164i		−	1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−20.2164	+	20.2164i	−0.863601	+	0.863601i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	−1.92481	+	38.3997i	−0.0809774	+	1.61549i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.7164	+	15.5581i	0.530309	+	0.648816i
$576$		0			0
$577$	−18.5251	−	18.5251i	−0.771208	−	0.771208i	0.207109	−	0.978318i	$-0.433594\pi$
$577$	−18.5251	−	18.5251i	−0.771208	−	0.771208i	−0.978318	+	0.207109i	$0.933594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0501	−	12.0501i	0.499065	−	0.499065i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.6834	−	20.6834i	−0.853694	−	0.853694i	0.136892	−	0.990586i	$-0.456289\pi$
$587$	−20.6834	−	20.6834i	−0.853694	−	0.853694i	−0.990586	+	0.136892i	$0.956289\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−5.89975	−	5.89975i	−0.242478	−	0.242478i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.0000i			1.47092i	0.677568	+	0.735460i	$0.263034\pi$
$599$			36.0000i			1.47092i	−0.677568	+	0.735460i	$0.736966\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−18.2414	+	16.5000i	−0.741620	+	0.670820i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.73350	+	7.73350i	0.311339	+	0.311339i	0.845428	−	0.534089i	$-0.179345\pi$
$617$	7.73350	+	7.73350i	0.311339	+	0.311339i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$		−	1.00000i		−	0.0401934i	−0.999798	−	0.0200967i	$-0.993603\pi$
$619$		−	1.00000i		−	0.0401934i	0.999798	−	0.0200967i	$-0.00639741\pi$
$620$	29.8496	+	33.0000i	1.19879	+	1.32531i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5000	−	4.97494i	−0.980000	−	0.198997i
$626$		0			0
$627$		0			0
$628$	−32.9499	+	32.9499i	−1.31484	+	1.31484i
$629$		0			0
$630$		0			0
$631$	7.00000			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.2164			−0.916992			−0.458496	−	0.888697i	$-0.651612\pi$
$641$	−23.2164			−0.916992			−0.458496	+	0.888697i	$0.651612\pi$
$642$		0			0
$643$	−5.57519	+	5.57519i	−0.219864	+	0.219864i	−0.808441	−	0.588577i	$-0.799688\pi$
$643$	−5.57519	+	5.57519i	−0.219864	+	0.219864i	0.588577	+	0.808441i	$0.299688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.05806	−	8.05806i	−0.316795	−	0.316795i	0.530740	−	0.847535i	$-0.321914\pi$
$647$	−8.05806	−	8.05806i	−0.316795	−	0.316795i	−0.847535	+	0.530740i	$0.821914\pi$
$648$		0			0
$649$		−	11.0000i		−	0.431788i
$650$		0			0
$651$		0			0
$652$	−35.8997	−	35.8997i	−1.40594	−	1.40594i
$653$	27.1583	−	27.1583i	1.06279	−	1.06279i	0.0648948	−	0.997892i	$-0.479329\pi$
$653$	27.1583	−	27.1583i	1.06279	−	1.06279i	0.997892	−	0.0648948i	$-0.0206712\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		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
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.2164	+	35.2164i	−1.34752	+	1.34752i	−0.459167	+	0.888350i	$0.651852\pi$
$683$	−35.2164	+	35.2164i	−1.34752	+	1.34752i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−31.9248	−	1.60025i	−1.21978	−	0.0611425i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	17.0000			0.646710			0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000			0.646710			0.323355	+	0.946278i	$0.395189\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			26.5330i			1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			19.0000i			0.713560i	0.934188	+	0.356780i	$0.116125\pi$
$709$			19.0000i			0.713560i	−0.934188	+	0.356780i	$0.883875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	28.2744	−	28.2744i	1.05889	−	1.05889i
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$			51.0000i			1.90198i	0.309223	+	0.950990i	$0.399931\pi$
$719$			51.0000i			1.90198i	−0.309223	+	0.950990i	$0.600069\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		−	19.8997i		−	0.739568i
$725$		0			0
$726$		0			0
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.982831	+	0.184510i	$0.0590699\pi$
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.184510	+	0.982831i	$0.440930\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−38.0581	−	38.0581i	−1.40189	−	1.40189i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	0.467002	−	9.31662i	0.0171673	−	0.342486i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$	−37.2665	−	37.2665i	−1.35897	−	1.35897i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.899749	−	0.899749i	−0.0327019	−	0.0327019i	0.690567	−	0.723269i	$-0.257361\pi$
$757$	−0.899749	−	0.899749i	−0.0327019	−	0.0327019i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		−	46.4327i		−	1.67988i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.6332	+	33.6332i	−1.20970	+	1.20970i	−0.238581	+	0.971123i	$0.576682\pi$
$773$	−33.6332	+	33.6332i	−1.20970	+	1.20970i	−0.971123	+	0.238581i	$0.923318\pi$
$774$		0			0
$775$	−4.97494	+	49.5000i	−0.178705	+	1.77809i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.94987			−0.356034
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$	−52.0330	−	2.60819i	−1.85714	−	0.0930902i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	39.7995			1.41066
$797$	26.6913	+	26.6913i	0.945455	+	0.945455i	0.998587	−	0.0531327i	$-0.0169206\pi$
$797$	26.6913	+	26.6913i	0.945455	+	0.945455i	−0.0531327	+	0.998587i	$0.516921\pi$
$798$		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			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.84169	−	56.6913i	0.0995400	−	1.98581i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	39.4248	−	39.4248i	1.37426	−	1.37426i	0.520246	−	0.854016i	$-0.325840\pi$
$823$	39.4248	−	39.4248i	1.37426	−	1.37426i	0.854016	−	0.520246i	$-0.174160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$			29.0000i			1.00721i	0.863934	+	0.503606i	$0.167994\pi$
$829$			29.0000i			1.00721i	−0.863934	+	0.503606i	$0.832006\pi$
$830$		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
$839$			36.4829i			1.25953i	0.776786	+	0.629764i	$0.216849\pi$
$839$			36.4829i			1.25953i	−0.776786	+	0.629764i	$0.783151\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	19.5000	+	21.5581i	0.670820	+	0.741620i
$846$		0			0
$847$		0			0
$848$	14.5330	−	14.5330i	0.499065	−	0.499065i
$849$		0			0
$850$		0			0
$851$	−8.38262			−0.287353
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.21637	+	5.21637i	−0.177567	+	0.177567i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	−5.21637	+	5.21637i	−0.177567	+	0.177567i	0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$	−22.0000	+	19.8997i	−0.741620	+	0.670820i
$881$	−57.0000			−1.92038			−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000			−1.92038			−0.960189	+	0.279350i	$0.909881\pi$
$882$		0			0
$883$	−37.9499	+	37.9499i	−1.27711	+	1.27711i	−0.334840	+	0.942275i	$0.608682\pi$
$883$	−37.9499	+	37.9499i	−1.27711	+	1.27711i	−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	28.8496	+	28.8496i	0.965957	+	0.965957i
$893$		0			0
$894$		0			0
$895$	−31.5000	−	34.8246i	−1.05293	−	1.16406i
$896$		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
$905$	16.5000	−	14.9248i	0.548479	−	0.496118i
$906$		0			0
$907$	33.8496	+	33.8496i	1.12396	+	1.12396i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	33.8496	+	33.8496i	1.12396	+	1.12396i	0.132818	+	0.991140i	$0.457597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.63325			−0.219769			−0.109885	−	0.993944i	$-0.535048\pi$
$911$	−6.63325			−0.219769			−0.109885	+	0.993944i	$0.535048\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−59.6992			−1.97252
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.07519	−	6.60025i	0.265511	−	0.217015i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			53.0660i			1.74104i	0.492134	+	0.870519i	$0.336217\pi$
$929$			53.0660i			1.74104i	−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$	2.94987	−	58.8496i	0.0962143	−	1.91946i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		−	13.2665i		−	0.431788i
$945$		0			0
$946$		0			0
$947$	40.1082	+	40.1082i	1.30334	+	1.30334i	0.926126	+	0.377215i	$0.123118\pi$
$947$	40.1082	+	40.1082i	1.30334	+	1.30334i	0.377215	+	0.926126i	$0.376882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	38.5000	−	34.8246i	1.24583	−	1.12690i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	68.0000			2.19355
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43.1161			1.38366			0.691831	−	0.722059i	$-0.256804\pi$
$971$	43.1161			1.38366			0.691831	+	0.722059i	$0.256804\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41.6913	+	41.6913i	1.33382	+	1.33382i	0.901920	+	0.431903i	$0.142158\pi$
$977$	41.6913	+	41.6913i	1.33382	+	1.33382i	0.431903	+	0.901920i	$0.357842\pi$
$978$		0			0
$979$			29.8496i			0.953998i
$980$	23.2164	−	21.0000i	0.741620	−	0.670820i
$981$		0			0
$982$		0			0
$983$	43.7414	−	43.7414i	1.39514	−	1.39514i	0.581811	−	0.813324i	$-0.302344\pi$
$983$	43.7414	−	43.7414i	1.39514	−	1.39514i	0.813324	−	0.581811i	$-0.197656\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−59.6992			−1.89641			−0.948205	−	0.317660i	$-0.897103\pi$
$991$	−59.6992			−1.89641			−0.948205	+	0.317660i	$0.897103\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	29.8496	+	33.0000i	0.946297	+	1.04617i
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.k.a.307.1		4
3.2	odd	2		55.2.e.b.32.1	✓	4
5.3	odd	4	inner	495.2.k.a.208.1		4
11.10	odd	2	CM	495.2.k.a.307.1		4
12.11	even	2		880.2.bd.d.417.2		4
15.2	even	4		275.2.e.a.43.2		4
15.8	even	4		55.2.e.b.43.1	yes	4
15.14	odd	2		275.2.e.a.32.2		4
33.2	even	10		605.2.m.a.282.2		16
33.5	odd	10		605.2.m.a.602.1		16
33.8	even	10		605.2.m.a.112.2		16
33.14	odd	10		605.2.m.a.112.2		16
33.17	even	10		605.2.m.a.602.1		16
33.20	odd	10		605.2.m.a.282.2		16
33.26	odd	10		605.2.m.a.457.2		16
33.29	even	10		605.2.m.a.457.2		16
33.32	even	2		55.2.e.b.32.1	✓	4
55.43	even	4	inner	495.2.k.a.208.1		4
60.23	odd	4		880.2.bd.d.593.2		4
132.131	odd	2		880.2.bd.d.417.2		4
165.8	odd	20		605.2.m.a.233.2		16
165.32	odd	4		275.2.e.a.43.2		4
165.38	even	20		605.2.m.a.118.2		16
165.53	even	20		605.2.m.a.403.1		16
165.68	odd	20		605.2.m.a.403.1		16
165.83	odd	20		605.2.m.a.118.2		16
165.98	odd	4		55.2.e.b.43.1	yes	4
165.113	even	20		605.2.m.a.233.2		16
165.128	odd	20		605.2.m.a.578.2		16
165.158	even	20		605.2.m.a.578.2		16
165.164	even	2		275.2.e.a.32.2		4
660.263	even	4		880.2.bd.d.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.1	✓	4	3.2	odd	2
55.2.e.b.32.1	✓	4	33.32	even	2
55.2.e.b.43.1	yes	4	15.8	even	4
55.2.e.b.43.1	yes	4	165.98	odd	4
275.2.e.a.32.2		4	15.14	odd	2
275.2.e.a.32.2		4	165.164	even	2
275.2.e.a.43.2		4	15.2	even	4
275.2.e.a.43.2		4	165.32	odd	4
495.2.k.a.208.1		4	5.3	odd	4	inner
495.2.k.a.208.1		4	55.43	even	4	inner
495.2.k.a.307.1		4	1.1	even	1	trivial
495.2.k.a.307.1		4	11.10	odd	2	CM
605.2.m.a.112.2		16	33.8	even	10
605.2.m.a.112.2		16	33.14	odd	10
605.2.m.a.118.2		16	165.38	even	20
605.2.m.a.118.2		16	165.83	odd	20
605.2.m.a.233.2		16	165.8	odd	20
605.2.m.a.233.2		16	165.113	even	20
605.2.m.a.282.2		16	33.2	even	10
605.2.m.a.282.2		16	33.20	odd	10
605.2.m.a.403.1		16	165.53	even	20
605.2.m.a.403.1		16	165.68	odd	20
605.2.m.a.457.2		16	33.26	odd	10
605.2.m.a.457.2		16	33.29	even	10
605.2.m.a.578.2		16	165.128	odd	20
605.2.m.a.578.2		16	165.158	even	20
605.2.m.a.602.1		16	33.5	odd	10
605.2.m.a.602.1		16	33.17	even	10
880.2.bd.d.417.2		4	12.11	even	2
880.2.bd.d.417.2		4	132.131	odd	2
880.2.bd.d.593.2		4	60.23	odd	4
880.2.bd.d.593.2		4	660.263	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -3.31662 q^{11} -4.00000 q^{16} +(-3.00000 - 3.31662i) q^{20} +(-2.84169 + 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} -9.94987 q^{31} +(1.47494 + 1.47494i) q^{37} -6.63325i q^{44} +(9.31662 + 9.31662i) q^{47} +7.00000i q^{49} +(-3.63325 + 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} +3.31662i q^{59} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +3.00000 q^{71} +(6.63325 - 6.00000i) q^{80} -9.00000i q^{89} +(-5.68338 - 5.68338i) q^{92} +(-3.52506 - 3.52506i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{16} - 12 q^{20} - 18 q^{23} + 2 q^{25} - 14 q^{37} + 24 q^{47} + 12 q^{53} + 22 q^{55} + 26 q^{67} + 12 q^{71} - 36 q^{92} - 34 q^{97}+O(q^{100}) \) 4 * q - 16 * q^16 - 12 * q^20 - 18 * q^23 + 2 * q^25 - 14 * q^37 + 24 * q^47 + 12 * q^53 + 22 * q^55 + 26 * q^67 + 12 * q^71 - 36 * q^92 - 34 * q^97

Introduction

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