LMFDB - Embedded newform 605.2.b.a.364.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(364,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("605.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.83094932229$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			364.1
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	605.364
Dual form			605.2.b.a.364.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31662i q^{3} +2.00000 q^{4} +(1.50000 - 1.65831i) q^{5} -8.00000 q^{9} +O(q^{10})$	$q-3.31662i q^{3} +2.00000 q^{4} +(1.50000 - 1.65831i) q^{5} -8.00000 q^{9} -6.63325i q^{12} +(-5.50000 - 4.97494i) q^{15} +4.00000 q^{16} +(3.00000 - 3.31662i) q^{20} +3.31662i q^{23} +(-0.500000 - 4.97494i) q^{25} +16.5831i q^{27} +5.00000 q^{31} -16.0000 q^{36} +9.94987i q^{37} +(-12.0000 + 13.2665i) q^{45} +6.63325i q^{47} -13.2665i q^{48} +7.00000 q^{49} -13.2665i q^{53} -15.0000 q^{59} +(-11.0000 - 9.94987i) q^{60} +8.00000 q^{64} -9.94987i q^{67} +11.0000 q^{69} -3.00000 q^{71} +(-16.5000 + 1.65831i) q^{75} +(6.00000 - 6.63325i) q^{80} +31.0000 q^{81} +9.00000 q^{89} +6.63325i q^{92} -16.5831i q^{93} +9.94987i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9}+O(q^{10}) $ 2 * q + 4 * q^4 + 3 * q^5 - 16 * q^9	$ 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9} - 11 q^{15} + 8 q^{16} + 6 q^{20} - q^{25} + 10 q^{31} - 32 q^{36} - 24 q^{45} + 14 q^{49} - 30 q^{59} - 22 q^{60} + 16 q^{64} + 22 q^{69} - 6 q^{71} - 33 q^{75} + 12 q^{80} + 62 q^{81} + 18 q^{89}+O(q^{100}) $ 2 * q + 4 * q^4 + 3 * q^5 - 16 * q^9 - 11 * q^15 + 8 * q^16 + 6 * q^20 - q^25 + 10 * q^31 - 32 * q^36 - 24 * q^45 + 14 * q^49 - 30 * q^59 - 22 * q^60 + 16 * q^64 + 22 * q^69 - 6 * q^71 - 33 * q^75 + 12 * q^80 + 62 * q^81 + 18 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$		−	3.31662i		−	1.91485i	−0.288675	−	0.957427i	$-0.593215\pi$
$3$		−	3.31662i		−	1.91485i	0.288675	−	0.957427i	$-0.406785\pi$
$4$	2.00000			1.00000
$5$	1.50000	−	1.65831i	0.670820	−	0.741620i
$5$	1.50000	−	1.65831i	0.670820	−	0.741620i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−8.00000			−2.66667
$10$		0			0
$11$		0			0
$11$		0			0
$12$		−	6.63325i		−	1.91485i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−5.50000	−	4.97494i	−1.42009	−	1.28452i
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		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
$23$			3.31662i			0.691564i	0.938315	+	0.345782i	$0.112386\pi$
$23$			3.31662i			0.691564i	−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	−0.500000	−	4.97494i	−0.100000	−	0.994987i
$26$		0			0
$27$			16.5831i			3.19142i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−16.0000			−2.66667
$37$			9.94987i			1.63575i	0.575396	+	0.817875i	$0.304848\pi$
$37$			9.94987i			1.63575i	−0.575396	+	0.817875i	$0.695152\pi$
$38$		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
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−12.0000	+	13.2665i	−1.78885	+	1.97765i
$46$		0			0
$47$			6.63325i			0.967559i	0.875190	+	0.483779i	$0.160736\pi$
$47$			6.63325i			0.967559i	−0.875190	+	0.483779i	$0.839264\pi$
$48$		−	13.2665i		−	1.91485i
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.2665i		−	1.82229i	−0.412082	−	0.911147i	$-0.635198\pi$
$53$		−	13.2665i		−	1.82229i	0.412082	−	0.911147i	$-0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−15.0000			−1.95283			−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000			−1.95283			−0.976417	+	0.215894i	$0.930733\pi$
$60$	−11.0000	−	9.94987i	−1.42009	−	1.28452i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$		−	9.94987i		−	1.21557i	−0.794101	−	0.607785i	$-0.792058\pi$
$67$		−	9.94987i		−	1.21557i	0.794101	−	0.607785i	$-0.207942\pi$
$68$		0			0
$69$	11.0000			1.32424
$70$		0			0
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−16.5000	+	1.65831i	−1.90526	+	0.191485i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	6.00000	−	6.63325i	0.670820	−	0.741620i
$81$	31.0000			3.44444
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$		0			0
$92$			6.63325i			0.691564i
$93$		−	16.5831i		−	1.71959i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			9.94987i			1.01026i	0.863044	+	0.505128i	$0.168555\pi$
$97$			9.94987i			1.01026i	−0.863044	+	0.505128i	$0.831445\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000	−	9.94987i	−0.100000	−	0.994987i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			19.8997i			1.96078i	0.197066	+	0.980390i	$0.436859\pi$
$103$			19.8997i			1.96078i	−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			33.1662i			3.19142i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	33.0000			3.13222
$112$		0			0
$113$			3.31662i			0.312002i	0.987757	+	0.156001i	$0.0498603\pi$
$113$			3.31662i			0.312002i	−0.987757	+	0.156001i	$0.950140\pi$
$114$		0			0
$115$	5.50000	+	4.97494i	0.512878	+	0.463915i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	10.0000			0.898027
$125$	−9.00000	−	6.63325i	−0.804984	−	0.593296i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		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
$133$		0			0
$134$		0			0
$135$	27.5000	+	24.8747i	2.36682	+	2.14087i
$136$		0			0
$137$		−	23.2164i		−	1.98351i	−0.128154	−	0.991754i	$-0.540905\pi$
$137$		−	23.2164i		−	1.98351i	0.128154	−	0.991754i	$-0.459095\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	22.0000			1.85273
$142$		0			0
$143$		0			0
$144$	−32.0000			−2.66667
$145$		0			0
$146$		0			0
$147$		−	23.2164i		−	1.91485i
$148$			19.8997i			1.63575i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.50000	−	8.29156i	0.602414	−	0.665994i
$156$		0			0
$157$			9.94987i			0.794086i	0.917800	+	0.397043i	$0.129964\pi$
$157$			9.94987i			0.794086i	−0.917800	+	0.397043i	$0.870036\pi$
$158$		0			0
$159$	−44.0000			−3.48943
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	19.8997i		−	1.55867i	−0.626608	−	0.779334i	$-0.715557\pi$
$163$		−	19.8997i		−	1.55867i	0.626608	−	0.779334i	$-0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			49.7494i			3.73939i
$178$		0			0
$179$	−21.0000			−1.56961			−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000			−1.56961			−0.784807	+	0.619740i	$0.787238\pi$
$180$	−24.0000	+	26.5330i	−1.78885	+	1.97765i
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.5000	+	14.9248i	1.21310	+	1.09729i
$186$		0			0
$187$		0			0
$188$			13.2665i			0.967559i
$189$		0			0
$190$		0			0
$191$	15.0000			1.08536			0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000			1.08536			0.542681	+	0.839939i	$0.317409\pi$
$192$		−	26.5330i		−	1.91485i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	14.0000			1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−33.0000			−2.32764
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	26.5330i		−	1.84417i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		−	26.5330i		−	1.82229i
$213$			9.94987i			0.681754i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			29.8496i			1.99888i	0.0334825	+	0.999439i	$0.489340\pi$
$223$			29.8496i			1.99888i	−0.0334825	+	0.999439i	$0.510660\pi$
$224$		0			0
$225$	4.00000	+	39.7995i	0.266667	+	2.65330i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−5.00000			−0.330409			−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000			−0.330409			−0.165205	+	0.986259i	$0.552828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	11.0000	+	9.94987i	0.717561	+	0.649058i
$236$	−30.0000			−1.95283
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−22.0000	−	19.8997i	−1.42009	−	1.28452i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		−	53.0660i		−	3.40419i
$244$		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
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		−	26.5330i		−	1.65508i	−0.561405	−	0.827541i	$-0.689739\pi$
$257$		−	26.5330i		−	1.65508i	0.561405	−	0.827541i	$-0.310261\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−22.0000	−	19.8997i	−1.35145	−	1.22243i
$266$		0			0
$267$		−	29.8496i		−	1.82677i
$268$		−	19.8997i		−	1.21557i
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	22.0000			1.32424
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−40.0000			−2.39474
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	33.0000			1.93449
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	−22.5000	+	24.8747i	−1.31000	+	1.44826i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−33.0000	+	3.31662i	−1.90526	+	0.191485i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	66.0000			3.75461
$310$		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
$313$			29.8496i			1.68720i	0.536972	+	0.843600i	$0.319568\pi$
$313$			29.8496i			1.68720i	−0.536972	+	0.843600i	$0.680432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			23.2164i			1.30396i	0.758236	+	0.651981i	$0.226062\pi$
$317$			23.2164i			1.30396i	−0.758236	+	0.651981i	$0.773938\pi$
$318$		0			0
$319$		0			0
$320$	12.0000	−	13.2665i	0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	62.0000			3.44444
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−35.0000			−1.92377			−0.961887	−	0.273447i	$-0.911836\pi$
$331$	−35.0000			−1.92377			−0.961887	+	0.273447i	$0.911836\pi$
$332$		0			0
$333$		−	79.5990i		−	4.36200i
$334$		0			0
$335$	−16.5000	−	14.9248i	−0.901491	−	0.815430i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	11.0000			0.597438
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	16.5000	−	18.2414i	0.888330	−	0.982086i
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		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
$353$			36.4829i			1.94179i	0.239511	+	0.970894i	$0.423013\pi$
$353$			36.4829i			1.94179i	−0.239511	+	0.970894i	$0.576987\pi$
$354$		0			0
$355$	−4.50000	+	4.97494i	−0.238835	+	0.264042i
$356$	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
$367$		−	9.94987i		−	0.519379i	−0.965692	−	0.259690i	$-0.916380\pi$
$367$		−	9.94987i		−	0.519379i	0.965692	−	0.259690i	$-0.0836203\pi$
$368$			13.2665i			0.691564i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		−	33.1662i		−	1.71959i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−22.0000	+	29.8496i	−1.13608	+	1.54143i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−25.0000			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.31662i		−	0.169472i	−0.996403	−	0.0847358i	$-0.972995\pi$
$383$		−	3.31662i		−	0.169472i	0.996403	−	0.0847358i	$-0.0270046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$			19.8997i			1.01026i
$389$	−15.0000			−0.760530			−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000			−0.760530			−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
$397$		−	39.7995i		−	1.99748i	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$		−	39.7995i		−	1.99748i	0.0501886	−	0.998740i	$-0.484018\pi$
$398$		0			0
$399$		0			0
$400$	−2.00000	−	19.8997i	−0.100000	−	0.994987i
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	46.5000	−	51.4077i	2.31060	−	2.55447i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−77.0000			−3.79813
$412$			39.7995i			1.96078i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		−	53.0660i		−	2.58016i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$			66.3325i			3.19142i
$433$			29.8496i			1.43448i	0.696826	+	0.717241i	$0.254595\pi$
$433$			29.8496i			1.43448i	−0.696826	+	0.717241i	$0.745405\pi$
$434$		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
$441$	−56.0000			−2.66667
$442$		0			0
$443$			36.4829i			1.73335i	0.498870	+	0.866677i	$0.333748\pi$
$443$			36.4829i			1.73335i	−0.498870	+	0.866677i	$0.666252\pi$
$444$	66.0000			3.13222
$445$	13.5000	−	14.9248i	0.639961	−	0.707504i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.0000			−1.84052			−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000			−1.84052			−0.920262	+	0.391303i	$0.872024\pi$
$450$		0			0
$451$		0			0
$452$			6.63325i			0.312002i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$	11.0000	+	9.94987i	0.512878	+	0.463915i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			29.8496i			1.38723i	0.720346	+	0.693615i	$0.243983\pi$
$463$			29.8496i			1.38723i	−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	−27.5000	−	24.8747i	−1.27528	−	1.15354i
$466$		0			0
$467$			43.1161i			1.99518i	0.0694117	+	0.997588i	$0.477888\pi$
$467$			43.1161i			1.99518i	−0.0694117	+	0.997588i	$0.522112\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	33.0000			1.52056
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			106.132i			4.85945i
$478$		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
$484$		0			0
$485$	16.5000	+	14.9248i	0.749226	+	0.677701i
$486$		0			0
$487$		−	9.94987i		−	0.450872i	−0.974258	−	0.225436i	$-0.927619\pi$
$487$		−	9.94987i		−	0.450872i	0.974258	−	0.225436i	$-0.0723806\pi$
$488$		0			0
$489$	−66.0000			−2.98462
$490$		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
$496$	20.0000			0.898027
$497$		0			0
$498$		0			0
$499$	40.0000			1.79065			0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000			1.79065			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		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	43.1161i		−	1.91485i
$508$		0			0
$509$	−45.0000			−1.99459			−0.997295	−	0.0735034i	$-0.976582\pi$
$509$	−45.0000			−1.99459			−0.997295	+	0.0735034i	$0.976582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	33.0000	+	29.8496i	1.45415	+	1.31533i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.0000			0.657162			0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000			0.657162			0.328581	+	0.944476i	$0.393430\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12.0000			0.521739
$530$		0			0
$531$	120.000			5.20756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$			69.6491i			3.00558i
$538$		0			0
$539$		0			0
$540$	55.0000	+	49.7494i	2.36682	+	2.14087i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		−	82.9156i		−	3.55825i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		−	46.4327i		−	1.98351i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	49.5000	−	54.7243i	2.10116	−	2.32292i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	44.0000			1.85273
$565$	5.50000	+	4.97494i	0.231387	+	0.209297i
$566$		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			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	49.7494i		−	2.07831i
$574$		0			0
$575$	16.5000	−	1.65831i	0.688098	−	0.0691564i
$576$	−64.0000			−2.66667
$577$		−	9.94987i		−	0.414219i	−0.978318	−	0.207109i	$-0.933594\pi$
$577$		−	9.94987i		−	0.414219i	0.978318	−	0.207109i	$-0.0664056\pi$
$578$		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
$587$		−	6.63325i		−	0.273784i	−0.990586	−	0.136892i	$-0.956289\pi$
$587$		−	6.63325i		−	0.273784i	0.990586	−	0.136892i	$-0.0437113\pi$
$588$		−	46.4327i		−	1.91485i
$589$		0			0
$590$		0			0
$591$		0			0
$592$			39.7995i			1.63575i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			66.3325i			2.71481i
$598$		0			0
$599$	36.0000			1.47092			0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000			1.47092			0.735460	+	0.677568i	$0.236966\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$			79.5990i			3.24152i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.5330i		−	1.06818i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$		−	26.5330i		−	1.06818i	0.845428	−	0.534089i	$-0.179345\pi$
$618$		0			0
$619$	1.00000			0.0401934			0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000			0.0401934			0.0200967	+	0.999798i	$0.493603\pi$
$620$	15.0000	−	16.5831i	0.602414	−	0.665994i
$621$	−55.0000			−2.20707
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5000	+	4.97494i	−0.980000	+	0.198997i
$626$		0			0
$627$		0			0
$628$			19.8997i			0.794086i
$629$		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
$636$	−88.0000			−3.48943
$637$		0			0
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	45.0000			1.77739			0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000			1.77739			0.888697	+	0.458496i	$0.151612\pi$
$642$		0			0
$643$		−	29.8496i		−	1.17715i	−0.808441	−	0.588577i	$-0.799688\pi$
$643$		−	29.8496i		−	1.17715i	0.808441	−	0.588577i	$-0.200312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.1161i		−	1.69507i	−0.530740	−	0.847535i	$-0.678086\pi$
$647$		−	43.1161i		−	1.69507i	0.530740	−	0.847535i	$-0.321914\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		−	39.7995i		−	1.55867i
$653$			3.31662i			0.129790i	0.997892	+	0.0648948i	$0.0206712\pi$
$653$			3.31662i			0.129790i	−0.997892	+	0.0648948i	$0.979329\pi$
$654$		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	$-0.418642\pi$
$661$	13.0000			0.505641			0.252821	+	0.967513i	$0.418642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	99.0000			3.82756
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	82.5000	−	8.29156i	3.17543	−	0.319142i
$676$	26.0000			1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			46.4327i			1.77670i	0.459167	+	0.888350i	$0.348148\pi$
$683$			46.4327i			1.77670i	−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	−38.5000	−	34.8246i	−1.47101	−	1.33058i
$686$		0			0
$687$			16.5831i			0.632686i
$688$		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			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	33.0000	−	36.4829i	1.24285	−	1.37402i
$706$		0			0
$707$		0			0
$708$			99.4987i			3.73939i
$709$	−19.0000			−0.713560			−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000			−0.713560			−0.356780	+	0.934188i	$0.616125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16.5831i			0.621043i
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$	−51.0000			−1.90198			−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000			−1.90198			−0.950990	+	0.309223i	$0.899931\pi$
$720$	−48.0000	+	53.0660i	−1.78885	+	1.97765i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	50.0000			1.85824
$725$		0			0
$726$		0			0
$727$			9.94987i			0.369020i	0.982831	+	0.184510i	$0.0590699\pi$
$727$			9.94987i			0.369020i	−0.982831	+	0.184510i	$0.940930\pi$
$728$		0			0
$729$	−83.0000			−3.07407
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−38.5000	−	34.8246i	−1.42009	−	1.28452i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	33.0000	+	29.8496i	1.21310	+	1.09729i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		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	$0.637844\pi$
$752$			26.5330i			0.967559i
$753$			89.5489i			3.26334i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	39.7995i		−	1.44654i	−0.690567	−	0.723269i	$-0.742639\pi$
$757$		−	39.7995i		−	1.44654i	0.690567	−	0.723269i	$-0.257361\pi$
$758$		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
$763$		0			0
$764$	30.0000			1.08536
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	53.0660i		−	1.91485i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−88.0000			−3.16924
$772$		0			0
$773$			13.2665i			0.477163i	0.971123	+	0.238581i	$0.0766824\pi$
$773$			13.2665i			0.477163i	−0.971123	+	0.238581i	$0.923318\pi$
$774$		0			0
$775$	−2.50000	−	24.8747i	−0.0898027	−	0.893525i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	16.5000	+	14.9248i	0.588910	+	0.532689i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−66.0000	+	72.9657i	−2.34078	+	2.58783i
$796$	−40.0000			−1.41776
$797$		−	56.3826i		−	1.99717i	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$		−	56.3826i		−	1.99717i	0.0531327	−	0.998587i	$-0.483079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−72.0000			−2.54399
$802$		0			0
$803$		0			0
$804$	−66.0000			−2.32764
$805$		0			0
$806$		0			0
$807$		−	99.4987i		−	3.50252i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.0000	−	29.8496i	−1.15594	−	1.04559i
$816$		0			0
$817$		0			0
$818$		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
$823$		−	29.8496i		−	1.04049i	−0.854016	−	0.520246i	$-0.825840\pi$
$823$		−	29.8496i		−	1.04049i	0.854016	−	0.520246i	$-0.174160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		−	53.0660i		−	1.84417i
$829$	−29.0000			−1.00721			−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000			−1.00721			−0.503606	+	0.863934i	$0.667994\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$			82.9156i			2.86598i
$838$		0			0
$839$	−45.0000			−1.55357			−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000			−1.55357			−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
$848$		−	53.0660i		−	1.82229i
$849$		0			0
$850$		0			0
$851$	−33.0000			−1.13123
$852$			19.8997i			0.681754i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	31.0000			1.05771			0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000			1.05771			0.528853	+	0.848713i	$0.322622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46.4327i			1.58059i	0.612727	+	0.790295i	$0.290072\pi$
$863$			46.4327i			1.58059i	−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	56.3826i		−	1.91485i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	79.5990i		−	2.69402i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$		−	19.8997i		−	0.669680i	−0.942275	−	0.334840i	$-0.891318\pi$
$883$		−	19.8997i		−	0.669680i	0.942275	−	0.334840i	$-0.108682\pi$
$884$		0			0
$885$	82.5000	+	74.6241i	2.77321	+	2.50846i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			59.6992i			1.99888i
$893$		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
$900$	8.00000	+	79.5990i	0.266667	+	2.65330i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	37.5000	−	41.4578i	1.24654	−	1.37810i
$906$		0			0
$907$		−	59.6992i		−	1.98228i	−0.132818	−	0.991140i	$-0.542403\pi$
$907$		−	59.6992i		−	1.98228i	0.132818	−	0.991140i	$-0.457597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	60.0000			1.98789			0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000			1.98789			0.993944	+	0.109885i	$0.0350482\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−10.0000			−0.330409
$917$		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
$925$	49.5000	−	4.97494i	1.62755	−	0.163575i
$926$		0			0
$927$		−	159.198i		−	5.22875i
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			39.7995i			1.30298i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	99.0000			3.23074
$940$	22.0000	+	19.8997i	0.717561	+	0.649058i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−60.0000			−1.95283
$945$		0			0
$946$		0			0
$947$		−	23.2164i		−	0.754431i	−0.926126	−	0.377215i	$-0.876882\pi$
$947$		−	23.2164i		−	0.754431i	0.926126	−	0.377215i	$-0.123118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	77.0000			2.49690
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	22.5000	−	24.8747i	0.728083	−	0.804926i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−44.0000	−	39.7995i	−1.42009	−	1.28452i
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	45.0000			1.44412			0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000			1.44412			0.722059	+	0.691831i	$0.243196\pi$
$972$		−	106.132i		−	3.40419i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			56.3826i			1.80384i	0.431903	+	0.901920i	$0.357842\pi$
$977$			56.3826i			1.80384i	−0.431903	+	0.901920i	$0.642158\pi$
$978$		0			0
$979$		0			0
$980$	21.0000	−	23.2164i	0.670820	−	0.741620i
$981$		0			0
$982$		0			0
$983$			36.4829i			1.16362i	0.813324	+	0.581811i	$0.197656\pi$
$983$			36.4829i			1.16362i	−0.813324	+	0.581811i	$0.802344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$			116.082i			3.68375i
$994$		0			0
$995$	−30.0000	+	33.1662i	−0.951064	+	1.05144i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−165.000			−5.22037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.b.a.364.1	✓	2
5.2	odd	4		3025.2.a.k.1.1		2
5.3	odd	4		3025.2.a.k.1.2		2
5.4	even	2	inner	605.2.b.a.364.2	yes	2
11.2	odd	10		605.2.j.b.444.1		8
11.3	even	5		605.2.j.b.9.2		8
11.4	even	5		605.2.j.b.269.1		8
11.5	even	5		605.2.j.b.124.2		8
11.6	odd	10		605.2.j.b.124.2		8
11.7	odd	10		605.2.j.b.269.1		8
11.8	odd	10		605.2.j.b.9.2		8
11.9	even	5		605.2.j.b.444.1		8
11.10	odd	2	CM	605.2.b.a.364.1	✓	2
55.4	even	10		605.2.j.b.269.2		8
55.9	even	10		605.2.j.b.444.2		8
55.14	even	10		605.2.j.b.9.1		8
55.19	odd	10		605.2.j.b.9.1		8
55.24	odd	10		605.2.j.b.444.2		8
55.29	odd	10		605.2.j.b.269.2		8
55.32	even	4		3025.2.a.k.1.1		2
55.39	odd	10		605.2.j.b.124.1		8
55.43	even	4		3025.2.a.k.1.2		2
55.49	even	10		605.2.j.b.124.1		8
55.54	odd	2	inner	605.2.b.a.364.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.a.364.1	✓	2	1.1	even	1	trivial
605.2.b.a.364.1	✓	2	11.10	odd	2	CM
605.2.b.a.364.2	yes	2	5.4	even	2	inner
605.2.b.a.364.2	yes	2	55.54	odd	2	inner
605.2.j.b.9.1		8	55.14	even	10
605.2.j.b.9.1		8	55.19	odd	10
605.2.j.b.9.2		8	11.3	even	5
605.2.j.b.9.2		8	11.8	odd	10
605.2.j.b.124.1		8	55.39	odd	10
605.2.j.b.124.1		8	55.49	even	10
605.2.j.b.124.2		8	11.5	even	5
605.2.j.b.124.2		8	11.6	odd	10
605.2.j.b.269.1		8	11.4	even	5
605.2.j.b.269.1		8	11.7	odd	10
605.2.j.b.269.2		8	55.4	even	10
605.2.j.b.269.2		8	55.29	odd	10
605.2.j.b.444.1		8	11.2	odd	10
605.2.j.b.444.1		8	11.9	even	5
605.2.j.b.444.2		8	55.9	even	10
605.2.j.b.444.2		8	55.24	odd	10
3025.2.a.k.1.1		2	5.2	odd	4
3025.2.a.k.1.1		2	55.32	even	4
3025.2.a.k.1.2		2	5.3	odd	4
3025.2.a.k.1.2		2	55.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.31662i q^{3} +2.00000 q^{4} +(1.50000 - 1.65831i) q^{5} -8.00000 q^{9} +O(q^{10})\)	\(q-3.31662i q^{3} +2.00000 q^{4} +(1.50000 - 1.65831i) q^{5} -8.00000 q^{9} -6.63325i q^{12} +(-5.50000 - 4.97494i) q^{15} +4.00000 q^{16} +(3.00000 - 3.31662i) q^{20} +3.31662i q^{23} +(-0.500000 - 4.97494i) q^{25} +16.5831i q^{27} +5.00000 q^{31} -16.0000 q^{36} +9.94987i q^{37} +(-12.0000 + 13.2665i) q^{45} +6.63325i q^{47} -13.2665i q^{48} +7.00000 q^{49} -13.2665i q^{53} -15.0000 q^{59} +(-11.0000 - 9.94987i) q^{60} +8.00000 q^{64} -9.94987i q^{67} +11.0000 q^{69} -3.00000 q^{71} +(-16.5000 + 1.65831i) q^{75} +(6.00000 - 6.63325i) q^{80} +31.0000 q^{81} +9.00000 q^{89} +6.63325i q^{92} -16.5831i q^{93} +9.94987i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9}+O(q^{10}) \) 2 * q + 4 * q^4 + 3 * q^5 - 16 * q^9	\( 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9} - 11 q^{15} + 8 q^{16} + 6 q^{20} - q^{25} + 10 q^{31} - 32 q^{36} - 24 q^{45} + 14 q^{49} - 30 q^{59} - 22 q^{60} + 16 q^{64} + 22 q^{69} - 6 q^{71} - 33 q^{75} + 12 q^{80} + 62 q^{81} + 18 q^{89}+O(q^{100}) \) 2 * q + 4 * q^4 + 3 * q^5 - 16 * q^9 - 11 * q^15 + 8 * q^16 + 6 * q^20 - q^25 + 10 * q^31 - 32 * q^36 - 24 * q^45 + 14 * q^49 - 30 * q^59 - 22 * q^60 + 16 * q^64 + 22 * q^69 - 6 * q^71 - 33 * q^75 + 12 * q^80 + 62 * q^81 + 18 * q^89

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