LMFDB - Embedded newform 253.2.b.a.252.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [253,2,Mod(252,253)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("253.252"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(253, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 253 = 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	253.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.02021517114$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			252.1
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	253.252
Dual form			253.2.b.a.252.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000 q^{4} -3.31662i q^{5} -2.00000 q^{9} -3.31662i q^{11} -2.00000 q^{12} +3.31662i q^{15} +4.00000 q^{16} -6.63325i q^{20} +(4.50000 - 1.65831i) q^{23} -6.00000 q^{25} +5.00000 q^{27} +5.00000 q^{31} +3.31662i q^{33} -4.00000 q^{36} +9.94987i q^{37} -6.63325i q^{44} +6.63325i q^{45} -12.0000 q^{47} -4.00000 q^{48} -7.00000 q^{49} +13.2665i q^{53} -11.0000 q^{55} +15.0000 q^{59} +6.63325i q^{60} +8.00000 q^{64} +9.94987i q^{67} +(-4.50000 + 1.65831i) q^{69} +3.00000 q^{71} +6.00000 q^{75} -13.2665i q^{80} +1.00000 q^{81} -16.5831i q^{89} +(9.00000 - 3.31662i) q^{92} -5.00000 q^{93} +9.94987i q^{97} +6.63325i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{4} - 4 q^{9} - 4 q^{12} + 8 q^{16} + 9 q^{23} - 12 q^{25} + 10 q^{27} + 10 q^{31} - 8 q^{36} - 24 q^{47} - 8 q^{48} - 14 q^{49} - 22 q^{55} + 30 q^{59} + 16 q^{64} - 9 q^{69} + 6 q^{71}+ \cdots - 10 q^{93}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^4 - 4 * q^9 - 4 * q^12 + 8 * q^16 + 9 * q^23 - 12 * q^25 + 10 * q^27 + 10 * q^31 - 8 * q^36 - 24 * q^47 - 8 * q^48 - 14 * q^49 - 22 * q^55 + 30 * q^59 + 16 * q^64 - 9 * q^69 + 6 * q^71 + 12 * q^75 + 2 * q^81 + 18 * q^92 - 10 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/253\mathbb{Z}\right)^\times$.

$n$	$24$	$166$
$\chi(n)$	$-1$	$-1$

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			$\pi$
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$	2.00000			1.00000
$5$		−	3.31662i		−	1.48324i	−0.670820	−	0.741620i	$-0.734058\pi$
$5$		−	3.31662i		−	1.48324i	0.670820	−	0.741620i	$-0.265942\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$	−2.00000			−0.577350
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$			3.31662i			0.856349i
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	6.63325i		−	1.48324i
$21$		0			0
$22$		0			0
$23$	4.50000	−	1.65831i	0.938315	−	0.345782i
$23$	4.50000	−	1.65831i	0.938315	−	0.345782i
$24$		0			0
$25$	−6.00000			−1.20000
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		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
$33$			3.31662i			0.577350i
$34$		0			0
$35$		0			0
$36$	−4.00000			−0.666667
$37$			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		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		−	6.63325i		−	1.00000i
$45$			6.63325i			0.988826i
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$	−4.00000			−0.577350
$49$	−7.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			13.2665i			1.82229i	0.412082	+	0.911147i	$0.364802\pi$
$53$			13.2665i			1.82229i	−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−11.0000			−1.48324
$56$		0			0
$57$		0			0
$58$		0			0
$59$	15.0000			1.95283			0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000			1.95283			0.976417	+	0.215894i	$0.0692665\pi$
$60$			6.63325i			0.856349i
$61$		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	$0.207942\pi$
$67$			9.94987i			1.21557i	−0.794101	+	0.607785i	$0.792058\pi$
$68$		0			0
$69$	−4.50000	+	1.65831i	−0.541736	+	0.199637i
$70$		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		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	6.00000			0.692820
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		−	13.2665i		−	1.48324i
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.5831i		−	1.75781i	−0.476999	−	0.878904i	$-0.658275\pi$
$89$		−	16.5831i		−	1.75781i	0.476999	−	0.878904i	$-0.341725\pi$
$90$		0			0
$91$		0			0
$92$	9.00000	−	3.31662i	0.938315	−	0.345782i
$93$	−5.00000			−0.518476
$94$		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
$99$			6.63325i			0.666667i
$100$	−12.0000			−1.20000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	19.8997i		−	1.96078i	−0.197066	−	0.980390i	$-0.563141\pi$
$103$		−	19.8997i		−	1.96078i	0.197066	−	0.980390i	$-0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	10.0000			0.962250
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		−	9.94987i		−	0.944400i
$112$		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	−	14.9248i	−0.512878	−	1.39175i
$116$		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
$124$	10.0000			0.898027
$125$			3.31662i			0.296648i
$126$		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		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$			6.63325i			0.577350i
$133$		0			0
$134$		0			0
$135$		−	16.5831i		−	1.42725i
$136$		0			0
$137$			23.2164i			1.98351i	0.128154	+	0.991754i	$0.459095\pi$
$137$			23.2164i			1.98351i	−0.128154	+	0.991754i	$0.540905\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$		0			0
$144$	−8.00000			−0.666667
$145$		0			0
$146$		0			0
$147$	7.00000			0.577350
$148$			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		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	16.5831i		−	1.33199i
$156$		0			0
$157$		−	9.94987i		−	0.794086i	−0.917800	−	0.397043i	$-0.870036\pi$
$157$		−	9.94987i		−	0.794086i	0.917800	−	0.397043i	$-0.129964\pi$
$158$		0			0
$159$		−	13.2665i		−	1.05210i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$	11.0000			0.856349
$166$		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
$176$		−	13.2665i		−	1.00000i
$177$	−15.0000			−1.12747
$178$		0			0
$179$	−21.0000			−1.56961			−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000			−1.56961			−0.784807	+	0.619740i	$0.787238\pi$
$180$			13.2665i			0.988826i
$181$		−	9.94987i		−	0.739568i	−0.929118	−	0.369784i	$-0.879432\pi$
$181$		−	9.94987i		−	0.739568i	0.929118	−	0.369784i	$-0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	33.0000			2.42621
$186$		0			0
$187$		0			0
$188$	−24.0000			−1.75038
$189$		0			0
$190$		0			0
$191$		−	23.2164i		−	1.67988i	−0.542681	−	0.839939i	$-0.682591\pi$
$191$		−	23.2164i		−	1.67988i	0.542681	−	0.839939i	$-0.317409\pi$
$192$	−8.00000			−0.577350
$193$		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
$199$			19.8997i			1.41066i	0.708881	+	0.705328i	$0.249200\pi$
$199$			19.8997i			1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		−	9.94987i		−	0.701810i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−9.00000	+	3.31662i	−0.625543	+	0.230521i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			26.5330i			1.82229i
$213$	−3.00000			−0.205557
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−22.0000			−1.48324
$221$		0			0
$222$		0			0
$223$	1.00000			0.0669650			0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000			0.0669650			0.0334825	+	0.999439i	$0.489340\pi$
$224$		0			0
$225$	12.0000			0.800000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		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		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$			39.7995i			2.59623i
$236$	30.0000			1.95283
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$			13.2665i			0.856349i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−16.0000			−1.02640
$244$		0			0
$245$			23.2164i			1.48324i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	16.5831i		−	1.04672i	−0.852112	−	0.523359i	$-0.824679\pi$
$251$		−	16.5831i		−	1.04672i	0.852112	−	0.523359i	$-0.175321\pi$
$252$		0			0
$253$	−5.50000	−	14.9248i	−0.345782	−	0.938315i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−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			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	44.0000			2.70290
$266$		0			0
$267$			16.5831i			1.01487i
$268$			19.8997i			1.21557i
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			19.8997i			1.20000i
$276$	−9.00000	+	3.31662i	−0.541736	+	0.199637i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	9.94987i		−	0.583272i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		−	49.7494i		−	2.89652i
$296$		0			0
$297$		−	16.5831i		−	0.962250i
$298$		0			0
$299$		0			0
$300$	12.0000			0.692820
$301$		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
$309$			19.8997i			1.13206i
$310$		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	$-0.680432\pi$
$313$		−	29.8496i		−	1.68720i	0.536972	−	0.843600i	$-0.319568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.0000			−1.51647			−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000			−1.51647			−0.758236	+	0.651981i	$0.773938\pi$
$318$		0			0
$319$		0			0
$320$		−	26.5330i		−	1.48324i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	2.00000			0.111111
$325$		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	$-0.0881639\pi$
$331$	35.0000			1.92377			0.961887	+	0.273447i	$0.0881639\pi$
$332$		0			0
$333$		−	19.8997i		−	1.09050i
$334$		0			0
$335$	33.0000			1.80298
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		−	3.31662i		−	0.180134i
$340$		0			0
$341$		−	16.5831i		−	0.898027i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	5.50000	+	14.9248i	0.296110	+	0.803525i
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000			0.479022			0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000			0.479022			0.239511	+	0.970894i	$0.423013\pi$
$354$		0			0
$355$		−	9.94987i		−	0.528085i
$356$		−	33.1662i		−	1.75781i
$357$		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
$363$	11.0000			0.577350
$364$		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	$-0.0836203\pi$
$368$	18.0000	−	6.63325i	0.938315	−	0.345782i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−10.0000			−0.518476
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		−	3.31662i		−	0.171270i
$376$		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
$383$		−	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
$389$			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
$396$			13.2665i			0.666667i
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$	−24.0000			−1.20000
$401$			26.5330i			1.32499i	0.749064	+	0.662497i	$0.230503\pi$
$401$			26.5330i			1.32499i	−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	3.31662i		−	0.164804i
$406$		0			0
$407$	33.0000			1.63575
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		−	23.2164i		−	1.14518i
$412$		−	39.7995i		−	1.96078i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			33.1662i			1.62028i	0.586238	+	0.810139i	$0.300608\pi$
$419$			33.1662i			1.62028i	−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$		−	39.7995i		−	1.93971i	−0.243685	−	0.969854i	$-0.578356\pi$
$421$		−	39.7995i		−	1.93971i	0.243685	−	0.969854i	$-0.421644\pi$
$422$		0			0
$423$	24.0000			1.16692
$424$		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	$0.5\pi$
$432$	20.0000			0.962250
$433$		−	29.8496i		−	1.43448i	−0.696826	−	0.717241i	$-0.745405\pi$
$433$		−	29.8496i		−	1.43448i	0.696826	−	0.717241i	$-0.254595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	14.0000			0.666667
$442$		0			0
$443$	−21.0000			−0.997740			−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000			−0.997740			−0.498870	+	0.866677i	$0.666252\pi$
$444$		−	19.8997i		−	0.944400i
$445$	−55.0000			−2.60725
$446$		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			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$	−11.0000	−	29.8496i	−0.512878	−	1.39175i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−31.0000			−1.44069			−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000			−1.44069			−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$			16.5831i			0.769024i
$466$		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
$471$			9.94987i			0.458466i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	26.5330i		−	1.21486i
$478$		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
$484$	−22.0000			−1.00000
$485$	33.0000			1.49845
$486$		0			0
$487$	43.0000			1.94852			0.974258	−	0.225436i	$-0.0723806\pi$
$487$	43.0000			1.94852			0.974258	+	0.225436i	$0.0723806\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	22.0000			0.988826
$496$	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	$0.146945\pi$
$500$			6.63325i			0.296648i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.0000			−0.577350
$508$		0			0
$509$	45.0000			1.99459			0.997295	−	0.0735034i	$-0.0234180\pi$
$509$	45.0000			1.99459			0.997295	+	0.0735034i	$0.0234180\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−66.0000			−2.90831
$516$		0			0
$517$			39.7995i			1.75038i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			43.1161i			1.88895i	0.328581	+	0.944476i	$0.393430\pi$
$521$			43.1161i			1.88895i	−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$			13.2665i			0.577350i
$529$	17.5000	−	14.9248i	0.760870	−	0.648905i
$530$		0			0
$531$	−30.0000			−1.30189
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	21.0000			0.906217
$538$		0			0
$539$			23.2164i			1.00000i
$540$		−	33.1662i		−	1.42725i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$			9.94987i			0.426990i
$544$		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
$555$	−33.0000			−1.40077
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	24.0000			1.01058
$565$	11.0000			0.462773
$566$		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
$573$			23.2164i			0.969878i
$574$		0			0
$575$	−27.0000	+	9.94987i	−1.12598	+	0.414938i
$576$	−16.0000			−0.666667
$577$	−47.0000			−1.95664			−0.978318	−	0.207109i	$-0.933594\pi$
$577$	−47.0000			−1.95664			−0.978318	+	0.207109i	$0.933594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	44.0000			1.82229
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−48.0000			−1.98117			−0.990586	−	0.136892i	$-0.956289\pi$
$587$	−48.0000			−1.98117			−0.990586	+	0.136892i	$0.956289\pi$
$588$	14.0000			0.577350
$589$		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
$597$		−	19.8997i		−	0.814442i
$598$		0			0
$599$	−36.0000			−1.47092			−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000			−1.47092			−0.735460	+	0.677568i	$0.763034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	19.8997i		−	0.810380i
$604$		0			0
$605$			36.4829i			1.48324i
$606$		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			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.5330i			1.06818i	0.845428	+	0.534089i	$0.179345\pi$
$617$			26.5330i			1.06818i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$			49.7494i			1.99960i	0.0200967	+	0.999798i	$0.493603\pi$
$619$			49.7494i			1.99960i	−0.0200967	+	0.999798i	$0.506397\pi$
$620$		−	33.1662i		−	1.33199i
$621$	22.5000	−	8.29156i	0.902894	−	0.332729i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		−	19.8997i		−	0.794086i
$629$		0			0
$630$		0			0
$631$		−	49.7494i		−	1.98049i	−0.139333	−	0.990246i	$-0.544496\pi$
$631$		−	49.7494i		−	1.98049i	0.139333	−	0.990246i	$-0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		−	26.5330i		−	1.05210i
$637$		0			0
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$		−	23.2164i		−	0.916992i	−0.888697	−	0.458496i	$-0.848388\pi$
$641$		−	23.2164i		−	0.916992i	0.888697	−	0.458496i	$-0.151612\pi$
$642$		0			0
$643$			29.8496i			1.17715i	0.808441	+	0.588577i	$0.200312\pi$
$643$			29.8496i			1.17715i	−0.808441	+	0.588577i	$0.799688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27.0000			1.06148			0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000			1.06148			0.530740	+	0.847535i	$0.321914\pi$
$648$		0			0
$649$		−	49.7494i		−	1.95283i
$650$		0			0
$651$		0			0
$652$	32.0000			1.25322
$653$	−51.0000			−1.99578			−0.997892	−	0.0648948i	$-0.979329\pi$
$653$	−51.0000			−1.99578			−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	$0.5\pi$
$660$	22.0000			0.856349
$661$			49.7494i			1.93503i	0.252821	+	0.967513i	$0.418642\pi$
$661$			49.7494i			1.93503i	−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
$669$	−1.00000			−0.0386622
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−30.0000			−1.15470
$676$	26.0000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	77.0000			2.94202
$686$		0			0
$687$		−	29.8496i		−	1.13883i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−17.0000			−0.646710			−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000			−0.646710			−0.323355	+	0.946278i	$0.604811\pi$
$692$		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
$704$		−	26.5330i		−	1.00000i
$705$		−	39.7995i		−	1.49894i
$706$		0			0
$707$		0			0
$708$	−30.0000			−1.12747
$709$		−	49.7494i		−	1.86838i	−0.356780	−	0.934188i	$-0.616125\pi$
$709$		−	49.7494i		−	1.86838i	0.356780	−	0.934188i	$-0.383875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.5000	−	8.29156i	0.842632	−	0.310521i
$714$		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	$-0.100069\pi$
$719$	51.0000			1.90198			0.950990	+	0.309223i	$0.100069\pi$
$720$			26.5330i			0.988826i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		−	19.8997i		−	0.739568i
$725$		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
$729$	13.0000			0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		−	23.2164i		−	0.856349i
$736$		0			0
$737$	33.0000			1.21557
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	66.0000			2.42621
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			49.7494i			1.81538i	0.419641	+	0.907690i	$0.362156\pi$
$751$			49.7494i			1.81538i	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−48.0000			−1.75038
$753$			16.5831i			0.604323i
$754$		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
$759$	5.50000	+	14.9248i	0.199637	+	0.541736i
$760$		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
$768$	−16.0000			−0.577350
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		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
$775$	−30.0000			−1.07763
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	9.94987i		−	0.356034i
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$	−33.0000			−1.17782
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−44.0000			−1.56052
$796$			39.7995i			1.41066i
$797$		−	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
$801$			33.1662i			1.17187i
$802$		0			0
$803$		0			0
$804$		−	19.8997i		−	0.701810i
$805$		0			0
$806$		0			0
$807$	30.0000			1.05605
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	53.0660i		−	1.85882i
$816$		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
$823$	−49.0000			−1.70803			−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000			−1.70803			−0.854016	+	0.520246i	$0.825840\pi$
$824$		0			0
$825$		−	19.8997i		−	0.692820i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−18.0000	+	6.63325i	−0.625543	+	0.230521i
$829$	29.0000			1.00721			0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000			1.00721			0.503606	+	0.863934i	$0.332006\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	25.0000			0.864126
$838$		0			0
$839$		−	36.4829i		−	1.25953i	−0.776786	−	0.629764i	$-0.783151\pi$
$839$		−	36.4829i		−	1.25953i	0.776786	−	0.629764i	$-0.216849\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	43.1161i		−	1.48324i
$846$		0			0
$847$		0			0
$848$			53.0660i			1.82229i
$849$		0			0
$850$		0			0
$851$	16.5000	+	44.7744i	0.565613	+	1.53485i
$852$	−6.00000			−0.205557
$853$		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	$-0.677378\pi$
$859$	−31.0000			−1.05771			−0.528853	+	0.848713i	$0.677378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	17.0000			0.577350
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	19.8997i		−	0.673504i
$874$		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
$880$	−44.0000			−1.48324
$881$			16.5831i			0.558700i	0.960189	+	0.279350i	$0.0901189\pi$
$881$			16.5831i			0.558700i	−0.960189	+	0.279350i	$0.909881\pi$
$882$		0			0
$883$	56.0000			1.88455			0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000			1.88455			0.942275	+	0.334840i	$0.108682\pi$
$884$		0			0
$885$			49.7494i			1.67231i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	3.31662i		−	0.111111i
$892$	2.00000			0.0669650
$893$		0			0
$894$		0			0
$895$			69.6491i			2.32811i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	24.0000			0.800000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−33.0000			−1.09696
$906$		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
$911$		−	6.63325i		−	0.219769i	−0.993944	−	0.109885i	$-0.964952\pi$
$911$		−	6.63325i		−	0.219769i	0.993944	−	0.109885i	$-0.0350482\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$			59.6992i			1.97252i
$917$		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
$925$		−	59.6992i		−	1.96290i
$926$		0			0
$927$			39.7995i			1.30719i
$928$		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
$933$	12.0000			0.392862
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$			29.8496i			0.974106i
$940$			79.5990i			2.59623i
$941$		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
$947$	−57.0000			−1.85225			−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−57.0000			−1.85225			−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	27.0000			0.875535
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−77.0000			−2.49166
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$			26.5330i			0.856349i
$961$	−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
$971$		−	43.1161i		−	1.38366i	−0.722059	−	0.691831i	$-0.756804\pi$
$971$		−	43.1161i		−	1.38366i	0.722059	−	0.691831i	$-0.243196\pi$
$972$	−32.0000			−1.02640
$973$		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
$979$	−55.0000			−1.75781
$980$			46.4327i			1.48324i
$981$		0			0
$982$		0			0
$983$		−	36.4829i		−	1.16362i	−0.813324	−	0.581811i	$-0.802344\pi$
$983$		−	36.4829i		−	1.16362i	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
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$	−35.0000			−1.11069
$994$		0			0
$995$	66.0000			2.09234
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$			49.7494i			1.57400i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	253.2.b.a.252.1	✓	2
11.10	odd	2	CM	253.2.b.a.252.1	✓	2
23.22	odd	2	inner	253.2.b.a.252.2	yes	2
253.252	even	2	inner	253.2.b.a.252.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.b.a.252.1	✓	2	1.1	even	1	trivial
253.2.b.a.252.1	✓	2	11.10	odd	2	CM
253.2.b.a.252.2	yes	2	23.22	odd	2	inner
253.2.b.a.252.2	yes	2	253.252	even	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	253.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000 q^{4} -3.31662i q^{5} -2.00000 q^{9} -3.31662i q^{11} -2.00000 q^{12} +3.31662i q^{15} +4.00000 q^{16} -6.63325i q^{20} +(4.50000 - 1.65831i) q^{23} -6.00000 q^{25} +5.00000 q^{27} +5.00000 q^{31} +3.31662i q^{33} -4.00000 q^{36} +9.94987i q^{37} -6.63325i q^{44} +6.63325i q^{45} -12.0000 q^{47} -4.00000 q^{48} -7.00000 q^{49} +13.2665i q^{53} -11.0000 q^{55} +15.0000 q^{59} +6.63325i q^{60} +8.00000 q^{64} +9.94987i q^{67} +(-4.50000 + 1.65831i) q^{69} +3.00000 q^{71} +6.00000 q^{75} -13.2665i q^{80} +1.00000 q^{81} -16.5831i q^{89} +(9.00000 - 3.31662i) q^{92} -5.00000 q^{93} +9.94987i q^{97} +6.63325i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{4} - 4 q^{9} - 4 q^{12} + 8 q^{16} + 9 q^{23} - 12 q^{25} + 10 q^{27} + 10 q^{31} - 8 q^{36} - 24 q^{47} - 8 q^{48} - 14 q^{49} - 22 q^{55} + 30 q^{59} + 16 q^{64} - 9 q^{69} + 6 q^{71}+ \cdots - 10 q^{93}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^4 - 4 * q^9 - 4 * q^12 + 8 * q^16 + 9 * q^23 - 12 * q^25 + 10 * q^27 + 10 * q^31 - 8 * q^36 - 24 * q^47 - 8 * q^48 - 14 * q^49 - 22 * q^55 + 30 * q^59 + 16 * q^64 - 9 * q^69 + 6 * q^71 + 12 * q^75 + 2 * q^81 + 18 * q^92 - 10 * q^93

Introduction

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