LMFDB - Embedded newform 2112.2.m.f.1121.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,2,Mod(1121,2112)] 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("2112.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-12,0,0,0,-10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,44,0,0,0, 0,0,0,0,-22,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1121.1
Root			$-0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	2112.1121
Dual form			2112.2.m.f.1121.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.26217 - 1.18614i) q^{3} -4.37228 q^{5} +(0.186141 + 2.99422i) q^{9} +3.31662 q^{11} +(5.51856 + 5.18614i) q^{15} +1.62772i q^{23} +14.1168 q^{25} +(3.31662 - 4.00000i) q^{27} +0.644810 q^{31} +(-4.18614 - 3.93398i) q^{33} -11.0371i q^{37} +(-0.813859 - 13.0916i) q^{45} +12.0000i q^{47} +7.00000 q^{49} -6.00000 q^{53} -14.5012 q^{55} -11.3321 q^{59} +15.1168i q^{67} +(1.93070 - 2.05446i) q^{69} -15.8614i q^{71} +(-17.8178 - 16.7446i) q^{75} +(-8.93070 + 1.11469i) q^{81} -16.0858i q^{89} +(-0.813859 - 0.764836i) q^{93} -17.1168 q^{97} +(0.617359 + 9.93070i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{5} - 10 q^{9} + 44 q^{25} - 22 q^{33} - 18 q^{45} + 56 q^{49} - 48 q^{53} - 42 q^{69} - 14 q^{81} - 18 q^{93} - 68 q^{97}+O(q^{100}) $ 8 * q - 12 * q^5 - 10 * q^9 + 44 * q^25 - 22 * q^33 - 18 * q^45 + 56 * q^49 - 48 * q^53 - 42 * q^69 - 14 * q^81 - 18 * q^93 - 68 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$2$		0			0
$3$	−1.26217	−	1.18614i	−0.728714	−	0.684819i
$3$	−1.26217	−	1.18614i	−0.728714	−	0.684819i
$4$		0			0
$5$	−4.37228			−1.95534			−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−4.37228			−1.95534			−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	0.186141	+	2.99422i	0.0620469	+	0.998073i
$10$		0			0
$11$	3.31662			1.00000
$11$	3.31662			1.00000
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	5.51856	+	5.18614i	1.42489	+	1.33906i
$16$		0			0
$17$		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	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.62772i			0.339403i	0.985496	+	0.169701i	$0.0542803\pi$
$23$			1.62772i			0.339403i	−0.985496	+	0.169701i	$0.945720\pi$
$24$		0			0
$25$	14.1168			2.82337
$26$		0			0
$27$	3.31662	−	4.00000i	0.638285	−	0.769800i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	0.644810			0.115811			0.0579057	−	0.998322i	$-0.481558\pi$
$31$	0.644810			0.115811			0.0579057	+	0.998322i	$0.481558\pi$
$32$		0			0
$33$	−4.18614	−	3.93398i	−0.728714	−	0.684819i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	11.0371i		−	1.81449i	−0.420602	−	0.907245i	$-0.638181\pi$
$37$		−	11.0371i		−	1.81449i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		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			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−0.813859	−	13.0916i	−0.121323	−	1.95158i
$46$		0			0
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−14.5012			−1.95534
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.3321			−1.47531			−0.737655	−	0.675178i	$-0.764067\pi$
$59$	−11.3321			−1.47531			−0.737655	+	0.675178i	$0.764067\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			15.1168i			1.84682i	0.383819	+	0.923408i	$0.374609\pi$
$67$			15.1168i			1.84682i	−0.383819	+	0.923408i	$0.625391\pi$
$68$		0			0
$69$	1.93070	−	2.05446i	0.232429	−	0.247327i
$70$		0			0
$71$		−	15.8614i		−	1.88240i	−0.337846	−	0.941201i	$-0.609698\pi$
$71$		−	15.8614i		−	1.88240i	0.337846	−	0.941201i	$-0.390302\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−17.8178	−	16.7446i	−2.05743	−	1.93350i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		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
$89$		−	16.0858i		−	1.70509i	−0.522654	−	0.852545i	$-0.675058\pi$
$89$		−	16.0858i		−	1.70509i	0.522654	−	0.852545i	$-0.324942\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.813859	−	0.764836i	−0.0843933	−	0.0793098i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−17.1168			−1.73795			−0.868976	−	0.494854i	$-0.835222\pi$
$97$	−17.1168			−1.73795			−0.868976	+	0.494854i	$0.835222\pi$
$98$		0			0
$99$	0.617359	+	9.93070i	0.0620469	+	0.998073i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−19.8997			−1.96078			−0.980390	−	0.197066i	$-0.936859\pi$
$103$	−19.8997			−1.96078			−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−13.0916	+	13.9307i	−1.24260	+	1.32224i
$112$		0			0
$113$		−	19.8448i		−	1.86685i	−0.358778	−	0.933423i	$-0.616806\pi$
$113$		−	19.8448i		−	1.86685i	0.358778	−	0.933423i	$-0.383194\pi$
$114$		0			0
$115$		−	7.11684i		−	0.663649i
$116$		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
$124$		0			0
$125$	−39.8614			−3.56531
$126$		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			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−14.5012	+	17.4891i	−1.24807	+	1.50522i
$136$		0			0
$137$		−	14.2063i		−	1.21372i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$		−	14.2063i		−	1.21372i	0.794808	−	0.606861i	$-0.207572\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	14.2337	−	15.1460i	1.19869	−	1.27553i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.83518	−	8.30298i	−0.728714	−	0.684819i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.81929			−0.226451
$156$		0			0
$157$			24.8935i			1.98672i	0.115050	+	0.993360i	$0.463297\pi$
$157$			24.8935i			1.98672i	−0.115050	+	0.993360i	$0.536703\pi$
$158$		0			0
$159$	7.57301	+	7.11684i	0.600579	+	0.564402i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$		0			0
$165$	18.3030	+	17.2005i	1.42489	+	1.33906i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		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
$177$	14.3030	+	13.4414i	1.07508	+	1.01032i
$178$		0			0
$179$	26.4781			1.97907			0.989533	−	0.144308i	$-0.0460955\pi$
$179$	26.4781			1.97907			0.989533	+	0.144308i	$0.0460955\pi$
$180$		0			0
$181$		−	16.6757i		−	1.23949i	−0.784801	−	0.619747i	$-0.787235\pi$
$181$		−	16.6757i		−	1.23949i	0.784801	−	0.619747i	$-0.212765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			48.2574i			3.54795i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.6060i		−	0.912136i	−0.889945	−	0.456068i	$-0.849257\pi$
$191$		−	12.6060i		−	0.912136i	0.889945	−	0.456068i	$-0.150743\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	19.8997			1.41066			0.705328	−	0.708881i	$-0.250800\pi$
$199$	19.8997			1.41066			0.705328	+	0.708881i	$0.250800\pi$
$200$		0			0
$201$	17.9307	−	19.0800i	1.26473	−	1.34580i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−4.87375	+	0.302985i	−0.338749	+	0.0210589i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$	−18.8139	+	20.0198i	−1.28910	+	1.37173i
$214$		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
$223$	−15.7908			−1.05743			−0.528716	−	0.848799i	$-0.677326\pi$
$223$	−15.7908			−1.05743			−0.528716	+	0.848799i	$0.677326\pi$
$224$		0			0
$225$	2.62772	+	42.2689i	0.175181	+	2.81793i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		−	19.2549i		−	1.27240i	−0.771523	−	0.636201i	$-0.780505\pi$
$229$		−	19.2549i		−	1.27240i	0.771523	−	0.636201i	$-0.219495\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		−	52.4674i		−	3.42259i
$236$		0			0
$237$		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
$243$	12.5942	+	9.18614i	0.807921	+	0.589291i
$244$		0			0
$245$	−30.6060			−1.95534
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.0911			−0.952543			−0.476272	−	0.879298i	$-0.658012\pi$
$251$	−15.0911			−0.952543			−0.476272	+	0.879298i	$0.658012\pi$
$252$		0			0
$253$			5.39853i			0.339403i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	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			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	26.2337			1.61152
$266$		0			0
$267$	−19.0800	+	20.3030i	−1.16768	+	1.24252i
$268$		0			0
$269$	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		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	46.8203			2.82337
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	0.120025	+	1.93070i	0.00718573	+	0.115588i
$280$		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	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	21.6043	+	20.3030i	1.26647	+	1.19018i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	49.5470			2.88474
$296$		0			0
$297$	11.0000	−	13.2665i	0.638285	−	0.769800i
$298$		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			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	25.1168	+	23.6039i	1.42885	+	1.34278i
$310$		0			0
$311$			12.0000i			0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$			12.0000i			0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$		0			0
$313$	−35.3505			−1.99813			−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−35.3505			−1.99813			−0.999065	+	0.0432311i	$0.986235\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.60597			−0.371028			−0.185514	−	0.982642i	$-0.559395\pi$
$317$	−6.60597			−0.371028			−0.185514	+	0.982642i	$0.559395\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	8.88316i		−	0.488262i	−0.969742	−	0.244131i	$-0.921497\pi$
$331$		−	8.88316i		−	0.488262i	0.969742	−	0.244131i	$-0.0785028\pi$
$332$		0			0
$333$	33.0475	−	2.05446i	1.81099	−	0.112583i
$334$		0			0
$335$		−	66.0951i		−	3.61116i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−23.5388	+	25.0475i	−1.27845	+	1.36040i
$340$		0			0
$341$	2.13859			0.115811
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.44158	+	8.98266i	−0.454479	+	0.483610i
$346$		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
$353$		−	10.4472i		−	0.556049i	−0.960574	−	0.278024i	$-0.910320\pi$
$353$		−	10.4472i		−	0.556049i	0.960574	−	0.278024i	$-0.0896796\pi$
$354$		0			0
$355$			69.3505i			3.68074i
$356$		0			0
$357$		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
$363$	−13.8839	−	13.0475i	−0.728714	−	0.684819i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	27.0680			1.41294			0.706469	−	0.707744i	$-0.250287\pi$
$367$	27.0680			1.41294			0.706469	+	0.707744i	$0.250287\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	50.3118	+	47.2812i	2.59809	+	2.44159i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			13.3505i			0.685771i	0.939377	+	0.342885i	$0.111404\pi$
$379$			13.3505i			0.685771i	−0.939377	+	0.342885i	$0.888596\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	22.3723i		−	1.14317i	−0.820543	−	0.571585i	$-0.806329\pi$
$383$		−	22.3723i		−	1.14317i	0.820543	−	0.571585i	$-0.193671\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.0951			−1.22167			−0.610835	−	0.791758i	$-0.709166\pi$
$389$	−24.0951			−1.22167			−0.610835	+	0.791758i	$0.709166\pi$
$390$		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
$400$		0			0
$401$		−	26.5330i		−	1.32499i	−0.749064	−	0.662497i	$-0.769497\pi$
$401$		−	26.5330i		−	1.32499i	0.749064	−	0.662497i	$-0.230503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	39.0475	−	4.87375i	1.94029	−	0.242178i
$406$		0			0
$407$		−	36.6060i		−	1.81449i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−16.8506	+	17.9307i	−0.831180	+	0.884456i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−33.1662			−1.62028			−0.810139	−	0.586238i	$-0.800608\pi$
$419$	−33.1662			−1.62028			−0.810139	+	0.586238i	$0.800608\pi$
$420$		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
$423$	−35.9306	+	2.23369i	−1.74701	+	0.108606i
$424$		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	$0.5\pi$
$432$		0			0
$433$	−11.3505			−0.545472			−0.272736	−	0.962089i	$-0.587929\pi$
$433$	−11.3505			−0.545472			−0.272736	+	0.962089i	$0.587929\pi$
$434$		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
$441$	1.30298	+	20.9595i	0.0620469	+	0.998073i
$442$		0			0
$443$	0.0549029			0.00260851			0.00130426	−	0.999999i	$-0.499585\pi$
$443$	0.0549029			0.00260851			0.00130426	+	0.999999i	$0.499585\pi$
$444$		0			0
$445$			70.3316i			3.33404i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.4834i			1.20264i	0.799009	+	0.601319i	$0.205358\pi$
$449$			25.4834i			1.20264i	−0.799009	+	0.601319i	$0.794642\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		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
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−11.9220			−0.554061			−0.277031	−	0.960861i	$-0.589350\pi$
$463$	−11.9220			−0.554061			−0.277031	+	0.960861i	$0.589350\pi$
$464$		0			0
$465$	3.55842	+	3.34408i	0.165018	+	0.155078i
$466$		0			0
$467$	−18.9600			−0.877363			−0.438682	−	0.898642i	$-0.644554\pi$
$467$	−18.9600			−0.877363			−0.438682	+	0.898642i	$0.644554\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	29.5272	−	31.4198i	1.36054	−	1.44775i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.11684	−	17.9653i	−0.0511368	−	0.822575i
$478$		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
$485$	74.8397			3.39829
$486$		0			0
$487$	42.2140			1.91290			0.956450	−	0.291896i	$-0.0942860\pi$
$487$	42.2140			1.91290			0.956450	+	0.291896i	$0.0942860\pi$
$488$		0			0
$489$	−18.9783	+	20.1947i	−0.858226	+	0.913236i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−2.69927	−	43.4198i	−0.121323	−	1.95158i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			40.0000i			1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$			40.0000i			1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$		0			0
$501$		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
$507$	16.4082	+	15.4198i	0.728714	+	0.684819i
$508$		0			0
$509$	19.6277			0.869983			0.434992	−	0.900434i	$-0.356751\pi$
$509$	19.6277			0.869983			0.434992	+	0.900434i	$0.356751\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	87.0073			3.83400
$516$		0			0
$517$			39.7995i			1.75038i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			8.56768i			0.375357i	0.982231	+	0.187678i	$0.0600963\pi$
$521$			8.56768i			0.375357i	−0.982231	+	0.187678i	$0.939904\pi$
$522$		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
$528$		0			0
$529$	20.3505			0.884806
$530$		0			0
$531$	−2.10936	−	33.9307i	−0.0915384	−	1.47247i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−33.4198	−	31.4067i	−1.44217	−	1.35530i
$538$		0			0
$539$	23.2164			1.00000
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	−19.7797	+	21.0475i	−0.848829	+	0.903237i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	57.2400	−	60.9090i	2.42970	−	2.58544i
$556$		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			$\pi$
$564$		0			0
$565$			86.7672i			3.65033i
$566$		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
$573$	−14.9525	+	15.9109i	−0.624648	+	0.664686i
$574$		0			0
$575$			22.9783i			0.958259i
$576$		0			0
$577$	−14.8832			−0.619594			−0.309797	−	0.950803i	$-0.600261\pi$
$577$	−14.8832			−0.619594			−0.309797	+	0.950803i	$0.600261\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−19.8997			−0.824163
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.63325			0.273784			0.136892	−	0.990586i	$-0.456289\pi$
$587$	6.63325			0.273784			0.136892	+	0.990586i	$0.456289\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−25.1168	−	23.6039i	−1.02796	−	0.966043i
$598$		0			0
$599$		−	36.0000i		−	1.47092i	−0.677568	−	0.735460i	$-0.736966\pi$
$599$		−	36.0000i		−	1.47092i	0.677568	−	0.735460i	$-0.263034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−45.2632	+	2.81386i	−1.84326	+	0.114589i
$604$		0			0
$605$	−48.0951			−1.95534
$606$		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
$619$		−	43.5842i		−	1.75180i	−0.482495	−	0.875899i	$-0.660269\pi$
$619$		−	43.5842i		−	1.75180i	0.482495	−	0.875899i	$-0.339731\pi$
$620$		0			0
$621$	6.51087	+	5.39853i	0.261272	+	0.216636i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	103.701			4.14804
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−30.9369			−1.23158			−0.615789	−	0.787911i	$-0.711162\pi$
$631$	−30.9369			−1.23158			−0.615789	+	0.787911i	$0.711162\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	47.4925	−	2.95245i	1.87878	−	0.116797i
$640$		0			0
$641$			27.3630i			1.08077i	0.841417	+	0.540386i	$0.181722\pi$
$641$			27.3630i			1.08077i	−0.841417	+	0.540386i	$0.818278\pi$
$642$		0			0
$643$		−	5.35053i		−	0.211004i	−0.994419	−	0.105502i	$-0.966355\pi$
$643$		−	5.35053i		−	0.211004i	0.994419	−	0.105502i	$-0.0336450\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			50.8397i			1.99871i	0.0358667	+	0.999357i	$0.488581\pi$
$647$			50.8397i			1.99871i	−0.0358667	+	0.999357i	$0.511419\pi$
$648$		0			0
$649$	−37.5842			−1.47531
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.3723			−1.11029			−0.555147	−	0.831753i	$-0.687338\pi$
$653$	−28.3723			−1.11029			−0.555147	+	0.831753i	$0.687338\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	13.6164i		−	0.529615i	−0.964301	−	0.264807i	$-0.914692\pi$
$661$		−	13.6164i		−	0.529615i	0.964301	−	0.264807i	$-0.0853084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	19.9307	+	18.7302i	0.770566	+	0.724150i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	46.8203	−	56.4674i	1.80211	−	2.17343i
$676$		0			0
$677$		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
$683$	−46.4327			−1.77670			−0.888350	−	0.459167i	$-0.848148\pi$
$683$	−46.4327			−1.77670			−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$			62.1138i			2.37325i
$686$		0			0
$687$	−22.8391	+	24.3030i	−0.871365	+	0.927217i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			51.5842i			1.96236i	0.193105	+	0.981178i	$0.438144\pi$
$691$			51.5842i			1.96236i	−0.193105	+	0.981178i	$0.561856\pi$
$692$		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
$704$		0			0
$705$	−62.2337	+	66.2227i	−2.34386	+	2.49409i
$706$		0			0
$707$		0			0
$708$		0			0
$709$			41.3292i			1.55215i	0.630641	+	0.776075i	$0.282792\pi$
$709$			41.3292i			1.55215i	−0.630641	+	0.776075i	$0.717208\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.04957i			0.0393067i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	39.8614i		−	1.48658i	−0.668970	−	0.743290i	$-0.733264\pi$
$719$		−	39.8614i		−	1.48658i	0.668970	−	0.743290i	$-0.266736\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−40.9244			−1.51780			−0.758901	−	0.651206i	$-0.774263\pi$
$727$	−40.9244			−1.51780			−0.758901	+	0.651206i	$0.774263\pi$
$728$		0			0
$729$	−5.00000	−	26.5330i	−0.185185	−	0.982704i
$730$		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
$735$	38.6299	+	36.3030i	1.42489	+	1.33906i
$736$		0			0
$737$			50.1369i			1.84682i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		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
$751$	44.7933			1.63453			0.817265	−	0.576262i	$-0.195489\pi$
$751$	44.7933			1.63453			0.817265	+	0.576262i	$0.195489\pi$
$752$		0			0
$753$	19.0475	+	17.9002i	0.694131	+	0.652319i
$754$		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
$759$	6.40342	−	6.81386i	0.232429	−	0.247327i
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−31.4719	+	33.4891i	−1.13343	+	1.20608i
$772$		0			0
$773$	−54.0000			−1.94225			−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000			−1.94225			−0.971123	+	0.238581i	$0.923318\pi$
$774$		0			0
$775$	9.10268			0.326978
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	52.6063i		−	1.88240i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	108.841i		−	3.88472i
$786$		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
$795$	−33.1113	−	31.1168i	−1.17434	−	1.10360i
$796$		0			0
$797$	50.3288			1.78274			0.891368	−	0.453279i	$-0.149746\pi$
$797$	50.3288			1.78274			0.891368	+	0.453279i	$0.149746\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	48.1644	−	2.99422i	1.70181	−	0.105796i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−37.8651	−	35.5842i	−1.33291	−	1.25262i
$808$		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
$815$			69.9565i			2.45047i
$816$		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
$823$	−57.3601			−1.99945			−0.999723	−	0.0235383i	$-0.992507\pi$
$823$	−57.3601			−1.99945			−0.999723	+	0.0235383i	$0.992507\pi$
$824$		0			0
$825$	−59.0951	−	55.5354i	−2.05743	−	1.93350i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	0.240051i		−	0.00833731i	−0.999991	−	0.00416865i	$-0.998673\pi$
$829$		−	0.240051i		−	0.00833731i	0.999991	−	0.00416865i	$-0.00132693\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.13859	−	2.57924i	0.0739206	−	0.0891516i
$838$		0			0
$839$		−	54.0951i		−	1.86757i	−0.357834	−	0.933785i	$-0.616485\pi$
$839$		−	54.0951i		−	1.86757i	0.357834	−	0.933785i	$-0.383515\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	56.8397			1.95534
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	17.9653			0.615843
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			27.5842i			0.941161i	0.882357	+	0.470581i	$0.155956\pi$
$859$			27.5842i			0.941161i	−0.882357	+	0.470581i	$0.844044\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	21.4569	+	20.1644i	0.728714	+	0.684819i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−3.18614	−	51.2516i	−0.107835	−	1.73460i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	57.6550i		−	1.94245i	−0.238171	−	0.971223i	$-0.576548\pi$
$881$		−	57.6550i		−	1.94245i	0.238171	−	0.971223i	$-0.423452\pi$
$882$		0			0
$883$		−	56.0000i		−	1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$		−	56.0000i		−	1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$		0			0
$885$	−62.5367	−	58.7697i	−2.10215	−	1.97552i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−29.6198	+	3.69702i	−0.992300	+	0.123855i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−115.770			−3.86975
$896$		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
$905$			72.9108i			2.42364i
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			60.0000i			1.98789i	0.109885	+	0.993944i	$0.464952\pi$
$911$			60.0000i			1.98789i	−0.109885	+	0.993944i	$0.535048\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	155.809i		−	5.12298i
$926$		0			0
$927$	−3.70415	−	59.5842i	−0.121660	−	1.95700i
$928$		0			0
$929$		−	53.0660i		−	1.74104i	−0.492134	−	0.870519i	$-0.663783\pi$
$929$		−	53.0660i		−	1.74104i	0.492134	−	0.870519i	$-0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	14.2337	−	15.1460i	0.465990	−	0.495859i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	44.6183	+	41.9307i	1.45606	+	1.36836i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−37.7553			−1.22688			−0.613441	−	0.789741i	$-0.710215\pi$
$947$	−37.7553			−1.22688			−0.613441	+	0.789741i	$0.710215\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	8.33785	+	7.83561i	0.270373	+	0.254087i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$			55.1168i			1.78354i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−30.5842			−0.986588
$962$		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
$971$	−60.5292			−1.94247			−0.971237	−	0.238114i	$-0.923471\pi$
$971$	−60.5292			−1.94247			−0.971237	+	0.238114i	$0.923471\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.80862i			0.153842i	0.997037	+	0.0769208i	$0.0245089\pi$
$977$			4.80862i			0.153842i	−0.997037	+	0.0769208i	$0.975491\pi$
$978$		0			0
$979$		−	53.3505i		−	1.70509i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	6.09509i		−	0.194403i	−0.995265	−	0.0972017i	$-0.969011\pi$
$983$		−	6.09509i		−	0.194403i	0.995265	−	0.0972017i	$-0.0309892\pi$
$984$		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	$-0.102897\pi$
$991$	59.6992			1.89641			0.948205	+	0.317660i	$0.102897\pi$
$992$		0			0
$993$	−10.5367	+	11.2120i	−0.334371	+	0.355803i
$994$		0			0
$995$	−87.0073			−2.75832
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	−44.1485	−	36.6060i	−1.39680	−	1.15816i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.m.f.1121.1	✓	8
3.2	odd	2		2112.2.m.k.1121.7	yes	8
4.3	odd	2	inner	2112.2.m.f.1121.8	yes	8
8.3	odd	2		2112.2.m.k.1121.1	yes	8
8.5	even	2		2112.2.m.k.1121.8	yes	8
11.10	odd	2	CM	2112.2.m.f.1121.1	✓	8
12.11	even	2		2112.2.m.k.1121.2	yes	8
24.5	odd	2	inner	2112.2.m.f.1121.2	yes	8
24.11	even	2	inner	2112.2.m.f.1121.7	yes	8
33.32	even	2		2112.2.m.k.1121.7	yes	8
44.43	even	2	inner	2112.2.m.f.1121.8	yes	8
88.21	odd	2		2112.2.m.k.1121.8	yes	8
88.43	even	2		2112.2.m.k.1121.1	yes	8
132.131	odd	2		2112.2.m.k.1121.2	yes	8
264.131	odd	2	inner	2112.2.m.f.1121.7	yes	8
264.197	even	2	inner	2112.2.m.f.1121.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.m.f.1121.1	✓	8	1.1	even	1	trivial
2112.2.m.f.1121.1	✓	8	11.10	odd	2	CM
2112.2.m.f.1121.2	yes	8	24.5	odd	2	inner
2112.2.m.f.1121.2	yes	8	264.197	even	2	inner
2112.2.m.f.1121.7	yes	8	24.11	even	2	inner
2112.2.m.f.1121.7	yes	8	264.131	odd	2	inner
2112.2.m.f.1121.8	yes	8	4.3	odd	2	inner
2112.2.m.f.1121.8	yes	8	44.43	even	2	inner
2112.2.m.k.1121.1	yes	8	8.3	odd	2
2112.2.m.k.1121.1	yes	8	88.43	even	2
2112.2.m.k.1121.2	yes	8	12.11	even	2
2112.2.m.k.1121.2	yes	8	132.131	odd	2
2112.2.m.k.1121.7	yes	8	3.2	odd	2
2112.2.m.k.1121.7	yes	8	33.32	even	2
2112.2.m.k.1121.8	yes	8	8.5	even	2
2112.2.m.k.1121.8	yes	8	88.21	odd	2

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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.26217 - 1.18614i) q^{3} -4.37228 q^{5} +(0.186141 + 2.99422i) q^{9} +3.31662 q^{11} +(5.51856 + 5.18614i) q^{15} +1.62772i q^{23} +14.1168 q^{25} +(3.31662 - 4.00000i) q^{27} +0.644810 q^{31} +(-4.18614 - 3.93398i) q^{33} -11.0371i q^{37} +(-0.813859 - 13.0916i) q^{45} +12.0000i q^{47} +7.00000 q^{49} -6.00000 q^{53} -14.5012 q^{55} -11.3321 q^{59} +15.1168i q^{67} +(1.93070 - 2.05446i) q^{69} -15.8614i q^{71} +(-17.8178 - 16.7446i) q^{75} +(-8.93070 + 1.11469i) q^{81} -16.0858i q^{89} +(-0.813859 - 0.764836i) q^{93} -17.1168 q^{97} +(0.617359 + 9.93070i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{5} - 10 q^{9} + 44 q^{25} - 22 q^{33} - 18 q^{45} + 56 q^{49} - 48 q^{53} - 42 q^{69} - 14 q^{81} - 18 q^{93} - 68 q^{97}+O(q^{100}) \) 8 * q - 12 * q^5 - 10 * q^9 + 44 * q^25 - 22 * q^33 - 18 * q^45 + 56 * q^49 - 48 * q^53 - 42 * q^69 - 14 * q^81 - 18 * q^93 - 68 * q^97

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