LMFDB - Embedded newform 2112.2.m.d.1121.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1121,2112)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1121.3
Root			$1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2112.1121
Dual form			2112.2.m.d.1121.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.65831 - 0.500000i) q^{3} +3.00000 q^{5} +(2.50000 - 1.65831i) q^{9} +O(q^{10})$	$q+(1.65831 - 0.500000i) q^{3} +3.00000 q^{5} +(2.50000 - 1.65831i) q^{9} +3.31662 q^{11} +(4.97494 - 1.50000i) q^{15} -9.00000i q^{23} +4.00000 q^{25} +(3.31662 - 4.00000i) q^{27} -9.94987 q^{31} +(5.50000 - 1.65831i) q^{33} +9.94987i q^{37} +(7.50000 - 4.97494i) q^{45} +12.0000i q^{47} +7.00000 q^{49} -6.00000 q^{53} +9.94987 q^{55} -3.31662 q^{59} -13.0000i q^{67} +(-4.50000 - 14.9248i) q^{69} +3.00000i q^{71} +(6.63325 - 2.00000i) q^{75} +(3.50000 - 8.29156i) q^{81} +16.5831i q^{89} +(-16.5000 + 4.97494i) q^{93} +17.0000 q^{97} +(8.29156 - 5.50000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} + 10 q^{9}+O(q^{10}) $ 4 * q + 12 * q^5 + 10 * q^9	$ 4 q + 12 q^{5} + 10 q^{9} + 16 q^{25} + 22 q^{33} + 30 q^{45} + 28 q^{49} - 24 q^{53} - 18 q^{69} + 14 q^{81} - 66 q^{93} + 68 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 + 10 * q^9 + 16 * q^25 + 22 * q^33 + 30 * q^45 + 28 * q^49 - 24 * q^53 - 18 * q^69 + 14 * q^81 - 66 * q^93 + 68 * q^97

Display 100 coefficients 10 coefficients

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
$3$	1.65831	−	0.500000i	0.957427	−	0.288675i
$3$	1.65831	−	0.500000i	0.957427	−	0.288675i
$4$		0			0
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	2.50000	−	1.65831i	0.833333	−	0.552771i
$10$		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
$15$	4.97494	−	1.50000i	1.28452	−	0.387298i
$16$		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
$23$		−	9.00000i		−	1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$		−	9.00000i		−	1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		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
$31$	−9.94987			−1.78705			−0.893525	−	0.449013i	$-0.851776\pi$
$31$	−9.94987			−1.78705			−0.893525	+	0.449013i	$0.851776\pi$
$32$		0			0
$33$	5.50000	−	1.65831i	0.957427	−	0.288675i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			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			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	7.50000	−	4.97494i	1.11803	−	0.741620i
$46$		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
$55$	9.94987			1.34164
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.31662			−0.431788			−0.215894	−	0.976417i	$-0.569267\pi$
$59$	−3.31662			−0.431788			−0.215894	+	0.976417i	$0.569267\pi$
$60$		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
$67$		−	13.0000i		−	1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$		−	13.0000i		−	1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$		0			0
$69$	−4.50000	−	14.9248i	−0.541736	−	1.79674i
$70$		0			0
$71$			3.00000i			0.356034i	0.984027	+	0.178017i	$0.0569683\pi$
$71$			3.00000i			0.356034i	−0.984027	+	0.178017i	$0.943032\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	6.63325	−	2.00000i	0.765942	−	0.230940i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	3.50000	−	8.29156i	0.388889	−	0.921285i
$82$		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
$89$			16.5831i			1.75781i	0.476999	+	0.878904i	$0.341725\pi$
$89$			16.5831i			1.75781i	−0.476999	+	0.878904i	$0.658275\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−16.5000	+	4.97494i	−1.71097	+	0.515877i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	17.0000			1.72609			0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000			1.72609			0.863044	+	0.505128i	$0.168555\pi$
$98$		0			0
$99$	8.29156	−	5.50000i	0.833333	−	0.552771i
$100$		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
$111$	4.97494	+	16.5000i	0.472200	+	1.56611i
$112$		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
$115$		−	27.0000i		−	2.51776i
$116$		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
$125$	−3.00000			−0.268328
$126$		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
$135$	9.94987	−	12.0000i	0.856349	−	1.03280i
$136$		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			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	6.00000	+	19.8997i	0.505291	+	1.67586i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	11.6082	−	3.50000i	0.957427	−	0.288675i
$148$		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
$155$	−29.8496			−2.39758
$156$		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
$159$	−9.94987	+	3.00000i	−0.789076	+	0.237915i
$160$		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
$165$	16.5000	−	4.97494i	1.28452	−	0.387298i
$166$		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
$177$	−5.50000	+	1.65831i	−0.413405	+	0.124646i
$178$		0			0
$179$	−16.5831			−1.23948			−0.619740	−	0.784807i	$-0.712762\pi$
$179$	−16.5831			−1.23948			−0.619740	+	0.784807i	$0.712762\pi$
$180$		0			0
$181$		−	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
$185$			29.8496i			2.19459i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.0000i		−	1.08536i	−0.839939	−	0.542681i	$-0.817409\pi$
$191$		−	15.0000i		−	1.08536i	0.839939	−	0.542681i	$-0.182591\pi$
$192$		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
$201$	−6.50000	−	21.5581i	−0.458475	−	1.52059i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−14.9248	−	22.5000i	−1.03735	−	1.56386i
$208$		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
$213$	1.50000	+	4.97494i	0.102778	+	0.340877i
$214$		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
$223$	29.8496			1.99888			0.999439	−	0.0334825i	$-0.0106598\pi$
$223$	29.8496			1.99888			0.999439	+	0.0334825i	$0.0106598\pi$
$224$		0			0
$225$	10.0000	−	6.63325i	0.666667	−	0.442217i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		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			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			36.0000i			2.34838i
$236$		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
$243$	1.65831	−	15.5000i	0.106381	−	0.994325i
$244$		0			0
$245$	21.0000			1.34164
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.5831			−1.04672			−0.523359	−	0.852112i	$-0.675321\pi$
$251$	−16.5831			−1.04672			−0.523359	+	0.852112i	$0.675321\pi$
$252$		0			0
$253$		−	29.8496i		−	1.87663i
$254$		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
$265$	−18.0000			−1.10573
$266$		0			0
$267$	8.29156	+	27.5000i	0.507435	+	1.68297i
$268$		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
$275$	13.2665			0.800000
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	−24.8747	+	16.5000i	−1.48921	+	0.987829i
$280$		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
$291$	28.1913	−	8.50000i	1.65260	−	0.498279i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	−9.94987			−0.579304
$296$		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
$309$	−33.0000	+	9.94987i	−1.87730	+	0.566029i
$310$		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
$313$	19.0000			1.07394			0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000			1.07394			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
$320$		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
$331$			35.0000i			1.92377i	0.273447	+	0.961887i	$0.411836\pi$
$331$			35.0000i			1.92377i	−0.273447	+	0.961887i	$0.588164\pi$
$332$		0			0
$333$	16.5000	+	24.8747i	0.904194	+	1.36312i
$334$		0			0
$335$		−	39.0000i		−	2.13080i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	1.65831	+	5.50000i	0.0900672	+	0.298719i
$340$		0			0
$341$	−33.0000			−1.78705
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−13.5000	−	44.7744i	−0.726816	−	2.41057i
$346$		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
$355$			9.00000i			0.477670i
$356$		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
$363$	18.2414	−	5.50000i	0.957427	−	0.288675i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	9.94987			0.519379			0.259690	−	0.965692i	$-0.416380\pi$
$367$	9.94987			0.519379			0.259690	+	0.965692i	$0.416380\pi$
$368$		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
$375$	−4.97494	+	1.50000i	−0.256905	+	0.0774597i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			25.0000i			1.28416i	0.766636	+	0.642082i	$0.221929\pi$
$379$			25.0000i			1.28416i	−0.766636	+	0.642082i	$0.778071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			39.0000i			1.99281i	0.0847358	+	0.996403i	$0.472995\pi$
$383$			39.0000i			1.99281i	−0.0847358	+	0.996403i	$0.527005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−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
$400$		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
$405$	10.5000	−	24.8747i	0.521749	−	1.23603i
$406$		0			0
$407$			33.0000i			1.63575i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	11.6082	+	38.5000i	0.572590	+	1.89906i
$412$		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
$423$	19.8997	+	30.0000i	0.967559	+	1.45865i
$424$		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
$433$	−29.0000			−1.39365			−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000			−1.39365			−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		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	17.5000	−	11.6082i	0.833333	−	0.552771i
$442$		0			0
$443$	−36.4829			−1.73335			−0.866677	−	0.498870i	$-0.833748\pi$
$443$	−36.4829			−1.73335			−0.866677	+	0.498870i	$0.833748\pi$
$444$		0			0
$445$			49.7494i			2.35835i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16.5831i			0.782606i	0.920262	+	0.391303i	$0.127976\pi$
$449$			16.5831i			0.782606i	−0.920262	+	0.391303i	$0.872024\pi$
$450$		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
$463$	−29.8496			−1.38723			−0.693615	−	0.720346i	$-0.743983\pi$
$463$	−29.8496			−1.38723			−0.693615	+	0.720346i	$0.743983\pi$
$464$		0			0
$465$	−49.5000	+	14.9248i	−2.29551	+	0.692122i
$466$		0			0
$467$	43.1161			1.99518			0.997588	−	0.0694117i	$-0.0221122\pi$
$467$	43.1161			1.99518			0.997588	+	0.0694117i	$0.0221122\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.97494	−	16.5000i	−0.229233	−	0.760280i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15.0000	+	9.94987i	−0.686803	+	0.455573i
$478$		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
$485$	51.0000			2.31579
$486$		0			0
$487$	−9.94987			−0.450872			−0.225436	−	0.974258i	$-0.572381\pi$
$487$	−9.94987			−0.450872			−0.225436	+	0.974258i	$0.572381\pi$
$488$		0			0
$489$	−8.00000	−	26.5330i	−0.361773	−	1.19986i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	24.8747	−	16.5000i	1.11803	−	0.741620i
$496$		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
$507$	−21.5581	+	6.50000i	−0.957427	+	0.288675i
$508$		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
$515$	−59.6992			−2.63066
$516$		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	$-0.606570\pi$
$521$		−	43.1161i		−	1.88895i	0.328581	−	0.944476i	$-0.393430\pi$
$522$		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
$529$	−58.0000			−2.52174
$530$		0			0
$531$	−8.29156	+	5.50000i	−0.359823	+	0.238680i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−27.5000	+	8.29156i	−1.18671	+	0.357807i
$538$		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
$543$	−4.97494	−	16.5000i	−0.213495	−	0.708083i
$544$		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
$555$	14.9248	+	49.5000i	0.633523	+	2.10116i
$556$		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
$565$			9.94987i			0.418594i
$566$		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
$573$	−7.50000	−	24.8747i	−0.313317	−	1.03915i
$574$		0			0
$575$		−	36.0000i		−	1.50130i
$576$		0			0
$577$	47.0000			1.95664			0.978318	−	0.207109i	$-0.0664056\pi$
$577$	47.0000			1.95664			0.978318	+	0.207109i	$0.0664056\pi$
$578$		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
$597$	33.0000	−	9.94987i	1.35060	−	0.407221i
$598$		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
$603$	−21.5581	−	32.5000i	−0.877912	−	1.32350i
$604$		0			0
$605$	33.0000			1.34164
$606$		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
$619$			1.00000i			0.0401934i	0.999798	+	0.0200967i	$0.00639741\pi$
$619$			1.00000i			0.0401934i	−0.999798	+	0.0200967i	$0.993603\pi$
$620$		0			0
$621$	−36.0000	−	29.8496i	−1.44463	−	1.19782i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	49.7494			1.98049			0.990246	−	0.139333i	$-0.0444958\pi$
$631$	49.7494			1.98049			0.990246	+	0.139333i	$0.0444958\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	4.97494	+	7.50000i	0.196805	+	0.296695i
$640$		0			0
$641$			23.2164i			0.916992i	0.888697	+	0.458496i	$0.151612\pi$
$641$			23.2164i			0.916992i	−0.888697	+	0.458496i	$0.848388\pi$
$642$		0			0
$643$		−	41.0000i		−	1.61688i	−0.588577	−	0.808441i	$-0.700312\pi$
$643$		−	41.0000i		−	1.61688i	0.588577	−	0.808441i	$-0.299688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	27.0000i		−	1.06148i	−0.847535	−	0.530740i	$-0.821914\pi$
$647$		−	27.0000i		−	1.06148i	0.847535	−	0.530740i	$-0.178086\pi$
$648$		0			0
$649$	−11.0000			−0.431788
$650$		0			0
$651$		0			0
$652$		0			0
$653$	51.0000			1.99578			0.997892	−	0.0648948i	$-0.0206712\pi$
$653$	51.0000			1.99578			0.997892	+	0.0648948i	$0.0206712\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			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
$669$	49.5000	−	14.9248i	1.91378	−	0.577027i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	13.2665	−	16.0000i	0.510628	−	0.615840i
$676$		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
$685$			69.6491i			2.66116i
$686$		0			0
$687$	14.9248	+	49.5000i	0.569417	+	1.88854i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	17.0000i		−	0.646710i	−0.946278	−	0.323355i	$-0.895189\pi$
$691$		−	17.0000i		−	0.646710i	0.946278	−	0.323355i	$-0.104811\pi$
$692$		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
$705$	18.0000	+	59.6992i	0.677919	+	2.24840i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	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
$713$			89.5489i			3.35363i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			51.0000i			1.90198i	0.309223	+	0.950990i	$0.399931\pi$
$719$			51.0000i			1.90198i	−0.309223	+	0.950990i	$0.600069\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−9.94987			−0.369020			−0.184510	−	0.982831i	$-0.559070\pi$
$727$	−9.94987			−0.369020			−0.184510	+	0.982831i	$0.559070\pi$
$728$		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
$735$	34.8246	−	10.5000i	1.28452	−	0.387298i
$736$		0			0
$737$		−	43.1161i		−	1.58820i
$738$		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
$751$	−49.7494			−1.81538			−0.907690	−	0.419641i	$-0.862156\pi$
$751$	−49.7494			−1.81538			−0.907690	+	0.419641i	$0.862156\pi$
$752$		0			0
$753$	−27.5000	+	8.29156i	−1.00216	+	0.302161i
$754$		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
$759$	−14.9248	−	49.5000i	−0.541736	−	1.79674i
$760$		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
$771$	−13.2665	−	44.0000i	−0.477781	−	1.58462i
$772$		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
$775$	−39.7995			−1.42964
$776$		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
$784$		0			0
$785$		−	29.8496i		−	1.06538i
$786$		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
$795$	−29.8496	+	9.00000i	−1.05866	+	0.319197i
$796$		0			0
$797$	−3.00000			−0.106265			−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000			−0.106265			−0.0531327	+	0.998587i	$0.516921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	27.5000	+	41.4578i	0.971665	+	1.46484i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	49.7494	−	15.0000i	1.75126	−	0.528025i
$808$		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
$815$		−	48.0000i		−	1.68137i
$816$		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
$823$	29.8496			1.04049			0.520246	−	0.854016i	$-0.325840\pi$
$823$	29.8496			1.04049			0.520246	+	0.854016i	$0.325840\pi$
$824$		0			0
$825$	22.0000	−	6.63325i	0.765942	−	0.230940i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	49.7494i		−	1.72787i	−0.503606	−	0.863934i	$-0.667994\pi$
$829$		−	49.7494i		−	1.72787i	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
$837$	−33.0000	+	39.7995i	−1.14065	+	1.37567i
$838$		0			0
$839$			45.0000i			1.55357i	0.629764	+	0.776786i	$0.283151\pi$
$839$			45.0000i			1.55357i	−0.629764	+	0.776786i	$0.716849\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−39.0000			−1.34164
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	89.5489			3.06970
$852$		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
$859$			31.0000i			1.05771i	0.848713	+	0.528853i	$0.177378\pi$
$859$			31.0000i			1.05771i	−0.848713	+	0.528853i	$0.822622\pi$
$860$		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
$867$	−28.1913	+	8.50000i	−0.957427	+	0.288675i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	42.5000	−	28.1913i	1.43841	−	0.954131i
$874$		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
$881$			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
$883$		−	56.0000i		−	1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$		−	56.0000i		−	1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$		0			0
$885$	−16.5000	+	4.97494i	−0.554641	+	0.167231i
$886$		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
$891$	11.6082	−	27.5000i	0.388889	−	0.921285i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−49.7494			−1.66294
$896$		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
$905$		−	29.8496i		−	0.992235i
$906$		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
$925$			39.7995i			1.30860i
$926$		0			0
$927$	−49.7494	+	33.0000i	−1.63398	+	1.08386i
$928$		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
$933$	6.00000	+	19.8997i	0.196431	+	0.651489i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	31.5079	−	9.50000i	1.02822	−	0.310021i
$940$		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
$947$	−23.2164			−0.754431			−0.377215	−	0.926126i	$-0.623118\pi$
$947$	−23.2164			−0.754431			−0.377215	+	0.926126i	$0.623118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−44.7744	+	13.5000i	−1.45191	+	0.437767i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		−	45.0000i		−	1.45617i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	68.0000			2.19355
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43.1161			1.38366			0.691831	−	0.722059i	$-0.256804\pi$
$971$	43.1161			1.38366			0.691831	+	0.722059i	$0.256804\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	56.3826i		−	1.80384i	−0.431903	−	0.901920i	$-0.642158\pi$
$977$		−	56.3826i		−	1.80384i	0.431903	−	0.901920i	$-0.357842\pi$
$978$		0			0
$979$			55.0000i			1.75781i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	51.0000i		−	1.62665i	−0.581811	−	0.813324i	$-0.697656\pi$
$983$		−	51.0000i		−	1.62665i	0.581811	−	0.813324i	$-0.302344\pi$
$984$		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
$993$	17.5000	+	58.0409i	0.555346	+	1.84187i
$994$		0			0
$995$	59.6992			1.89259
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	39.7995	+	33.0000i	1.25920	+	1.04407i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.m.d.1121.3	yes	4
3.2	odd	2		2112.2.m.a.1121.1	✓	4
4.3	odd	2	inner	2112.2.m.d.1121.2	yes	4
8.3	odd	2		2112.2.m.a.1121.3	yes	4
8.5	even	2		2112.2.m.a.1121.2	yes	4
11.10	odd	2	CM	2112.2.m.d.1121.3	yes	4
12.11	even	2		2112.2.m.a.1121.4	yes	4
24.5	odd	2	inner	2112.2.m.d.1121.4	yes	4
24.11	even	2	inner	2112.2.m.d.1121.1	yes	4
33.32	even	2		2112.2.m.a.1121.1	✓	4
44.43	even	2	inner	2112.2.m.d.1121.2	yes	4
88.21	odd	2		2112.2.m.a.1121.2	yes	4
88.43	even	2		2112.2.m.a.1121.3	yes	4
132.131	odd	2		2112.2.m.a.1121.4	yes	4
264.131	odd	2	inner	2112.2.m.d.1121.1	yes	4
264.197	even	2	inner	2112.2.m.d.1121.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.m.a.1121.1	✓	4	3.2	odd	2
2112.2.m.a.1121.1	✓	4	33.32	even	2
2112.2.m.a.1121.2	yes	4	8.5	even	2
2112.2.m.a.1121.2	yes	4	88.21	odd	2
2112.2.m.a.1121.3	yes	4	8.3	odd	2
2112.2.m.a.1121.3	yes	4	88.43	even	2
2112.2.m.a.1121.4	yes	4	12.11	even	2
2112.2.m.a.1121.4	yes	4	132.131	odd	2
2112.2.m.d.1121.1	yes	4	24.11	even	2	inner
2112.2.m.d.1121.1	yes	4	264.131	odd	2	inner
2112.2.m.d.1121.2	yes	4	4.3	odd	2	inner
2112.2.m.d.1121.2	yes	4	44.43	even	2	inner
2112.2.m.d.1121.3	yes	4	1.1	even	1	trivial
2112.2.m.d.1121.3	yes	4	11.10	odd	2	CM
2112.2.m.d.1121.4	yes	4	24.5	odd	2	inner
2112.2.m.d.1121.4	yes	4	264.197	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.65831 - 0.500000i) q^{3} +3.00000 q^{5} +(2.50000 - 1.65831i) q^{9} +O(q^{10})\)	\(q+(1.65831 - 0.500000i) q^{3} +3.00000 q^{5} +(2.50000 - 1.65831i) q^{9} +3.31662 q^{11} +(4.97494 - 1.50000i) q^{15} -9.00000i q^{23} +4.00000 q^{25} +(3.31662 - 4.00000i) q^{27} -9.94987 q^{31} +(5.50000 - 1.65831i) q^{33} +9.94987i q^{37} +(7.50000 - 4.97494i) q^{45} +12.0000i q^{47} +7.00000 q^{49} -6.00000 q^{53} +9.94987 q^{55} -3.31662 q^{59} -13.0000i q^{67} +(-4.50000 - 14.9248i) q^{69} +3.00000i q^{71} +(6.63325 - 2.00000i) q^{75} +(3.50000 - 8.29156i) q^{81} +16.5831i q^{89} +(-16.5000 + 4.97494i) q^{93} +17.0000 q^{97} +(8.29156 - 5.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} + 10 q^{9}+O(q^{10}) \) 4 * q + 12 * q^5 + 10 * q^9	\( 4 q + 12 q^{5} + 10 q^{9} + 16 q^{25} + 22 q^{33} + 30 q^{45} + 28 q^{49} - 24 q^{53} - 18 q^{69} + 14 q^{81} - 66 q^{93} + 68 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 + 10 * q^9 + 16 * q^25 + 22 * q^33 + 30 * q^45 + 28 * q^49 - 24 * q^53 - 18 * q^69 + 14 * q^81 - 66 * q^93 + 68 * q^97

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