LMFDB - Embedded newform 2112.2.b.f.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(65,2112)]

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, 0, 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("2112.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.b (of order $2$, degree $1$, not 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:	$2$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			65.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	2112.65
Dual form			2112.2.b.f.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} +O(q^{10})$	$q+(0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} -3.31662i q^{11} +(-5.50000 - 1.65831i) q^{15} -3.31662i q^{23} -6.00000 q^{25} +(-4.00000 + 3.31662i) q^{27} -5.00000 q^{31} +(-5.50000 - 1.65831i) q^{33} +7.00000 q^{37} +(-5.50000 + 8.29156i) q^{45} +6.63325i q^{47} +7.00000 q^{49} -13.2665i q^{53} -11.0000 q^{55} +3.31662i q^{59} -13.0000 q^{67} +(-5.50000 - 1.65831i) q^{69} +16.5831i q^{71} +(-3.00000 + 9.94987i) q^{75} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} +(-2.50000 + 8.29156i) q^{93} +17.0000 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 5 q^{9}+O(q^{10}) $ 2 * q + q^3 - 5 * q^9	$ 2 q + q^{3} - 5 q^{9} - 11 q^{15} - 12 q^{25} - 8 q^{27} - 10 q^{31} - 11 q^{33} + 14 q^{37} - 11 q^{45} + 14 q^{49} - 22 q^{55} - 26 q^{67} - 11 q^{69} - 6 q^{75} + 7 q^{81} - 5 q^{93} + 34 q^{97} - 11 q^{99}+O(q^{100}) $ 2 * q + q^3 - 5 * q^9 - 11 * q^15 - 12 * q^25 - 8 * q^27 - 10 * q^31 - 11 * q^33 + 14 * q^37 - 11 * q^45 + 14 * q^49 - 22 * q^55 - 26 * q^67 - 11 * q^69 - 6 * q^75 + 7 * q^81 - 5 * q^93 + 34 * q^97 - 11 * q^99

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
$3$	0.500000	−	1.65831i	0.288675	−	0.957427i
$3$	0.500000	−	1.65831i	0.288675	−	0.957427i
$4$		0			0
$5$		−	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		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.50000	−	1.65831i	−0.833333	−	0.552771i
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−5.50000	−	1.65831i	−1.42009	−	0.428174i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.31662i		−	0.691564i	−0.938315	−	0.345782i	$-0.887614\pi$
$23$		−	3.31662i		−	0.691564i	0.938315	−	0.345782i	$-0.112386\pi$
$24$		0			0
$25$	−6.00000			−1.20000
$26$		0			0
$27$	−4.00000	+	3.31662i	−0.769800	+	0.638285i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$	−5.50000	−	1.65831i	−0.957427	−	0.288675i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−5.50000	+	8.29156i	−0.819892	+	1.23603i
$46$		0			0
$47$			6.63325i			0.967559i	0.875190	+	0.483779i	$0.160736\pi$
$47$			6.63325i			0.967559i	−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.2665i		−	1.82229i	−0.412082	−	0.911147i	$-0.635198\pi$
$53$		−	13.2665i		−	1.82229i	0.412082	−	0.911147i	$-0.364802\pi$
$54$		0			0
$55$	−11.0000			−1.48324
$56$		0			0
$57$		0			0
$58$		0			0
$59$			3.31662i			0.431788i	0.976417	+	0.215894i	$0.0692665\pi$
$59$			3.31662i			0.431788i	−0.976417	+	0.215894i	$0.930733\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−13.0000			−1.58820			−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000			−1.58820			−0.794101	+	0.607785i	$0.792058\pi$
$68$		0			0
$69$	−5.50000	−	1.65831i	−0.662122	−	0.199637i
$70$		0			0
$71$			16.5831i			1.96805i	0.178017	+	0.984027i	$0.443032\pi$
$71$			16.5831i			1.96805i	−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−3.00000	+	9.94987i	−0.346410	+	1.14891i
$76$		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			−	1.00000i	$-0.5\pi$
$83$		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
$92$		0			0
$93$	−2.50000	+	8.29156i	−0.259238	+	0.859795i
$94$		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
$99$	−5.50000	+	8.29156i	−0.552771	+	0.833333i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			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	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	3.50000	−	11.6082i	0.332205	−	1.10180i
$112$		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
$115$	−11.0000			−1.02576
$116$		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
$125$			3.31662i			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			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	11.0000	+	13.2665i	0.946729	+	1.14180i
$136$		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
$141$	11.0000	+	3.31662i	0.926367	+	0.279310i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.50000	−	11.6082i	0.288675	−	0.957427i
$148$		0			0
$149$		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
$157$	−23.0000			−1.83560			−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000			−1.83560			−0.917800	+	0.397043i	$0.870036\pi$
$158$		0			0
$159$	−22.0000	−	6.63325i	−1.74471	−	0.526051i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$	−5.50000	+	18.2414i	−0.428174	+	1.42009i
$166$		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			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.50000	+	1.65831i	0.413405	+	0.124646i
$178$		0			0
$179$		−	16.5831i		−	1.23948i	−0.784807	−	0.619740i	$-0.787238\pi$
$179$		−	16.5831i		−	1.23948i	0.784807	−	0.619740i	$-0.212762\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	23.2164i		−	1.70690i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		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	$-0.317409\pi$
$192$		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			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−6.50000	+	21.5581i	−0.458475	+	1.52059i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.50000	+	8.29156i	−0.382276	+	0.576303i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	27.5000	+	8.29156i	1.88427	+	0.568128i
$214$		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
$223$	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
$225$	15.0000	+	9.94987i	1.00000	+	0.663325i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−5.00000			−0.330409			−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000			−0.330409			−0.165205	+	0.986259i	$0.552828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	22.0000			1.43512
$236$		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
$243$	15.5000	−	1.65831i	0.994325	−	0.106381i
$244$		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
$253$	−11.0000			−0.691564
$254$		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
$265$	−44.0000			−2.70290
$266$		0			0
$267$	−27.5000	−	8.29156i	−1.68297	−	0.507435i
$268$		0			0
$269$		−	13.2665i		−	0.808873i	−0.914566	−	0.404436i	$-0.867468\pi$
$269$		−	13.2665i		−	0.808873i	0.914566	−	0.404436i	$-0.132532\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			19.8997i			1.20000i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	12.5000	+	8.29156i	0.748355	+	0.496403i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	8.50000	−	28.1913i	0.498279	−	1.65260i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	11.0000			0.640445
$296$		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		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	2.00000	−	6.63325i	0.113776	−	0.377352i
$310$		0			0
$311$		−	33.1662i		−	1.88069i	−0.340229	−	0.940343i	$-0.610505\pi$
$311$		−	33.1662i		−	1.88069i	0.340229	−	0.940343i	$-0.389495\pi$
$312$		0			0
$313$	−19.0000			−1.07394			−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000			−1.07394			−0.536972	+	0.843600i	$0.680432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	23.2164i		−	1.30396i	−0.758236	−	0.651981i	$-0.773938\pi$
$317$		−	23.2164i		−	1.30396i	0.758236	−	0.651981i	$-0.226062\pi$
$318$		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
$331$	35.0000			1.92377			0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000			1.92377			0.961887	+	0.273447i	$0.0881639\pi$
$332$		0			0
$333$	−17.5000	−	11.6082i	−0.958994	−	0.636125i
$334$		0			0
$335$			43.1161i			2.35569i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	5.50000	+	1.65831i	0.298719	+	0.0900672i
$340$		0			0
$341$			16.5831i			0.898027i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−5.50000	+	18.2414i	−0.296110	+	0.982086i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	36.4829i		−	1.94179i	−0.239511	−	0.970894i	$-0.576987\pi$
$353$		−	36.4829i		−	1.94179i	0.239511	−	0.970894i	$-0.423013\pi$
$354$		0			0
$355$	55.0000			2.91910
$356$		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
$363$	−5.50000	+	18.2414i	−0.288675	+	0.957427i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	37.0000			1.93138			0.965692	−	0.259690i	$-0.0836203\pi$
$367$	37.0000			1.93138			0.965692	+	0.259690i	$0.0836203\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	5.50000	+	1.65831i	0.284019	+	0.0856349i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−25.0000			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.31662i		−	0.169472i	−0.996403	−	0.0847358i	$-0.972995\pi$
$383$		−	3.31662i		−	0.169472i	0.996403	−	0.0847358i	$-0.0270046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			36.4829i			1.84976i	0.380265	+	0.924878i	$0.375833\pi$
$389$			36.4829i			1.84976i	−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		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
$405$	27.5000	−	11.6082i	1.36649	−	0.576815i
$406$		0			0
$407$		−	23.2164i		−	1.15079i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	38.5000	+	11.6082i	1.89906	+	0.572590i
$412$		0			0
$413$		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
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	11.0000	−	16.5831i	0.534838	−	0.806299i
$424$		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	$-0.254595\pi$
$433$	29.0000			1.39365			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
$441$	−17.5000	−	11.6082i	−0.833333	−	0.552771i
$442$		0			0
$443$		−	36.4829i		−	1.73335i	−0.498870	−	0.866677i	$-0.666252\pi$
$443$		−	36.4829i		−	1.73335i	0.498870	−	0.866677i	$-0.333748\pi$
$444$		0			0
$445$	−55.0000			−2.60725
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.5831i		−	0.782606i	−0.920262	−	0.391303i	$-0.872024\pi$
$449$		−	16.5831i		−	0.782606i	0.920262	−	0.391303i	$-0.127976\pi$
$450$		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			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	31.0000			1.44069			0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000			1.44069			0.720346	+	0.693615i	$0.243983\pi$
$464$		0			0
$465$	27.5000	+	8.29156i	1.27528	+	0.384512i
$466$		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
$471$	−11.5000	+	38.1412i	−0.529892	+	1.75745i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−22.0000	+	33.1662i	−1.00731	+	1.51858i
$478$		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
$485$		−	56.3826i		−	2.56020i
$486$		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
$489$	−8.00000	+	26.5330i	−0.361773	+	1.19986i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	27.5000	+	18.2414i	1.23603	+	0.819892i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−40.0000			−1.79065			−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000			−1.79065			−0.895323	+	0.445418i	$0.853055\pi$
$500$		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
$507$	6.50000	−	21.5581i	0.288675	−	0.957427i
$508$		0			0
$509$		−	3.31662i		−	0.147007i	−0.997295	−	0.0735034i	$-0.976582\pi$
$509$		−	3.31662i		−	0.147007i	0.997295	−	0.0735034i	$-0.0234180\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	13.2665i		−	0.584592i
$516$		0			0
$517$	22.0000			0.967559
$518$		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		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12.0000			0.521739
$530$		0			0
$531$	5.50000	−	8.29156i	0.238680	−	0.359823i
$532$		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
$539$		−	23.2164i		−	1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	12.5000	−	41.4578i	0.536426	−	1.77912i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−38.5000	−	11.6082i	−1.63423	−	0.492740i
$556$		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	$0.5\pi$
$564$		0			0
$565$	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		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−38.5000	−	11.6082i	−1.60836	−	0.484939i
$574$		0			0
$575$			19.8997i			0.829877i
$576$		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
$583$	−44.0000			−1.82229
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	6.63325i		−	0.273784i	−0.990586	−	0.136892i	$-0.956289\pi$
$587$		−	6.63325i		−	0.273784i	0.990586	−	0.136892i	$-0.0437113\pi$
$588$		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
$597$	−10.0000	+	33.1662i	−0.409273	+	1.35740i
$598$		0			0
$599$		−	33.1662i		−	1.35514i	−0.735460	−	0.677568i	$-0.763034\pi$
$599$		−	33.1662i		−	1.35514i	0.735460	−	0.677568i	$-0.236966\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	32.5000	+	21.5581i	1.32350	+	0.877912i
$604$		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		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.5330i		−	1.06818i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$		−	26.5330i		−	1.06818i	0.845428	−	0.534089i	$-0.179345\pi$
$618$		0			0
$619$	−1.00000			−0.0401934			−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000			−0.0401934			−0.0200967	+	0.999798i	$0.506397\pi$
$620$		0			0
$621$	11.0000	+	13.2665i	0.441415	+	0.532366i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	7.00000			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	27.5000	−	41.4578i	1.08788	−	1.64005i
$640$		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
$643$	41.0000			1.61688			0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000			1.61688			0.808441	+	0.588577i	$0.200312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.1161i		−	1.69507i	−0.530740	−	0.847535i	$-0.678086\pi$
$647$		−	43.1161i		−	1.69507i	0.530740	−	0.847535i	$-0.321914\pi$
$648$		0			0
$649$	11.0000			0.431788
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	3.31662i		−	0.129790i	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$		−	3.31662i		−	0.129790i	0.997892	−	0.0648948i	$-0.0206712\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	13.0000			0.505641			0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000			0.505641			0.252821	+	0.967513i	$0.418642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.500000	−	1.65831i	0.0193311	−	0.0641141i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	24.0000	−	19.8997i	0.923760	−	0.765942i
$676$		0			0
$677$		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
$683$		−	46.4327i		−	1.77670i	−0.459167	−	0.888350i	$-0.651852\pi$
$683$		−	46.4327i		−	1.77670i	0.459167	−	0.888350i	$-0.348148\pi$
$684$		0			0
$685$	77.0000			2.94202
$686$		0			0
$687$	−2.50000	+	8.29156i	−0.0953809	+	0.316343i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	17.0000			0.646710			0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000			0.646710			0.323355	+	0.946278i	$0.395189\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	11.0000	−	36.4829i	0.414284	−	1.37402i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	19.0000			0.713560			0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000			0.713560			0.356780	+	0.934188i	$0.383875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16.5831i			0.621043i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			16.5831i			0.618446i	0.950990	+	0.309223i	$0.100069\pi$
$719$			16.5831i			0.618446i	−0.950990	+	0.309223i	$0.899931\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−53.0000			−1.96566			−0.982831	−	0.184510i	$-0.940930\pi$
$727$	−53.0000			−1.96566			−0.982831	+	0.184510i	$0.940930\pi$
$728$		0			0
$729$	5.00000	−	26.5330i	0.185185	−	0.982704i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−38.5000	−	11.6082i	−1.42009	−	0.428174i
$736$		0			0
$737$			43.1161i			1.58820i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		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
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$		0			0
$753$	−27.5000	−	8.29156i	−1.00216	−	0.302161i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$	−5.50000	+	18.2414i	−0.199637	+	0.662122i
$760$		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
$771$	−44.0000	−	13.2665i	−1.58462	−	0.477781i
$772$		0			0
$773$		−	13.2665i		−	0.477163i	−0.971123	−	0.238581i	$-0.923318\pi$
$773$		−	13.2665i		−	0.477163i	0.971123	−	0.238581i	$-0.0766824\pi$
$774$		0			0
$775$	30.0000			1.07763
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	55.0000			1.96805
$782$		0			0
$783$		0			0
$784$		0			0
$785$			76.2824i			2.72263i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−22.0000	+	72.9657i	−0.780260	+	2.58783i
$796$		0			0
$797$			56.3826i			1.99717i	0.0531327	+	0.998587i	$0.483079\pi$
$797$			56.3826i			1.99717i	−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
$807$	−22.0000	−	6.63325i	−0.774437	−	0.233501i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			53.0660i			1.85882i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	49.0000			1.70803			0.854016	−	0.520246i	$-0.174160\pi$
$823$	49.0000			1.70803			0.854016	+	0.520246i	$0.174160\pi$
$824$		0			0
$825$	33.0000	+	9.94987i	1.14891	+	0.346410i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−29.0000			−1.00721			−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000			−1.00721			−0.503606	+	0.863934i	$0.667994\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	20.0000	−	16.5831i	0.691301	−	0.573197i
$838$		0			0
$839$			36.4829i			1.25953i	0.776786	+	0.629764i	$0.216849\pi$
$839$			36.4829i			1.25953i	−0.776786	+	0.629764i	$0.783151\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	43.1161i		−	1.48324i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	23.2164i		−	0.795847i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		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
$863$			46.4327i			1.58059i	0.612727	+	0.790295i	$0.290072\pi$
$863$			46.4327i			1.58059i	−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−8.50000	+	28.1913i	−0.288675	+	0.957427i
$868$		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		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	16.5831i		−	0.558700i	−0.960189	−	0.279350i	$-0.909881\pi$
$881$		−	16.5831i		−	0.558700i	0.960189	−	0.279350i	$-0.0901189\pi$
$882$		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
$885$	5.50000	−	18.2414i	0.184880	−	0.613179i
$886$		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
$891$	27.5000	−	11.6082i	0.921285	−	0.388889i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−55.0000			−1.83845
$896$		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
$905$		−	82.9156i		−	2.75621i
$906$		0			0
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			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	$0.0350482\pi$
$911$			6.63325i			0.219769i	−0.993944	+	0.109885i	$0.964952\pi$
$912$		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
$925$	−42.0000			−1.38095
$926$		0			0
$927$	−10.0000	−	6.63325i	−0.328443	−	0.217865i
$928$		0			0
$929$			53.0660i			1.74104i	0.492134	+	0.870519i	$0.336217\pi$
$929$			53.0660i			1.74104i	−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−55.0000	−	16.5831i	−1.80062	−	0.542907i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−9.50000	+	31.5079i	−0.310021	+	1.02822i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			23.2164i			0.754431i	0.926126	+	0.377215i	$0.123118\pi$
$947$			23.2164i			0.754431i	−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−38.5000	−	11.6082i	−1.24845	−	0.376421i
$952$		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
$960$		0			0
$961$	−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	$0.243196\pi$
$971$			43.1161i			1.38366i	−0.722059	+	0.691831i	$0.756804\pi$
$972$		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
$979$	−55.0000			−1.75781
$980$		0			0
$981$		0			0
$982$		0			0
$983$			36.4829i			1.16362i	0.813324	+	0.581811i	$0.197656\pi$
$983$			36.4829i			1.16362i	−0.813324	+	0.581811i	$0.802344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$	17.5000	−	58.0409i	0.555346	−	1.84187i
$994$		0			0
$995$			66.3325i			2.10288i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−28.0000	+	23.2164i	−0.885881	+	0.734534i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.b.f.65.1		2
3.2	odd	2	inner	2112.2.b.f.65.2		2
4.3	odd	2		2112.2.b.e.65.2		2
8.3	odd	2		33.2.d.a.32.1	✓	2
8.5	even	2		528.2.b.a.65.2		2
11.10	odd	2	CM	2112.2.b.f.65.1		2
12.11	even	2		2112.2.b.e.65.1		2
24.5	odd	2		528.2.b.a.65.1		2
24.11	even	2		33.2.d.a.32.2	yes	2
33.32	even	2	inner	2112.2.b.f.65.2		2
40.3	even	4		825.2.d.a.824.4		4
40.19	odd	2		825.2.f.a.626.2		2
40.27	even	4		825.2.d.a.824.1		4
44.43	even	2		2112.2.b.e.65.2		2
72.11	even	6		891.2.g.a.296.2		4
72.43	odd	6		891.2.g.a.296.1		4
72.59	even	6		891.2.g.a.593.1		4
72.67	odd	6		891.2.g.a.593.2		4
88.3	odd	10		363.2.f.c.233.2		8
88.19	even	10		363.2.f.c.233.2		8
88.21	odd	2		528.2.b.a.65.2		2
88.27	odd	10		363.2.f.c.239.2		8
88.35	even	10		363.2.f.c.161.1		8
88.43	even	2		33.2.d.a.32.1	✓	2
88.51	even	10		363.2.f.c.215.1		8
88.59	odd	10		363.2.f.c.215.1		8
88.75	odd	10		363.2.f.c.161.1		8
88.83	even	10		363.2.f.c.239.2		8
120.59	even	2		825.2.f.a.626.1		2
120.83	odd	4		825.2.d.a.824.2		4
120.107	odd	4		825.2.d.a.824.3		4
132.131	odd	2		2112.2.b.e.65.1		2
264.35	odd	10		363.2.f.c.161.2		8
264.59	even	10		363.2.f.c.215.2		8
264.83	odd	10		363.2.f.c.239.1		8
264.107	odd	10		363.2.f.c.233.1		8
264.131	odd	2		33.2.d.a.32.2	yes	2
264.179	even	10		363.2.f.c.233.1		8
264.197	even	2		528.2.b.a.65.1		2
264.203	even	10		363.2.f.c.239.1		8
264.227	odd	10		363.2.f.c.215.2		8
264.251	even	10		363.2.f.c.161.2		8
440.43	odd	4		825.2.d.a.824.4		4
440.219	even	2		825.2.f.a.626.2		2
440.307	odd	4		825.2.d.a.824.1		4
792.43	even	6		891.2.g.a.296.1		4
792.131	odd	6		891.2.g.a.593.1		4
792.571	even	6		891.2.g.a.593.2		4
792.659	odd	6		891.2.g.a.296.2		4
1320.659	odd	2		825.2.f.a.626.1		2
1320.923	even	4		825.2.d.a.824.2		4
1320.1187	even	4		825.2.d.a.824.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.d.a.32.1	✓	2	8.3	odd	2
33.2.d.a.32.1	✓	2	88.43	even	2
33.2.d.a.32.2	yes	2	24.11	even	2
33.2.d.a.32.2	yes	2	264.131	odd	2
363.2.f.c.161.1		8	88.35	even	10
363.2.f.c.161.1		8	88.75	odd	10
363.2.f.c.161.2		8	264.35	odd	10
363.2.f.c.161.2		8	264.251	even	10
363.2.f.c.215.1		8	88.51	even	10
363.2.f.c.215.1		8	88.59	odd	10
363.2.f.c.215.2		8	264.59	even	10
363.2.f.c.215.2		8	264.227	odd	10
363.2.f.c.233.1		8	264.107	odd	10
363.2.f.c.233.1		8	264.179	even	10
363.2.f.c.233.2		8	88.3	odd	10
363.2.f.c.233.2		8	88.19	even	10
363.2.f.c.239.1		8	264.83	odd	10
363.2.f.c.239.1		8	264.203	even	10
363.2.f.c.239.2		8	88.27	odd	10
363.2.f.c.239.2		8	88.83	even	10
528.2.b.a.65.1		2	24.5	odd	2
528.2.b.a.65.1		2	264.197	even	2
528.2.b.a.65.2		2	8.5	even	2
528.2.b.a.65.2		2	88.21	odd	2
825.2.d.a.824.1		4	40.27	even	4
825.2.d.a.824.1		4	440.307	odd	4
825.2.d.a.824.2		4	120.83	odd	4
825.2.d.a.824.2		4	1320.923	even	4
825.2.d.a.824.3		4	120.107	odd	4
825.2.d.a.824.3		4	1320.1187	even	4
825.2.d.a.824.4		4	40.3	even	4
825.2.d.a.824.4		4	440.43	odd	4
825.2.f.a.626.1		2	120.59	even	2
825.2.f.a.626.1		2	1320.659	odd	2
825.2.f.a.626.2		2	40.19	odd	2
825.2.f.a.626.2		2	440.219	even	2
891.2.g.a.296.1		4	72.43	odd	6
891.2.g.a.296.1		4	792.43	even	6
891.2.g.a.296.2		4	72.11	even	6
891.2.g.a.296.2		4	792.659	odd	6
891.2.g.a.593.1		4	72.59	even	6
891.2.g.a.593.1		4	792.131	odd	6
891.2.g.a.593.2		4	72.67	odd	6
891.2.g.a.593.2		4	792.571	even	6
2112.2.b.e.65.1		2	12.11	even	2
2112.2.b.e.65.1		2	132.131	odd	2
2112.2.b.e.65.2		2	4.3	odd	2
2112.2.b.e.65.2		2	44.43	even	2
2112.2.b.f.65.1		2	1.1	even	1	trivial
2112.2.b.f.65.1		2	11.10	odd	2	CM
2112.2.b.f.65.2		2	3.2	odd	2	inner
2112.2.b.f.65.2		2	33.32	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} -3.31662i q^{11} +(-5.50000 - 1.65831i) q^{15} -3.31662i q^{23} -6.00000 q^{25} +(-4.00000 + 3.31662i) q^{27} -5.00000 q^{31} +(-5.50000 - 1.65831i) q^{33} +7.00000 q^{37} +(-5.50000 + 8.29156i) q^{45} +6.63325i q^{47} +7.00000 q^{49} -13.2665i q^{53} -11.0000 q^{55} +3.31662i q^{59} -13.0000 q^{67} +(-5.50000 - 1.65831i) q^{69} +16.5831i q^{71} +(-3.00000 + 9.94987i) q^{75} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} +(-2.50000 + 8.29156i) q^{93} +17.0000 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 5 q^{9}+O(q^{10}) \) 2 * q + q^3 - 5 * q^9	\( 2 q + q^{3} - 5 q^{9} - 11 q^{15} - 12 q^{25} - 8 q^{27} - 10 q^{31} - 11 q^{33} + 14 q^{37} - 11 q^{45} + 14 q^{49} - 22 q^{55} - 26 q^{67} - 11 q^{69} - 6 q^{75} + 7 q^{81} - 5 q^{93} + 34 q^{97} - 11 q^{99}+O(q^{100}) \) 2 * q + q^3 - 5 * q^9 - 11 * q^15 - 12 * q^25 - 8 * q^27 - 10 * q^31 - 11 * q^33 + 14 * q^37 - 11 * q^45 + 14 * q^49 - 22 * q^55 - 26 * q^67 - 11 * q^69 - 6 * q^75 + 7 * q^81 - 5 * q^93 + 34 * q^97 - 11 * q^99

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