LMFDB - Embedded newform 99.2.g.a.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,2,Mod(32,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("99.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	99.g (of order $6$, degree $2$, 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:	$0.790518980011$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			65.2
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	99.65
Dual form			99.2.g.a.32.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.18614 - 1.26217i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-0.813859 + 0.469882i) q^{5} +(-0.186141 - 2.99422i) q^{9} +(-2.87228 - 1.65831i) q^{11} +(3.37228 + 0.792287i) q^{12} +(-0.372281 + 1.58457i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(-1.62772 - 0.939764i) q^{20} +(-2.87228 + 1.65831i) q^{23} +(-2.05842 + 3.56529i) q^{25} +(-4.00000 - 3.31662i) q^{27} +(-3.05842 - 5.29734i) q^{31} +(-5.50000 + 1.65831i) q^{33} +(5.00000 - 3.31662i) q^{36} +12.1168 q^{37} -6.63325i q^{44} +(1.55842 + 2.34941i) q^{45} +(11.8723 + 6.85446i) q^{47} +(2.00000 + 6.63325i) q^{48} +(-3.50000 - 6.06218i) q^{49} +11.8294i q^{53} +3.11684 q^{55} +(12.6861 - 7.32435i) q^{59} +(-3.11684 + 0.939764i) q^{60} -8.00000 q^{64} +(-7.55842 - 13.0916i) q^{67} +(-1.31386 + 5.59230i) q^{69} -10.8896i q^{71} +(2.05842 + 6.82701i) q^{75} -3.75906i q^{80} +(-8.93070 + 1.11469i) q^{81} +16.5831i q^{89} +(-5.74456 - 3.31662i) q^{92} +(-10.3139 - 2.42315i) q^{93} +(-0.0584220 + 0.101190i) q^{97} +(-4.43070 + 8.90892i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{4} - 9 q^{5} + 5 q^{9} + 2 q^{12} + 10 q^{15} - 8 q^{16} - 18 q^{20} + 9 q^{25} - 16 q^{27} + 5 q^{31} - 22 q^{33} + 20 q^{36} + 14 q^{37} - 11 q^{45} + 36 q^{47} + 8 q^{48} - 14 q^{49}+ \cdots + 11 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^4 - 9 * q^5 + 5 * q^9 + 2 * q^12 + 10 * q^15 - 8 * q^16 - 18 * q^20 + 9 * q^25 - 16 * q^27 + 5 * q^31 - 22 * q^33 + 20 * q^36 + 14 * q^37 - 11 * q^45 + 36 * q^47 + 8 * q^48 - 14 * q^49 - 22 * q^55 + 45 * q^59 + 22 * q^60 - 32 * q^64 - 13 * q^67 - 11 * q^69 - 9 * q^75 - 7 * q^81 - 47 * q^93 + 17 * q^97 + 11 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−0.813859	+	0.469882i	−0.363969	+	0.210138i	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−0.813859	+	0.469882i	−0.363969	+	0.210138i	0.306851	+	0.951757i	$0.400725\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		0			0
$9$	−0.186141	−	2.99422i	−0.0620469	−	0.998073i
$10$		0			0
$11$	−2.87228	−	1.65831i	−0.866025	−	0.500000i
$11$	−2.87228	−	1.65831i	−0.866025	−	0.500000i
$12$	3.37228	+	0.792287i	0.973494	+	0.228714i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−0.372281	+	1.58457i	−0.0961226	+	0.409135i
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		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			$\pi$
$20$	−1.62772	−	0.939764i	−0.363969	−	0.210138i
$21$		0			0
$22$		0			0
$23$	−2.87228	+	1.65831i	−0.598912	+	0.345782i	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	−2.87228	+	1.65831i	−0.598912	+	0.345782i	0.169701	+	0.985496i	$0.445720\pi$
$24$		0			0
$25$	−2.05842	+	3.56529i	−0.411684	+	0.713058i
$26$		0			0
$27$	−4.00000	−	3.31662i	−0.769800	−	0.638285i
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−3.05842	−	5.29734i	−0.549309	−	0.951431i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−3.05842	−	5.29734i	−0.549309	−	0.951431i	0.449013	−	0.893525i	$-0.351776\pi$
$32$		0			0
$33$	−5.50000	+	1.65831i	−0.957427	+	0.288675i
$34$		0			0
$35$		0			0
$36$	5.00000	−	3.31662i	0.833333	−	0.552771i
$37$	12.1168			1.99200			0.995998	−	0.0893706i	$-0.0284856\pi$
$37$	12.1168			1.99200			0.995998	+	0.0893706i	$0.0284856\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		−	6.63325i		−	1.00000i
$45$	1.55842	+	2.34941i	0.232316	+	0.350229i
$46$		0			0
$47$	11.8723	+	6.85446i	1.73175	+	0.999826i	0.875190	+	0.483779i	$0.160736\pi$
$47$	11.8723	+	6.85446i	1.73175	+	0.999826i	0.856560	+	0.516047i	$0.172597\pi$
$48$	2.00000	+	6.63325i	0.288675	+	0.957427i
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.8294i			1.62489i	0.583036	+	0.812447i	$0.301865\pi$
$53$			11.8294i			1.62489i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		0			0
$55$	3.11684			0.420275
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.6861	−	7.32435i	1.65159	−	0.953549i	0.675178	−	0.737655i	$-0.264067\pi$
$59$	12.6861	−	7.32435i	1.65159	−	0.953549i	0.976417	−	0.215894i	$-0.0692665\pi$
$60$	−3.11684	+	0.939764i	−0.402383	+	0.121323i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−7.55842	−	13.0916i	−0.923408	−	1.59939i	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−7.55842	−	13.0916i	−0.923408	−	1.59939i	−0.129307	−	0.991605i	$-0.541275\pi$
$68$		0			0
$69$	−1.31386	+	5.59230i	−0.158170	+	0.673233i
$70$		0			0
$71$		−	10.8896i		−	1.29236i	−0.763184	−	0.646181i	$-0.776365\pi$
$71$		−	10.8896i		−	1.29236i	0.763184	−	0.646181i	$-0.223635\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	2.05842	+	6.82701i	0.237686	+	0.788316i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		−	3.75906i		−	0.420275i
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		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
$92$	−5.74456	−	3.31662i	−0.598912	−	0.345782i
$93$	−10.3139	−	2.42315i	−1.06950	−	0.251269i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.0584220	+	0.101190i	−0.00593185	+	0.0102743i	−0.868976	−	0.494854i	$-0.835222\pi$
$97$	−0.0584220	+	0.101190i	−0.00593185	+	0.0102743i	0.863044	+	0.505128i	$0.168555\pi$
$98$		0			0
$99$	−4.43070	+	8.90892i	−0.445302	+	0.895380i
$100$	−8.23369			−0.823369
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	7.61684	+	13.1928i	0.750510	+	1.29992i	0.947576	+	0.319531i	$0.103525\pi$
$103$	7.61684	+	13.1928i	0.750510	+	1.29992i	−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	1.74456	−	10.2448i	0.167871	−	0.985809i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	14.3723	−	15.2935i	1.36416	−	1.45160i
$112$		0			0
$113$	−14.3139	+	8.26411i	−1.34653	+	0.777422i	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−14.3139	+	8.26411i	−1.34653	+	0.777422i	−0.358778	+	0.933423i	$0.616806\pi$
$114$		0			0
$115$	1.55842	−	2.69927i	0.145324	−	0.251708i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	6.11684	−	10.5947i	0.549309	−	0.951431i
$125$		−	8.56768i		−	0.766317i
$126$		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		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$	−8.37228	−	7.86797i	−0.728714	−	0.684819i
$133$		0			0
$134$		0			0
$135$	4.81386	+	0.819738i	0.414311	+	0.0705519i
$136$		0			0
$137$	−7.80298	−	4.50506i	−0.666654	−	0.384893i	0.128154	−	0.991754i	$-0.459095\pi$
$137$	−7.80298	−	4.50506i	−0.666654	−	0.384893i	−0.794808	+	0.606861i	$0.792428\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	22.7337	−	6.85446i	1.91452	−	0.577250i
$142$		0			0
$143$		0			0
$144$	10.7446	+	5.34363i	0.895380	+	0.445302i
$145$		0			0
$146$		0			0
$147$	−11.8030	−	2.77300i	−0.973494	−	0.228714i
$148$	12.1168	+	20.9870i	0.995998	+	1.72512i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.97825	+	2.87419i	0.399863	+	0.230861i
$156$		0			0
$157$	1.44158	+	2.49689i	0.115050	+	0.199273i	0.917800	−	0.397043i	$-0.129964\pi$
$157$	1.44158	+	2.49689i	0.115050	+	0.199273i	−0.802749	+	0.596316i	$0.796630\pi$
$158$		0			0
$159$	14.9307	+	14.0313i	1.18408	+	1.11276i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−9.23369			−0.723238			−0.361619	−	0.932326i	$-0.617776\pi$
$163$	−9.23369			−0.723238			−0.361619	+	0.932326i	$0.617776\pi$
$164$		0			0
$165$	3.69702	−	3.93398i	0.287812	−	0.306260i
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	11.4891	−	6.63325i	0.866025	−	0.500000i
$177$	5.80298	−	24.6998i	0.436179	−	1.85655i
$178$		0			0
$179$		−	26.4781i		−	1.97907i	−0.144308	−	0.989533i	$-0.546095\pi$
$179$		−	26.4781i		−	1.97907i	0.144308	−	0.989533i	$-0.453905\pi$
$180$	−2.51087	+	5.04868i	−0.187150	+	0.376306i
$181$	21.1168			1.56960			0.784801	−	0.619747i	$-0.212765\pi$
$181$	21.1168			1.56960			0.784801	+	0.619747i	$0.212765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.86141	+	5.69349i	−0.725025	+	0.418593i
$186$		0			0
$187$		0			0
$188$			27.4179i			1.99965i
$189$		0			0
$190$		0			0
$191$	−21.3030	−	12.2993i	−1.54143	−	0.889945i	−0.998749	−	0.0500060i	$-0.984076\pi$
$191$	−21.3030	−	12.2993i	−1.54143	−	0.889945i	−0.542681	−	0.839939i	$-0.682591\pi$
$192$	−9.48913	+	10.0974i	−0.684819	+	0.728714i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	7.00000	−	12.1244i	0.500000	−	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−27.2337			−1.93054			−0.965272	−	0.261245i	$-0.915867\pi$
$199$	−27.2337			−1.93054			−0.965272	+	0.261245i	$0.915867\pi$
$200$		0			0
$201$	−25.4891	−	5.98844i	−1.79786	−	0.422392i
$202$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	−20.4891	+	11.8294i	−1.40720	+	0.812447i
$213$	−13.7446	−	12.9166i	−0.941762	−	0.885034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	3.11684	+	5.39853i	0.210138	+	0.363969i
$221$		0			0
$222$		0			0
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	0.882281	+	0.470723i	$0.156007\pi$
$224$		0			0
$225$	11.0584	+	5.49972i	0.737228	+	0.366648i
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	0.771523	−	0.636201i	$-0.219495\pi$
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	−0.936728	+	0.350058i	$0.886162\pi$
$230$		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
$235$	−12.8832			−0.840404
$236$	25.3723	+	14.6487i	1.65159	+	0.953549i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$	−4.74456	−	4.45877i	−0.306260	−	0.287812i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	−9.18614	+	12.5942i	−0.589291	+	0.807921i
$244$		0			0
$245$	5.69702	+	3.28917i	0.363969	+	0.210138i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			16.5831i			1.04672i	0.852112	+	0.523359i	$0.175321\pi$
$251$			16.5831i			1.04672i	−0.852112	+	0.523359i	$0.824679\pi$
$252$		0			0
$253$	11.0000			0.691564
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	22.9783	−	13.2665i	1.43334	−	0.827541i	0.435970	−	0.899961i	$-0.356405\pi$
$257$	22.9783	−	13.2665i	1.43334	−	0.827541i	0.997374	+	0.0724199i	$0.0230722\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−5.55842	−	9.62747i	−0.341451	−	0.591411i
$266$		0			0
$267$	20.9307	+	19.6699i	1.28094	+	1.20378i
$268$	15.1168	−	26.1831i	0.923408	−	1.59939i
$269$		−	19.3475i		−	1.17964i	−0.807535	−	0.589819i	$-0.799199\pi$
$269$		−	19.3475i		−	1.17964i	0.807535	−	0.589819i	$-0.200801\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.8247	−	6.82701i	0.713058	−	0.411684i
$276$	−11.0000	+	3.31662i	−0.662122	+	0.199637i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	−15.2921	+	10.1436i	−0.915515	+	0.607284i
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	18.8614	−	10.8896i	1.11922	−	0.646181i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	0.0584220	+	0.193764i	0.00342476	+	0.0113586i
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	−6.88316	+	11.9220i	−0.400753	+	0.694124i
$296$		0			0
$297$	5.98913	+	16.1595i	0.347524	+	0.937671i
$298$		0			0
$299$		0			0
$300$	−9.76631	+	10.3923i	−0.563858	+	0.600000i
$301$		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
$309$	25.6861	+	6.03473i	1.46123	+	0.343304i
$310$		0			0
$311$	5.36141	−	3.09541i	0.304017	−	0.175525i	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	5.36141	−	3.09541i	0.304017	−	0.175525i	0.644246	+	0.764818i	$0.277171\pi$
$312$		0			0
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	0.999065	−	0.0432311i	$-0.0137652\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.1060	+	11.6082i	1.12926	+	0.651981i	0.943750	−	0.330661i	$-0.107272\pi$
$317$	20.1060	+	11.6082i	1.12926	+	0.651981i	0.185514	+	0.982642i	$0.440605\pi$
$318$		0			0
$319$		0			0
$320$	6.51087	−	3.75906i	0.363969	−	0.210138i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−10.8614	−	14.3537i	−0.603411	−	0.797430i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.44158	−	7.69304i	0.244131	−	0.422848i	−0.717756	−	0.696295i	$-0.754831\pi$
$331$	4.44158	−	7.69304i	0.244131	−	0.422848i	0.961887	+	0.273447i	$0.0881639\pi$
$332$		0			0
$333$	−2.25544	−	36.2805i	−0.123597	−	1.98816i
$334$		0			0
$335$	12.3030	+	7.10313i	0.672184	+	0.388086i
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$	−6.54755	+	27.8689i	−0.355614	+	1.51363i
$340$		0			0
$341$			20.2873i			1.09862i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.55842	−	5.16870i	−0.0839026	−	0.278274i
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−31.5951	−	18.2414i	−1.68164	−	0.970894i	−0.960574	−	0.278024i	$-0.910320\pi$
$353$	−31.5951	−	18.2414i	−1.68164	−	0.970894i	−0.721063	−	0.692869i	$-0.756346\pi$
$354$		0			0
$355$	5.11684	+	8.86263i	0.271574	+	0.470380i
$356$	−28.7228	+	16.5831i	−1.52231	+	0.878904i
$357$		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
$363$	18.5475	+	4.35758i	0.973494	+	0.228714i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.5584	+	23.4839i	−0.707744	+	1.22585i	0.257948	+	0.966159i	$0.416954\pi$
$367$	−13.5584	+	23.4839i	−0.707744	+	1.22585i	−0.965692	+	0.259690i	$0.916380\pi$
$368$		−	13.2665i		−	0.691564i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6.11684	−	20.2873i	−0.317144	−	1.05185i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	−10.8139	−	10.1625i	−0.558425	−	0.524788i
$376$		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
$383$	−27.8139	+	16.0583i	−1.42122	+	0.820543i	−0.996403	−	0.0847358i	$-0.972995\pi$
$383$	−27.8139	+	16.0583i	−1.42122	+	0.820543i	−0.424818	+	0.905279i	$0.639662\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.233688			−0.0118637
$389$	4.54755	+	2.62553i	0.230570	+	0.133120i	0.610835	−	0.791758i	$-0.290834\pi$
$389$	4.54755	+	2.62553i	0.230570	+	0.133120i	−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
$396$	−19.8614	+	1.23472i	−0.998073	+	0.0620469i
$397$	33.4674			1.67968			0.839840	−	0.542834i	$-0.182649\pi$
$397$	33.4674			1.67968			0.839840	+	0.542834i	$0.182649\pi$
$398$		0			0
$399$		0			0
$400$	−8.23369	−	14.2612i	−0.411684	−	0.713058i
$401$	−33.9891	+	19.6236i	−1.69734	+	0.979957i	−0.749064	+	0.662497i	$0.769497\pi$
$401$	−33.9891	+	19.6236i	−1.69734	+	0.979957i	−0.948272	+	0.317460i	$0.897170\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	6.74456	−	5.10358i	0.335140	−	0.253599i
$406$		0			0
$407$	−34.8030	−	20.0935i	−1.72512	−	0.995998i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$	−14.9416	+	4.50506i	−0.737014	+	0.222218i
$412$	−15.2337	+	26.3855i	−0.750510	+	1.29992i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	32.3614	−	18.6839i	1.58096	−	0.912767i	0.586238	−	0.810139i	$-0.300608\pi$
$419$	32.3614	−	18.6839i	1.58096	−	0.912767i	0.994720	−	0.102628i	$-0.0327251\pi$
$420$		0			0
$421$	−19.7337	+	34.1798i	−0.961761	+	1.66582i	−0.243685	+	0.969854i	$0.578356\pi$
$421$	−19.7337	+	34.1798i	−0.961761	+	1.66582i	−0.718076	+	0.695965i	$0.754977\pi$
$422$		0			0
$423$	18.3139	−	36.8241i	0.890450	−	1.79045i
$424$		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	$0.5\pi$
$432$	19.4891	−	7.22316i	0.937671	−	0.347524i
$433$	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		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−17.5000	+	11.6082i	−0.833333	+	0.552771i
$442$		0			0
$443$	31.5475	+	18.2140i	1.49887	+	0.865373i	0.999999	−	0.00130426i	$-0.000415158\pi$
$443$	31.5475	+	18.2140i	1.49887	+	0.865373i	0.498870	+	0.866677i	$0.333748\pi$
$444$	40.8614	+	9.60002i	1.93920	+	0.455597i
$445$	−7.79211	−	13.4963i	−0.369381	−	0.639787i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	42.0666i		−	1.98524i	−0.121253	−	0.992622i	$-0.538691\pi$
$449$		−	42.0666i		−	1.98524i	0.121253	−	0.992622i	$-0.461309\pi$
$450$		0			0
$451$		0			0
$452$	−28.6277	−	16.5282i	−1.34653	−	0.777422i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$	6.23369			0.290647
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	15.5000	+	26.8468i	0.720346	+	1.24768i	0.960861	+	0.277031i	$0.0893503\pi$
$463$	15.5000	+	26.8468i	0.720346	+	1.24768i	−0.240515	+	0.970645i	$0.577316\pi$
$464$		0			0
$465$	9.53262	−	2.87419i	0.442065	−	0.133288i
$466$		0			0
$467$			18.9600i			0.877363i	0.898642	+	0.438682i	$0.144554\pi$
$467$			18.9600i			0.877363i	−0.898642	+	0.438682i	$0.855446\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4.86141	+	1.14214i	0.224002	+	0.0526272i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	35.4198	−	2.20193i	1.62176	−	0.100820i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		−	0.109806i		−	0.00498602i
$486$		0			0
$487$	30.1168			1.36472			0.682362	−	0.731014i	$-0.260953\pi$
$487$	30.1168			1.36472			0.682362	+	0.731014i	$0.260953\pi$
$488$		0			0
$489$	−10.9525	+	11.6545i	−0.495287	+	0.527034i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.580171	−	9.33252i	−0.0260768	−	0.419465i
$496$	24.4674			1.09862
$497$		0			0
$498$		0			0
$499$	−1.38316	−	2.39570i	−0.0619186	−	0.107246i	0.833404	−	0.552664i	$-0.186389\pi$
$499$	−1.38316	−	2.39570i	−0.0619186	−	0.107246i	−0.895323	+	0.445418i	$0.853055\pi$
$500$	14.8397	−	8.56768i	0.663650	−	0.383158i
$501$		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
$509$	−2.87228	+	1.65831i	−0.127312	+	0.0735034i	−0.562303	−	0.826931i	$-0.690085\pi$
$509$	−2.87228	+	1.65831i	−0.127312	+	0.0735034i	0.434992	+	0.900434i	$0.356751\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.3981	−	7.15803i	−0.546325	−	0.315421i
$516$		0			0
$517$	−22.7337	−	39.3759i	−0.999826	−	1.73175i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			34.5484i			1.51359i	0.653650	+	0.756797i	$0.273237\pi$
$521$			34.5484i			1.51359i	−0.653650	+	0.756797i	$0.726763\pi$
$522$		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
$528$	5.25544	−	22.3692i	0.228714	−	0.973494i
$529$	−6.00000	+	10.3923i	−0.260870	+	0.451839i
$530$		0			0
$531$	−24.2921	−	36.6217i	−1.05419	−	1.58925i
$532$		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
$539$			23.2164i			1.00000i
$540$	3.39403	+	9.15759i	0.146056	+	0.394080i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	25.0475	−	26.6530i	1.07489	−	1.14379i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		−	18.0202i		−	0.769786i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.51087	+	19.2000i	−0.191476	+	0.814996i
$556$		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		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	34.6060	+	32.5214i	1.45717	+	1.36940i
$565$	7.76631	−	13.4516i	0.326731	−	0.565915i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	−40.7921	+	12.2993i	−1.70411	+	0.513810i
$574$		0			0
$575$		−	13.6540i		−	0.569412i
$576$	1.48913	+	23.9538i	0.0620469	+	0.998073i
$577$	−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
$583$	19.6168	−	33.9774i	0.812447	−	1.40720i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	38.8723	+	22.4429i	1.60443	+	0.926319i	0.990586	+	0.136892i	$0.0437113\pi$
$587$	38.8723	+	22.4429i	1.60443	+	0.926319i	0.613845	+	0.789427i	$0.289622\pi$
$588$	−7.00000	−	23.2164i	−0.288675	−	0.957427i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−24.2337	+	41.9740i	−0.995998	+	1.72512i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−32.3030	+	34.3735i	−1.32207	+	1.40681i
$598$		0			0
$599$	−28.7228	+	16.5831i	−1.17358	+	0.677568i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−28.7228	+	16.5831i	−1.17358	+	0.677568i	−0.219061	+	0.975711i	$0.570299\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	−37.7921	+	25.0684i	−1.53901	+	1.02087i
$604$		0			0
$605$	−8.95245	−	5.16870i	−0.363969	−	0.210138i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		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
$617$	20.0109	−	11.5533i	0.805607	−	0.465118i	−0.0398207	−	0.999207i	$-0.512679\pi$
$617$	20.0109	−	11.5533i	0.805607	−	0.465118i	0.845428	+	0.534089i	$0.179345\pi$
$618$		0			0
$619$	21.2921	−	36.8790i	0.855802	−	1.48229i	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	21.2921	−	36.8790i	0.855802	−	1.48229i	0.875899	−	0.482495i	$-0.160269\pi$
$620$			11.4968i			0.461722i
$621$	16.9891	+	2.89303i	0.681750	+	0.116093i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−6.26631	−	10.8536i	−0.250652	−	0.434143i
$626$		0			0
$627$		0			0
$628$	−2.88316	+	4.99377i	−0.115050	+	0.199273i
$629$		0			0
$630$		0			0
$631$	−39.5842			−1.57582			−0.787911	−	0.615789i	$-0.788838\pi$
$631$	−39.5842			−1.57582			−0.787911	+	0.615789i	$0.788838\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−9.37228	+	39.8921i	−0.371635	+	1.58182i
$637$		0			0
$638$		0			0
$639$	−32.6060	+	2.02700i	−1.28987	+	0.0801871i
$640$		0			0
$641$	20.1060	+	11.6082i	0.794138	+	0.458496i	0.841417	−	0.540386i	$-0.181722\pi$
$641$	20.1060	+	11.6082i	0.794138	+	0.458496i	−0.0472793	+	0.998882i	$0.515055\pi$
$642$		0			0
$643$	−20.5000	−	35.5070i	−0.808441	−	1.40026i	−0.913943	−	0.405842i	$-0.866978\pi$
$643$	−20.5000	−	35.5070i	−0.808441	−	1.40026i	0.105502	−	0.994419i	$-0.466355\pi$
$644$		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
$649$	−48.5842			−1.90710
$650$		0			0
$651$		0			0
$652$	−9.23369	−	15.9932i	−0.361619	−	0.626343i
$653$	39.6861	−	22.9128i	1.55304	−	0.896647i	0.555147	−	0.831753i	$-0.312662\pi$
$653$	39.6861	−	22.9128i	1.55304	−	0.896647i	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		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$	10.5109	+	2.46943i	0.409135	+	0.0961226i
$661$	18.2921	+	31.6829i	0.711481	+	1.23232i	0.964301	+	0.264807i	$0.0853084\pi$
$661$	18.2921	+	31.6829i	0.711481	+	1.23232i	−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
$669$	−0.500000	−	1.65831i	−0.0193311	−	0.0641141i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	20.0584	−	7.43415i	0.772049	−	0.286141i
$676$	−26.0000			−1.00000
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	2.43176i		−	0.0930489i	−0.998917	−	0.0465244i	$-0.985185\pi$
$683$		−	2.43176i		−	0.0930489i	0.998917	−	0.0465244i	$-0.0148145\pi$
$684$		0			0
$685$	8.46738			0.323522
$686$		0			0
$687$	−8.43070	−	1.98072i	−0.321651	−	0.0755691i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	25.7921	−	44.6732i	0.981178	−	1.69945i	0.323355	−	0.946278i	$-0.395189\pi$
$691$	25.7921	−	44.6732i	0.981178	−	1.69945i	0.657823	−	0.753173i	$-0.271478\pi$
$692$		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
$704$	22.9783	+	13.2665i	0.866025	+	0.500000i
$705$	−15.2812	+	16.2607i	−0.575525	+	0.612414i
$706$		0			0
$707$		0			0
$708$	48.5842	−	14.6487i	1.82591	−	0.550532i
$709$	16.7921	−	29.0848i	0.630641	−	1.09230i	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	16.7921	−	29.0848i	0.630641	−	1.09230i	0.987421	−	0.158114i	$-0.0505412\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.5693	+	10.1436i	0.657975	+	0.379882i
$714$		0			0
$715$		0			0
$716$	45.8614	−	26.4781i	1.71392	−	0.989533i
$717$		0			0
$718$		0			0
$719$			35.8757i			1.33794i	0.743290	+	0.668970i	$0.233264\pi$
$719$			35.8757i			1.33794i	−0.743290	+	0.668970i	$0.766736\pi$
$720$	−11.2554	+	0.699713i	−0.419465	+	0.0260768i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	21.1168	+	36.5754i	0.784801	+	1.35932i
$725$		0			0
$726$		0			0
$727$	8.94158	−	15.4873i	0.331625	−	0.574391i	−0.651206	−	0.758901i	$-0.725737\pi$
$727$	8.94158	−	15.4873i	0.331625	−	0.574391i	0.982831	+	0.184510i	$0.0590699\pi$
$728$		0			0
$729$	5.00000	+	26.5330i	0.185185	+	0.982704i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	10.9090	−	3.28917i	0.402383	−	0.121323i
$736$		0			0
$737$			50.1369i			1.84682i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−19.7228	−	11.3870i	−0.725025	−	0.418593i
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	27.2921	+	47.2713i	0.995903	+	1.72496i	0.576262	+	0.817265i	$0.304511\pi$
$751$	27.2921	+	47.2713i	0.995903	+	1.72496i	0.419641	+	0.907690i	$0.362156\pi$
$752$	−47.4891	+	27.4179i	−1.73175	+	0.999826i
$753$	20.9307	+	19.6699i	0.762757	+	0.716812i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	15.4674			0.562171			0.281086	−	0.959683i	$-0.409305\pi$
$757$	15.4674			0.562171			0.281086	+	0.959683i	$0.409305\pi$
$758$		0			0
$759$	13.0475	−	13.8839i	0.473596	−	0.503952i
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		−	49.1971i		−	1.77989i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−26.9783	−	6.33830i	−0.973494	−	0.228714i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	10.5109	−	44.7384i	0.378540	−	1.61121i
$772$		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
$775$	25.1821			0.904567
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−18.0584	+	31.2781i	−0.646181	+	1.11922i
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	−2.34648	−	1.35474i	−0.0837496	−	0.0483528i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−18.7446	−	4.40387i	−0.664801	−	0.156189i
$796$	−27.2337	−	47.1701i	−0.965272	−	1.67190i
$797$	−26.6644	+	15.3947i	−0.944501	+	0.545308i	−0.891368	−	0.453279i	$-0.850254\pi$
$797$	−26.6644	+	15.3947i	−0.944501	+	0.545308i	−0.0531327	+	0.998587i	$0.516921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	49.6535	−	3.08679i	1.75442	−	0.109066i
$802$		0			0
$803$		0			0
$804$	−15.1168	−	50.1369i	−0.533130	−	1.76819i
$805$		0			0
$806$		0			0
$807$	−24.4198	−	22.9489i	−0.859619	−	0.807839i
$808$		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
$815$	7.51492	−	4.33874i	0.263236	−	0.151980i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	24.5000	+	42.4352i	0.854016	+	1.47920i	0.877555	+	0.479477i	$0.159174\pi$
$823$	24.5000	+	42.4352i	0.854016	+	1.47920i	−0.0235383	+	0.999723i	$0.507493\pi$
$824$		0			0
$825$	5.40895	−	23.0226i	0.188316	−	0.801544i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−8.86141	+	17.8178i	−0.307955	+	0.619213i
$829$	−57.5842			−1.99998			−0.999991	−	0.00416865i	$-0.998673\pi$
$829$	−57.5842			−1.99998			−0.999991	+	0.00416865i	$0.998673\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.33561	+	31.3330i	−0.184426	+	1.08303i
$838$		0			0
$839$	−31.5951	−	18.2414i	−1.09078	−	0.629764i	−0.156999	−	0.987599i	$-0.550182\pi$
$839$	−31.5951	−	18.2414i	−1.09078	−	0.629764i	−0.933785	+	0.357834i	$0.883515\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	12.2169i		−	0.420275i
$846$		0			0
$847$		0			0
$848$	−40.9783	−	23.6588i	−1.40720	−	0.812447i
$849$		0			0
$850$		0			0
$851$	−34.8030	+	20.0935i	−1.19303	+	0.688797i
$852$	8.62772	−	36.7229i	0.295581	−	1.25811i
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	13.7921	+	23.8886i	0.470581	+	0.815070i	0.999434	−	0.0336436i	$-0.0107111\pi$
$859$	13.7921	+	23.8886i	0.470581	+	0.815070i	−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
$867$	−20.1644	+	21.4569i	−0.684819	+	0.728714i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.313859	+	0.156093i	0.0106225	+	0.00528294i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$	−6.23369	+	10.7971i	−0.210138	+	0.363969i
$881$		−	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
$883$	−45.2337			−1.52223			−0.761117	−	0.648614i	$-0.775349\pi$
$883$	−45.2337			−1.52223			−0.761117	+	0.648614i	$0.775349\pi$
$884$		0			0
$885$	6.88316	+	22.8288i	0.231375	+	0.767383i
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	27.5000	+	11.6082i	0.921285	+	0.388889i
$892$	2.00000			0.0669650
$893$		0			0
$894$		0			0
$895$	12.4416	+	21.5494i	0.415876	+	0.720319i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.53262	+	24.6535i	0.0510875	+	0.821782i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.1861	+	9.92242i	−0.571287	+	0.329832i
$906$		0			0
$907$	−4.00000	+	6.92820i	−0.132818	+	0.230047i	−0.924762	−	0.380547i	$-0.875736\pi$
$907$	−4.00000	+	6.92820i	−0.132818	+	0.230047i	0.791944	+	0.610594i	$0.209069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.1277	−	24.3224i	−1.39575	−	0.805839i	−0.401809	−	0.915723i	$-0.631619\pi$
$911$	−42.1277	−	24.3224i	−1.39575	−	0.805839i	−0.993944	+	0.109885i	$0.964952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	5.00000	−	8.66025i	0.165205	−	0.286143i
$917$		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
$925$	−24.9416	+	43.2001i	−0.820074	+	1.42041i
$926$		0			0
$927$	38.0842	−	25.2622i	1.25085	−	0.829720i
$928$		0			0
$929$	−0.478251	−	0.276118i	−0.0156909	−	0.00905914i	0.492134	−	0.870519i	$-0.336217\pi$
$929$	−0.478251	−	0.276118i	−0.0156909	−	0.00905914i	−0.507825	+	0.861460i	$0.669550\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	2.45245	−	10.4386i	0.0802897	−	0.341744i
$934$		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
$940$	−12.8832	−	22.3143i	−0.420202	−	0.727812i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$			58.5948i			1.90710i
$945$		0			0
$946$		0			0
$947$	32.6970	+	18.8776i	1.06251	+	0.613441i	0.926126	−	0.377215i	$-0.123118\pi$
$947$	32.6970	+	18.8776i	1.06251	+	0.613441i	0.136385	+	0.990656i	$0.456452\pi$
$948$		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
$955$	23.1168			0.748044
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	2.97825	−	12.6766i	0.0961226	−	0.409135i
$961$	−3.20789	+	5.55623i	−0.103480	+	0.179233i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	43.1161i		−	1.38366i	−0.722059	−	0.691831i	$-0.756804\pi$
$971$		−	43.1161i		−	1.38366i	0.722059	−	0.691831i	$-0.243196\pi$
$972$	−31.0000	−	3.31662i	−0.994325	−	0.106381i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	48.8288	−	28.1913i	1.56217	−	0.901920i	0.565134	−	0.824999i	$-0.308824\pi$
$977$	48.8288	−	28.1913i	1.56217	−	0.901920i	0.997037	−	0.0769208i	$-0.0245089\pi$
$978$		0			0
$979$	27.5000	−	47.6314i	0.878904	−	1.52231i
$980$			13.1567i			0.420275i
$981$		0			0
$982$		0			0
$983$	−22.4525	−	12.9629i	−0.716122	−	0.413453i	0.0972017	−	0.995265i	$-0.469011\pi$
$983$	−22.4525	−	12.9629i	−0.716122	−	0.413453i	−0.813324	+	0.581811i	$0.802344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$	−4.44158	−	14.7310i	−0.140949	−	0.467476i
$994$		0			0
$995$	22.1644	−	12.7966i	0.702658	−	0.405680i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	−48.4674	−	40.1870i	−1.53344	−	1.27146i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.2.g.a.65.2	yes	4
3.2	odd	2		297.2.g.a.197.1		4
9.2	odd	6		891.2.d.a.890.3		4
9.4	even	3		297.2.g.a.98.1		4
9.5	odd	6	inner	99.2.g.a.32.2	✓	4
9.7	even	3		891.2.d.a.890.2		4
11.10	odd	2	CM	99.2.g.a.65.2	yes	4
33.32	even	2		297.2.g.a.197.1		4
99.32	even	6	inner	99.2.g.a.32.2	✓	4
99.43	odd	6		891.2.d.a.890.2		4
99.65	even	6		891.2.d.a.890.3		4
99.76	odd	6		297.2.g.a.98.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.g.a.32.2	✓	4	9.5	odd	6	inner
99.2.g.a.32.2	✓	4	99.32	even	6	inner
99.2.g.a.65.2	yes	4	1.1	even	1	trivial
99.2.g.a.65.2	yes	4	11.10	odd	2	CM
297.2.g.a.98.1		4	9.4	even	3
297.2.g.a.98.1		4	99.76	odd	6
297.2.g.a.197.1		4	3.2	odd	2
297.2.g.a.197.1		4	33.32	even	2
891.2.d.a.890.2		4	9.7	even	3
891.2.d.a.890.2		4	99.43	odd	6
891.2.d.a.890.3		4	9.2	odd	6
891.2.d.a.890.3		4	99.65	even	6

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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.18614 - 1.26217i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-0.813859 + 0.469882i) q^{5} +(-0.186141 - 2.99422i) q^{9} +(-2.87228 - 1.65831i) q^{11} +(3.37228 + 0.792287i) q^{12} +(-0.372281 + 1.58457i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(-1.62772 - 0.939764i) q^{20} +(-2.87228 + 1.65831i) q^{23} +(-2.05842 + 3.56529i) q^{25} +(-4.00000 - 3.31662i) q^{27} +(-3.05842 - 5.29734i) q^{31} +(-5.50000 + 1.65831i) q^{33} +(5.00000 - 3.31662i) q^{36} +12.1168 q^{37} -6.63325i q^{44} +(1.55842 + 2.34941i) q^{45} +(11.8723 + 6.85446i) q^{47} +(2.00000 + 6.63325i) q^{48} +(-3.50000 - 6.06218i) q^{49} +11.8294i q^{53} +3.11684 q^{55} +(12.6861 - 7.32435i) q^{59} +(-3.11684 + 0.939764i) q^{60} -8.00000 q^{64} +(-7.55842 - 13.0916i) q^{67} +(-1.31386 + 5.59230i) q^{69} -10.8896i q^{71} +(2.05842 + 6.82701i) q^{75} -3.75906i q^{80} +(-8.93070 + 1.11469i) q^{81} +16.5831i q^{89} +(-5.74456 - 3.31662i) q^{92} +(-10.3139 - 2.42315i) q^{93} +(-0.0584220 + 0.101190i) q^{97} +(-4.43070 + 8.90892i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{4} - 9 q^{5} + 5 q^{9} + 2 q^{12} + 10 q^{15} - 8 q^{16} - 18 q^{20} + 9 q^{25} - 16 q^{27} + 5 q^{31} - 22 q^{33} + 20 q^{36} + 14 q^{37} - 11 q^{45} + 36 q^{47} + 8 q^{48} - 14 q^{49}+ \cdots + 11 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^4 - 9 * q^5 + 5 * q^9 + 2 * q^12 + 10 * q^15 - 8 * q^16 - 18 * q^20 + 9 * q^25 - 16 * q^27 + 5 * q^31 - 22 * q^33 + 20 * q^36 + 14 * q^37 - 11 * q^45 + 36 * q^47 + 8 * q^48 - 14 * q^49 - 22 * q^55 + 45 * q^59 + 22 * q^60 - 32 * q^64 - 13 * q^67 - 11 * q^69 - 9 * q^75 - 7 * q^81 - 47 * q^93 + 17 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more