LMFDB - Embedded newform 99.2.g.a.65.1

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			65.1
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	99.65
Dual form			99.2.g.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.68614 + 0.396143i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-3.68614 + 2.12819i) q^{5} +(2.68614 - 1.33591i) q^{9} +O(q^{10})$	$q+(-1.68614 + 0.396143i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-3.68614 + 2.12819i) q^{5} +(2.68614 - 1.33591i) q^{9} +(2.87228 + 1.65831i) q^{11} +(-2.37228 - 2.52434i) q^{12} +(5.37228 - 5.04868i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(-7.37228 - 4.25639i) q^{20} +(2.87228 - 1.65831i) q^{23} +(6.55842 - 11.3595i) q^{25} +(-4.00000 + 3.31662i) q^{27} +(5.55842 + 9.62747i) q^{31} +(-5.50000 - 1.65831i) q^{33} +(5.00000 + 3.31662i) q^{36} -5.11684 q^{37} +6.63325i q^{44} +(-7.05842 + 10.6410i) q^{45} +(6.12772 + 3.53784i) q^{47} +(2.00000 - 6.63325i) q^{48} +(-3.50000 - 6.06218i) q^{49} -1.43710i q^{53} -14.1168 q^{55} +(9.81386 - 5.66603i) q^{59} +(14.1168 + 4.25639i) q^{60} -8.00000 q^{64} +(1.05842 + 1.83324i) q^{67} +(-4.18614 + 3.93398i) q^{69} +5.69349i q^{71} +(-6.55842 + 21.7518i) q^{75} -17.0256i q^{80} +(5.43070 - 7.17687i) q^{81} -16.5831i q^{89} +(5.74456 + 3.31662i) q^{92} +(-13.1861 - 14.0313i) q^{93} +(8.55842 - 14.8236i) q^{97} +(9.93070 + 0.617359i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{4} - 9 q^{5} + 5 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^4 - 9 * q^5 + 5 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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	$0.166667\pi$
$3$	−1.68614	+	0.396143i	−0.973494	+	0.228714i
$3$	−1.68614	+	0.396143i	−0.973494	+	0.228714i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−3.68614	+	2.12819i	−1.64849	+	0.951757i	−0.670820	+	0.741620i	$0.734058\pi$
$5$	−3.68614	+	2.12819i	−1.64849	+	0.951757i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		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
$9$	2.68614	−	1.33591i	0.895380	−	0.445302i
$10$		0			0
$11$	2.87228	+	1.65831i	0.866025	+	0.500000i
$11$	2.87228	+	1.65831i	0.866025	+	0.500000i
$12$	−2.37228	−	2.52434i	−0.684819	−	0.728714i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	5.37228	−	5.04868i	1.38712	−	1.30356i
$16$	−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			$\pi$
$20$	−7.37228	−	4.25639i	−1.64849	−	0.951757i
$21$		0			0
$22$		0			0
$23$	2.87228	−	1.65831i	0.598912	−	0.345782i	−0.169701	−	0.985496i	$-0.554280\pi$
$23$	2.87228	−	1.65831i	0.598912	−	0.345782i	0.768613	+	0.639713i	$0.220947\pi$
$24$		0			0
$25$	6.55842	−	11.3595i	1.31168	−	2.27190i
$26$		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
$31$	5.55842	+	9.62747i	0.998322	+	1.72914i	0.549309	+	0.835619i	$0.314891\pi$
$31$	5.55842	+	9.62747i	0.998322	+	1.72914i	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
$37$	−5.11684			−0.841204			−0.420602	−	0.907245i	$-0.638181\pi$
$37$	−5.11684			−0.841204			−0.420602	+	0.907245i	$0.638181\pi$
$38$		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
$45$	−7.05842	+	10.6410i	−1.05221	+	1.58626i
$46$		0			0
$47$	6.12772	+	3.53784i	0.893820	+	0.516047i	0.875190	−	0.483779i	$-0.160736\pi$
$47$	6.12772	+	3.53784i	0.893820	+	0.516047i	0.0186297	+	0.999826i	$0.494070\pi$
$48$	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
$53$		−	1.43710i		−	0.197400i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$		−	1.43710i		−	0.197400i	0.995117	−	0.0987002i	$-0.0314685\pi$
$54$		0			0
$55$	−14.1168			−1.90351
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.81386	−	5.66603i	1.27766	−	0.737655i	0.301239	−	0.953549i	$-0.402600\pi$
$59$	9.81386	−	5.66603i	1.27766	−	0.737655i	0.976417	+	0.215894i	$0.0692665\pi$
$60$	14.1168	+	4.25639i	1.82248	+	0.549497i
$61$		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
$67$	1.05842	+	1.83324i	0.129307	+	0.223966i	0.923408	−	0.383819i	$-0.125391\pi$
$67$	1.05842	+	1.83324i	0.129307	+	0.223966i	−0.794101	+	0.607785i	$0.792058\pi$
$68$		0			0
$69$	−4.18614	+	3.93398i	−0.503952	+	0.473596i
$70$		0			0
$71$			5.69349i			0.675692i	0.941201	+	0.337846i	$0.109698\pi$
$71$			5.69349i			0.675692i	−0.941201	+	0.337846i	$0.890302\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−6.55842	+	21.7518i	−0.757301	+	2.51168i
$76$		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	$0.666667\pi$
$80$		−	17.0256i		−	1.90351i
$81$	5.43070	−	7.17687i	0.603411	−	0.797430i
$82$		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	$-0.658275\pi$
$89$		−	16.5831i		−	1.75781i	0.476999	−	0.878904i	$-0.341725\pi$
$90$		0			0
$91$		0			0
$92$	5.74456	+	3.31662i	0.598912	+	0.345782i
$93$	−13.1861	−	14.0313i	−1.36734	−	1.45498i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.55842	−	14.8236i	0.868976	−	1.50511i	0.00593185	−	0.999982i	$-0.498112\pi$
$97$	8.55842	−	14.8236i	0.868976	−	1.50511i	0.863044	−	0.505128i	$-0.168555\pi$
$98$		0			0
$99$	9.93070	+	0.617359i	0.998073	+	0.0620469i
$100$	26.2337			2.62337
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−9.61684	−	16.6569i	−0.947576	−	1.64125i	−0.750510	−	0.660859i	$-0.770192\pi$
$103$	−9.61684	−	16.6569i	−0.947576	−	1.64125i	−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	$0.5\pi$
$108$	−9.74456	−	3.61158i	−0.937671	−	0.347524i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	8.62772	−	2.02700i	0.818907	−	0.192395i
$112$		0			0
$113$	−17.1861	+	9.92242i	−1.61674	+	0.933423i	−0.628979	+	0.777422i	$0.716527\pi$
$113$	−17.1861	+	9.92242i	−1.61674	+	0.933423i	−0.987757	+	0.156001i	$0.950140\pi$
$114$		0			0
$115$	−7.05842	+	12.2255i	−0.658201	+	1.14004i
$116$		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
$124$	−11.1168	+	19.2549i	−0.998322	+	1.72914i
$125$			34.5484i			3.09011i
$126$		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	$0.833333\pi$
$132$	−2.62772	−	11.1846i	−0.228714	−	0.973494i
$133$		0			0
$134$		0			0
$135$	7.68614	−	20.7383i	0.661518	−	1.78487i
$136$		0			0
$137$	12.3030	+	7.10313i	1.05111	+	0.606861i	0.922961	−	0.384893i	$-0.125762\pi$
$137$	12.3030	+	7.10313i	1.05111	+	0.606861i	0.128154	+	0.991754i	$0.459095\pi$
$138$		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
$141$	−11.7337	−	3.53784i	−0.988155	−	0.297940i
$142$		0			0
$143$		0			0
$144$	−0.744563	+	11.9769i	−0.0620469	+	0.998073i
$145$		0			0
$146$		0			0
$147$	8.30298	+	8.83518i	0.684819	+	0.728714i
$148$	−5.11684	−	8.86263i	−0.420602	−	0.728504i
$149$		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
$155$	−40.9783	−	23.6588i	−3.29145	−	1.90032i
$156$		0			0
$157$	10.0584	+	17.4217i	0.802749	+	1.39040i	0.917800	+	0.397043i	$0.129964\pi$
$157$	10.0584	+	17.4217i	0.802749	+	1.39040i	−0.115050	+	0.993360i	$0.536703\pi$
$158$		0			0
$159$	0.569297	+	2.42315i	0.0451482	+	0.192168i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	25.2337			1.97645			0.988227	−	0.152992i	$-0.0488907\pi$
$163$	25.2337			1.97645			0.988227	+	0.152992i	$0.0488907\pi$
$164$		0			0
$165$	23.8030	−	5.59230i	1.85306	−	0.435360i
$166$		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
$177$	−14.3030	+	13.4414i	−1.07508	+	1.01032i
$178$		0			0
$179$		−	9.89497i		−	0.739585i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$		−	9.89497i		−	0.739585i	0.929114	−	0.369792i	$-0.120571\pi$
$180$	−25.4891	−	1.58457i	−1.89985	−	0.118107i
$181$	3.88316			0.288633			0.144316	−	0.989532i	$-0.453902\pi$
$181$	3.88316			0.288633			0.144316	+	0.989532i	$0.453902\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.8614	−	10.8896i	1.38672	−	0.800622i
$186$		0			0
$187$		0			0
$188$			14.1514i			1.03209i
$189$		0			0
$190$		0			0
$191$	−1.19702	−	0.691097i	−0.0866130	−	0.0500060i	0.456068	−	0.889945i	$-0.349257\pi$
$191$	−1.19702	−	0.691097i	−0.0866130	−	0.0500060i	−0.542681	+	0.839939i	$0.682591\pi$
$192$	13.4891	−	3.16915i	0.973494	−	0.228714i
$193$		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
$199$	7.23369			0.512783			0.256391	−	0.966573i	$-0.417466\pi$
$199$	7.23369			0.512783			0.256391	+	0.966573i	$0.417466\pi$
$200$		0			0
$201$	−2.51087	−	2.67181i	−0.177103	−	0.188455i
$202$		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	$0.333333\pi$
$212$	2.48913	−	1.43710i	0.170954	−	0.0987002i
$213$	−2.25544	−	9.60002i	−0.154540	−	0.657782i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−14.1168	−	24.4511i	−0.951757	−	1.64849i
$221$		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
$225$	2.44158	−	39.2747i	0.162772	−	2.61831i
$226$		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
$235$	−30.1168			−1.96461
$236$	19.6277	+	11.3321i	1.27766	+	0.737655i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$	6.74456	+	28.7075i	0.435360	+	1.85306i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	−6.31386	+	14.2525i	−0.405034	+	0.914302i
$244$		0			0
$245$	25.8030	+	14.8974i	1.64849	+	0.951757i
$246$		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
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−22.9783	+	13.2665i	−1.43334	+	0.827541i	−0.997374	−	0.0724199i	$-0.976928\pi$
$257$	−22.9783	+	13.2665i	−1.43334	+	0.827541i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		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
$265$	3.05842	+	5.29734i	0.187877	+	0.325413i
$266$		0			0
$267$	6.56930	+	27.9615i	0.402034	+	1.71121i
$268$	−2.11684	+	3.66648i	−0.129307	+	0.223966i
$269$		−	32.6140i		−	1.98851i	−0.107031	−	0.994256i	$-0.534134\pi$
$269$		−	32.6140i		−	1.98851i	0.107031	−	0.994256i	$-0.465866\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	37.6753	−	21.7518i	2.27190	−	1.31168i
$276$	−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
$279$	27.7921	+	18.4352i	1.66387	+	1.10369i
$280$		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	$0.333333\pi$
$284$	−9.86141	+	5.69349i	−0.585167	+	0.337846i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	−8.55842	+	28.3851i	−0.501704	+	1.66396i
$292$		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
$295$	−24.1168	+	41.7716i	−1.40414	+	2.43204i
$296$		0			0
$297$	−16.9891	+	2.89303i	−0.985809	+	0.167871i
$298$		0			0
$299$		0			0
$300$	−44.2337	+	10.3923i	−2.55383	+	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
$309$	22.8139	+	24.2762i	1.29784	+	1.38102i
$310$		0			0
$311$	−23.3614	+	13.4877i	−1.32470	+	0.764818i	−0.984475	−	0.175525i	$-0.943838\pi$
$311$	−23.3614	+	13.4877i	−1.32470	+	0.764818i	−0.340229	+	0.940343i	$0.610505\pi$
$312$		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	−0.185514	−	0.982642i	$-0.559395\pi$
$317$	−20.1060	−	11.6082i	−1.12926	−	0.651981i	−0.943750	+	0.330661i	$0.892728\pi$
$318$		0			0
$319$		0			0
$320$	29.4891	−	17.0256i	1.64849	−	0.951757i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	17.8614	+	2.22938i	0.992300	+	0.123855i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.0584	−	22.6179i	0.717756	−	1.24319i	−0.244131	−	0.969742i	$-0.578503\pi$
$331$	13.0584	−	22.6179i	0.717756	−	1.24319i	0.961887	−	0.273447i	$-0.0881639\pi$
$332$		0			0
$333$	−13.7446	+	6.83563i	−0.753197	+	0.374590i
$334$		0			0
$335$	−7.80298	−	4.50506i	−0.426323	−	0.246137i
$336$		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
$339$	25.0475	−	23.5388i	1.36040	−	1.27845i
$340$		0			0
$341$			36.8704i			1.99664i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	7.05842	−	23.4101i	0.380013	−	1.26036i
$346$		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	$0.0896796\pi$
$353$	31.5951	+	18.2414i	1.68164	+	0.970894i	0.721063	+	0.692869i	$0.243654\pi$
$354$		0			0
$355$	−12.1168	−	20.9870i	−0.643095	−	1.11387i
$356$	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
$363$	−13.0475	−	13.8839i	−0.684819	−	0.728714i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−4.94158	+	8.55906i	−0.257948	+	0.446780i	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−4.94158	+	8.55906i	−0.257948	+	0.446780i	0.707744	+	0.706469i	$0.249713\pi$
$368$			13.2665i			0.691564i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	11.1168	−	36.8704i	0.576382	−	1.91164i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	−13.6861	−	58.2535i	−0.706749	−	3.00820i
$376$		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
$383$	−30.6861	+	17.7167i	−1.56799	+	0.905279i	−0.571585	+	0.820543i	$0.693671\pi$
$383$	−30.6861	+	17.7167i	−1.56799	+	0.905279i	−0.996403	+	0.0847358i	$0.972995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	34.2337			1.73795
$389$	−27.0475	−	15.6159i	−1.37137	−	0.791758i	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−27.0475	−	15.6159i	−1.37137	−	0.791758i	−0.991100	+	0.133120i	$0.957501\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	8.86141	+	17.8178i	0.445302	+	0.895380i
$397$	−35.4674			−1.78006			−0.890028	−	0.455905i	$-0.849316\pi$
$397$	−35.4674			−1.78006			−0.890028	+	0.455905i	$0.849316\pi$
$398$		0			0
$399$		0			0
$400$	26.2337	+	45.4381i	1.31168	+	2.27190i
$401$	−11.0109	+	6.35713i	−0.549857	+	0.317460i	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−11.0109	+	6.35713i	−0.549857	+	0.317460i	0.199207	+	0.979957i	$0.436163\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−4.74456	+	38.0125i	−0.235759	+	1.88886i
$406$		0			0
$407$	−14.6970	−	8.48533i	−0.728504	−	0.420602i
$408$		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
$411$	−23.5584	−	7.10313i	−1.16205	−	0.350372i
$412$	19.2337	−	33.3137i	0.947576	−	1.64125i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.63859	−	2.10074i	0.177757	−	0.102628i	−0.408481	−	0.912767i	$-0.633942\pi$
$419$	3.63859	−	2.10074i	0.177757	−	0.102628i	0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	14.7337	−	25.5195i	0.718076	−	1.24374i	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	14.7337	−	25.5195i	0.718076	−	1.24374i	0.961761	−	0.273890i	$-0.0883103\pi$
$422$		0			0
$423$	21.1861	+	1.31707i	1.03011	+	0.0640382i
$424$		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	$0.5\pi$
$432$	−3.48913	−	20.4897i	−0.167871	−	0.985809i
$433$	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
$443$	−0.0475473	−	0.0274514i	−0.00225904	−	0.00130426i	0.498870	−	0.866677i	$-0.333748\pi$
$443$	−0.0475473	−	0.0274514i	−0.00225904	−	0.00130426i	−0.501129	+	0.865373i	$0.667082\pi$
$444$	12.1386	+	12.9166i	0.576072	+	0.612997i
$445$	35.2921	+	61.1277i	1.67301	+	2.89773i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.4834i		−	1.20264i	−0.799009	−	0.601319i	$-0.794642\pi$
$449$		−	25.4834i		−	1.20264i	0.799009	−	0.601319i	$-0.205358\pi$
$450$		0			0
$451$		0			0
$452$	−34.3723	−	19.8448i	−1.61674	−	0.933423i
$453$		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
$460$	−28.2337			−1.31640
$461$		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
$465$	78.4674	+	23.6588i	3.63884	+	1.09715i
$466$		0			0
$467$		−	24.1561i		−	1.11781i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$		−	24.1561i		−	1.11781i	0.829231	−	0.558906i	$-0.188779\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−23.8614	−	25.3909i	−1.09948	−	1.16995i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.91983	−	3.86025i	−0.0879029	−	0.176748i
$478$		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
$485$			72.8559i			3.30822i
$486$		0			0
$487$	12.8832			0.583792			0.291896	−	0.956450i	$-0.405714\pi$
$487$	12.8832			0.583792			0.291896	+	0.956450i	$0.405714\pi$
$488$		0			0
$489$	−42.5475	+	9.99616i	−1.92407	+	0.452042i
$490$		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
$495$	−37.9198	+	18.8588i	−1.70437	+	0.847640i
$496$	−44.4674			−1.99664
$497$		0			0
$498$		0			0
$499$	−18.6168	−	32.2453i	−0.833404	−	1.44350i	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−18.6168	−	32.2453i	−0.833404	−	1.44350i	0.0619186	−	0.998081i	$-0.480278\pi$
$500$	−59.8397	+	34.5484i	−2.67611	+	1.54505i
$501$		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	−0.434992	−	0.900434i	$-0.643249\pi$
$509$	2.87228	−	1.65831i	0.127312	−	0.0735034i	0.562303	+	0.826931i	$0.309915\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	70.8981	+	40.9330i	3.12414	+	1.80372i
$516$		0			0
$517$	11.7337	+	20.3233i	0.516047	+	0.893820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.56768i		−	0.375357i	−0.982231	−	0.187678i	$-0.939904\pi$
$521$		−	8.56768i		−	0.375357i	0.982231	−	0.187678i	$-0.0600963\pi$
$522$		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
$528$	16.7446	−	15.7359i	0.728714	−	0.684819i
$529$	−6.00000	+	10.3923i	−0.260870	+	0.451839i
$530$		0			0
$531$	18.7921	−	28.3302i	0.815508	−	1.22942i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.91983	+	16.6843i	0.169153	+	0.719981i
$538$		0			0
$539$		−	23.2164i		−	1.00000i
$540$	43.6060	−	7.42554i	1.87650	−	0.319544i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−6.54755	+	1.53829i	−0.280982	+	0.0660142i
$544$		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	$0.666667\pi$
$548$			28.4125i			1.21372i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−27.4891	+	25.8333i	−1.16685	+	1.09656i
$556$		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	$0.833333\pi$
$564$	−5.60597	−	23.8612i	−0.236054	−	1.00474i
$565$	42.2337	−	73.1509i	1.77678	−	3.07748i
$566$		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
$573$	2.29211	+	0.691097i	0.0957543	+	0.0288710i
$574$		0			0
$575$		−	43.5036i		−	1.81423i
$576$	−21.4891	+	10.6873i	−0.895380	+	0.445302i
$577$	−32.1168			−1.33704			−0.668521	−	0.743693i	$-0.733072\pi$
$577$	−32.1168			−1.33704			−0.668521	+	0.743693i	$0.733072\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.38316	−	4.12775i	0.0987002	−	0.170954i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.1277	+	19.1263i	1.36733	+	0.789427i	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	33.1277	+	19.1263i	1.36733	+	0.789427i	0.376741	+	0.926319i	$0.377045\pi$
$588$	−7.00000	+	23.2164i	−0.288675	+	0.957427i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	10.2337	−	17.7253i	0.420602	−	0.728504i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.1970	+	2.86558i	−0.499191	+	0.117280i
$598$		0			0
$599$	28.7228	−	16.5831i	1.17358	−	0.677568i	0.219061	−	0.975711i	$-0.429701\pi$
$599$	28.7228	−	16.5831i	1.17358	−	0.677568i	0.954521	+	0.298143i	$0.0963673\pi$
$600$		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
$603$	5.29211	+	3.51039i	0.215511	+	0.142954i
$604$		0			0
$605$	−40.5475	−	23.4101i	−1.64849	−	0.951757i
$606$		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
$617$	42.9891	−	24.8198i	1.73068	−	0.999207i	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.9891	−	24.8198i	1.73068	−	0.999207i	0.885249	−	0.465118i	$-0.153988\pi$
$618$		0			0
$619$	−21.7921	+	37.7450i	−0.875899	+	1.51710i	−0.0200967	+	0.999798i	$0.506397\pi$
$619$	−21.7921	+	37.7450i	−0.875899	+	1.51710i	−0.855802	+	0.517303i	$0.826936\pi$
$620$		−	94.6352i		−	3.80064i
$621$	−5.98913	+	16.1595i	−0.240335	+	0.648460i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−40.7337	−	70.5528i	−1.62935	−	2.82211i
$626$		0			0
$627$		0			0
$628$	−20.1168	+	34.8434i	−0.802749	+	1.39040i
$629$		0			0
$630$		0			0
$631$	46.5842			1.85449			0.927244	−	0.374457i	$-0.122171\pi$
$631$	46.5842			1.85449			0.927244	+	0.374457i	$0.122171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−3.62772	+	3.40920i	−0.143848	+	0.135184i
$637$		0			0
$638$		0			0
$639$	7.60597	+	15.2935i	0.300887	+	0.605002i
$640$		0			0
$641$	−20.1060	−	11.6082i	−0.794138	−	0.458496i	0.0472793	−	0.998882i	$-0.484945\pi$
$641$	−20.1060	−	11.6082i	−0.794138	−	0.458496i	−0.841417	+	0.540386i	$0.818278\pi$
$642$		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	$0.321914\pi$
$647$			43.1161i			1.69507i	−0.530740	+	0.847535i	$0.678086\pi$
$648$		0			0
$649$	37.5842			1.47531
$650$		0			0
$651$		0			0
$652$	25.2337	+	43.7060i	0.988227	+	1.71166i
$653$	36.8139	−	21.2545i	1.44064	−	0.831753i	0.442746	−	0.896647i	$-0.354005\pi$
$653$	36.8139	−	21.2545i	1.44064	−	0.831753i	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	$0.166667\pi$
$660$	33.4891	+	35.6357i	1.30356	+	1.38712i
$661$	−24.7921	−	42.9412i	−0.964301	−	1.67022i	−0.711481	−	0.702706i	$-0.751975\pi$
$661$	−24.7921	−	42.9412i	−0.964301	−	1.67022i	−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
$675$	11.4416	+	67.1899i	0.440387	+	2.58614i
$676$	−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
$683$			44.0010i			1.68365i	0.539750	+	0.841825i	$0.318519\pi$
$683$			44.0010i			1.68365i	−0.539750	+	0.841825i	$0.681481\pi$
$684$		0			0
$685$	−60.4674			−2.31034
$686$		0			0
$687$	5.93070	+	6.31084i	0.226270	+	0.240774i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−17.2921	+	29.9508i	−0.657823	+	1.13938i	0.323355	+	0.946278i	$0.395189\pi$
$691$	−17.2921	+	29.9508i	−0.657823	+	1.13938i	−0.981178	+	0.193105i	$0.938144\pi$
$692$		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
$705$	50.7812	−	11.9306i	1.91253	−	0.449332i
$706$		0			0
$707$		0			0
$708$	−37.5842	−	11.3321i	−1.41250	−	0.425885i
$709$	−26.2921	+	45.5393i	−0.987421	+	1.71026i	−0.356780	+	0.934188i	$0.616125\pi$
$709$	−26.2921	+	45.5393i	−0.987421	+	1.71026i	−0.630641	+	0.776075i	$0.717208\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31.9307	+	18.4352i	1.19581	+	0.690404i
$714$		0			0
$715$		0			0
$716$	17.1386	−	9.89497i	0.640499	−	0.369792i
$717$		0			0
$718$		0			0
$719$			52.4589i			1.95639i	0.207700	+	0.978193i	$0.433402\pi$
$719$			52.4589i			1.95639i	−0.207700	+	0.978193i	$0.566598\pi$
$720$	−22.7446	−	45.7330i	−0.847640	−	1.70437i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	3.88316	+	6.72582i	0.144316	+	0.249963i
$725$		0			0
$726$		0			0
$727$	17.5584	−	30.4121i	0.651206	−	1.12792i	−0.331625	−	0.943411i	$-0.607597\pi$
$727$	17.5584	−	30.4121i	0.651206	−	1.12792i	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
$735$	−49.4090	−	14.8974i	−1.82248	−	0.549497i
$736$		0			0
$737$			7.02078i			0.258614i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	37.7228	+	21.7793i	1.38672	+	0.800622i
$741$		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
$751$	−15.7921	−	27.3527i	−0.576262	−	0.998116i	−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	−15.7921	−	27.3527i	−0.576262	−	0.998116i	0.419641	−	0.907690i	$-0.362156\pi$
$752$	−24.5109	+	14.1514i	−0.893820	+	0.516047i
$753$	6.56930	+	27.9615i	0.239398	+	1.01897i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−53.4674			−1.94330			−0.971652	−	0.236414i	$-0.924028\pi$
$757$	−53.4674			−1.94330			−0.971652	+	0.236414i	$0.924028\pi$
$758$		0			0
$759$	−18.5475	+	4.35758i	−0.673233	+	0.158170i
$760$		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
$764$		−	2.76439i		−	0.100012i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	18.9783	+	20.1947i	0.684819	+	0.728714i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	33.4891	−	31.4719i	1.20608	−	1.13343i
$772$		0			0
$773$			13.2665i			0.477163i	0.971123	+	0.238581i	$0.0766824\pi$
$773$			13.2665i			0.477163i	−0.971123	+	0.238581i	$0.923318\pi$
$774$		0			0
$775$	145.818			5.23793
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.44158	+	16.3533i	−0.337846	+	0.585167i
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	−74.1535	−	42.8126i	−2.64665	−	1.52805i
$786$		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
$795$	−7.25544	−	7.72049i	−0.257324	−	0.273818i
$796$	7.23369	+	12.5291i	0.256391	+	0.444083i
$797$	22.1644	−	12.7966i	0.785103	−	0.453279i	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	22.1644	−	12.7966i	0.785103	−	0.453279i	0.838236	+	0.545308i	$0.183587\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−22.1535	−	44.5446i	−0.782756	−	1.57391i
$802$		0			0
$803$		0			0
$804$	2.11684	−	7.02078i	0.0746553	−	0.247604i
$805$		0			0
$806$		0			0
$807$	12.9198	+	54.9918i	0.454800	+	1.93580i
$808$		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
$815$	−93.0149	+	53.7022i	−3.25817	+	1.88111i
$816$		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
$825$	−54.9090	+	51.6014i	−1.91168	+	1.79653i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	19.8614	+	1.23472i	0.690232	+	0.0429094i
$829$	28.5842			0.992771			0.496385	−	0.868102i	$-0.334660\pi$
$829$	28.5842			0.992771			0.496385	+	0.868102i	$0.334660\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−54.1644	−	20.0747i	−1.87220	−	0.693882i
$838$		0			0
$839$	31.5951	+	18.2414i	1.09078	+	0.629764i	0.933785	−	0.357834i	$-0.116485\pi$
$839$	31.5951	+	18.2414i	1.09078	+	0.629764i	0.156999	+	0.987599i	$0.449818\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	55.3331i		−	1.90351i
$846$		0			0
$847$		0			0
$848$	4.97825	+	2.87419i	0.170954	+	0.0987002i
$849$		0			0
$850$		0			0
$851$	−14.6970	+	8.48533i	−0.503807	+	0.290873i
$852$	14.3723	−	13.5065i	0.492386	−	0.462727i
$853$		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
$859$	−29.2921	−	50.7354i	−0.999434	−	1.73107i	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−29.2921	−	50.7354i	−0.999434	−	1.73107i	−0.470581	−	0.882357i	$-0.655956\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	46.4327i		−	1.58059i	−0.612727	−	0.790295i	$-0.709928\pi$
$863$		−	46.4327i		−	1.58059i	0.612727	−	0.790295i	$-0.290072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	28.6644	−	6.73444i	0.973494	−	0.228714i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	3.18614	−	51.2516i	0.107835	−	1.73460i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$	28.2337	−	48.9022i	0.951757	−	1.64849i
$881$		−	41.0719i		−	1.38375i	−0.722019	−	0.691873i	$-0.756786\pi$
$881$		−	41.0719i		−	1.38375i	0.722019	−	0.691873i	$-0.243214\pi$
$882$		0			0
$883$	−10.7663			−0.362315			−0.181158	−	0.983454i	$-0.557984\pi$
$883$	−10.7663			−0.362315			−0.181158	+	0.983454i	$0.557984\pi$
$884$		0			0
$885$	24.1168	−	79.9865i	0.810679	−	2.68872i
$886$		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
$895$	21.0584	+	36.4743i	0.703905	+	1.21920i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	70.4674	−	35.0458i	2.34891	−	1.16819i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.3139	+	8.26411i	−0.475809	+	0.274708i
$906$		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
$911$	−47.8723	−	27.6391i	−1.58608	−	0.915723i	−0.993944	−	0.109885i	$-0.964952\pi$
$911$	−47.8723	−	27.6391i	−1.58608	−	0.915723i	−0.592135	−	0.805839i	$-0.701715\pi$
$912$		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
$925$	−33.5584	+	58.1249i	−1.10339	+	1.91113i
$926$		0			0
$927$	−48.0842	−	31.8955i	−1.57929	−	1.04758i
$928$		0			0
$929$	45.4783	+	26.2569i	1.49209	+	0.861460i	0.999959	−	0.00905914i	$-0.00288365\pi$
$929$	45.4783	+	26.2569i	1.49209	+	0.861460i	0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	34.0475	−	31.9967i	1.11467	−	1.04752i
$934$		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
$940$	−30.1168	−	52.1639i	−0.982303	−	1.70140i
$941$		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
$944$			45.3283i			1.47531i
$945$		0			0
$946$		0			0
$947$	52.8030	+	30.4858i	1.71587	+	0.990656i	0.926126	+	0.377215i	$0.123118\pi$
$947$	52.8030	+	30.4858i	1.71587	+	0.990656i	0.789741	+	0.613441i	$0.210215\pi$
$948$		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
$955$	5.88316			0.190374
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−42.9783	+	40.3894i	−1.38712	+	1.30356i
$961$	−46.2921	+	80.1803i	−1.49329	+	2.58646i
$962$		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	$0.243196\pi$
$971$			43.1161i			1.38366i	−0.722059	+	0.691831i	$0.756804\pi$
$972$	−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	$0.691176\pi$
$977$	−48.8288	+	28.1913i	−1.56217	+	0.901920i	−0.997037	+	0.0769208i	$0.975491\pi$
$978$		0			0
$979$	27.5000	−	47.6314i	0.878904	−	1.52231i
$980$			59.5894i			1.90351i
$981$		0			0
$982$		0			0
$983$	−54.0475	−	31.2044i	−1.72385	−	0.995265i	−0.910525	−	0.413453i	$-0.864323\pi$
$983$	−54.0475	−	31.2044i	−1.72385	−	0.995265i	−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
$993$	−13.0584	+	43.3099i	−0.414396	+	1.37440i
$994$		0			0
$995$	−26.6644	+	15.3947i	−0.845318	+	0.488045i
$996$		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
$999$	20.4674	−	16.9707i	0.647559	−	0.536928i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.2.g.a.65.1	yes	4
3.2	odd	2		297.2.g.a.197.2		4
9.2	odd	6		891.2.d.a.890.4		4
9.4	even	3		297.2.g.a.98.2		4
9.5	odd	6	inner	99.2.g.a.32.1	✓	4
9.7	even	3		891.2.d.a.890.1		4
11.10	odd	2	CM	99.2.g.a.65.1	yes	4
33.32	even	2		297.2.g.a.197.2		4
99.32	even	6	inner	99.2.g.a.32.1	✓	4
99.43	odd	6		891.2.d.a.890.1		4
99.65	even	6		891.2.d.a.890.4		4
99.76	odd	6		297.2.g.a.98.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.g.a.32.1	✓	4	9.5	odd	6	inner
99.2.g.a.32.1	✓	4	99.32	even	6	inner
99.2.g.a.65.1	yes	4	1.1	even	1	trivial
99.2.g.a.65.1	yes	4	11.10	odd	2	CM
297.2.g.a.98.2		4	9.4	even	3
297.2.g.a.98.2		4	99.76	odd	6
297.2.g.a.197.2		4	3.2	odd	2
297.2.g.a.197.2		4	33.32	even	2
891.2.d.a.890.1		4	9.7	even	3
891.2.d.a.890.1		4	99.43	odd	6
891.2.d.a.890.4		4	9.2	odd	6
891.2.d.a.890.4		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)

\(f(q)\)	\(=\)	\(q+(-1.68614 + 0.396143i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-3.68614 + 2.12819i) q^{5} +(2.68614 - 1.33591i) q^{9} +O(q^{10})\)	\(q+(-1.68614 + 0.396143i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-3.68614 + 2.12819i) q^{5} +(2.68614 - 1.33591i) q^{9} +(2.87228 + 1.65831i) q^{11} +(-2.37228 - 2.52434i) q^{12} +(5.37228 - 5.04868i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(-7.37228 - 4.25639i) q^{20} +(2.87228 - 1.65831i) q^{23} +(6.55842 - 11.3595i) q^{25} +(-4.00000 + 3.31662i) q^{27} +(5.55842 + 9.62747i) q^{31} +(-5.50000 - 1.65831i) q^{33} +(5.00000 + 3.31662i) q^{36} -5.11684 q^{37} +6.63325i q^{44} +(-7.05842 + 10.6410i) q^{45} +(6.12772 + 3.53784i) q^{47} +(2.00000 - 6.63325i) q^{48} +(-3.50000 - 6.06218i) q^{49} -1.43710i q^{53} -14.1168 q^{55} +(9.81386 - 5.66603i) q^{59} +(14.1168 + 4.25639i) q^{60} -8.00000 q^{64} +(1.05842 + 1.83324i) q^{67} +(-4.18614 + 3.93398i) q^{69} +5.69349i q^{71} +(-6.55842 + 21.7518i) q^{75} -17.0256i q^{80} +(5.43070 - 7.17687i) q^{81} -16.5831i q^{89} +(5.74456 + 3.31662i) q^{92} +(-13.1861 - 14.0313i) q^{93} +(8.55842 - 14.8236i) q^{97} +(9.93070 + 0.617359i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{4} - 9 q^{5} + 5 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^4 - 9 * q^5 + 5 * q^9	\( 4 q - q^{3} + 4 q^{4} - 9 q^{5} + 5 q^{9} + 2 q^{12} + 10 q^{15} - 8 q^{16} - 18 q^{20} + 9 q^{25} - 16 q^{27} + 5 q^{31} - 22 q^{33} + 20 q^{36} + 14 q^{37} - 11 q^{45} + 36 q^{47} + 8 q^{48} - 14 q^{49} - 22 q^{55} + 45 q^{59} + 22 q^{60} - 32 q^{64} - 13 q^{67} - 11 q^{69} - 9 q^{75} - 7 q^{81} - 47 q^{93} + 17 q^{97} + 11 q^{99}+O(q^{100}) \) 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

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