LMFDB - Embedded newform 399.2.br.a.2.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(2,399)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 6, 1]))

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

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

chi := DirichletCharacter("399.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.br (of order $18$, degree $6$, 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:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			2.1
Root			$0.939693 + 0.342020i$ of defining polynomial
Character	$\chi$	$=$	399.2
Dual form			399.2.br.a.200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70574 - 0.300767i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.64543 + 0.0412527i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(-1.70574 - 0.300767i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.64543 + 0.0412527i) q^{7} +(2.81908 + 1.02606i) q^{9} +3.46410i q^{12} +(-4.23783 - 5.05044i) q^{13} +(-3.75877 + 1.36808i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-0.837496 - 5.22481i) q^{28} +(-9.12108 - 5.26606i) q^{31} +(1.04189 - 5.90885i) q^{36} +(8.60014 + 4.96529i) q^{37} +(5.70961 + 9.88933i) q^{39} +(3.72416 - 1.35548i) q^{43} +(6.82295 - 1.20307i) q^{48} +(6.99660 + 0.218262i) q^{49} +(-8.47565 + 10.1009i) q^{52} +(-2.15523 + 7.23567i) q^{57} +(11.8931 - 9.97946i) q^{61} +(7.41534 + 2.83067i) q^{63} +(4.00000 + 6.92820i) q^{64} +(-5.60994 - 6.68566i) q^{67} +(-2.87686 + 16.3155i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-8.70233 + 0.519030i) q^{76} +(-2.37598 - 6.52796i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-0.142903 + 9.16404i) q^{84} +(-11.0025 - 13.5354i) q^{91} +(13.9743 + 11.7258i) q^{93} +(18.7631 + 3.30844i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 6 q^{13} + 3 q^{19} - 27 q^{21} - 27 q^{27} + 24 q^{43} - 12 q^{52} + 39 q^{61} + 24 q^{64} + 15 q^{67} - 21 q^{73} + 45 q^{75} + 39 q^{79} - 51 q^{91} + 54 q^{93}+O(q^{100}) $ 6 * q - 6 * q^13 + 3 * q^19 - 27 * q^21 - 27 * q^27 + 24 * q^43 - 12 * q^52 + 39 * q^61 + 24 * q^64 + 15 * q^67 - 21 * q^73 + 45 * q^75 + 39 * q^79 - 51 * q^91 + 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{18}\right)$

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		0.642788	−	0.766044i	$-0.277778\pi$
$2$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$3$	−1.70574	−	0.300767i	−0.984808	−	0.173648i
$3$	−1.70574	−	0.300767i	−0.984808	−	0.173648i
$4$	−0.347296	−	1.96962i	−0.173648	−	0.984808i
$5$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$5$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$6$		0			0
$7$	2.64543	+	0.0412527i	0.999878	+	0.0155920i
$7$	2.64543	+	0.0412527i	0.999878	+	0.0155920i
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			3.46410i			1.00000i
$13$	−4.23783	−	5.05044i	−1.17536	−	1.40074i	−0.898011	−	0.439972i	$-0.854988\pi$
$13$	−4.23783	−	5.05044i	−1.17536	−	1.40074i	−0.277350	−	0.960769i	$-0.589456\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	+	1.36808i	−0.939693	+	0.342020i
$17$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$17$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$18$		0			0
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$20$		0			0
$21$	−4.50000	−	0.866025i	−0.981981	−	0.188982i
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		0			0
$25$	−4.69846	−	1.71010i	−0.939693	−	0.342020i
$26$		0			0
$27$	−4.50000	−	2.59808i	−0.866025	−	0.500000i
$28$	−0.837496	−	5.22481i	−0.158272	−	0.987396i
$29$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$29$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$30$		0			0
$31$	−9.12108	−	5.26606i	−1.63819	−	0.945812i	−0.981455	−	0.191695i	$-0.938602\pi$
$31$	−9.12108	−	5.26606i	−1.63819	−	0.945812i	−0.656740	−	0.754117i	$-0.728065\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.04189	−	5.90885i	0.173648	−	0.984808i
$37$	8.60014	+	4.96529i	1.41385	+	0.816289i	0.995749	−	0.0921098i	$-0.0293611\pi$
$37$	8.60014	+	4.96529i	1.41385	+	0.816289i	0.418105	+	0.908399i	$0.362694\pi$
$38$		0			0
$39$	5.70961	+	9.88933i	0.914269	+	1.58356i
$40$		0			0
$41$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$41$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$42$		0			0
$43$	3.72416	−	1.35548i	0.567928	−	0.206709i	−0.0420659	−	0.999115i	$-0.513394\pi$
$43$	3.72416	−	1.35548i	0.567928	−	0.206709i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$47$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$48$	6.82295	−	1.20307i	0.984808	−	0.173648i
$49$	6.99660	+	0.218262i	0.999514	+	0.0311803i
$50$		0			0
$51$		0			0
$52$	−8.47565	+	10.1009i	−1.17536	+	1.40074i
$53$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$53$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.15523	+	7.23567i	−0.285467	+	0.958388i
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	11.8931	−	9.97946i	1.52275	−	1.27774i	0.690510	−	0.723323i	$-0.257386\pi$
$61$	11.8931	−	9.97946i	1.52275	−	1.27774i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$	7.41534	+	2.83067i	0.934246	+	0.356630i
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	−5.60994	−	6.68566i	−0.685363	−	0.816784i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	−5.60994	−	6.68566i	−0.685363	−	0.816784i	−0.990787	+	0.135433i	$0.956757\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$71$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$72$		0			0
$73$	−2.87686	+	16.3155i	−0.336711	+	1.90958i	0.0729331	+	0.997337i	$0.476764\pi$
$73$	−2.87686	+	16.3155i	−0.336711	+	1.90958i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	−8.70233	+	0.519030i	−0.998226	+	0.0595368i
$77$		0			0
$78$		0			0
$79$	−2.37598	−	6.52796i	−0.267319	−	0.734452i	−0.998626	−	0.0524041i	$-0.983312\pi$
$79$	−2.37598	−	6.52796i	−0.267319	−	0.734452i	0.731307	−	0.682048i	$-0.238911\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	−0.142903	+	9.16404i	−0.0155920	+	0.999878i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$89$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$90$		0			0
$91$	−11.0025	−	13.5354i	−1.15338	−	1.41890i
$92$		0			0
$93$	13.9743	+	11.7258i	1.44907	+	1.21591i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.7631	+	3.30844i	1.90510	+	0.335921i	0.996631	−	0.0820195i	$-0.0261370\pi$
$97$	18.7631	+	3.30844i	1.90510	+	0.335921i	0.908474	+	0.417941i	$0.137248\pi$
$98$		0			0
$99$		0			0
$100$	−1.73648	+	9.84808i	−0.173648	+	0.984808i
$101$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$101$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$102$		0			0
$103$	−16.5364	+	9.54731i	−1.62938	+	0.940724i	−0.645105	+	0.764094i	$0.723187\pi$
$103$	−16.5364	+	9.54731i	−1.62938	+	0.940724i	−0.984277	+	0.176631i	$0.943480\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−3.55438	+	9.76557i	−0.342020	+	0.939693i
$109$	13.3601	−	15.9219i	1.27966	−	1.52504i	0.566458	−	0.824090i	$-0.308313\pi$
$109$	13.3601	−	15.9219i	1.27966	−	1.52504i	0.713206	−	0.700954i	$-0.247242\pi$
$110$		0			0
$111$	−13.1762	−	11.0561i	−1.25063	−	1.04940i
$112$	−10.0000	+	3.46410i	−0.944911	+	0.327327i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−6.76470	−	18.5859i	−0.625397	−	1.71826i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$	−7.20439	+	19.7939i	−0.646974	+	1.77755i
$125$		0			0
$126$		0			0
$127$	7.79339	+	9.28780i	0.691551	+	0.824159i	0.991542	−	0.129783i	$-0.0414282\pi$
$127$	7.79339	+	9.28780i	0.691551	+	0.824159i	−0.299991	+	0.953942i	$0.596984\pi$
$128$		0			0
$129$	−6.76011	+	1.19199i	−0.595195	+	0.104949i
$130$		0			0
$131$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$131$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$132$		0			0
$133$	1.50134	−	11.4344i	0.130183	−	0.991490i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$137$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	15.1951	−	12.7502i	1.28883	−	1.08146i	0.296866	−	0.954919i	$-0.404058\pi$
$139$	15.1951	−	12.7502i	1.28883	−	1.08146i	0.991962	−	0.126536i	$-0.0403860\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	−11.8687	−	2.47665i	−0.978915	−	0.204270i
$148$	6.79292	−	18.6634i	0.558374	−	1.53412i
$149$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$149$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$150$		0			0
$151$	7.50000	−	4.33013i	0.610341	−	0.352381i	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	7.50000	−	4.33013i	0.610341	−	0.352381i	0.773099	+	0.634285i	$0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	17.4953	−	14.6803i	1.40074	−	1.17536i
$157$	2.01889	+	1.69405i	0.161125	+	0.135200i	0.719785	−	0.694197i	$-0.244240\pi$
$157$	2.01889	+	1.69405i	0.161125	+	0.135200i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.388003	+	0.672042i	0.0303908	+	0.0526384i	0.880821	−	0.473450i	$-0.156991\pi$
$163$	0.388003	+	0.672042i	0.0303908	+	0.0526384i	−0.850430	+	0.526088i	$0.823658\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$167$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$168$		0			0
$169$	−5.29039	+	30.0033i	−0.406953	+	2.30795i
$170$		0			0
$171$	5.85251	−	11.6939i	0.447553	−	0.894258i
$172$	−3.96316	−	6.86440i	−0.302188	−	0.523406i
$173$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$173$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$174$		0			0
$175$	−12.3589	−	4.71778i	−0.934246	−	0.356630i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−6.82295	+	1.20307i	−0.507146	+	0.0894235i	−0.421366	−	0.906891i	$-0.638449\pi$
$181$	−6.82295	+	1.20307i	−0.507146	+	0.0894235i	−0.0857797	+	0.996314i	$0.527338\pi$
$182$		0			0
$183$	−23.2879	+	13.4453i	−1.72149	+	0.993904i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−11.7973	−	7.05866i	−0.858124	−	0.513442i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−4.73917	−	13.0208i	−0.342020	−	0.939693i
$193$	19.6407	+	3.46318i	1.41377	+	0.249285i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	19.6407	+	3.46318i	1.41377	+	0.249285i	0.585979	+	0.810326i	$0.300710\pi$
$194$		0			0
$195$		0			0
$196$	−2.00000	−	13.8564i	−0.142857	−	0.989743i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−1.11128	+	0.932476i	−0.0787767	+	0.0661015i	−0.681326	−	0.731980i	$-0.738596\pi$
$199$	−1.11128	+	0.932476i	−0.0787767	+	0.0661015i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$	7.55825	+	13.0913i	0.533118	+	0.923387i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	22.8384	+	13.1858i	1.58356	+	0.914269i
$209$		0			0
$210$		0			0
$211$	−19.6643	−	3.46735i	−1.35375	−	0.238702i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−19.6643	−	3.46735i	−1.35375	−	0.238702i	−0.803005	+	0.595973i	$0.796767\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−23.9119	−	14.3073i	−1.62325	−	0.971240i
$218$		0			0
$219$	9.81433	−	26.9647i	0.663191	−	1.82210i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.87851	+	24.3935i	0.594549	+	1.63351i	0.761961	+	0.647623i	$0.224237\pi$
$223$	8.87851	+	24.3935i	0.594549	+	1.63351i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	−11.4907	−	9.64181i	−0.766044	−	0.642788i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	15.0000	+	1.73205i	0.993399	+	0.114708i
$229$	−1.74644	+	3.02493i	−0.115408	+	0.199893i	−0.917943	−	0.396713i	$-0.870151\pi$
$229$	−1.74644	+	3.02493i	−0.115408	+	0.199893i	0.802535	+	0.596606i	$0.203484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$233$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.08940	+	11.8496i	0.135721	+	0.769714i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−4.18298	+	11.4926i	−0.269450	+	0.740306i	0.728993	+	0.684521i	$0.239989\pi$
$241$	−4.18298	+	11.4926i	−0.269450	+	0.740306i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	−10.0201	−	11.9415i	−0.642788	−	0.766044i
$244$	−23.7861	−	19.9589i	−1.52275	−	1.27774i
$245$		0			0
$246$		0			0
$247$	−23.9880	+	15.8251i	−1.52632	+	1.00693i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$251$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$252$	3.00000	−	15.5885i	0.188982	−	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	−	10.2846i	0.766044	−	0.642788i
$257$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$257$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$258$		0			0
$259$	22.5462	+	13.4901i	1.40095	+	0.838235i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−11.2199	+	13.3713i	−0.685363	+	0.816784i
$269$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$269$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$270$		0			0
$271$	5.03580	−	28.5594i	0.305903	−	1.73486i	−0.313321	−	0.949647i	$-0.601442\pi$
$271$	5.03580	−	28.5594i	0.305903	−	1.73486i	0.619224	−	0.785214i	$-0.287447\pi$
$272$		0			0
$273$	14.6964	+	26.3971i	0.889467	+	1.59762i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	−20.3097	−	24.2042i	−1.21591	−	1.44907i
$280$		0			0
$281$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$281$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$282$		0			0
$283$	6.57785	−	2.39414i	0.391012	−	0.142317i	−0.139030	−	0.990288i	$-0.544398\pi$
$283$	6.57785	−	2.39414i	0.391012	−	0.142317i	0.530042	+	0.847971i	$0.322176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0228	−	10.9274i	0.766044	−	0.642788i
$290$		0			0
$291$	−31.0099	−	11.2867i	−1.81783	−	0.661636i
$292$	33.1343			1.93904
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	5.92396	−	16.2760i	0.342020	−	0.939693i
$301$	9.90791	−	3.43220i	0.571082	−	0.197829i
$302$		0			0
$303$		0			0
$304$	4.04458	+	16.9600i	0.231972	+	0.972722i
$305$		0			0
$306$		0			0
$307$	20.0401	−	23.8829i	1.14375	−	1.36307i	0.222112	−	0.975021i	$-0.428705\pi$
$307$	20.0401	−	23.8829i	1.14375	−	1.36307i	0.921639	−	0.388048i	$-0.126851\pi$
$308$		0			0
$309$	31.0783	−	11.3116i	1.76798	−	0.643493i
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−20.6732	−	7.52444i	−1.16852	−	0.425307i	−0.316387	−	0.948630i	$-0.602470\pi$
$313$	−20.6732	−	7.52444i	−1.16852	−	0.425307i	−0.852134	+	0.523324i	$0.824692\pi$
$314$		0			0
$315$		0			0
$316$	−12.0324	+	6.94691i	−0.676875	+	0.390794i
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	−	15.5885i	0.500000	−	0.866025i
$325$	11.2745	+	30.9764i	0.625397	+	1.71826i
$326$		0			0
$327$	−27.5776	+	23.1404i	−1.52504	+	1.27966i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−29.3041	+	16.9187i	−1.61070	+	0.929938i	−0.621492	+	0.783420i	$0.713473\pi$
$331$	−29.3041	+	16.9187i	−1.61070	+	0.929938i	−0.989208	+	0.146518i	$0.953193\pi$
$332$		0			0
$333$	19.1498	+	22.8218i	1.04940	+	1.25063i
$334$		0			0
$335$		0			0
$336$	18.0993	−	2.90117i	0.987396	−	0.158272i
$337$	3.84436	+	10.5623i	0.209416	+	0.575365i	0.999281	−	0.0379157i	$-0.0120718\pi$
$337$	3.84436	+	10.5623i	0.209416	+	0.575365i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5000	+	0.866025i	0.998906	+	0.0467610i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$347$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$348$		0			0
$349$	15.8418	+	27.4389i	0.847994	+	1.46877i	0.882996	+	0.469381i	$0.155523\pi$
$349$	15.8418	+	27.4389i	0.847994	+	1.46877i	−0.0350017	+	0.999387i	$0.511144\pi$
$350$		0			0
$351$	5.94878	+	33.7372i	0.317522	+	1.80076i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$359$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$	−18.7631	−	3.30844i	−0.984808	−	0.173648i
$364$	−22.8384	+	26.3715i	−1.19706	+	1.38224i
$365$		0			0
$366$		0			0
$367$	25.8957	+	9.42528i	1.35175	+	0.491996i	0.913493	−	0.406855i	$-0.133375\pi$
$367$	25.8957	+	9.42528i	1.35175	+	0.491996i	0.438254	+	0.898851i	$0.355597\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	18.2422	−	31.5964i	0.945812	−	1.63819i
$373$			29.4449i			1.52460i	0.647225	+	0.762299i	$0.275929\pi$
$373$			29.4449i			1.52460i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			14.4953i			0.744576i	0.928117	+	0.372288i	$0.121427\pi$
$379$			14.4953i			0.744576i	−0.928117	+	0.372288i	$0.878573\pi$
$380$		0			0
$381$	−10.5000	−	18.1865i	−0.537931	−	0.931724i
$382$		0			0
$383$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$383$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	11.8895			0.604377
$388$		−	38.1051i		−	1.93449i
$389$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$389$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.1573	+	9.36208i	0.559968	+	0.469869i	0.878300	−	0.478110i	$-0.158678\pi$
$397$	11.1573	+	9.36208i	0.559968	+	0.469869i	−0.318332	+	0.947979i	$0.603123\pi$
$398$		0			0
$399$	−6.00000	+	19.0526i	−0.300376	+	0.953821i
$400$	20.0000			1.00000
$401$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$401$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$402$		0			0
$403$	12.0576	+	68.3822i	0.600633	+	3.40636i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−16.7001	−	19.9024i	−0.825767	−	0.984111i	0.174232	−	0.984705i	$-0.444256\pi$
$409$	−16.7001	−	19.9024i	−0.825767	−	0.984111i	−1.00000		0.000593299i	$0.999811\pi$
$410$		0			0
$411$		0			0
$412$	24.5476	+	29.2547i	1.20937	+	1.44127i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−29.7536	+	17.1783i	−1.45704	+	0.841223i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	23.3802	−	27.8634i	1.13948	−	1.35798i	0.215060	−	0.976601i	$-0.431005\pi$
$421$	23.3802	−	27.8634i	1.13948	−	1.35798i	0.924419	−	0.381377i	$-0.124550\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	31.8739	−	25.9093i	1.54249	−	1.25384i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$431$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$432$	20.4688	+	3.60921i	0.984808	+	0.173648i
$433$	−3.14249	+	8.63393i	−0.151019	+	0.414920i	−0.992015	−	0.126121i	$-0.959747\pi$
$433$	−3.14249	+	8.63393i	−0.151019	+	0.414920i	0.840996	+	0.541041i	$0.181970\pi$
$434$		0			0
$435$		0			0
$436$	−36.0000	−	20.7846i	−1.72409	−	0.995402i
$437$		0			0
$438$		0			0
$439$	12.6759	−	15.1065i	0.604986	−	0.720995i	−0.373425	−	0.927660i	$-0.621817\pi$
$439$	12.6759	−	15.1065i	0.604986	−	0.720995i	0.978412	+	0.206666i	$0.0662612\pi$
$440$		0			0
$441$	19.5000	+	7.79423i	0.928571	+	0.371154i
$442$		0			0
$443$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$443$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$444$	−17.2003	+	29.7917i	−0.816289	+	1.41385i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	10.2959	+	18.4931i	0.486436	+	0.873716i
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−14.0954	+	5.13030i	−0.662259	+	0.241043i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.41029	−	4.17475i	0.112749	−	0.195286i	−0.804129	−	0.594455i	$-0.797368\pi$
$457$	2.41029	−	4.17475i	0.112749	−	0.195286i	0.916878	+	0.399169i	$0.130701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$	−20.4996	−	35.5063i	−0.952697	−	1.65012i	−0.739553	−	0.673098i	$-0.764963\pi$
$463$	−20.4996	−	35.5063i	−0.952697	−	1.65012i	−0.213144	−	0.977021i	$-0.568370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	−34.2576	+	19.7787i	−1.58356	+	0.914269i
$469$	−14.5649	−	17.9179i	−0.672544	−	0.827371i
$470$		0			0
$471$	−2.93417	−	3.49681i	−0.135200	−	0.161125i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.75418	+	19.4899i	−0.447553	+	0.894258i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$479$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$480$		0			0
$481$	−11.3690	−	64.4766i	−0.518380	−	2.93988i
$482$		0			0
$483$		0			0
$484$	−3.82026	−	21.6658i	−0.173648	−	0.984808i
$485$		0			0
$486$		0			0
$487$	3.00000	+	1.73205i	0.135943	+	0.0784867i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	3.00000	+	1.73205i	0.135943	+	0.0784867i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−0.459704	−	1.26302i	−0.0207885	−	0.0571160i
$490$		0			0
$491$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$491$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	41.4884	+	7.31553i	1.86289	+	0.328477i
$497$		0			0
$498$		0			0
$499$	4.36665	−	24.7645i	0.195478	−	1.10861i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	4.36665	−	24.7645i	0.195478	−	1.10861i	0.911736	−	0.410776i	$-0.134742\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$503$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	18.0480	−	49.5866i	0.801541	−	2.20222i
$508$	15.5868	−	18.5756i	0.691551	−	0.824159i
$509$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$509$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$510$		0			0
$511$	−8.28359	+	43.0428i	−0.366444	+	1.90410i
$512$		0			0
$513$	−13.5000	+	18.1865i	−0.596040	+	0.802955i
$514$		0			0
$515$		0			0
$516$	4.69553	+	12.9009i	0.206709	+	0.567928i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−15.4093	+	42.3366i	−0.673800	+	1.85125i	−0.174908	+	0.984585i	$0.555963\pi$
$523$	−15.4093	+	42.3366i	−0.673800	+	1.85125i	−0.498892	+	0.866664i	$0.666259\pi$
$524$		0			0
$525$	19.6621	+	11.7644i	0.858124	+	0.513442i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.99391	−	22.6506i	0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$	−23.0428	+	1.01406i	−0.999033	+	0.0439652i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−38.3923	−	13.9737i	−1.65062	−	0.600775i	−0.661768	−	0.749708i	$-0.730194\pi$
$541$	−38.3923	−	13.9737i	−1.65062	−	0.600775i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	12.0000			0.514969
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.1471	−	41.6163i	0.647642	−	1.77938i	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	15.1471	−	41.6163i	0.647642	−	1.77938i	0.626264	−	0.779611i	$-0.284583\pi$
$548$		0			0
$549$	43.7670	−	15.9299i	1.86793	−	0.679871i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−6.01620	−	17.3673i	−0.255835	−	0.738531i
$554$		0			0
$555$		0			0
$556$	−30.3901	−	25.5003i	−1.28883	−	1.08146i
$557$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$557$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$558$		0			0
$559$	−22.6281	−	13.0643i	−0.957067	−	0.552563i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$		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
$571$	−8.34507	−	14.4541i	−0.349231	−	0.604885i	0.636882	−	0.770961i	$-0.280224\pi$
$571$	−8.34507	−	14.4541i	−0.349231	−	0.604885i	−0.986113	+	0.166076i	$0.946890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	4.16756	+	23.6354i	0.173648	+	0.984808i
$577$	17.5000	−	30.3109i	0.728535	−	1.26186i	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	17.5000	−	30.3109i	0.728535	−	1.26186i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	−32.4602	−	11.8146i	−1.34900	−	0.490996i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$587$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$588$	−0.756082	+	24.2369i	−0.0311803	+	0.999514i
$589$	−27.3632	+	36.8624i	−1.12748	+	1.51889i
$590$		0			0
$591$		0			0
$592$	−39.1189	−	6.89771i	−1.60778	−	0.283494i
$593$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$593$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.17601	−	1.25632i	0.0890583	−	0.0514178i
$598$		0			0
$599$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$599$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$600$		0			0
$601$	−28.4259	+	16.4117i	−1.15952	+	0.669448i	−0.951188	−	0.308611i	$-0.900136\pi$
$601$	−28.4259	+	16.4117i	−1.15952	+	0.669448i	−0.208329	+	0.978059i	$0.566802\pi$
$602$		0			0
$603$	−8.95496	−	24.6035i	−0.364674	−	1.00193i
$604$	−11.1334	−	13.2683i	−0.453012	−	0.539879i
$605$		0			0
$606$		0			0
$607$	−10.2851	−	5.93810i	−0.417459	−	0.241020i	0.276531	−	0.961005i	$-0.410815\pi$
$607$	−10.2851	−	5.93810i	−0.417459	−	0.241020i	−0.693990	+	0.719985i	$0.744149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−36.0041	−	30.2110i	−1.45419	−	1.22021i	−0.929449	−	0.368950i	$-0.879717\pi$
$613$	−36.0041	−	30.2110i	−1.45419	−	1.22021i	−0.524742	−	0.851261i	$-0.675838\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$617$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$	−43.0830			−1.73165			−0.865825	−	0.500347i	$-0.833206\pi$
$619$	−43.0830			−1.73165			−0.865825	+	0.500347i	$0.833206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−34.9905	−	29.3605i	−1.40074	−	1.17536i
$625$	19.1511	+	16.0697i	0.766044	+	0.642788i
$626$		0			0
$627$		0			0
$628$	2.63547	−	4.56476i	0.105167	−	0.182154i
$629$		0			0
$630$		0			0
$631$	−15.2604	−	5.55434i	−0.607508	−	0.221115i	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	−15.2604	−	5.55434i	−0.607508	−	0.221115i	−0.627412	+	0.778687i	$0.715886\pi$
$632$		0			0
$633$	32.4993	+	11.8288i	1.29173	+	0.470151i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−28.5480	−	36.2609i	−1.13111	−	1.43671i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$641$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$642$		0			0
$643$	36.9623	+	31.0150i	1.45765	+	1.22311i	0.926750	+	0.375680i	$0.122591\pi$
$643$	36.9623	+	31.0150i	1.45765	+	1.22311i	0.530901	+	0.847434i	$0.321854\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	36.4843	+	31.5964i	1.42993	+	1.23836i
$652$	1.18891	−	0.997615i	0.0465614	−	0.0390696i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−24.8508	+	43.0428i	−0.969520	+	1.67926i
$658$		0			0
$659$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$659$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$660$		0			0
$661$	49.4664	−	8.72226i	1.92402	−	0.339256i	0.924862	−	0.380303i	$-0.124180\pi$
$661$	49.4664	−	8.72226i	1.92402	−	0.339256i	0.999157	+	0.0410470i	$0.0130693\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−7.80763	−	44.2793i	−0.301860	−	1.71194i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	40.5596	+	23.4171i	1.56346	+	0.902664i	0.996903	+	0.0786409i	$0.0250581\pi$
$673$	40.5596	+	23.4171i	1.56346	+	0.902664i	0.566557	+	0.824023i	$0.308275\pi$
$674$		0			0
$675$	16.7001	+	19.9024i	0.642788	+	0.766044i
$676$	60.9323			2.34355
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	49.5000	+	9.52628i	1.89964	+	0.365585i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$	−25.0651	−	7.46594i	−0.958388	−	0.285467i
$685$		0			0
$686$		0			0
$687$	3.88877	−	4.63446i	0.148366	−	0.176816i
$688$	−12.1438	+	10.1899i	−0.462979	+	0.388486i
$689$		0			0
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5.00000	+	25.9808i	−0.188982	+	0.981981i
$701$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$701$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$	25.8004	−	34.7570i	0.973081	−	1.31089i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.47019	+	36.6943i	0.242993	+	1.37808i	0.825108	+	0.564975i	$0.191114\pi$
$709$	6.47019	+	36.6943i	0.242993	+	1.37808i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$		−	20.8407i		−	0.781588i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$719$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$720$		0			0
$721$	−44.1398	+	24.5746i	−1.64385	+	0.915205i
$722$		0			0
$723$	10.5917	−	18.3453i	0.393909	−	0.682270i
$724$	4.73917	+	13.0208i	0.176130	+	0.483913i
$725$		0			0
$726$		0			0
$727$	−12.8417	+	4.67399i	−0.476271	+	0.173349i	−0.568991	−	0.822344i	$-0.692666\pi$
$727$	−12.8417	+	4.67399i	−0.476271	+	0.173349i	0.0927199	+	0.995692i	$0.470444\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	34.5699	+	41.1988i	1.27774	+	1.52275i
$733$	−25.0000	−	43.3013i	−0.923396	−	1.59937i	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−25.0000	−	43.3013i	−0.923396	−	1.59937i	−0.129275	−	0.991609i	$-0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−50.6969	+	18.4522i	−1.86491	+	0.678773i	−0.890096	+	0.455773i	$0.849363\pi$
$739$	−50.6969	+	18.4522i	−1.86491	+	0.678773i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$	45.6769	−	19.7787i	1.67798	−	0.726587i
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.5435	−	50.9480i	0.676663	−	1.85912i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	18.5435	−	50.9480i	0.676663	−	1.85912i	0.475965	−	0.879464i	$-0.342099\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−9.80571	+	25.6875i	−0.356630	+	0.934246i
$757$	−40.0453	−	33.6020i	−1.45547	−	1.22129i	−0.928461	−	0.371429i	$-0.878868\pi$
$757$	−40.0453	−	33.6020i	−1.45547	−	1.22129i	−0.527011	−	0.849858i	$-0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	36.0000	−	41.5692i	1.30329	−	1.50491i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	+	13.8564i	−0.866025	+	0.500000i
$769$	9.41803	+	53.4123i	0.339623	+	1.92610i	0.375684	+	0.926748i	$0.377408\pi$
$769$	9.41803	+	53.4123i	0.339623	+	1.92610i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$		−	39.8873i		−	1.43558i
$773$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$773$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$774$		0			0
$775$	33.8496	+	40.3404i	1.21591	+	1.44907i
$776$		0			0
$777$	−34.4005	−	29.7917i	−1.23411	−	1.06877i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.5972	+	8.75151i	−0.949900	+	0.312554i
$785$		0			0
$786$		0			0
$787$		−	22.4083i		−	0.798770i	−0.916783	−	0.399385i	$-0.869224\pi$
$787$		−	22.4083i		−	0.798770i	0.916783	−	0.399385i	$-0.130776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−100.801	−	17.7740i	−3.57956	−	0.631173i
$794$		0			0
$795$		0			0
$796$	2.22256	+	1.86495i	0.0787767	+	0.0661015i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	23.1598	−	19.4334i	0.816784	−	0.685363i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	27.8335	+	33.1707i	0.977367	+	1.16478i	0.986324	+	0.164821i	$0.0527046\pi$
$811$	27.8335	+	33.1707i	0.977367	+	1.16478i	−0.00895645	+	0.999960i	$0.502851\pi$
$812$		0			0
$813$	−17.1795	+	47.2003i	−0.602511	+	1.65539i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.00733	−	16.8038i	−0.140199	−	0.587891i
$818$		0			0
$819$	−17.1288	−	49.4467i	−0.598529	−	1.72781i
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$	4.69846	+	1.71010i	0.163778	+	0.0596104i	0.422608	−	0.906313i	$-0.361115\pi$
$823$	4.69846	+	1.71010i	0.163778	+	0.0596104i	−0.258830	+	0.965923i	$0.583337\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$827$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$828$		0			0
$829$	44.4414	+	25.6583i	1.54351	+	0.891148i	0.998613	+	0.0526472i	$0.0167659\pi$
$829$	44.4414	+	25.6583i	1.54351	+	0.891148i	0.544900	+	0.838501i	$0.316567\pi$
$830$		0			0
$831$	44.3492	+	7.81995i	1.53846	+	0.271271i
$832$	18.0392	−	49.5623i	0.625397	−	1.71826i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	27.3632	+	47.3945i	0.945812	+	1.63819i
$838$		0			0
$839$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$839$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$840$		0			0
$841$	27.2511	−	9.91858i	0.939693	−	0.342020i
$842$		0			0
$843$		0			0
$844$			39.9353i			1.37463i
$845$		0			0
$846$		0			0
$847$	29.0997	+	0.453779i	0.999878	+	0.0155920i
$848$		0			0
$849$	−11.9402	+	2.10537i	−0.409785	+	0.0722562i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	9.87568	−	56.0077i	0.338137	−	1.91767i	−0.0556158	−	0.998452i	$-0.517712\pi$
$853$	9.87568	−	56.0077i	0.338137	−	1.91767i	0.393753	−	0.919216i	$-0.371177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$857$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$858$		0			0
$859$	−24.3335	+	20.4182i	−0.830247	+	0.696660i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−24.3335	+	20.4182i	−0.830247	+	0.696660i	0.125101	+	0.992144i	$0.460075\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.5000	+	14.7224i	−0.866025	+	0.500000i
$868$	−19.8753	+	52.0662i	−0.674611	+	1.76724i
$869$		0			0
$870$		0			0
$871$	−9.99163	+	56.6654i	−0.338553	+	1.92003i
$872$		0			0
$873$	49.5000	+	28.5788i	1.67532	+	0.967247i
$874$		0			0
$875$		0			0
$876$	−56.5185	−	9.96573i	−1.90958	−	0.336711i
$877$	5.23168	+	14.3739i	0.176661	+	0.485373i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	5.23168	+	14.3739i	0.176661	+	0.485373i	−0.819483	+	0.573103i	$0.805739\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	4.80294	+	27.2388i	0.161632	+	0.916659i	0.952470	+	0.304633i	$0.0985339\pi$
$883$	4.80294	+	27.2388i	0.161632	+	0.916659i	−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$	20.2337	+	24.8917i	0.678617	+	0.834841i
$890$		0			0
$891$		0			0
$892$	44.9623	−	25.9590i	1.50545	−	0.869172i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−15.0000	+	25.9808i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−17.9326	+	2.87445i	−0.596759	+	0.0956559i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−28.9469	+	34.4975i	−0.961165	+	1.14547i	0.0281394	+	0.999604i	$0.491042\pi$
$907$	−28.9469	+	34.4975i	−0.961165	+	1.14547i	−0.989304	+	0.145868i	$0.953403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−1.79797	−	30.1458i	−0.0595368	−	0.998226i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	6.56448	+	2.38928i	0.216897	+	0.0789439i
$917$		0			0
$918$		0			0
$919$	19.7942			0.652950			0.326475	−	0.945206i	$-0.394139\pi$
$919$	19.7942			0.652950			0.326475	+	0.945206i	$0.394139\pi$
$920$		0			0
$921$	−41.3664	+	34.7105i	−1.36307	+	1.14375i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−31.9163	−	38.0363i	−1.04940	−	1.25063i
$926$		0			0
$927$	−56.4136	+	9.94724i	−1.85287	+	0.326710i
$928$		0			0
$929$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$929$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$930$		0			0
$931$	4.44340	−	30.1870i	0.145627	−	0.989340i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.3549	+	32.1836i	−1.25300	+	1.05139i	−0.256608	+	0.966516i	$0.582605\pi$
$937$	−38.3549	+	32.1836i	−1.25300	+	1.05139i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$	33.0000	+	19.0526i	1.07691	+	0.621757i
$940$		0			0
$941$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$941$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$947$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$948$	22.6135	−	8.23064i	0.734452	−	0.267319i
$949$	94.5920	−	54.6127i	3.07059	−	1.77280i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$953$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	39.9628	+	69.2175i	1.28912	+	2.23282i
$962$		0			0
$963$		0			0
$964$	24.0888	+	4.24751i	0.775849	+	0.136803i
$965$		0			0
$966$		0			0
$967$	−9.23365	+	52.3666i	−0.296934	+	1.68400i	0.362301	+	0.932061i	$0.381991\pi$
$967$	−9.23365	+	52.3666i	−0.296934	+	1.68400i	−0.659236	+	0.751936i	$0.729120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$971$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$972$	−20.0401	+	23.8829i	−0.642788	+	0.766044i
$973$	40.7234	−	33.1028i	1.30553	−	1.06123i
$974$		0			0
$975$	−9.91463	−	56.2287i	−0.317522	−	1.80076i
$976$	−31.0506	+	53.7812i	−0.993904	+	1.72149i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	54.0000	−	31.1769i	1.72409	−	0.995402i
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	39.5003	+	41.7511i	1.25667	+	1.32828i
$989$		0			0
$990$		0			0
$991$	−6.12045	−	1.07920i	−0.194423	−	0.0342819i	0.0755888	−	0.997139i	$-0.475916\pi$
$991$	−6.12045	−	1.07920i	−0.194423	−	0.0342819i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	55.0737	−	20.0452i	1.74771	−	0.636115i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.13840	−	7.66803i	0.289416	−	0.242849i	−0.486507	−	0.873677i	$-0.661729\pi$
$997$	9.13840	−	7.66803i	0.289416	−	0.242849i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	−25.8004	−	44.6876i	−0.816289	−	1.41385i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.br.a.2.1	✓	6
3.2	odd	2	CM	399.2.br.a.2.1	✓	6
7.4	even	3		399.2.bv.a.116.1	yes	6
19.10	odd	18		399.2.bv.a.86.1	yes	6
21.11	odd	6		399.2.bv.a.116.1	yes	6
57.29	even	18		399.2.bv.a.86.1	yes	6
133.67	odd	18	inner	399.2.br.a.200.1	yes	6
399.200	even	18	inner	399.2.br.a.200.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.br.a.2.1	✓	6	1.1	even	1	trivial
399.2.br.a.2.1	✓	6	3.2	odd	2	CM
399.2.br.a.200.1	yes	6	133.67	odd	18	inner
399.2.br.a.200.1	yes	6	399.200	even	18	inner
399.2.bv.a.86.1	yes	6	19.10	odd	18
399.2.bv.a.86.1	yes	6	57.29	even	18
399.2.bv.a.116.1	yes	6	7.4	even	3
399.2.bv.a.116.1	yes	6	21.11	odd	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:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.br (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70574 - 0.300767i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.64543 + 0.0412527i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(-1.70574 - 0.300767i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.64543 + 0.0412527i) q^{7} +(2.81908 + 1.02606i) q^{9} +3.46410i q^{12} +(-4.23783 - 5.05044i) q^{13} +(-3.75877 + 1.36808i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-0.837496 - 5.22481i) q^{28} +(-9.12108 - 5.26606i) q^{31} +(1.04189 - 5.90885i) q^{36} +(8.60014 + 4.96529i) q^{37} +(5.70961 + 9.88933i) q^{39} +(3.72416 - 1.35548i) q^{43} +(6.82295 - 1.20307i) q^{48} +(6.99660 + 0.218262i) q^{49} +(-8.47565 + 10.1009i) q^{52} +(-2.15523 + 7.23567i) q^{57} +(11.8931 - 9.97946i) q^{61} +(7.41534 + 2.83067i) q^{63} +(4.00000 + 6.92820i) q^{64} +(-5.60994 - 6.68566i) q^{67} +(-2.87686 + 16.3155i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-8.70233 + 0.519030i) q^{76} +(-2.37598 - 6.52796i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-0.142903 + 9.16404i) q^{84} +(-11.0025 - 13.5354i) q^{91} +(13.9743 + 11.7258i) q^{93} +(18.7631 + 3.30844i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 6 q^{13} + 3 q^{19} - 27 q^{21} - 27 q^{27} + 24 q^{43} - 12 q^{52} + 39 q^{61} + 24 q^{64} + 15 q^{67} - 21 q^{73} + 45 q^{75} + 39 q^{79} - 51 q^{91} + 54 q^{93}+O(q^{100}) \) 6 * q - 6 * q^13 + 3 * q^19 - 27 * q^21 - 27 * q^27 + 24 * q^43 - 12 * q^52 + 39 * q^61 + 24 * q^64 + 15 * q^67 - 21 * q^73 + 45 * q^75 + 39 * q^79 - 51 * q^91 + 54 * q^93

Introduction

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