LMFDB - Embedded newform 399.2.br.a.128.1

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			128.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	399.128
Dual form			399.2.br.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.592396 + 1.62760i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-1.35844 + 2.27038i) q^{7} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})$	$q+(0.592396 + 1.62760i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-1.35844 + 2.27038i) q^{7} +(-2.29813 + 1.92836i) q^{9} +3.46410i q^{12} +(-0.754900 + 0.133109i) q^{13} +(3.06418 + 2.57115i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-4.10607 + 3.33770i) q^{28} +(1.84864 + 1.06731i) q^{31} +(-5.63816 + 2.05212i) q^{36} +(0.970437 + 0.560282i) q^{37} +(-0.663848 - 1.14982i) q^{39} +(9.81180 + 8.23308i) q^{43} +(-2.36959 + 6.51038i) q^{48} +(-3.30928 - 6.16836i) q^{49} +(-1.50980 - 0.266219i) q^{52} +(7.34389 - 1.75135i) q^{57} +(-1.08899 - 6.17598i) q^{61} +(-1.25624 - 7.83721i) q^{63} +(4.00000 + 6.92820i) q^{64} +(7.51501 - 1.32510i) q^{67} +(-11.1912 + 4.07326i) q^{73} +(7.50000 + 4.33013i) q^{75} +(3.90167 - 7.79596i) q^{76} +(11.3414 - 13.5161i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-7.86484 - 4.70578i) q^{84} +(0.723278 - 1.89473i) q^{91} +(-0.642026 + 3.64111i) q^{93} +(-6.51636 - 17.9035i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{7}{18}\right)$

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		−0.984808	−	0.173648i	$-0.944444\pi$
$2$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$3$	0.592396	+	1.62760i	0.342020	+	0.939693i
$3$	0.592396	+	1.62760i	0.342020	+	0.939693i
$4$	1.87939	+	0.684040i	0.939693	+	0.342020i
$5$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$5$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$6$		0			0
$7$	−1.35844	+	2.27038i	−0.513442	+	0.858124i
$7$	−1.35844	+	2.27038i	−0.513442	+	0.858124i
$8$		0			0
$9$	−2.29813	+	1.92836i	−0.766044	+	0.642788i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			3.46410i			1.00000i
$13$	−0.754900	+	0.133109i	−0.209372	+	0.0369179i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−0.754900	+	0.133109i	−0.209372	+	0.0369179i	0.0679785	+	0.997687i	$0.478345\pi$
$14$		0			0
$15$		0			0
$16$	3.06418	+	2.57115i	0.766044	+	0.642788i
$17$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$17$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$18$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$23$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$24$		0			0
$25$	3.83022	−	3.21394i	0.766044	−	0.642788i
$26$		0			0
$27$	−4.50000	−	2.59808i	−0.866025	−	0.500000i
$28$	−4.10607	+	3.33770i	−0.775974	+	0.630765i
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$	1.84864	+	1.06731i	0.332026	+	0.191695i	0.656740	−	0.754117i	$-0.271935\pi$
$31$	1.84864	+	1.06731i	0.332026	+	0.191695i	−0.324714	+	0.945812i	$0.605268\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.63816	+	2.05212i	−0.939693	+	0.342020i
$37$	0.970437	+	0.560282i	0.159539	+	0.0921098i	0.577644	−	0.816289i	$-0.303972\pi$
$37$	0.970437	+	0.560282i	0.159539	+	0.0921098i	−0.418105	+	0.908399i	$0.637306\pi$
$38$		0			0
$39$	−0.663848	−	1.14982i	−0.106301	−	0.184118i
$40$		0			0
$41$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$	9.81180	+	8.23308i	1.49629	+	1.25553i	0.886292	+	0.463127i	$0.153273\pi$
$43$	9.81180	+	8.23308i	1.49629	+	1.25553i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$47$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$48$	−2.36959	+	6.51038i	−0.342020	+	0.939693i
$49$	−3.30928	−	6.16836i	−0.472754	−	0.881194i
$50$		0			0
$51$		0			0
$52$	−1.50980	−	0.266219i	−0.209372	−	0.0369179i
$53$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$53$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.34389	−	1.75135i	0.972722	−	0.231972i
$58$		0			0
$59$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$59$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$60$		0			0
$61$	−1.08899	−	6.17598i	−0.139431	−	0.790754i	−0.971671	−	0.236338i	$-0.924053\pi$
$61$	−1.08899	−	6.17598i	−0.139431	−	0.790754i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$	−1.25624	−	7.83721i	−0.158272	−	0.987396i
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	7.51501	−	1.32510i	0.918105	−	0.161887i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	7.51501	−	1.32510i	0.918105	−	0.161887i	0.612682	+	0.790330i	$0.290091\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$71$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$72$		0			0
$73$	−11.1912	+	4.07326i	−1.30983	+	0.476739i	−0.900186	−	0.435506i	$-0.856569\pi$
$73$	−11.1912	+	4.07326i	−1.30983	+	0.476739i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	3.90167	−	7.79596i	0.447553	−	0.894258i
$77$		0			0
$78$		0			0
$79$	11.3414	−	13.5161i	1.27600	−	1.52068i	0.544696	−	0.838633i	$-0.316645\pi$
$79$	11.3414	−	13.5161i	1.27600	−	1.52068i	0.731307	−	0.682048i	$-0.238911\pi$
$80$		0			0
$81$	1.56283	−	8.86327i	0.173648	−	0.984808i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	−7.86484	−	4.70578i	−0.858124	−	0.513442i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$89$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$90$		0			0
$91$	0.723278	−	1.89473i	0.0758201	−	0.198622i
$92$		0			0
$93$	−0.642026	+	3.64111i	−0.0665750	+	0.377566i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.51636	−	17.9035i	−0.661636	−	1.81783i	−0.569346	−	0.822098i	$-0.692804\pi$
$97$	−6.51636	−	17.9035i	−0.661636	−	1.81783i	−0.0922897	−	0.995732i	$-0.529419\pi$
$98$		0			0
$99$		0			0
$100$	9.39693	−	3.42020i	0.939693	−	0.342020i
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$	3.10488	−	1.79261i	0.305933	−	0.176631i	−0.339172	−	0.940724i	$-0.610147\pi$
$103$	3.10488	−	1.79261i	0.305933	−	0.176631i	0.645105	+	0.764094i	$0.276813\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−6.68004	−	7.96097i	−0.642788	−	0.766044i
$109$	−20.4688	−	3.60921i	−1.96056	−	0.345700i	−0.996912	−	0.0785223i	$-0.974980\pi$
$109$	−20.4688	−	3.60921i	−1.96056	−	0.345700i	−0.963647	−	0.267177i	$-0.913909\pi$
$110$		0			0
$111$	−0.337029	+	1.91139i	−0.0319894	+	0.181421i
$112$	−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
$117$	1.47818	−	1.76162i	0.136658	−	0.162862i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$	2.74422	+	3.27044i	0.246438	+	0.293694i
$125$		0			0
$126$		0			0
$127$	−11.9402	+	2.10537i	−1.05952	+	0.186822i	−0.676142	−	0.736771i	$-0.736350\pi$
$127$	−11.9402	+	2.10537i	−1.05952	+	0.186822i	−0.383375	+	0.923593i	$0.625238\pi$
$128$		0			0
$129$	−7.58765	+	20.8469i	−0.668055	+	1.83547i
$130$		0			0
$131$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$131$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$132$		0			0
$133$	9.15183	+	7.01741i	0.793564	+	0.608487i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$137$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$138$		0			0
$139$	−3.63950	−	20.6406i	−0.308698	−	1.75072i	−0.605564	−	0.795796i	$-0.707053\pi$
$139$	−3.63950	−	20.6406i	−0.308698	−	1.75072i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	8.07919	−	9.04028i	0.666361	−	0.745630i
$148$	1.44057	+	1.71680i	0.118414	+	0.141120i
$149$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$149$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$150$		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
$156$	−0.461104	−	2.61505i	−0.0369179	−	0.209372i
$157$	−3.97653	+	22.5520i	−0.317362	+	1.79985i	0.241299	+	0.970451i	$0.422426\pi$
$157$	−3.97653	+	22.5520i	−0.317362	+	1.79985i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.8576	+	18.8059i	0.850430	+	1.47299i	0.880821	+	0.473450i	$0.156991\pi$
$163$	10.8576	+	18.8059i	0.850430	+	1.47299i	−0.0303908	+	0.999538i	$0.509675\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	−11.6638	+	4.24529i	−0.897219	+	0.326561i
$170$		0			0
$171$	7.20099	+	10.9154i	0.550673	+	0.834721i
$172$	12.8084	+	22.1848i	0.976631	+	1.69158i
$173$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$173$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$174$		0			0
$175$	2.09374	+	13.0620i	0.158272	+	0.987396i
$176$		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
$181$	2.36959	−	6.51038i	0.176130	−	0.483913i	−0.819943	−	0.572444i	$-0.805995\pi$
$181$	2.36959	−	6.51038i	0.176130	−	0.483913i	0.996073	+	0.0885316i	$0.0282174\pi$
$182$		0			0
$183$	9.40689	−	5.43107i	0.695377	−	0.401476i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	12.0116	−	6.68739i	0.873716	−	0.486436i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−8.90673	+	10.6146i	−0.642788	+	0.766044i
$193$	−2.31954	−	6.37290i	−0.166964	−	0.458731i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−2.31954	−	6.37290i	−0.166964	−	0.458731i	−0.994753	+	0.102310i	$0.967377\pi$
$194$		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
$199$	4.36319	+	24.7449i	0.309298	+	1.75412i	0.602549	+	0.798082i	$0.294152\pi$
$199$	4.36319	+	24.7449i	0.309298	+	1.75412i	−0.293251	+	0.956036i	$0.594737\pi$
$200$		0			0
$201$	6.60859	+	11.4464i	0.466134	+	0.807368i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2.65539	−	1.53309i	−0.184118	−	0.106301i
$209$		0			0
$210$		0			0
$211$	−9.66503	−	26.5545i	−0.665368	−	1.82808i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−9.66503	−	26.5545i	−0.665368	−	1.82808i	−0.114625	−	0.993409i	$-0.536567\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.93448	+	2.74724i	−0.334974	+	0.186495i
$218$		0			0
$219$	−13.2592	−	15.8017i	−0.895976	−	1.06778i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−16.5646	+	19.7410i	−1.10925	+	1.32195i	−0.167412	+	0.985887i	$0.553541\pi$
$223$	−16.5646	+	19.7410i	−1.10925	+	1.32195i	−0.941838	+	0.336066i	$0.890903\pi$
$224$		0			0
$225$	−2.60472	+	14.7721i	−0.173648	+	0.984808i
$226$		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
$229$	−12.1446	+	21.0350i	−0.802535	+	1.39003i	0.115408	+	0.993318i	$0.463182\pi$
$229$	−12.1446	+	21.0350i	−0.802535	+	1.39003i	−0.917943	+	0.396713i	$0.870151\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$233$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	28.7173	+	10.4523i	1.86539	+	0.678947i
$238$		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
$241$	−11.9556	−	14.2481i	−0.770127	−	0.917802i	0.228316	−	0.973587i	$-0.426678\pi$
$241$	−11.9556	−	14.2481i	−0.770127	−	0.917802i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	15.3516	−	2.70691i	0.984808	−	0.173648i
$244$	2.17799	−	12.3520i	0.139431	−	0.790754i
$245$		0			0
$246$		0			0
$247$	0.198930	+	3.33537i	0.0126576	+	0.212224i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$251$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$252$	3.00000	−	15.5885i	0.188982	−	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	2.77837	+	15.7569i	0.173648	+	0.984808i
$257$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$257$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$258$		0			0
$259$	−2.59034	+	1.44215i	−0.160956	+	0.0896111i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$263$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	15.0300	+	2.65020i	0.918105	+	0.161887i
$269$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$269$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$270$		0			0
$271$	−27.2511	+	9.91858i	−1.65539	+	0.602511i	−0.989628	−	0.143657i	$-0.954114\pi$
$271$	−27.2511	+	9.91858i	−1.65539	+	0.602511i	−0.665758	+	0.746168i	$0.731892\pi$
$272$		0			0
$273$	3.51233	+	0.0547710i	0.212576	+	0.00331489i
$274$		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
$279$	−6.30659	+	1.11202i	−0.377566	+	0.0665750i
$280$		0			0
$281$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$281$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$282$		0			0
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	0.469344	−	0.883016i	$-0.344491\pi$
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	−0.788100	+	0.615547i	$0.788935\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.95202	+	16.7417i	0.173648	+	0.984808i
$290$		0			0
$291$	25.2795	−	21.2120i	1.48191	−	1.24347i
$292$	−23.8188			−1.39389
$293$		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
$300$	11.1334	+	13.2683i	0.642788	+	0.766044i
$301$	−32.0210	+	11.0924i	−1.84566	+	0.639355i
$302$		0			0
$303$		0			0
$304$	12.6655	−	11.9827i	0.726416	−	0.687255i
$305$		0			0
$306$		0			0
$307$	−30.7033	−	5.41381i	−1.75233	−	0.308983i	−0.796879	−	0.604139i	$-0.793517\pi$
$307$	−30.7033	−	5.41381i	−1.75233	−	0.308983i	−0.955449	+	0.295156i	$0.904628\pi$
$308$		0			0
$309$	4.75696	+	3.99156i	0.270614	+	0.227072i
$310$		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
$313$	16.8530	−	14.1413i	0.952587	−	0.799315i	−0.0271446	−	0.999632i	$-0.508641\pi$
$313$	16.8530	−	14.1413i	0.952587	−	0.799315i	0.979731	+	0.200316i	$0.0641970\pi$
$314$		0			0
$315$		0			0
$316$	30.5604	−	17.6440i	1.71915	−	0.992554i
$317$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$317$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$318$		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
$325$	−2.46363	+	2.93604i	−0.136658	+	0.162862i
$326$		0			0
$327$	−6.25133	−	35.4531i	−0.345700	−	1.96056i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	24.6871	−	14.2531i	1.35692	−	0.783420i	0.367716	−	0.929938i	$-0.380140\pi$
$331$	24.6871	−	14.2531i	1.35692	−	0.783420i	0.989208	+	0.146518i	$0.0468065\pi$
$332$		0			0
$333$	−3.31062	+	0.583752i	−0.181421	+	0.0319894i
$334$		0			0
$335$		0			0
$336$	−11.5621	−	14.2238i	−0.630765	−	0.775974i
$337$	−23.0694	+	27.4930i	−1.25667	+	1.49764i	−0.466805	+	0.884361i	$0.654595\pi$
$337$	−23.0694	+	27.4930i	−1.25667	+	1.49764i	−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		0.173648	−	0.984808i	$-0.444444\pi$
$347$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$348$		0			0
$349$	0.653886	+	1.13256i	0.0350017	+	0.0606247i	0.882996	−	0.469381i	$-0.155523\pi$
$349$	0.653886	+	1.13256i	0.0350017	+	0.0606247i	−0.847994	+	0.530006i	$0.822190\pi$
$350$		0			0
$351$	3.74288	+	1.36230i	0.199780	+	0.0727140i
$352$		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		0.173648	−	0.984808i	$-0.444444\pi$
$359$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$	6.51636	+	17.9035i	0.342020	+	0.939693i
$364$	2.65539	−	3.06618i	0.139180	−	0.160712i
$365$		0			0
$366$		0			0
$367$	28.2147	−	23.6749i	1.47279	−	1.23582i	0.559301	−	0.828965i	$-0.311070\pi$
$367$	28.2147	−	23.6749i	1.47279	−	1.23582i	0.913493	−	0.406855i	$-0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.69728	+	6.40388i	−0.191695	+	0.332026i
$373$			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
$379$		−	38.5433i		−	1.97983i	−0.141648	−	0.989917i	$-0.545240\pi$
$379$		−	38.5433i		−	1.97983i	0.141648	−	0.989917i	$-0.454760\pi$
$380$		0			0
$381$	−10.5000	−	18.1865i	−0.537931	−	0.931724i
$382$		0			0
$383$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$383$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−38.4252			−1.95326
$388$		−	38.1051i		−	1.93449i
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.31356	−	24.4634i	0.216491	−	1.22778i	−0.661809	−	0.749673i	$-0.730211\pi$
$397$	4.31356	−	24.4634i	0.216491	−	1.22778i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$	−6.00000	+	19.0526i	−0.300376	+	0.953821i
$400$	20.0000			1.00000
$401$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$401$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$402$		0			0
$403$	−1.53761	−	0.559644i	−0.0765937	−	0.0278778i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	25.5861	−	4.51151i	1.26515	−	0.223080i	0.499486	−	0.866322i	$-0.333522\pi$
$409$	25.5861	−	4.51151i	1.26515	−	0.223080i	0.765663	+	0.643242i	$0.222411\pi$
$410$		0			0
$411$		0			0
$412$	7.06149	−	1.24513i	0.347895	−	0.0613432i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	31.4386	−	18.1511i	1.53955	−	0.888861i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−35.8205	−	6.31612i	−1.74578	−	0.307829i	−0.792492	−	0.609882i	$-0.791217\pi$
$421$	−35.8205	−	6.31612i	−1.74578	−	0.307829i	−0.953291	+	0.302053i	$0.902328\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.5012	+	5.91728i	0.750155	+	0.286357i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$431$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$432$	−7.10876	−	19.5311i	−0.342020	−	0.939693i
$433$	25.5484	+	30.4475i	1.22778	+	1.46321i	0.840996	+	0.541041i	$0.181970\pi$
$433$	25.5484	+	30.4475i	1.22778	+	1.46321i	0.386784	+	0.922170i	$0.373586\pi$
$434$		0			0
$435$		0			0
$436$	−36.0000	−	20.7846i	−1.72409	−	0.995402i
$437$		0			0
$438$		0			0
$439$	41.2447	+	7.27255i	1.96850	+	0.347100i	0.990090	+	0.140434i	$0.0448499\pi$
$439$	41.2447	+	7.27255i	1.96850	+	0.347100i	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		−0.939693	−	0.342020i	$-0.888889\pi$
$443$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$444$	−1.94087	+	3.36169i	−0.0921098	+	0.159539i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−21.1634	−	0.330021i	−0.999878	−	0.0155920i
$449$		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
$453$	11.4907	+	9.64181i	0.539879	+	0.453012i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.1903	−	29.7745i	0.804129	−	1.39279i	−0.112749	−	0.993624i	$-0.535966\pi$
$457$	17.1903	−	29.7745i	0.804129	−	1.39279i	0.916878	−	0.399169i	$-0.130701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$	15.9133	+	27.5626i	0.739553	+	1.28094i	0.952697	+	0.303923i	$0.0982964\pi$
$463$	15.9133	+	27.5626i	0.739553	+	1.28094i	−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	$0.333333\pi$
$468$	3.98309	−	2.29964i	0.184118	−	0.106301i
$469$	−7.20022	+	18.8620i	−0.332475	+	0.870968i
$470$		0			0
$471$	−39.0612	+	6.88755i	−1.79985	+	0.317362i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−12.0016	−	18.1923i	−0.550673	−	0.834721i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$	−0.807162	−	0.293783i	−0.0368034	−	0.0133953i
$482$		0			0
$483$		0			0
$484$	20.6732	+	7.52444i	0.939693	+	0.342020i
$485$		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
$489$	−24.1763	+	28.8122i	−1.09329	+	1.30293i
$490$		0			0
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	2.92034	+	8.02357i	0.131127	+	0.360269i
$497$		0			0
$498$		0			0
$499$	−18.2366	+	6.63760i	−0.816384	+	0.297140i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−18.2366	+	6.63760i	−0.816384	+	0.297140i	−0.100126	+	0.994975i	$0.531925\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$503$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.8192	−	16.4691i	−0.613734	−	0.731420i
$508$	−23.8803	−	4.21074i	−1.05952	−	0.186822i
$509$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$509$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$510$		0			0
$511$	5.95471	−	30.9416i	0.263421	−	1.36877i
$512$		0			0
$513$	−13.5000	+	18.1865i	−0.596040	+	0.802955i
$514$		0			0
$515$		0			0
$516$	−28.5202	+	33.9891i	−1.25553	+	1.49629i
$517$		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
$523$	18.8692	+	22.4874i	0.825091	+	0.983306i	0.999999	−	0.00127919i	$-0.000407178\pi$
$523$	18.8692	+	22.4874i	0.825091	+	0.983306i	−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	−20.0194	+	11.1457i	−0.873716	+	0.486436i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.6129	+	7.86646i	−0.939693	+	0.342020i
$530$		0			0
$531$		0			0
$532$	12.3996	+	19.4486i	0.537591	+	0.843205i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.4054	+	25.5131i	−1.30723	+	1.09690i	−0.318382	+	0.947962i	$0.603140\pi$
$541$	−30.4054	+	25.5131i	−1.30723	+	1.09690i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	12.0000			0.514969
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−22.6143	−	26.9506i	−0.966917	−	1.15233i	−0.988295	−	0.152555i	$-0.951250\pi$
$547$	−22.6143	−	26.9506i	−0.966917	−	1.15233i	0.0213785	−	0.999771i	$-0.493195\pi$
$548$		0			0
$549$	14.4122	+	12.0933i	0.615097	+	0.516128i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	15.2802	+	44.1101i	0.649779	+	1.87575i
$554$		0			0
$555$		0			0
$556$	7.27900	−	41.2813i	0.308698	−	1.75072i
$557$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$557$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$558$		0			0
$559$	−8.50283	−	4.90911i	−0.359631	−	0.207633i
$560$		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
$571$	23.5638	+	40.8136i	0.986113	+	1.70800i	0.636882	+	0.770961i	$0.280224\pi$
$571$	23.5638	+	40.8136i	0.986113	+	1.70800i	0.349231	+	0.937037i	$0.386443\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−22.5526	−	8.20848i	−0.939693	−	0.342020i
$577$	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
$579$	8.99841	−	7.55056i	0.373961	−	0.313791i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$587$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$588$	21.3678	−	11.4637i	0.881194	−	0.472754i
$589$	5.54592	−	7.47119i	0.228516	−	0.307845i
$590$		0			0
$591$		0			0
$592$	1.53302	+	4.21194i	0.0630068	+	0.173110i
$593$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$593$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−37.6899	+	21.7603i	−1.54254	+	0.890589i
$598$		0			0
$599$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$599$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$600$		0			0
$601$	41.5301	−	23.9774i	1.69405	−	0.978059i	0.742859	−	0.669448i	$-0.233469\pi$
$601$	41.5301	−	23.9774i	1.69405	−	0.978059i	0.951188	−	0.308611i	$-0.0998642\pi$
$602$		0			0
$603$	−14.7152	+	17.5369i	−0.599251	+	0.714159i
$604$	17.0574	−	3.00767i	0.694055	−	0.122381i
$605$		0			0
$606$		0			0
$607$	−30.7240	−	17.7385i	−1.24705	−	0.719985i	−0.276531	−	0.961005i	$-0.589185\pi$
$607$	−30.7240	−	17.7385i	−1.24705	−	0.719985i	−0.970520	+	0.241020i	$0.922518\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8.16146	+	46.2860i	−0.329638	+	1.86947i	0.145204	+	0.989402i	$0.453616\pi$
$613$	−8.16146	+	46.2860i	−0.329638	+	1.86947i	−0.474843	+	0.880071i	$0.657495\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$617$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$618$		0			0
$619$	43.1029			1.73245			0.866226	−	0.499653i	$-0.166539\pi$
$619$	43.1029			1.73245			0.866226	+	0.499653i	$0.166539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.922208	−	5.23010i	0.0369179	−	0.209372i
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$	−22.8999	+	39.6638i	−0.913806	+	1.58276i
$629$		0			0
$630$		0			0
$631$	25.3200	−	21.2460i	1.00797	−	0.845790i	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	25.3200	−	21.2460i	1.00797	−	0.845790i	0.988069	+	0.154011i	$0.0492193\pi$
$632$		0			0
$633$	37.4944	−	31.4615i	1.49027	−	1.25048i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.31924	+	4.21600i	0.131513	+	0.167044i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$641$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$642$		0			0
$643$	−1.84096	+	10.4406i	−0.0726002	+	0.411736i	0.926750	+	0.375680i	$0.122591\pi$
$643$	−1.84096	+	10.4406i	−0.0726002	+	0.411736i	−0.999350	+	0.0360565i	$0.988520\pi$
$644$		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
$651$	−7.39456	−	6.40388i	−0.289816	−	0.250988i
$652$	7.54158	+	42.7705i	0.295351	+	1.67502i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	17.8641	−	30.9416i	0.696946	−	1.20715i
$658$		0			0
$659$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$659$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	−17.1795	+	47.2003i	−0.668205	+	1.83588i	−0.133078	+	0.991106i	$0.542486\pi$
$661$	−17.1795	+	47.2003i	−0.668205	+	1.83588i	−0.535126	+	0.844772i	$0.679736\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−41.9432	−	15.2661i	−1.62162	−	0.590220i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−3.53360	−	2.04012i	−0.136210	−	0.0786409i	0.430346	−	0.902664i	$-0.358391\pi$
$673$	−3.53360	−	2.04012i	−0.136210	−	0.0786409i	−0.566557	+	0.824023i	$0.691725\pi$
$674$		0			0
$675$	−25.5861	+	4.51151i	−0.984808	+	0.173648i
$676$	−24.8248			−0.954801
$677$		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	$0.833333\pi$
$684$	6.06687	+	25.4400i	0.231972	+	0.972722i
$685$		0			0
$686$		0			0
$687$	−41.4308	−	7.30537i	−1.58068	−	0.278717i
$688$	8.89662	+	50.4552i	0.339181	+	1.92359i
$689$		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		0.939693	−	0.342020i	$-0.111111\pi$
$701$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$702$		0			0
$703$	2.91131	−	3.92198i	0.109802	−	0.147920i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.4569	−	4.89792i	−0.505386	−	0.183945i	0.0767291	−	0.997052i	$-0.475552\pi$
$709$	−13.4569	−	4.89792i	−0.505386	−	0.183945i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$			52.9321i			1.98511i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$719$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$720$		0			0
$721$	−0.147900	+	9.48443i	−0.00550807	+	0.353219i
$722$		0			0
$723$	16.1077	−	27.8994i	0.599052	−	1.03759i
$724$	8.90673	−	10.6146i	0.331016	−	0.394489i
$725$		0			0
$726$		0			0
$727$	29.3730	+	24.6469i	1.08939	+	0.914103i	0.996666	−	0.0815889i	$-0.0259995\pi$
$727$	29.3730	+	24.6469i	1.08939	+	0.914103i	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
$732$	21.3942	−	3.77238i	0.790754	−	0.139431i
$733$	−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
$739$	−25.1316	−	21.0879i	−0.924481	−	0.775731i	0.0503375	−	0.998732i	$-0.483970\pi$
$739$	−25.1316	−	21.0879i	−0.924481	−	0.775731i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$	−5.31078	+	2.29964i	−0.195097	+	0.0844793i
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−21.8940	−	26.0922i	−0.798923	−	0.952119i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	−21.8940	−	26.0922i	−0.798923	−	0.952119i	−0.999621	+	0.0275338i	$0.991235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	27.1489	−	4.35176i	0.987396	−	0.158272i
$757$	7.12289	−	40.3959i	0.258886	−	1.46822i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	7.12289	−	40.3959i	0.258886	−	1.46822i	0.785897	−	0.618357i	$-0.212201\pi$
$758$		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
$769$	16.0474	+	5.84078i	0.578684	+	0.210624i	0.614745	−	0.788726i	$-0.289259\pi$
$769$	16.0474	+	5.84078i	0.578684	+	0.210624i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$		−	13.5638i		−	0.488171i
$773$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$773$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$774$		0			0
$775$	10.5110	−	1.85337i	0.377566	−	0.0665750i
$776$		0			0
$777$	−3.88175	−	3.36169i	−0.139257	−	0.120600i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	5.71957	−	27.4096i	0.204270	−	0.978915i
$785$		0			0
$786$		0			0
$787$		−	33.3425i		−	1.18853i	−0.804269	−	0.594265i	$-0.797443\pi$
$787$		−	33.3425i		−	1.18853i	0.804269	−	0.594265i	$-0.202557\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.64416	+	4.51730i	0.0583859	+	0.160414i
$794$		0			0
$795$		0			0
$796$	−8.72638	+	49.4897i	−0.309298	+	1.75412i
$797$		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
$804$	4.59028	+	26.0328i	0.161887	+	0.918105i
$805$		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
$811$	−42.6434	+	7.51919i	−1.49741	+	0.264034i	−0.861512	−	0.507736i	$-0.830482\pi$
$811$	−42.6434	+	7.51919i	−1.49741	+	0.264034i	−0.635901	+	0.771771i	$0.719371\pi$
$812$		0			0
$813$	−32.2869	−	38.4780i	−1.13235	−	1.34948i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	40.5562	−	38.3698i	1.41888	−	1.34239i
$818$		0			0
$819$	1.99154	+	5.74909i	0.0695902	+	0.200890i
$820$		0			0
$821$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$821$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$822$		0			0
$823$	−3.83022	+	3.21394i	−0.133513	+	0.112031i	−0.707099	−	0.707115i	$-0.749996\pi$
$823$	−3.83022	+	3.21394i	−0.133513	+	0.112031i	0.573586	+	0.819146i	$0.305552\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$827$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$828$		0			0
$829$	−41.8159	−	24.1424i	−1.45233	−	0.838501i	−0.453713	−	0.891148i	$-0.649901\pi$
$829$	−41.8159	−	24.1424i	−1.45233	−	0.838501i	−0.998613	+	0.0526472i	$0.983234\pi$
$830$		0			0
$831$	−15.4023	−	42.3175i	−0.534300	−	1.46798i
$832$	−3.94181	−	4.69766i	−0.136658	−	0.162862i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.54592	−	9.60582i	−0.191695	−	0.332026i
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$		0			0
$844$		−	56.5173i		−	1.94541i
$845$		0			0
$846$		0			0
$847$	−14.9428	+	24.9742i	−0.513442	+	0.858124i
$848$		0			0
$849$	4.14677	−	11.3932i	0.142317	−	0.391012i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	37.5663	−	13.6730i	1.28625	−	0.468155i	0.393753	−	0.919216i	$-0.371177\pi$
$853$	37.5663	−	13.6730i	1.28625	−	0.468155i	0.892493	+	0.451061i	$0.148954\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$857$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$858$		0			0
$859$	−4.65059	−	26.3748i	−0.158676	−	0.899896i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−4.65059	−	26.3748i	−0.158676	−	0.899896i	0.796672	−	0.604412i	$-0.206592\pi$
$860$		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
$868$	−11.1530	+	1.78774i	−0.378558	+	0.0606799i
$869$		0			0
$870$		0			0
$871$	−5.49670	+	2.00064i	−0.186249	+	0.0677890i
$872$		0			0
$873$	49.5000	+	28.5788i	1.67532	+	0.967247i
$874$		0			0
$875$		0			0
$876$	−14.1102	−	38.7674i	−0.476739	−	1.30983i
$877$	26.9360	−	32.1011i	0.909564	−	1.08398i	−0.0865807	−	0.996245i	$-0.527594\pi$
$877$	26.9360	−	32.1011i	0.909564	−	1.08398i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		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
$883$	−29.8120	−	10.8507i	−1.00325	−	0.365154i	−0.212415	−	0.977180i	$-0.568133\pi$
$883$	−29.8120	−	10.8507i	−1.00325	−	0.365154i	−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$887$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$888$		0			0
$889$	11.4400	−	29.9688i	0.383685	−	1.00512i
$890$		0			0
$891$		0			0
$892$	−44.6350	+	25.7700i	−1.49449	+	0.862844i
$893$		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
$903$	−37.0231	−	45.5461i	−1.23205	−	1.51568i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	44.3492	+	7.81995i	1.47259	+	0.259657i	0.851613	−	0.524171i	$-0.175625\pi$
$907$	44.3492	+	7.81995i	1.47259	+	0.259657i	0.620977	+	0.783829i	$0.286736\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	27.0060	+	13.5158i	0.894258	+	0.447553i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−37.2131	+	31.2255i	−1.22955	+	1.03172i
$917$		0			0
$918$		0			0
$919$	39.7330			1.31067			0.655335	−	0.755338i	$-0.272528\pi$
$919$	39.7330			1.31067			0.655335	+	0.755338i	$0.272528\pi$
$920$		0			0
$921$	−9.37700	−	53.1796i	−0.308983	−	1.75233i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	5.51770	−	0.972920i	0.181421	−	0.0319894i
$926$		0			0
$927$	−3.67864	+	10.1070i	−0.120823	+	0.331957i
$928$		0			0
$929$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$929$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$	−28.3644	+	11.2454i	−0.929607	+	0.368553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.950781	−	5.39215i	−0.0310607	−	0.176154i	0.965331	−	0.261029i	$-0.0840619\pi$
$937$	−0.950781	−	5.39215i	−0.0310607	−	0.176154i	−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		0.984808	−	0.173648i	$-0.0555556\pi$
$941$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$947$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$948$	46.8212	+	39.2876i	1.52068	+	1.27600i
$949$	7.90604	−	4.56455i	0.256641	−	0.148172i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$953$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−13.2217	−	22.9006i	−0.426506	−	0.738730i
$962$		0			0
$963$		0			0
$964$	−12.7229	−	34.9558i	−0.409776	−	1.12585i
$965$		0			0
$966$		0			0
$967$	−51.2340	+	18.6477i	−1.64757	+	0.599668i	−0.988339	−	0.152268i	$-0.951342\pi$
$967$	−51.2340	+	18.6477i	−1.64757	+	0.599668i	−0.659236	+	0.751936i	$0.729120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$971$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$972$	30.7033	+	5.41381i	0.984808	+	0.173648i
$973$	51.8062	+	19.7760i	1.66083	+	0.633990i
$974$		0			0
$975$	−6.23813	−	2.27049i	−0.199780	−	0.0727140i
$976$	12.5425	−	21.7243i	0.401476	−	0.695377i
$977$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$983$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.90766	+	6.40452i	−0.0606908	+	0.203755i
$989$		0			0
$990$		0			0
$991$	17.4948	+	48.0667i	0.555742	+	1.52689i	0.825753	+	0.564031i	$0.190750\pi$
$991$	17.4948	+	48.0667i	0.555742	+	1.52689i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	37.8228	+	31.7371i	1.20027	+	1.00715i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.29009	+	47.0154i	0.262550	+	1.48899i	0.775923	+	0.630828i	$0.217285\pi$
$997$	8.29009	+	47.0154i	0.262550	+	1.48899i	−0.513373	+	0.858166i	$0.671604\pi$
$998$		0			0
$999$	−2.91131	−	5.04254i	−0.0921098	−	0.159539i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.br.a.128.1	yes	6
3.2	odd	2	CM	399.2.br.a.128.1	yes	6
7.4	even	3		399.2.bv.a.242.1	yes	6
19.15	odd	18		399.2.bv.a.338.1	yes	6
21.11	odd	6		399.2.bv.a.242.1	yes	6
57.53	even	18		399.2.bv.a.338.1	yes	6
133.53	odd	18	inner	399.2.br.a.53.1	✓	6
399.53	even	18	inner	399.2.br.a.53.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.br.a.53.1	✓	6	133.53	odd	18	inner
399.2.br.a.53.1	✓	6	399.53	even	18	inner
399.2.br.a.128.1	yes	6	1.1	even	1	trivial
399.2.br.a.128.1	yes	6	3.2	odd	2	CM
399.2.bv.a.242.1	yes	6	7.4	even	3
399.2.bv.a.242.1	yes	6	21.11	odd	6
399.2.bv.a.338.1	yes	6	19.15	odd	18
399.2.bv.a.338.1	yes	6	57.53	even	18

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)

\(f(q)\)	\(=\)	\(q+(0.592396 + 1.62760i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-1.35844 + 2.27038i) q^{7} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})\)	\(q+(0.592396 + 1.62760i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-1.35844 + 2.27038i) q^{7} +(-2.29813 + 1.92836i) q^{9} +3.46410i q^{12} +(-0.754900 + 0.133109i) q^{13} +(3.06418 + 2.57115i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-4.10607 + 3.33770i) q^{28} +(1.84864 + 1.06731i) q^{31} +(-5.63816 + 2.05212i) q^{36} +(0.970437 + 0.560282i) q^{37} +(-0.663848 - 1.14982i) q^{39} +(9.81180 + 8.23308i) q^{43} +(-2.36959 + 6.51038i) q^{48} +(-3.30928 - 6.16836i) q^{49} +(-1.50980 - 0.266219i) q^{52} +(7.34389 - 1.75135i) q^{57} +(-1.08899 - 6.17598i) q^{61} +(-1.25624 - 7.83721i) q^{63} +(4.00000 + 6.92820i) q^{64} +(7.51501 - 1.32510i) q^{67} +(-11.1912 + 4.07326i) q^{73} +(7.50000 + 4.33013i) q^{75} +(3.90167 - 7.79596i) q^{76} +(11.3414 - 13.5161i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-7.86484 - 4.70578i) q^{84} +(0.723278 - 1.89473i) q^{91} +(-0.642026 + 3.64111i) q^{93} +(-6.51636 - 17.9035i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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