LMFDB - Embedded newform 399.2.cb.a.320.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(5,399)]

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, 15, 16]))

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

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

chi := DirichletCharacter("399.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.cb (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			320.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	399.320
Dual form			399.2.cb.a.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.11334 + 1.32683i) q^{3} +(1.53209 - 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +3.46410i q^{12} +(1.99273 - 5.47497i) q^{13} +(0.694593 - 3.93923i) q^{16} +(-0.500000 + 4.33013i) q^{19} +(4.50000 + 0.866025i) q^{21} +(-0.868241 - 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(-4.94356 - 1.88711i) q^{28} +(7.27244 + 4.19875i) q^{31} +(-4.59627 - 3.85673i) q^{36} +(2.54323 - 4.40501i) q^{37} +(5.04576 + 8.73951i) q^{39} +(1.53596 - 8.71086i) q^{43} +(4.45336 + 5.30731i) q^{48} +(-3.68732 + 5.95010i) q^{49} +(-3.98545 - 10.9499i) q^{52} +(-5.18866 - 5.48432i) q^{57} +(-4.30406 + 11.8253i) q^{61} +(-6.15910 + 5.00654i) q^{63} +(-4.00000 - 6.92820i) q^{64} +(-0.594922 - 0.216534i) q^{67} +(-10.5680 + 12.5945i) q^{73} +(7.50000 + 4.33013i) q^{75} +(4.80066 + 7.27693i) q^{76} +(-2.46538 + 13.9819i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(8.00774 - 4.45826i) q^{84} +(-15.2208 + 2.43977i) q^{91} +(-13.6677 + 4.97464i) q^{93} +(12.2467 - 14.5951i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

Character values

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

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{2}{9}\right)$

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		0.939693	−	0.342020i	$-0.111111\pi$
$2$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$3$	−1.11334	+	1.32683i	−0.642788	+	0.766044i
$3$	−1.11334	+	1.32683i	−0.642788	+	0.766044i
$4$	1.53209	−	1.28558i	0.766044	−	0.642788i
$5$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$5$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$6$		0			0
$7$	−1.28699	−	2.31164i	−0.486436	−	0.873716i
$7$	−1.28699	−	2.31164i	−0.486436	−	0.873716i
$8$		0			0
$9$	−0.520945	−	2.95442i	−0.173648	−	0.984808i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			3.46410i			1.00000i
$13$	1.99273	−	5.47497i	0.552683	−	1.51848i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	1.99273	−	5.47497i	0.552683	−	1.51848i	0.830033	−	0.557714i	$-0.188322\pi$
$14$		0			0
$15$		0			0
$16$	0.694593	−	3.93923i	0.173648	−	0.984808i
$17$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$17$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$18$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$23$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$24$		0			0
$25$	−0.868241	−	4.92404i	−0.173648	−	0.984808i
$26$		0			0
$27$	4.50000	+	2.59808i	0.866025	+	0.500000i
$28$	−4.94356	−	1.88711i	−0.934246	−	0.356630i
$29$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$29$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$30$		0			0
$31$	7.27244	+	4.19875i	1.30617	+	0.754117i	0.981455	−	0.191695i	$-0.0613985\pi$
$31$	7.27244	+	4.19875i	1.30617	+	0.754117i	0.324714	+	0.945812i	$0.394732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.59627	−	3.85673i	−0.766044	−	0.642788i
$37$	2.54323	−	4.40501i	0.418105	−	0.724179i	−0.577644	−	0.816289i	$-0.696028\pi$
$37$	2.54323	−	4.40501i	0.418105	−	0.724179i	0.995749	+	0.0921098i	$0.0293611\pi$
$38$		0			0
$39$	5.04576	+	8.73951i	0.807968	+	1.39944i
$40$		0			0
$41$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$41$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$42$		0			0
$43$	1.53596	−	8.71086i	0.234232	−	1.32839i	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	1.53596	−	8.71086i	0.234232	−	1.32839i	0.844226	−	0.535988i	$-0.180061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$	4.45336	+	5.30731i	0.642788	+	0.766044i
$49$	−3.68732	+	5.95010i	−0.526760	+	0.850014i
$50$		0			0
$51$		0			0
$52$	−3.98545	−	10.9499i	−0.552683	−	1.51848i
$53$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$53$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.18866	−	5.48432i	−0.687255	−	0.726416i
$58$		0			0
$59$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$59$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$60$		0			0
$61$	−4.30406	+	11.8253i	−0.551079	+	1.51408i	0.281161	+	0.959661i	$0.409281\pi$
$61$	−4.30406	+	11.8253i	−0.551079	+	1.51408i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	−6.15910	+	5.00654i	−0.775974	+	0.630765i
$64$	−4.00000	−	6.92820i	−0.500000	−	0.866025i
$65$		0			0
$66$		0			0
$67$	−0.594922	−	0.216534i	−0.0726813	−	0.0264538i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	−0.594922	−	0.216534i	−0.0726813	−	0.0264538i	−0.378105	+	0.925763i	$0.623424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$71$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$72$		0			0
$73$	−10.5680	+	12.5945i	−1.23690	+	1.47408i	−0.409644	+	0.912245i	$0.634347\pi$
$73$	−10.5680	+	12.5945i	−1.23690	+	1.47408i	−0.827252	+	0.561830i	$0.810097\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	4.80066	+	7.27693i	0.550673	+	0.834721i
$77$		0			0
$78$		0			0
$79$	−2.46538	+	13.9819i	−0.277377	+	1.57309i	0.453930	+	0.891038i	$0.350022\pi$
$79$	−2.46538	+	13.9819i	−0.277377	+	1.57309i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$	8.00774	−	4.45826i	0.873716	−	0.486436i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$89$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$90$		0			0
$91$	−15.2208	+	2.43977i	−1.59557	+	0.255757i
$92$		0			0
$93$	−13.6677	+	4.97464i	−1.41728	+	0.515846i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.2467	−	14.5951i	1.24347	−	1.48191i	0.427284	−	0.904117i	$-0.359470\pi$
$97$	12.2467	−	14.5951i	1.24347	−	1.48191i	0.816185	−	0.577791i	$-0.196085\pi$
$98$		0			0
$99$		0			0
$100$	−7.66044	−	6.42788i	−0.766044	−	0.642788i
$101$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$101$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$102$		0			0
$103$	−13.4315	+	7.75470i	−1.32345	+	0.764094i	−0.984277	−	0.176631i	$-0.943480\pi$
$103$	−13.4315	+	7.75470i	−1.32345	+	0.764094i	−0.339172	+	0.940724i	$0.610147\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	10.2344	−	1.80460i	0.984808	−	0.173648i
$109$	1.87939	−	0.684040i	0.180012	−	0.0655192i	−0.250441	−	0.968132i	$-0.580576\pi$
$109$	1.87939	−	0.684040i	0.180012	−	0.0655192i	0.430454	+	0.902613i	$0.358354\pi$
$110$		0			0
$111$	3.01320	+	8.27871i	0.286001	+	0.785780i
$112$	−10.0000	+	3.46410i	−0.944911	+	0.327327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−17.2135	−	3.03520i	−1.59139	−	0.280604i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$	16.5398	−	2.91642i	1.48532	−	0.261902i
$125$		0			0
$126$		0			0
$127$	17.8542	+	6.49838i	1.58430	+	0.576638i	0.976134	−	0.217171i	$-0.0696829\pi$
$127$	17.8542	+	6.49838i	1.58430	+	0.576638i	0.608167	+	0.793809i	$0.291905\pi$
$128$		0			0
$129$	9.84776	+	11.7361i	0.867047	+	1.03331i
$130$		0			0
$131$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$131$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$132$		0			0
$133$	10.6532	−	4.41701i	0.923747	−	0.383003i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$137$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$138$		0			0
$139$	8.05556	−	22.1325i	0.683264	−	1.87725i	0.296866	−	0.954919i	$-0.404058\pi$
$139$	8.05556	−	22.1325i	0.683264	−	1.87725i	0.386398	−	0.922332i	$-0.373719\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	−3.78952	−	11.5169i	−0.312554	−	0.949900i
$148$	−1.76651	−	10.0184i	−0.145206	−	0.823506i
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$		0			0
$151$	11.5000	+	19.9186i	0.935857	+	1.62095i	0.773099	+	0.634285i	$0.218706\pi$
$151$	11.5000	+	19.9186i	0.935857	+	1.62095i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	18.9659	+	6.90301i	1.51848	+	0.552683i
$157$	−5.04236	−	13.8538i	−0.402424	−	1.10565i	−0.961085	−	0.276254i	$-0.910907\pi$
$157$	−5.04236	−	13.8538i	−0.402424	−	1.10565i	0.558661	−	0.829396i	$-0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.2456	+	19.4779i	0.880821	+	1.52563i	0.850430	+	0.526088i	$0.176342\pi$
$163$	11.2456	+	19.4779i	0.880821	+	1.52563i	0.0303908	+	0.999538i	$0.490325\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$167$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$168$		0			0
$169$	−16.0458	−	13.4640i	−1.23429	−	1.03569i
$170$		0			0
$171$	13.0535	−	0.778544i	0.998226	−	0.0595368i
$172$	−8.84524	−	15.3204i	−0.674443	−	1.16817i
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$	−10.2652	+	8.34424i	−0.775974	+	0.630765i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−4.45336	−	5.30731i	−0.331016	−	0.394489i	0.574707	−	0.818359i	$-0.305116\pi$
$181$	−4.45336	−	5.30731i	−0.331016	−	0.394489i	−0.905723	+	0.423870i	$0.860671\pi$
$182$		0			0
$183$	−10.8983	−	18.8764i	−0.805623	−	1.39538i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.214355	−	13.7461i	0.0155920	−	0.999878i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	13.6459	+	2.40614i	0.984808	+	0.173648i
$193$	−5.82114	−	4.88451i	−0.419015	−	0.351595i	0.408773	−	0.912636i	$-0.365957\pi$
$193$	−5.82114	−	4.88451i	−0.419015	−	0.351595i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		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
$199$	−5.24809	+	14.4190i	−0.372027	+	1.02214i	0.602549	+	0.798082i	$0.294152\pi$
$199$	−5.24809	+	14.4190i	−0.372027	+	1.02214i	−0.974576	+	0.224055i	$0.928070\pi$
$200$		0			0
$201$	0.949655	−	0.548284i	0.0669835	−	0.0386729i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−20.1830	−	11.6527i	−1.39944	−	0.807968i
$209$		0			0
$210$		0			0
$211$	21.3293	+	17.8974i	1.46837	+	1.23211i	0.917630	+	0.397436i	$0.130100\pi$
$211$	21.3293	+	17.8974i	1.46837	+	1.23211i	0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.346419	−	22.2150i	0.0235165	−	1.50805i
$218$		0			0
$219$	−4.94491	−	28.0440i	−0.334146	−	1.89504i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.186137	+	0.0328209i	0.0124646	+	0.00219785i	0.179877	−	0.983689i	$-0.442430\pi$
$223$	0.186137	+	0.0328209i	0.0124646	+	0.00219785i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	−14.0954	+	5.13030i	−0.939693	+	0.342020i
$226$		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
$229$	10.3981	+	6.00335i	0.687126	+	0.396713i	0.802535	−	0.596606i	$-0.203484\pi$
$229$	10.3981	+	6.00335i	0.687126	+	0.396713i	−0.115408	+	0.993318i	$0.536818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$233$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−15.8068	−	18.8378i	−1.02676	−	1.22364i
$238$		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
$241$	30.3614	−	5.35354i	1.95575	−	0.344852i	0.957309	−	0.289066i	$-0.0933448\pi$
$241$	30.3614	−	5.35354i	1.95575	−	0.344852i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$	5.33157	−	14.6484i	0.342020	−	0.939693i
$244$	8.60813	+	23.6506i	0.551079	+	1.51408i
$245$		0			0
$246$		0			0
$247$	22.7110	+	11.3662i	1.44506	+	0.723217i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$251$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$252$	−3.00000	+	15.5885i	−0.188982	+	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.0351	−	5.47232i	−0.939693	−	0.342020i
$257$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$257$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$258$		0			0
$259$	−13.4559	−	0.209830i	−0.836108	−	0.0130382i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.18984	+	0.433068i	−0.0726813	+	0.0264538i
$269$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$269$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$270$		0			0
$271$	10.0201	−	11.9415i	0.608676	−	0.725391i	−0.370403	−	0.928871i	$-0.620781\pi$
$271$	10.0201	−	11.9415i	0.608676	−	0.725391i	0.979079	+	0.203479i	$0.0652250\pi$
$272$		0			0
$273$	13.7087	−	22.9116i	0.829690	−	1.38667i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	26.0000			1.56219			0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000			1.56219			0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$	8.61633	−	23.6732i	0.515846	−	1.41728i
$280$		0			0
$281$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$281$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$282$		0			0
$283$	−32.4090	−	5.71458i	−1.92652	−	0.339697i	−0.927130	−	0.374741i	$-0.877732\pi$
$283$	−32.4090	−	5.71458i	−1.92652	−	0.339697i	−0.999386	+	0.0350443i	$0.988843\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.9748	+	5.81434i	0.939693	+	0.342020i
$290$		0			0
$291$	5.73039	+	32.4987i	0.335921	+	1.90510i
$292$			32.8819i			1.92427i
$293$		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
$300$	17.0574	−	3.00767i	0.984808	−	0.173648i
$301$	−22.1131	+	7.66020i	−1.27458	+	0.441527i
$302$		0			0
$303$		0			0
$304$	16.7101	+	4.97729i	0.958388	+	0.285467i
$305$		0			0
$306$		0			0
$307$	10.6631	+	29.2967i	0.608577	+	1.67205i	0.733337	+	0.679865i	$0.237962\pi$
$307$	10.6631	+	29.2967i	0.608577	+	1.67205i	−0.124760	+	0.992187i	$0.539816\pi$
$308$		0			0
$309$	4.66473	−	26.4550i	0.265367	−	1.50497i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	27.2918	−	4.81228i	1.54262	−	0.272006i	0.663345	−	0.748314i	$-0.269136\pi$
$313$	27.2918	−	4.81228i	1.54262	−	0.272006i	0.879279	+	0.476308i	$0.158025\pi$
$314$		0			0
$315$		0			0
$316$	14.1976	+	24.5909i	0.798677	+	1.38335i
$317$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$317$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$318$		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
$325$	−28.6891	−	5.05867i	−1.59139	−	0.280604i
$326$		0			0
$327$	−1.18479	+	3.25519i	−0.0655192	+	0.180012i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−17.9971	−	31.1718i	−0.989208	−	1.71336i	−0.621492	−	0.783420i	$-0.713473\pi$
$331$	−17.9971	−	31.1718i	−0.989208	−	1.71336i	−0.367716	−	0.929938i	$-0.619860\pi$
$332$		0			0
$333$	−14.3391	−	5.21902i	−0.785780	−	0.286001i
$334$		0			0
$335$		0			0
$336$	6.53714	−	17.1250i	0.356630	−	0.934246i
$337$	−4.72503	+	26.7970i	−0.257389	+	1.45973i	0.532476	+	0.846445i	$0.321262\pi$
$337$	−4.72503	+	26.7970i	−0.257389	+	1.45973i	−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		0.939693	−	0.342020i	$-0.111111\pi$
$347$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$	−15.1879	+	8.76877i	−0.812992	+	0.469381i	−0.847994	−	0.530006i	$-0.822190\pi$
$349$	−15.1879	+	8.76877i	−0.812992	+	0.469381i	0.0350017	+	0.999387i	$0.488856\pi$
$350$		0			0
$351$	23.1917	−	19.4601i	1.23788	−	1.03870i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$	−12.2467	+	14.5951i	−0.642788	+	0.766044i
$364$	−20.1830	+	23.3054i	−1.05788	+	1.22153i
$365$		0			0
$366$		0			0
$367$	−36.6104	+	6.45540i	−1.91105	+	0.336969i	−0.997555	−	0.0698862i	$-0.977736\pi$
$367$	−36.6104	+	6.45540i	−1.91105	+	0.336969i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−14.5449	+	25.1925i	−0.754117	+	1.30617i
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−30.6219			−1.57294			−0.786469	−	0.617629i	$-0.788093\pi$
$379$	−30.6219			−1.57294			−0.786469	+	0.617629i	$0.788093\pi$
$380$		0			0
$381$	−28.5000	+	16.4545i	−1.46010	+	0.842989i
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−26.5357			−1.34889
$388$		−	38.1051i		−	1.93449i
$389$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$389$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.02915	+	5.57505i	0.101840	+	0.279804i	0.980140	−	0.198307i	$-0.0635442\pi$
$397$	2.02915	+	5.57505i	0.101840	+	0.279804i	−0.878300	+	0.478110i	$0.841322\pi$
$398$		0			0
$399$	−6.00000	+	19.0526i	−0.300376	+	0.953821i
$400$	−20.0000			−1.00000
$401$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$401$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$402$		0			0
$403$	37.4800	−	31.4494i	1.86701	−	1.56661i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.88594	+	24.4139i	−0.439382	+	1.20719i	0.500514	+	0.865729i	$0.333144\pi$
$409$	−8.88594	+	24.4139i	−0.439382	+	1.20719i	−0.939895	+	0.341463i	$0.889078\pi$
$410$		0			0
$411$		0			0
$412$	−10.6091	+	29.1482i	−0.522671	+	1.43603i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	20.3974	+	35.3293i	0.998865	+	1.73008i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	17.8542	−	6.49838i	0.870159	−	0.316712i	0.131927	−	0.991259i	$-0.457883\pi$
$421$	17.8542	−	6.49838i	0.870159	−	0.316712i	0.738231	+	0.674548i	$0.235661\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	32.8751	−	5.26963i	1.59094	−	0.255015i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$431$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$432$	13.3601	−	15.9219i	0.642788	−	0.766044i
$433$	−30.0940	+	5.30639i	−1.44623	+	0.255009i	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−30.0940	+	5.30639i	−1.44623	+	0.255009i	−0.605231	+	0.796050i	$0.706919\pi$
$434$		0			0
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$		0			0
$438$		0			0
$439$	7.57944	+	20.8243i	0.361747	+	0.993892i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	7.57944	+	20.8243i	0.361747	+	0.993892i	−0.616665	+	0.787226i	$0.711517\pi$
$440$		0			0
$441$	19.5000	+	7.79423i	0.928571	+	0.371154i
$442$		0			0
$443$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$443$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$444$	15.2594	+	8.81002i	0.724179	+	0.418105i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−10.8675	+	18.1631i	−0.513442	+	0.858124i
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−39.2320	−	6.91765i	−1.84328	−	0.325020i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.6006	−	33.9492i	0.916878	−	1.58808i	0.112749	−	0.993624i	$-0.464034\pi$
$457$	19.6006	−	33.9492i	0.916878	−	1.58808i	0.804129	−	0.594455i	$-0.202632\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$461$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$462$		0			0
$463$	4.58630	+	7.94371i	0.213144	+	0.369176i	0.952697	−	0.303923i	$-0.0982964\pi$
$463$	4.58630	+	7.94371i	0.213144	+	0.369176i	−0.739553	+	0.673098i	$0.764963\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$	−30.2746	+	17.4790i	−1.39944	+	0.807968i
$469$	0.265111	+	1.65392i	0.0122417	+	0.0763710i
$470$		0			0
$471$	23.9954	+	8.73362i	1.10565	+	0.402424i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	21.7558	−	1.29757i	0.998226	−	0.0595368i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$479$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$480$		0			0
$481$	−19.0493	−	22.7021i	−0.868574	−	1.03513i
$482$		0			0
$483$		0			0
$484$	16.8530	−	14.1413i	0.766044	−	0.642788i
$485$		0			0
$486$		0			0
$487$	22.0000	−	38.1051i	0.996915	−	1.72671i	0.430486	−	0.902597i	$-0.358342\pi$
$487$	22.0000	−	38.1051i	0.996915	−	1.72671i	0.566429	−	0.824110i	$-0.308325\pi$
$488$		0			0
$489$	−38.3640	−	6.76460i	−1.73488	−	0.305906i
$490$		0			0
$491$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$491$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	21.5912	−	25.7314i	0.969474	−	1.15537i
$497$		0			0
$498$		0			0
$499$	−34.1300	−	28.6385i	−1.52787	−	1.28203i	−0.811610	−	0.584199i	$-0.801409\pi$
$499$	−34.1300	−	28.6385i	−1.52787	−	1.28203i	−0.716258	−	0.697835i	$-0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$503$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	35.7288	−	6.29995i	1.58677	−	0.279791i
$508$	35.7083	−	12.9968i	1.58430	−	0.576638i
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	42.7149	+	8.22048i	1.88959	+	0.363653i
$512$		0			0
$513$	−13.5000	+	18.1865i	−0.596040	+	0.802955i
$514$		0			0
$515$		0			0
$516$	30.1753	+	5.32072i	1.32839	+	0.234232i
$517$		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
$523$	−15.4599	+	2.72600i	−0.676015	+	0.119200i	−0.501107	−	0.865385i	$-0.667074\pi$
$523$	−15.4599	+	2.72600i	−0.676015	+	0.119200i	−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	0.357259	−	22.9101i	0.0155920	−	0.999878i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	17.6190	+	14.7841i	0.766044	+	0.642788i
$530$		0			0
$531$		0			0
$532$	10.6432	−	20.4627i	0.461442	−	0.887171i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.202277	−	1.14717i	−0.00869658	−	0.0493207i	0.980151	−	0.198254i	$-0.0635271\pi$
$541$	−0.202277	−	1.14717i	−0.00869658	−	0.0493207i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	12.0000			0.514969
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.96720	+	45.1842i	0.340653	+	1.93194i	0.362031	+	0.932166i	$0.382083\pi$
$547$	7.96720	+	45.1842i	0.340653	+	1.93194i	−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$	37.1792	+	6.55569i	1.58677	+	0.279790i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	35.4940	−	12.2955i	1.50936	−	0.522857i
$554$		0			0
$555$		0			0
$556$	−16.1111	−	44.2649i	−0.683264	−	1.87725i
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$	−44.6309	−	25.7677i	−1.88769	−	1.08986i
$560$		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		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−15.2187	−	26.3595i	−0.636882	−	1.10311i	−0.986113	−	0.166076i	$-0.946890\pi$
$571$	−15.2187	−	26.3595i	−0.636882	−	1.10311i	0.349231	−	0.937037i	$-0.386443\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−18.3851	+	15.4269i	−0.766044	+	0.642788i
$577$	28.5000	+	16.4545i	1.18647	+	0.685009i	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	28.5000	+	16.4545i	1.18647	+	0.685009i	0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$	12.9618	−	2.28552i	0.538675	−	0.0949829i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$587$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$588$	−20.6117	−	12.7732i	−0.850014	−	0.526760i
$589$	−21.8173	+	29.3912i	−0.898967	+	1.21104i
$590$		0			0
$591$		0			0
$592$	−15.5858	−	13.0781i	−0.640574	−	0.537505i
$593$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$593$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.2886	−	23.0166i	−0.543868	−	0.942007i
$598$		0			0
$599$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$599$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$600$		0			0
$601$	13.1042	−	7.56570i	0.534530	−	0.308611i	−0.208329	−	0.978059i	$-0.566802\pi$
$601$	13.1042	−	7.56570i	0.534530	−	0.308611i	0.742859	+	0.669448i	$0.233469\pi$
$602$		0			0
$603$	−0.329812	+	1.87046i	−0.0134310	+	0.0761708i
$604$	43.2259	+	15.7329i	1.75884	+	0.640164i
$605$		0			0
$606$		0			0
$607$	−41.0091	−	23.6766i	−1.66451	−	0.961005i	−0.970520	−	0.241020i	$-0.922518\pi$
$607$	−41.0091	−	23.6766i	−1.66451	−	0.961005i	−0.693990	−	0.719985i	$-0.744149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−44.1656	+	16.0749i	−1.78383	+	0.649261i	−0.784245	+	0.620451i	$0.786949\pi$
$613$	−44.1656	+	16.0749i	−1.78383	+	0.649261i	−0.999585	+	0.0288097i	$0.990828\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$617$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$618$		0			0
$619$		−	49.7594i		−	2.00000i	−0.000400419	−	1.00000i	$-0.500127\pi$
$619$		−	49.7594i		−	2.00000i	0.000400419	−	1.00000i	$-0.499873\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	37.9317	−	13.8060i	1.51848	−	0.552683i
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$	−25.5354	−	14.7429i	−1.01897	−	0.588304i
$629$		0			0
$630$		0			0
$631$	8.55959	+	48.5439i	0.340752	+	1.93250i	0.360657	+	0.932699i	$0.382553\pi$
$631$	8.55959	+	48.5439i	0.340752	+	1.93250i	−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$	−47.4937	+	8.37441i	−1.88770	+	0.332853i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	25.2288	+	32.0449i	0.999601	+	1.26966i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$641$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$642$		0			0
$643$	−11.6213	−	31.9293i	−0.458301	−	1.25917i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−11.6213	−	31.9293i	−0.458301	−	1.25917i	0.468449	−	0.883491i	$-0.344813\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	29.0898	+	25.1925i	1.14012	+	0.987371i
$652$	42.2695	+	15.3848i	1.65540	+	0.602517i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	42.7149	+	24.6614i	1.66647	+	0.962135i
$658$		0			0
$659$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$659$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$660$		0			0
$661$	−32.2869	−	38.4780i	−1.25581	−	1.49662i	−0.791783	−	0.610802i	$-0.790847\pi$
$661$	−32.2869	−	38.4780i	−1.25581	−	1.49662i	−0.464031	−	0.885819i	$-0.653597\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.250781	+	0.210430i	−0.00969577	+	0.00813571i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	14.6977	−	25.4572i	0.566557	−	0.981305i	−0.430346	−	0.902664i	$-0.641609\pi$
$673$	14.6977	−	25.4572i	0.566557	−	0.981305i	0.996903	−	0.0786409i	$-0.0250581\pi$
$674$		0			0
$675$	8.88594	−	24.4139i	0.342020	−	0.939693i
$676$	−41.8925			−1.61125
$677$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	18.9982	−	17.9741i	0.726416	−	0.687255i
$685$		0			0
$686$		0			0
$687$	−19.5421	+	7.11272i	−0.745576	+	0.271367i
$688$	−33.2472	−	12.1010i	−1.26754	−	0.461346i
$689$		0			0
$690$		0			0
$691$		−	51.9615i		−	1.97671i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$		−	51.9615i		−	1.97671i	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		−0.766044	−	0.642788i	$-0.777778\pi$
$701$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$	17.8026	+	13.2150i	0.671439	+	0.498414i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	39.5133	−	33.1556i	1.48395	−	1.24518i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	39.5133	−	33.1556i	1.48395	−	1.24518i	0.901837	−	0.432077i	$-0.142219\pi$
$710$		0			0
$711$	42.5928			1.59735
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$719$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$720$		0			0
$721$	35.2123	+	21.0686i	1.31137	+	0.784636i
$722$		0			0
$723$	−26.6994	+	46.2447i	−0.992961	+	1.71986i
$724$	−13.6459	−	2.40614i	−0.507146	−	0.0894235i
$725$		0			0
$726$		0			0
$727$	14.0314	+	2.47411i	0.520395	+	0.0917597i	0.427675	−	0.903933i	$-0.359333\pi$
$727$	14.0314	+	2.47411i	0.520395	+	0.0917597i	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
$732$	−40.9641	−	14.9097i	−1.51408	−	0.551079i
$733$	18.0000	−	10.3923i	0.664845	−	0.383849i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	18.0000	−	10.3923i	0.664845	−	0.383849i	0.794121	+	0.607760i	$0.207932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	3.67153	−	20.8223i	0.135060	−	0.765961i	−0.839759	−	0.542960i	$-0.817304\pi$
$739$	3.67153	−	20.8223i	0.135060	−	0.765961i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$	−40.3661	+	17.4790i	−1.48288	+	0.642108i
$742$		0			0
$743$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$743$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8.85045	+	50.1934i	0.322958	+	1.83158i	0.523655	+	0.851930i	$0.324568\pi$
$751$	8.85045	+	50.1934i	0.322958	+	1.83158i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−17.3432	−	21.3357i	−0.630765	−	0.775974i
$757$	−10.5775	+	3.84991i	−0.384447	+	0.139927i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−10.5775	+	3.84991i	−0.384447	+	0.139927i	0.142564	+	0.989786i	$0.454465\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−4.00000	−	3.46410i	−0.144810	−	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	24.0000	−	13.8564i	0.866025	−	0.500000i
$769$	−26.4654	−	31.5403i	−0.954368	−	1.13737i	−0.990429	−	0.138022i	$-0.955925\pi$
$769$	−26.4654	−	31.5403i	−0.954368	−	1.13737i	0.0360609	−	0.999350i	$-0.488519\pi$
$770$		0			0
$771$		0			0
$772$	−15.1979			−0.546985
$773$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$773$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$774$		0			0
$775$	14.3606	−	39.4553i	0.515846	−	1.41728i
$776$		0			0
$777$	15.2594	−	17.6200i	0.547428	−	0.632115i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	20.8776	+	18.6581i	0.745630	+	0.666361i
$785$		0			0
$786$		0			0
$787$			55.7508i			1.98730i	0.112514	+	0.993650i	$0.464110\pi$
$787$			55.7508i			1.98730i	−0.112514	+	0.993650i	$0.535890\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	56.1664	+	47.1292i	1.99453	+	1.67361i
$794$		0			0
$795$		0			0
$796$	10.4962	+	28.8380i	0.372027	+	1.02214i
$797$		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
$804$	0.750096	−	2.06087i	0.0264538	−	0.0726813i
$805$		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
$811$	−14.8099	+	40.6899i	−0.520046	+	1.42882i	0.350423	+	0.936592i	$0.386038\pi$
$811$	−14.8099	+	40.6899i	−0.520046	+	1.42882i	−0.870469	+	0.492223i	$0.836184\pi$
$812$		0			0
$813$	4.68850	+	26.5898i	0.164433	+	0.932545i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	36.9511	+	11.0063i	1.29276	+	0.385063i
$818$		0			0
$819$	15.1373	+	43.6976i	0.528939	+	1.52692i
$820$		0			0
$821$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$821$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$822$		0			0
$823$	0.868241	+	4.92404i	0.0302650	+	0.171641i	0.996194	−	0.0871670i	$-0.0277814\pi$
$823$	0.868241	+	4.92404i	0.0302650	+	0.171641i	−0.965929	+	0.258808i	$0.916670\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$827$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$828$		0			0
$829$	2.62551	+	1.51584i	0.0911876	+	0.0526472i	0.544900	−	0.838501i	$-0.316567\pi$
$829$	2.62551	+	1.51584i	0.0911876	+	0.0526472i	−0.453713	+	0.891148i	$0.649901\pi$
$830$		0			0
$831$	−28.9469	+	34.4975i	−1.00416	+	1.19671i
$832$	−45.9026	+	8.09387i	−1.59139	+	0.280604i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21.8173	+	37.7887i	0.754117	+	1.30617i
$838$		0			0
$839$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$839$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$840$		0			0
$841$	5.03580	−	28.5594i	0.173648	−	0.984808i
$842$		0			0
$843$		0			0
$844$	55.6870			1.91682
$845$		0			0
$846$		0			0
$847$	−14.1569	−	25.4280i	−0.486436	−	0.873716i
$848$		0			0
$849$	43.6645	−	36.6389i	1.49856	−	1.25744i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−35.9420	+	42.8340i	−1.23063	+	1.46661i	−0.393753	+	0.919216i	$0.628823\pi$
$853$	−35.9420	+	42.8340i	−1.23063	+	1.46661i	−0.836877	+	0.547391i	$0.815621\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$857$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$858$		0			0
$859$	−0.984052	+	2.70366i	−0.0335754	+	0.0922477i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−0.984052	+	2.70366i	−0.0335754	+	0.0922477i	0.921772	+	0.387732i	$0.126741\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.5000	+	14.7224i	−0.866025	+	0.500000i
$868$	−28.0283	−	34.4807i	−0.951342	−	1.17035i
$869$		0			0
$870$		0			0
$871$	−2.37104	+	2.82569i	−0.0803395	+	0.0957448i
$872$		0			0
$873$	−49.5000	−	28.5788i	−1.67532	−	0.967247i
$874$		0			0
$875$		0			0
$876$	−43.6287	−	36.6088i	−1.47408	−	1.23690i
$877$	2.66766	−	15.1291i	0.0900805	−	0.510872i	−0.906064	−	0.423141i	$-0.860927\pi$
$877$	2.66766	−	15.1291i	0.0900805	−	0.510872i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$	−45.4910	+	38.1715i	−1.53089	+	1.28457i	−0.740055	+	0.672546i	$0.765201\pi$
$883$	−45.4910	+	38.1715i	−1.53089	+	1.28457i	−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$887$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$888$		0			0
$889$	−7.95621	−	49.6357i	−0.266843	−	1.66473i
$890$		0			0
$891$		0			0
$892$	0.327372	−	0.189008i	0.0109612	−	0.00632846i
$893$		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
$903$	14.4556	−	37.8687i	0.481054	−	1.26019i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−37.5877	+	13.6808i	−1.24808	+	0.454264i	−0.879752	−	0.475433i	$-0.842292\pi$
$907$	−37.5877	+	13.6808i	−1.24808	+	0.454264i	−0.368327	+	0.929696i	$0.620069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−25.2080	+	16.6300i	−0.834721	+	0.550673i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	23.6486	−	4.16988i	0.781371	−	0.137777i
$917$		0			0
$918$		0			0
$919$	−59.5271			−1.96362			−0.981810	−	0.189867i	$-0.939194\pi$
$919$	−59.5271			−1.96362			−0.981810	+	0.189867i	$0.939194\pi$
$920$		0			0
$921$	−50.7434	−	18.4691i	−1.67205	−	0.608577i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−23.8986	−	8.69837i	−0.785780	−	0.286001i
$926$		0			0
$927$	29.9078	+	35.6427i	0.982300	+	1.17066i
$928$		0			0
$929$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$929$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$930$		0			0
$931$	−23.9210	−	18.9416i	−0.783980	−	0.620786i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−8.80566	+	24.1933i	−0.287668	+	0.790362i	0.708723	+	0.705487i	$0.249271\pi$
$937$	−8.80566	+	24.1933i	−0.287668	+	0.790362i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$	−24.0000	+	41.5692i	−0.783210	+	1.35656i
$940$		0			0
$941$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$947$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$948$	−48.4347	−	8.54034i	−1.57309	−	0.277377i
$949$	47.8953	+	82.9572i	1.55475	+	2.69290i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$953$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.7589	+	34.2235i	0.637385	+	1.10398i
$962$		0			0
$963$		0			0
$964$	39.6340	−	47.2340i	1.27653	−	1.52130i
$965$		0			0
$966$		0			0
$967$	1.03234	+	0.866234i	0.0331977	+	0.0278562i	0.659236	−	0.751936i	$-0.270880\pi$
$967$	1.03234	+	0.866234i	0.0331977	+	0.0278562i	−0.626038	+	0.779793i	$0.715324\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$971$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$972$	−10.6631	−	29.2967i	−0.342020	−	0.939693i
$973$	−61.5296	+	9.86272i	−1.97255	+	0.316184i
$974$		0			0
$975$	38.6528	−	32.4335i	1.23788	−	1.03870i
$976$	43.5931	+	25.1685i	1.39538	+	0.805623i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−3.00000	−	5.19615i	−0.0957826	−	0.165900i
$982$		0			0
$983$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$983$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	49.4074	−	11.7825i	1.57186	−	0.374853i
$989$		0			0
$990$		0			0
$991$	19.8744	+	16.6766i	0.631331	+	0.529749i	0.901342	−	0.433108i	$-0.142583\pi$
$991$	19.8744	+	16.6766i	0.631331	+	0.529749i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	61.3965	+	10.8259i	1.94836	+	0.343548i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.07151	+	19.4288i	−0.223957	+	0.615317i	−0.999880	−	0.0155113i	$-0.995062\pi$
$997$	−7.07151	+	19.4288i	−0.223957	+	0.615317i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	22.8891	−	13.2150i	0.724179	−	0.418105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.cb.a.320.1	yes	6
3.2	odd	2	CM	399.2.cb.a.320.1	yes	6
7.3	odd	6		399.2.ci.a.206.1	yes	6
19.6	even	9		399.2.ci.a.215.1	yes	6
21.17	even	6		399.2.ci.a.206.1	yes	6
57.44	odd	18		399.2.ci.a.215.1	yes	6
133.101	odd	18	inner	399.2.cb.a.101.1	✓	6
399.101	even	18	inner	399.2.cb.a.101.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.cb.a.101.1	✓	6	133.101	odd	18	inner
399.2.cb.a.101.1	✓	6	399.101	even	18	inner
399.2.cb.a.320.1	yes	6	1.1	even	1	trivial
399.2.cb.a.320.1	yes	6	3.2	odd	2	CM
399.2.ci.a.206.1	yes	6	7.3	odd	6
399.2.ci.a.206.1	yes	6	21.17	even	6
399.2.ci.a.215.1	yes	6	19.6	even	9
399.2.ci.a.215.1	yes	6	57.44	odd	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.cb (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.11334 + 1.32683i) q^{3} +(1.53209 - 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +3.46410i q^{12} +(1.99273 - 5.47497i) q^{13} +(0.694593 - 3.93923i) q^{16} +(-0.500000 + 4.33013i) q^{19} +(4.50000 + 0.866025i) q^{21} +(-0.868241 - 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(-4.94356 - 1.88711i) q^{28} +(7.27244 + 4.19875i) q^{31} +(-4.59627 - 3.85673i) q^{36} +(2.54323 - 4.40501i) q^{37} +(5.04576 + 8.73951i) q^{39} +(1.53596 - 8.71086i) q^{43} +(4.45336 + 5.30731i) q^{48} +(-3.68732 + 5.95010i) q^{49} +(-3.98545 - 10.9499i) q^{52} +(-5.18866 - 5.48432i) q^{57} +(-4.30406 + 11.8253i) q^{61} +(-6.15910 + 5.00654i) q^{63} +(-4.00000 - 6.92820i) q^{64} +(-0.594922 - 0.216534i) q^{67} +(-10.5680 + 12.5945i) q^{73} +(7.50000 + 4.33013i) q^{75} +(4.80066 + 7.27693i) q^{76} +(-2.46538 + 13.9819i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(8.00774 - 4.45826i) q^{84} +(-15.2208 + 2.43977i) q^{91} +(-13.6677 + 4.97464i) q^{93} +(12.2467 - 14.5951i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{13} - 3 q^{19} + 27 q^{21} + 27 q^{27} - 24 q^{43} + 12 q^{52} - 39 q^{61} - 24 q^{64} + 15 q^{67} - 21 q^{73} + 45 q^{75} - 39 q^{79} - 51 q^{91} - 54 q^{93}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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