LMFDB - Embedded newform 399.2.br.a.212.1

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			212.1
Root			$-0.173648 - 0.984808i$ of defining polynomial
Character	$\chi$	$=$	399.212
Dual form			399.2.br.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11334 - 1.32683i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})$	$q+(1.11334 - 1.32683i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +3.46410i q^{12} +(1.99273 - 5.47497i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(0.868241 + 4.92404i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(4.94356 + 1.88711i) q^{28} +(7.27244 + 4.19875i) q^{31} +(4.59627 + 3.85673i) q^{36} +(-9.57057 - 5.52557i) q^{37} +(-5.04576 - 8.73951i) q^{39} +(-1.53596 + 8.71086i) q^{43} +(-4.45336 - 5.30731i) q^{48} +(-3.68732 + 5.95010i) q^{49} +(3.98545 + 10.9499i) q^{52} +(-5.18866 - 5.48432i) q^{57} +(8.69594 + 3.16506i) q^{61} +(-6.15910 + 5.00654i) q^{63} +(4.00000 + 6.92820i) q^{64} +(5.59492 - 15.3719i) q^{67} +(3.56805 + 2.99395i) q^{73} +(7.50000 + 4.33013i) q^{75} +(4.80066 + 7.27693i) q^{76} +(10.5346 + 1.85754i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(8.00774 - 4.45826i) q^{84} +(-15.2208 + 2.43977i) q^{91} +(13.6677 - 4.97464i) q^{93} +(-12.2467 + 14.5951i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+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 $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{13}{18}\right)$

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		−0.342020	−	0.939693i	$-0.611111\pi$
$2$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$3$	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		−0.766044	−	0.642788i	$-0.777778\pi$
$5$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$6$		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		0.173648	−	0.984808i	$-0.444444\pi$
$17$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$18$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		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
$37$	−9.57057	−	5.52557i	−1.57339	−	0.908399i	−0.995749	−	0.0921098i	$-0.970639\pi$
$37$	−9.57057	−	5.52557i	−1.57339	−	0.908399i	−0.577644	−	0.816289i	$-0.696028\pi$
$38$		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	$0.291172\pi$
$43$	−1.53596	+	8.71086i	−0.234232	+	1.32839i	−0.844226	+	0.535988i	$0.819939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$47$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$48$	−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		−0.642788	−	0.766044i	$-0.722222\pi$
$53$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$54$		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
$61$	8.69594	+	3.16506i	1.11340	+	0.405245i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	8.69594	+	3.16506i	1.11340	+	0.405245i	0.281161	+	0.959661i	$0.409281\pi$
$62$		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
$67$	5.59492	−	15.3719i	0.683529	−	1.87798i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.59492	−	15.3719i	0.683529	−	1.87798i	0.378105	−	0.925763i	$-0.376576\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$71$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$72$		0			0
$73$	3.56805	+	2.99395i	0.417608	+	0.350415i	0.827252	−	0.561830i	$-0.189903\pi$
$73$	3.56805	+	2.99395i	0.417608	+	0.350415i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		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
$79$	10.5346	+	1.85754i	1.18524	+	0.208989i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	10.5346	+	1.85754i	1.18524	+	0.208989i	0.453930	+	0.891038i	$0.350022\pi$
$80$		0			0
$81$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	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	$0.640530\pi$
$97$	−12.2467	+	14.5951i	−1.24347	+	1.48191i	−0.816185	+	0.577791i	$0.803915\pi$
$98$		0			0
$99$		0			0
$100$	−7.66044	−	6.42788i	−0.766044	−	0.642788i
$101$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$101$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$102$		0			0
$103$	13.4315	−	7.75470i	1.32345	−	0.764094i	0.339172	−	0.940724i	$-0.389853\pi$
$103$	13.4315	−	7.75470i	1.32345	−	0.764094i	0.984277	+	0.176631i	$0.0565198\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	10.2344	−	1.80460i	0.984808	−	0.173648i
$109$	7.10876	+	19.5311i	0.680895	+	1.87074i	0.430454	+	0.902613i	$0.358354\pi$
$109$	7.10876	+	19.5311i	0.680895	+	1.87074i	0.250441	+	0.968132i	$0.419424\pi$
$110$		0			0
$111$	−17.9868	+	6.54666i	−1.70723	+	0.621381i
$112$	−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
$117$	−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
$127$	4.14677	−	11.3932i	0.367967	−	1.01098i	−0.608167	−	0.793809i	$-0.708095\pi$
$127$	4.14677	−	11.3932i	0.367967	−	1.01098i	0.976134	−	0.217171i	$-0.0696829\pi$
$128$		0			0
$129$	9.84776	+	11.7361i	0.867047	+	1.03331i
$130$		0			0
$131$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$131$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$132$		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
$139$	−1.05556	−	0.384192i	−0.0895314	−	0.0325868i	0.296866	−	0.954919i	$-0.404058\pi$
$139$	−1.05556	−	0.384192i	−0.0895314	−	0.0325868i	−0.386398	+	0.922332i	$0.626281\pi$
$140$		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
$148$	21.7665	−	3.83802i	1.78920	−	0.315484i
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$		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
$156$	18.9659	+	6.90301i	1.51848	+	0.552683i
$157$	−19.0424	+	6.93085i	−1.51975	+	0.553142i	−0.961085	−	0.276254i	$-0.910907\pi$
$157$	−19.0424	+	6.93085i	−1.51975	+	0.553142i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		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	$-0.823658\pi$
$163$	−11.2456	−	19.4779i	−0.880821	−	1.52563i	−0.0303908	−	0.999538i	$-0.509675\pi$
$164$		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			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	4.45336	+	5.30731i	0.331016	+	0.394489i	0.905723	−	0.423870i	$-0.139329\pi$
$181$	4.45336	+	5.30731i	0.331016	+	0.394489i	−0.574707	+	0.818359i	$0.694884\pi$
$182$		0			0
$183$	13.8810	−	8.01422i	1.02612	−	0.592428i
$184$		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
$193$	17.1789	−	20.4730i	1.23656	−	1.47368i	0.408773	−	0.912636i	$-0.365957\pi$
$193$	17.1789	−	20.4730i	1.23656	−	1.47368i	0.827788	−	0.561041i	$-0.189599\pi$
$194$		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
$199$	22.2481	+	8.09764i	1.57713	+	0.574027i	0.974576	−	0.224055i	$-0.0719296\pi$
$199$	22.2481	+	8.09764i	1.57713	+	0.574027i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$	−14.1668	−	24.5377i	−0.999252	−	1.73076i
$202$		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
$211$	5.32934	−	6.35127i	0.366887	−	0.437239i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	5.32934	−	6.35127i	0.366887	−	0.437239i	0.917630	+	0.397436i	$0.130100\pi$
$212$		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
$219$	7.94491	−	1.40090i	0.536867	−	0.0946642i
$220$		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
$229$	13.8910	−	24.0599i	0.917943	−	1.58992i	0.115408	−	0.993318i	$-0.463182\pi$
$229$	13.8910	−	24.0599i	0.917943	−	1.58992i	0.802535	−	0.596606i	$-0.203484\pi$
$230$		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
$237$	14.1932	−	11.9095i	0.921951	−	0.773608i
$238$		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	$0.906655\pi$
$241$	−30.3614	+	5.35354i	−1.95575	+	0.344852i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	−5.33157	+	14.6484i	−0.342020	+	0.939693i
$244$	−17.3919	+	6.33012i	−1.11340	+	0.405245i
$245$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$251$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$252$	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
$259$	−0.455889	+	29.2350i	−0.0283276	+	1.81658i
$260$		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
$268$	11.1898	+	30.7438i	0.683529	+	1.87798i
$269$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$269$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$270$		0			0
$271$	22.2153	+	18.6408i	1.34948	+	1.13235i	0.979079	+	0.203479i	$0.0652250\pi$
$271$	22.2153	+	18.6408i	1.34948	+	1.13235i	0.370403	+	0.928871i	$0.379219\pi$
$272$		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	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	8.61633	−	23.6732i	0.515846	−	1.41728i
$280$		0			0
$281$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$281$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$282$		0			0
$283$	−1.21554	+	6.89365i	−0.0722562	+	0.409785i	0.927130	+	0.374741i	$0.122268\pi$
$283$	−1.21554	+	6.89365i	−0.0722562	+	0.409785i	−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
$292$	−9.31551			−0.545149
$293$		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		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	3.82026	+	21.6658i	0.215934	+	1.22462i	0.879279	+	0.476308i	$0.158025\pi$
$313$	3.82026	+	21.6658i	0.215934	+	1.22462i	−0.663345	+	0.748314i	$0.730864\pi$
$314$		0			0
$315$		0			0
$316$	−18.5280	+	10.6971i	−1.04228	+	0.601760i
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		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
$327$	33.8289	+	12.3127i	1.87074	+	0.680895i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.61705	−	2.66565i	0.253776	−	0.146518i	−0.367716	−	0.929938i	$-0.619860\pi$
$331$	4.61705	−	2.66565i	0.253776	−	0.146518i	0.621492	+	0.783420i	$0.286527\pi$
$332$		0			0
$333$	−11.3391	+	31.1540i	−0.621381	+	1.70723i
$334$		0			0
$335$		0			0
$336$	−6.53714	+	17.1250i	−0.356630	+	0.934246i
$337$	−24.2750	−	4.28033i	−1.32234	−	0.233165i	−0.532476	−	0.846445i	$-0.678738\pi$
$337$	−24.2750	−	4.28033i	−1.32234	−	0.233165i	−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
$349$	−16.4957	−	28.5714i	−0.882996	−	1.52939i	−0.847994	−	0.530006i	$-0.822190\pi$
$349$	−16.4957	−	28.5714i	−0.882996	−	1.52939i	−0.0350017	−	0.999387i	$-0.511144\pi$
$350$		0			0
$351$	−23.1917	+	19.4601i	−1.23788	+	1.03870i
$352$		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		0.939693	−	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
$367$	−1.61040	−	9.13306i	−0.0840624	−	0.476742i	−0.997555	−	0.0698862i	$-0.977736\pi$
$367$	−1.61040	−	9.13306i	−0.0840624	−	0.476742i	0.913493	−	0.406855i	$-0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−14.5449	+	25.1925i	−0.754117	+	1.30617i
$373$			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
$379$			24.0479i			1.23526i	0.786469	+	0.617629i	$0.211907\pi$
$379$			24.0479i			1.23526i	−0.786469	+	0.617629i	$0.788093\pi$
$380$		0			0
$381$	−10.5000	−	18.1865i	−0.537931	−	0.931724i
$382$		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
$397$	37.0292	−	13.4775i	1.85844	−	0.676417i	0.878300	−	0.478110i	$-0.158678\pi$
$397$	37.0292	−	13.4775i	1.85844	−	0.676417i	0.980140	−	0.198307i	$-0.0635442\pi$
$398$		0			0
$399$	−6.00000	+	19.0526i	−0.300376	+	0.953821i
$400$	20.0000			1.00000
$401$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$401$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$402$		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
$417$	−1.68495	+	0.972809i	−0.0825126	+	0.0476387i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	12.4403	+	34.1795i	0.606304	+	1.66581i	0.738231	+	0.674548i	$0.235661\pi$
$421$	12.4403	+	34.1795i	0.606304	+	1.66581i	−0.131927	+	0.991259i	$0.542117\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.87510	−	24.1752i	−0.187529	−	1.16992i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$431$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$432$	−13.3601	+	15.9219i	−0.642788	+	0.766044i
$433$	30.0940	−	5.30639i	1.44623	−	0.255009i	0.605231	−	0.796050i	$-0.293081\pi$
$433$	30.0940	−	5.30639i	1.44623	−	0.255009i	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
$439$	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	$0.777778\pi$
$444$	19.1411	−	33.1534i	0.908399	−	1.57339i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	10.8675	−	18.1631i	0.513442	−	0.858124i
$449$		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
$453$	2.60472	−	14.7721i	0.122381	−	0.694055i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.6006	+	33.9492i	−0.916878	+	1.58808i	−0.112749	+	0.993624i	$0.535966\pi$
$457$	−19.6006	+	33.9492i	−0.916878	+	1.58808i	−0.804129	+	0.594455i	$0.797368\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$461$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$462$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	30.2746	−	17.4790i	1.39944	−	0.807968i
$469$	−42.7349	+	6.85007i	−1.97331	+	0.316307i
$470$		0			0
$471$	−12.0046	+	32.9823i	−0.553142	+	1.51975i
$472$		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		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$	−49.3239	+	41.3877i	−2.24898	+	1.88711i
$482$		0			0
$483$		0			0
$484$	−16.8530	+	14.1413i	−0.766044	+	0.642788i
$485$		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
$489$	−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		0.766044	−	0.642788i	$-0.222222\pi$
$503$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−35.7288	+	6.29995i	−1.58677	+	0.279791i
$508$	8.29355	+	22.7863i	0.367967	+	1.01098i
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	2.32888	−	12.1012i	0.103023	−	0.535326i
$512$		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
$547$	8.96720	−	1.58116i	0.383410	−	0.0676055i	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	8.96720	−	1.58116i	0.383410	−	0.0676055i	0.362031	+	0.932166i	$0.382083\pi$
$548$		0			0
$549$	4.82083	−	27.3403i	0.205748	−	1.16686i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−9.26399	−	26.7428i	−0.393945	−	1.13722i
$554$		0			0
$555$		0			0
$556$	2.11112	−	0.768385i	0.0895314	−	0.0325868i
$557$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		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
$577$	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
$579$	−8.03818	−	45.5868i	−0.334055	−	1.89452i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$587$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$588$	−20.6117	−	12.7732i	−0.850014	−	0.526760i
$589$	21.8173	−	29.3912i	0.898967	−	1.21104i
$590$		0			0
$591$		0			0
$592$	−28.4142	+	33.8627i	−1.16781	+	1.39175i
$593$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$593$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	35.5139	−	20.5040i	1.45349	−	0.839171i
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$		0			0
$601$	−13.1042	+	7.56570i	−0.534530	+	0.308611i	−0.742859	−	0.669448i	$-0.766531\pi$
$601$	−13.1042	+	7.56570i	−0.534530	+	0.308611i	0.208329	+	0.978059i	$0.433198\pi$
$602$		0			0
$603$	−48.3298	−	8.52185i	−1.96814	−	0.347037i
$604$	−5.92396	+	16.2760i	−0.241043	+	0.662259i
$605$		0			0
$606$		0			0
$607$	41.0091	+	23.6766i	1.66451	+	0.961005i	0.970520	+	0.241020i	$0.0774820\pi$
$607$	41.0091	+	23.6766i	1.66451	+	0.961005i	0.693990	+	0.719985i	$0.255851\pi$
$608$		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	$-0.213051\pi$
$613$	44.1656	−	16.0749i	1.78383	−	0.649261i	0.999585	−	0.0288097i	$-0.00917168\pi$
$614$		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
$619$	−0.0199246			−0.000800838			−0.000400419	−	1.00000i	$-0.500127\pi$
$619$	−0.0199246			−0.000800838			−0.000400419		1.00000i	$0.500127\pi$
$620$		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
$628$	20.2645	−	35.0991i	0.808640	−	1.40060i
$629$		0			0
$630$		0			0
$631$	−8.55959	−	48.5439i	−0.340752	−	1.93250i	−0.360657	−	0.932699i	$-0.617447\pi$
$631$	−8.55959	−	48.5439i	−0.340752	−	1.93250i	0.0199047	−	0.999802i	$-0.493664\pi$
$632$		0			0
$633$	−2.49366	−	14.1422i	−0.0991141	−	0.562104i
$634$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$641$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$642$		0			0
$643$	35.3787	−	12.8768i	1.39520	−	0.507811i	0.468449	−	0.883491i	$-0.344813\pi$
$643$	35.3787	−	12.8768i	1.39520	−	0.507811i	0.926750	+	0.375680i	$0.122591\pi$
$644$		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
$651$	−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
$657$	6.98663	−	12.1012i	0.272575	−	0.472113i
$658$		0			0
$659$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$659$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$660$		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
$673$	−37.0261	−	21.3770i	−1.42725	−	0.824023i	−0.430346	−	0.902664i	$-0.641609\pi$
$673$	−37.0261	−	21.3770i	−1.42725	−	0.824023i	−0.996903	+	0.0786409i	$0.974942\pi$
$674$		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		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$	18.9982	−	17.9741i	0.726416	−	0.687255i
$685$		0			0
$686$		0			0
$687$	−16.4579	−	45.2178i	−0.627910	−	1.72517i
$688$	33.2472	+	12.1010i	1.26754	+	0.461346i
$689$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$701$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$	−28.7117	+	38.6790i	−1.08288	+	1.45881i
$704$		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	$0.697775\pi$
$709$	−39.5133	+	33.1556i	−1.48395	+	1.24518i	−0.901837	+	0.432077i	$0.857781\pi$
$710$		0			0
$711$		−	32.0914i		−	1.20352i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		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
$727$	−9.03137	+	51.2195i	−0.334955	+	1.89962i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	−9.03137	+	51.2195i	−0.334955	+	1.89962i	−0.427675	+	0.903933i	$0.640667\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	−10.9641	+	30.1236i	−0.405245	+	1.11340i
$733$	−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
$739$	−3.67153	+	20.8223i	−0.135060	+	0.765961i	0.839759	+	0.542960i	$0.182696\pi$
$739$	−3.67153	+	20.8223i	−0.135060	+	0.765961i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$	−40.3661	+	17.4790i	−1.48288	+	0.642108i
$742$		0			0
$743$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$743$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	19.8505	−	3.50017i	0.724353	−	0.127723i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	19.8505	−	3.50017i	0.724353	−	0.127723i	0.523655	+	0.851930i	$0.324568\pi$
$752$		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		0.500000	−	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
$769$	−28.4654	+	23.8853i	−1.02649	+	0.861327i	−0.990429	−	0.138022i	$-0.955925\pi$
$769$	−28.4654	+	23.8853i	−1.02649	+	0.861327i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$			53.4511i			1.92375i
$773$		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
$777$	38.2823	+	33.1534i	1.37337	+	1.18937i
$778$		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
$793$	34.6572	−	41.3029i	1.23071	−	1.46671i
$794$		0			0
$795$		0			0
$796$	−44.4962	+	16.1953i	−1.57713	+	0.574027i
$797$		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
$804$	53.2499	+	19.3814i	1.87798	+	0.683529i
$805$		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	$-0.613962\pi$
$811$	14.8099	−	40.6899i	0.520046	−	1.42882i	0.870469	−	0.492223i	$-0.163816\pi$
$812$		0			0
$813$	49.4664	−	8.72226i	1.73486	−	0.305903i
$814$		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	0.965929	−	0.258808i	$-0.0833297\pi$
$823$	−0.868241	−	4.92404i	−0.0302650	−	0.171641i	−0.996194	+	0.0871670i	$0.972219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$827$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$828$		0			0
$829$	−2.62551	−	1.51584i	−0.0911876	−	0.0526472i	0.453713	−	0.891148i	$-0.350099\pi$
$829$	−2.62551	−	1.51584i	−0.0911876	−	0.0526472i	−0.544900	+	0.838501i	$0.683433\pi$
$830$		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
$844$			16.5820i			0.570775i
$845$		0			0
$846$		0			0
$847$	−14.1569	−	25.4280i	−0.486436	−	0.873716i
$848$		0			0
$849$	7.79339	+	9.28780i	0.267468	+	0.318756i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−12.9420	−	10.8596i	−0.443125	−	0.371826i	0.393753	−	0.919216i	$-0.371177\pi$
$853$	−12.9420	−	10.8596i	−0.443125	−	0.371826i	−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
$859$	−55.0159	−	20.0242i	−1.87712	−	0.683216i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−55.0159	−	20.0242i	−1.87712	−	0.683216i	−0.921772	−	0.387732i	$-0.873259\pi$
$860$		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
$868$	28.0283	+	34.4807i	0.951342	+	1.17035i
$869$		0			0
$870$		0			0
$871$	−73.0117	−	61.2641i	−2.47391	−	2.07585i
$872$		0			0
$873$	49.5000	+	28.5788i	1.67532	+	0.967247i
$874$		0			0
$875$		0			0
$876$	−10.3713	+	12.3601i	−0.350415	+	0.417608i
$877$	56.3323	+	9.93291i	1.90221	+	0.335411i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	56.3323	+	9.93291i	1.90221	+	0.335411i	0.906064	+	0.423141i	$0.139073\pi$
$878$		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
$883$	−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
$889$	−31.6737	+	5.07705i	−1.06230	+	0.170279i
$890$		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
$907$	−15.4023	−	42.3175i	−0.511425	−	1.40513i	−0.879752	−	0.475433i	$-0.842292\pi$
$907$	−15.4023	−	42.3175i	−0.511425	−	1.40513i	0.368327	−	0.929696i	$-0.379931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−25.2080	+	16.6300i	−0.834721	+	0.550673i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	9.64858	+	54.7198i	0.318798	+	1.80799i
$917$		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
$925$	18.8986	−	51.9234i	0.621381	−	1.70723i
$926$		0			0
$927$	−29.9078	−	35.6427i	−0.982300	−	1.17066i
$928$		0			0
$929$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$929$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$930$		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
$937$	−52.1943	−	18.9972i	−1.70511	−	0.620611i	−0.708723	−	0.705487i	$-0.750729\pi$
$937$	−52.1943	−	18.9972i	−1.70511	−	0.620611i	−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		0.342020	−	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	$0.555556\pi$
$948$	−6.43470	+	36.4930i	−0.208989	+	1.18524i
$949$	23.5019	−	13.5688i	0.762905	−	0.440463i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$953$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$954$		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	0.626038	−	0.779793i	$-0.284676\pi$
$967$	−1.03234	−	0.866234i	−0.0331977	−	0.0278562i	−0.659236	+	0.751936i	$0.729120\pi$
$968$		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
$973$	0.470381	+	2.93452i	0.0150797	+	0.0940764i
$974$		0			0
$975$	38.6528	−	32.4335i	1.23788	−	1.03870i
$976$	18.5080	−	32.0569i	0.592428	−	1.02612i
$977$		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		−0.342020	−	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
$991$	−36.8744	+	43.9452i	−1.17135	+	1.39596i	−0.270011	+	0.962857i	$0.587027\pi$
$991$	−36.8744	+	43.9452i	−1.17135	+	1.39596i	−0.901342	+	0.433108i	$0.857417\pi$
$992$		0			0
$993$	1.60348	−	9.09381i	0.0508850	−	0.288583i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	56.0715	+	20.4084i	1.77580	+	0.646339i	0.999880	+	0.0155113i	$0.00493761\pi$
$997$	56.0715	+	20.4084i	1.77580	+	0.646339i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	28.7117	+	49.7302i	0.908399	+	1.57339i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.br.a.212.1	yes	6
3.2	odd	2	CM	399.2.br.a.212.1	yes	6
7.4	even	3		399.2.bv.a.326.1	yes	6
19.13	odd	18		399.2.bv.a.317.1	yes	6
21.11	odd	6		399.2.bv.a.326.1	yes	6
57.32	even	18		399.2.bv.a.317.1	yes	6
133.32	odd	18	inner	399.2.br.a.32.1	✓	6
399.32	even	18	inner	399.2.br.a.32.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.br.a.32.1	✓	6	133.32	odd	18	inner
399.2.br.a.32.1	✓	6	399.32	even	18	inner
399.2.br.a.212.1	yes	6	1.1	even	1	trivial
399.2.br.a.212.1	yes	6	3.2	odd	2	CM
399.2.bv.a.317.1	yes	6	19.13	odd	18
399.2.bv.a.317.1	yes	6	57.32	even	18
399.2.bv.a.326.1	yes	6	7.4	even	3
399.2.bv.a.326.1	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)

\(f(q)\)	\(=\)	\(q+(1.11334 - 1.32683i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})\)	\(q+(1.11334 - 1.32683i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-1.28699 - 2.31164i) q^{7} +(-0.520945 - 2.95442i) q^{9} +3.46410i q^{12} +(1.99273 - 5.47497i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-4.50000 - 0.866025i) q^{21} +(0.868241 + 4.92404i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(4.94356 + 1.88711i) q^{28} +(7.27244 + 4.19875i) q^{31} +(4.59627 + 3.85673i) q^{36} +(-9.57057 - 5.52557i) q^{37} +(-5.04576 - 8.73951i) q^{39} +(-1.53596 + 8.71086i) q^{43} +(-4.45336 - 5.30731i) q^{48} +(-3.68732 + 5.95010i) q^{49} +(3.98545 + 10.9499i) q^{52} +(-5.18866 - 5.48432i) q^{57} +(8.69594 + 3.16506i) q^{61} +(-6.15910 + 5.00654i) q^{63} +(4.00000 + 6.92820i) q^{64} +(5.59492 - 15.3719i) q^{67} +(3.56805 + 2.99395i) q^{73} +(7.50000 + 4.33013i) q^{75} +(4.80066 + 7.27693i) q^{76} +(10.5346 + 1.85754i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(8.00774 - 4.45826i) q^{84} +(-15.2208 + 2.43977i) q^{91} +(13.6677 - 4.97464i) q^{93} +(-12.2467 + 14.5951i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more