LMFDB - Embedded newform 576.2.r.c.481.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,2,Mod(97,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.97"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	576.r (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.59938315643$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			481.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	576.481
Dual form			576.2.r.c.97.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57313 - 0.724745i) q^{3} +(1.94949 + 2.28024i) q^{9} +(-0.476756 + 0.275255i) q^{11} +7.89898 q^{17} -6.34847i q^{19} +(-2.50000 - 4.33013i) q^{25} +(-1.41421 - 5.00000i) q^{27} +(0.949490 - 0.0874863i) q^{33} +(3.39898 - 5.88721i) q^{41} +(10.6941 - 6.17423i) q^{43} +(3.50000 - 6.06218i) q^{49} +(-12.4261 - 5.72474i) q^{51} +(-4.60102 + 9.98698i) q^{57} +(13.2047 + 7.62372i) q^{59} +(-0.301783 - 0.174235i) q^{67} -15.6969 q^{73} +(0.794593 + 8.62372i) q^{75} +(-1.39898 + 8.89060i) q^{81} +(-15.5885 + 9.00000i) q^{83} +18.0000 q^{89} +(-4.84847 - 8.39780i) q^{97} +(-1.55708 - 0.550510i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^9 + 24 * q^17 - 20 * q^25 - 12 * q^33 - 12 * q^41 + 28 * q^49 - 76 * q^57 - 8 * q^73 + 28 * q^81 + 144 * q^89 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$2$		0			0
$3$	−1.57313	−	0.724745i	−0.908248	−	0.418432i
$3$	−1.57313	−	0.724745i	−0.908248	−	0.418432i
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$	1.94949	+	2.28024i	0.649830	+	0.760080i
$10$		0			0
$11$	−0.476756	+	0.275255i	−0.143747	+	0.0829925i	−0.570149	−	0.821541i	$-0.693114\pi$
$11$	−0.476756	+	0.275255i	−0.143747	+	0.0829925i	0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.89898			1.91578			0.957892	−	0.287129i	$-0.0927008\pi$
$17$	7.89898			1.91578			0.957892	+	0.287129i	$0.0927008\pi$
$18$		0			0
$19$		−	6.34847i		−	1.45644i	−0.685344	−	0.728219i	$-0.740348\pi$
$19$		−	6.34847i		−	1.45644i	0.685344	−	0.728219i	$-0.259652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$26$		0			0
$27$	−1.41421	−	5.00000i	−0.272166	−	0.962250i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	0.949490	−	0.0874863i	0.165285	−	0.0152294i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.39898	−	5.88721i	0.530831	−	0.919427i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	3.39898	−	5.88721i	0.530831	−	0.919427i	0.999353	−	0.0359748i	$-0.0114536\pi$
$42$		0			0
$43$	10.6941	−	6.17423i	1.63083	−	0.941562i	0.646997	−	0.762493i	$-0.276025\pi$
$43$	10.6941	−	6.17423i	1.63083	−	0.941562i	0.983836	−	0.179069i	$-0.0573086\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$	−12.4261	−	5.72474i	−1.74001	−	0.801625i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.60102	+	9.98698i	−0.609420	+	1.32281i
$58$		0			0
$59$	13.2047	+	7.62372i	1.71910	+	0.992524i	0.920575	+	0.390567i	$0.127721\pi$
$59$	13.2047	+	7.62372i	1.71910	+	0.992524i	0.798528	+	0.601958i	$0.205612\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.301783	−	0.174235i	−0.0368687	−	0.0212861i	0.481452	−	0.876472i	$-0.340109\pi$
$67$	−0.301783	−	0.174235i	−0.0368687	−	0.0212861i	−0.518321	+	0.855186i	$0.673443\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−15.6969			−1.83719			−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−15.6969			−1.83719			−0.918594	+	0.395203i	$0.870674\pi$
$74$		0			0
$75$	0.794593	+	8.62372i	0.0917517	+	0.995782i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	−1.39898	+	8.89060i	−0.155442	+	0.987845i
$82$		0			0
$83$	−15.5885	+	9.00000i	−1.71106	+	0.987878i	−0.777913	+	0.628372i	$0.783721\pi$
$83$	−15.5885	+	9.00000i	−1.71106	+	0.987878i	−0.933143	+	0.359506i	$0.882945\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			0.953998	+	0.299813i	$0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.84847	−	8.39780i	−0.492287	−	0.852667i	0.507673	−	0.861550i	$-0.330506\pi$
$97$	−4.84847	−	8.39780i	−0.492287	−	0.852667i	−0.999961	+	0.00888289i	$0.997172\pi$
$98$		0			0
$99$	−1.55708	−	0.550510i	−0.156492	−	0.0553284i
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.1464i			1.36759i	0.729676	+	0.683793i	$0.239671\pi$
$107$			14.1464i			1.36759i	−0.729676	+	0.683793i	$0.760329\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	0.0375328	+	0.999295i	$0.488050\pi$
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	−0.884182	+	0.467143i	$0.845283\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.34847	+	9.26382i	−0.486224	+	0.842165i
$122$		0			0
$123$	−9.61377	+	6.79796i	−0.866844	+	0.612951i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	−21.2980	+	1.96240i	−1.87518	+	0.172780i
$130$		0			0
$131$	15.5885	+	9.00000i	1.36197	+	0.786334i	0.989886	−	0.141865i	$-0.0453100\pi$
$131$	15.5885	+	9.00000i	1.36197	+	0.786334i	0.372084	+	0.928199i	$0.378643\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.2980	−	19.5686i	−0.965250	−	1.67186i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	−11.2980	−	19.5686i	−0.965250	−	1.67186i	−0.256307	−	0.966595i	$-0.582506\pi$
$138$		0			0
$139$	15.8902	+	9.17423i	1.34779	+	0.778148i	0.987937	−	0.154859i	$-0.0494925\pi$
$139$	15.8902	+	9.17423i	1.34779	+	0.778148i	0.359856	+	0.933008i	$0.382826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.89949	+	7.00000i	−0.816497	+	0.577350i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$	15.3990	+	18.0116i	1.24493	+	1.45615i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	2.00000i		−	0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$		−	2.00000i		−	0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$	14.4760	−	12.3763i	1.10701	−	0.946437i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−15.2474	−	21.5631i	−1.14607	−	1.62079i
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.76588	+	2.17423i	−0.275389	+	0.158996i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	1.84847	−	3.20164i	0.133056	−	0.230459i	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	1.84847	−	3.20164i	0.133056	−	0.230459i	0.924853	+	0.380325i	$0.124188\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	0.348469	+	0.492810i	0.0245791	+	0.0347601i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.74745	+	3.02667i	0.120874	+	0.209359i
$210$		0			0
$211$	−12.1244	−	7.00000i	−0.834675	−	0.481900i	0.0207756	−	0.999784i	$-0.493386\pi$
$211$	−12.1244	−	7.00000i	−0.834675	−	0.481900i	−0.855451	+	0.517884i	$0.826720\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	24.6934	+	11.3763i	1.66862	+	0.768737i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$	5.00000	−	14.1421i	0.333333	−	0.942809i
$226$		0			0
$227$	−15.1117	+	8.72474i	−1.00300	+	0.579082i	−0.909134	−	0.416503i	$-0.863255\pi$
$227$	−15.1117	+	8.72474i	−1.00300	+	0.579082i	−0.0938647	+	0.995585i	$0.529922\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.1010			0.661740			0.330870	−	0.943676i	$-0.392658\pi$
$233$	10.1010			0.661740			0.330870	+	0.943676i	$0.392658\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	13.8485	+	23.9863i	0.892058	+	1.54509i	0.837404	+	0.546585i	$0.184072\pi$
$241$	13.8485	+	23.9863i	0.892058	+	1.54509i	0.0546547	+	0.998505i	$0.482594\pi$
$242$		0			0
$243$	8.64420	−	12.9722i	0.554526	−	0.832167i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	31.0454	−	2.86054i	1.96742	−	0.181279i
$250$		0			0
$251$		−	29.9444i		−	1.89007i	−0.326965	−	0.945036i	$-0.606026\pi$
$251$		−	29.9444i		−	1.89007i	0.326965	−	0.945036i	$-0.393974\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.3990	+	21.4757i	−0.773427	+	1.33962i	0.162247	+	0.986750i	$0.448126\pi$
$257$	−12.3990	+	21.4757i	−0.773427	+	1.33962i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−28.3164	−	13.0454i	−1.73293	−	0.798366i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.38378	+	1.37628i	0.143747	+	0.0829925i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	−19.0526	−	11.0000i	−1.13256	−	0.653882i	−0.187980	−	0.982173i	$-0.560194\pi$
$283$	−19.0526	−	11.0000i	−1.13256	−	0.653882i	−0.944577	+	0.328291i	$0.893527\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	45.3939			2.67023
$290$		0			0
$291$	1.54102	+	16.7247i	0.0903364	+	0.980422i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.05051	+	1.99451i	0.118983	+	0.115733i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			24.3485i			1.38964i	0.719183	+	0.694820i	$0.244516\pi$
$307$			24.3485i			1.38964i	−0.719183	+	0.694820i	$0.755484\pi$
$308$		0			0
$309$		0			0
$310$		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
$313$	17.1969	+	29.7860i	0.972028	+	1.68360i	0.689412	+	0.724370i	$0.257869\pi$
$313$	17.1969	+	29.7860i	0.972028	+	1.68360i	0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	10.2526	−	22.2542i	0.572242	−	1.24211i
$322$		0			0
$323$		−	50.1464i		−	2.79022i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−22.5167	+	13.0000i	−1.23763	+	0.714545i	−0.968609	−	0.248590i	$-0.920033\pi$
$331$	−22.5167	+	13.0000i	−1.23763	+	0.714545i	−0.269019	+	0.963135i	$0.586699\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.1969	−	19.3937i	0.609936	−	1.05644i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	11.1969	−	19.3937i	0.609936	−	1.05644i	0.991250	−	0.131995i	$-0.0421382\pi$
$338$		0			0
$339$	25.4558	−	18.0000i	1.38257	−	0.977626i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.9791	−	14.4217i	−1.34095	−	0.774197i	−0.354001	−	0.935245i	$-0.615179\pi$
$347$	−24.9791	−	14.4217i	−1.34095	−	0.774197i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.29796	+	3.98018i	0.122308	+	0.211844i	0.920677	−	0.390324i	$-0.127637\pi$
$353$	2.29796	+	3.98018i	0.122308	+	0.211844i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−21.3031			−1.12121
$362$		0			0
$363$	15.1278	−	10.6969i	0.794001	−	0.561444i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$	20.0505	−	3.72656i	1.04379	−	0.193997i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	11.6515i		−	0.598499i	−0.954175	−	0.299249i	$-0.903264\pi$
$379$		−	11.6515i		−	0.598499i	0.954175	−	0.299249i	$-0.0967363\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	34.9267	+	12.3485i	1.77543	+	0.627708i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−18.0000	−	25.4558i	−0.907980	−	1.28408i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.6464	+	32.2966i	−0.931158	+	1.61281i	−0.149813	+	0.988714i	$0.547867\pi$
$401$	−18.6464	+	32.2966i	−0.931158	+	1.61281i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−20.1969	+	34.9821i	−0.998674	+	1.72975i	−0.454759	+	0.890614i	$0.650275\pi$
$409$	−20.1969	+	34.9821i	−0.998674	+	1.72975i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	3.59091	+	38.9722i	0.177127	+	1.92236i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−18.3485	−	25.9487i	−0.898528	−	1.27071i
$418$		0			0
$419$	15.5885	+	9.00000i	0.761546	+	0.439679i	0.829851	−	0.557986i	$-0.188426\pi$
$419$	15.5885	+	9.00000i	0.761546	+	0.439679i	−0.0683046	+	0.997665i	$0.521759\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.7474	−	34.2036i	−0.957892	−	1.65912i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	33.6969			1.61937			0.809686	−	0.586864i	$-0.199638\pi$
$433$	33.6969			1.61937			0.809686	+	0.586864i	$0.199638\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	20.6464	−	3.83732i	0.983163	−	0.182729i
$442$		0			0
$443$	16.0652	−	9.27526i	0.763281	−	0.440681i	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	16.0652	−	9.27526i	0.763281	−	0.440681i	0.830473	+	0.557059i	$0.188070\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−25.8990			−1.22225			−0.611124	−	0.791535i	$-0.709282\pi$
$449$	−25.8990			−1.22225			−0.611124	+	0.791535i	$0.709282\pi$
$450$		0			0
$451$			3.74235i			0.176220i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.19694	−	14.1975i	−0.383437	−	0.664132i	0.608114	−	0.793849i	$-0.291926\pi$
$457$	−8.19694	−	14.1975i	−0.383437	−	0.664132i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	−11.1708	−	39.4949i	−0.521410	−	1.84346i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$463$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			11.9444i			0.552720i	0.961054	+	0.276360i	$0.0891283\pi$
$467$			11.9444i			0.552720i	−0.961054	+	0.276360i	$0.910872\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.39898	+	5.88721i	−0.156285	+	0.270694i
$474$		0			0
$475$	−27.4897	+	15.8712i	−1.26131	+	0.728219i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	−1.44949	+	3.14626i	−0.0655482	+	0.142279i
$490$		0			0
$491$	−28.7931	−	16.6237i	−1.29942	−	0.750218i	−0.319113	−	0.947717i	$-0.603385\pi$
$491$	−28.7931	−	16.6237i	−1.29942	−	0.750218i	−0.980303	+	0.197499i	$0.936718\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	37.8820	+	21.8712i	1.69583	+	0.979088i	0.949633	+	0.313363i	$0.101456\pi$
$499$	37.8820	+	21.8712i	1.69583	+	0.979088i	0.746197	+	0.665725i	$0.231878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.06594	+	22.4217i	0.0917517	+	0.995782i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−31.7423	+	8.97809i	−1.40146	+	0.396392i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−36.1918			−1.58559			−0.792797	−	0.609486i	$-0.791376\pi$
$521$	−36.1918			−1.58559			−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$			38.0000i			1.66162i	0.556553	+	0.830812i	$0.312124\pi$
$523$			38.0000i			1.66162i	−0.556553	+	0.830812i	$0.687876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	8.35847	+	44.9722i	0.362727	+	1.95163i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−13.0454	+	28.3164i	−0.562951	+	1.22194i
$538$		0			0
$539$			3.85357i			0.165985i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−26.2825	+	15.1742i	−1.12376	+	0.648803i	−0.942358	−	0.334606i	$-0.891397\pi$
$547$	−26.2825	+	15.1742i	−1.12376	+	0.648803i	−0.181402	+	0.983409i	$0.558064\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	7.50000	−	0.691053i	0.316650	−	0.0291763i
$562$		0			0
$563$	40.5675	+	23.4217i	1.70972	+	0.987106i	0.934892	+	0.354932i	$0.115496\pi$
$563$	40.5675	+	23.4217i	1.70972	+	0.987106i	0.774826	+	0.632175i	$0.217837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.2980	+	35.1571i	0.850935	+	1.47386i	0.880366	+	0.474295i	$0.157297\pi$
$569$	20.2980	+	35.1571i	0.850935	+	1.47386i	−0.0294311	+	0.999567i	$0.509370\pi$
$570$		0			0
$571$	−22.2935	−	12.8712i	−0.932955	−	0.538642i	−0.0452101	−	0.998978i	$-0.514396\pi$
$571$	−22.2935	−	12.8712i	−0.932955	−	0.538642i	−0.887745	+	0.460336i	$0.847729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−46.3939			−1.93140			−0.965701	−	0.259656i	$-0.916391\pi$
$577$	−46.3939			−1.93140			−0.965701	+	0.259656i	$0.916391\pi$
$578$		0			0
$579$	−5.22826	+	3.69694i	−0.217279	+	0.153640i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.6605	+	22.3207i	−1.59569	+	0.921272i	−0.603386	+	0.797449i	$0.706182\pi$
$587$	−38.6605	+	22.3207i	−1.59569	+	0.921272i	−0.992304	+	0.123823i	$0.960484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−4.15153	−	7.19066i	−0.169344	−	0.293313i	0.768845	−	0.639435i	$-0.220832\pi$
$601$	−4.15153	−	7.19066i	−0.169344	−	0.293313i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−0.191027	−	1.02781i	−0.00777921	−	0.0418555i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.64643	−	16.7081i	0.388351	−	0.672643i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	9.64643	−	16.7081i	0.388351	−	0.672643i	0.992228	+	0.124434i	$0.0397116\pi$
$618$		0			0
$619$	43.0781	−	24.8712i	1.73146	−	0.999657i	0.852631	−	0.522514i	$-0.175006\pi$
$619$	43.0781	−	24.8712i	1.73146	−	0.999657i	0.878826	−	0.477143i	$-0.158328\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	−0.555404	−	6.02781i	−0.0221807	−	0.240727i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	14.0000	+	19.7990i	0.556450	+	0.786939i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.74745	−	3.02667i	−0.0690201	−	0.119546i	0.829450	−	0.558581i	$-0.188654\pi$
$641$	−1.74745	−	3.02667i	−0.0690201	−	0.119546i	−0.898470	+	0.439034i	$0.855321\pi$
$642$		0			0
$643$	−15.2867	−	8.82577i	−0.602848	−	0.348054i	0.167313	−	0.985904i	$-0.446491\pi$
$643$	−15.2867	−	8.82577i	−0.602848	−	0.348054i	−0.770161	+	0.637850i	$0.779824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−8.39388			−0.329488
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−30.6010	−	35.7928i	−1.19386	−	1.39641i
$658$		0			0
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	−0.771885	−	0.635763i	$-0.780686\pi$
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	0.164644	+	0.986353i	$0.447352\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	0.753420	−	0.657539i	$-0.228403\pi$
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	−0.946156	+	0.323711i	$0.895069\pi$
$674$		0			0
$675$	−18.1151	+	18.6237i	−0.697251	+	0.716827i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	30.0959	−	2.77305i	1.15328	−	0.106263i
$682$		0			0
$683$			47.9444i			1.83454i	0.398265	+	0.917270i	$0.369613\pi$
$683$			47.9444i			1.83454i	−0.398265	+	0.917270i	$0.630387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	39.8372	−	23.0000i	1.51548	−	0.874961i	0.515642	−	0.856804i	$-0.327553\pi$
$691$	39.8372	−	23.0000i	1.51548	−	0.874961i	0.999835	−	0.0181572i	$-0.00577993\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	26.8485	−	46.5029i	1.01696	−	1.76142i
$698$		0			0
$699$	−15.8902	−	7.32066i	−0.601024	−	0.276893i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−4.40156	−	47.7702i	−0.163696	−	1.77659i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$	84.4724	−	48.7702i	3.12432	−	1.80383i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.191836			0.00706637
$738$		0			0
$739$		−	19.7423i		−	0.726234i	−0.931744	−	0.363117i	$-0.881713\pi$
$739$		−	19.7423i		−	0.726234i	0.931744	−	0.363117i	$-0.118287\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−50.9117	−	18.0000i	−1.86276	−	0.658586i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$	−21.7020	+	47.1065i	−0.790866	+	1.71666i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0000	−	46.7654i	0.978749	−	1.69524i	0.311787	−	0.950152i	$-0.399073\pi$
$761$	27.0000	−	46.7654i	0.978749	−	1.69524i	0.666962	−	0.745091i	$-0.267594\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	−0.596643	−	0.802507i	$-0.703499\pi$
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	0.993313	+	0.115454i	$0.0368323\pi$
$770$		0			0
$771$	35.0696	−	24.7980i	1.26300	−	0.893077i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−37.3747	−	21.5783i	−1.33909	−	0.773123i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	43.3013	+	25.0000i	1.54352	+	0.891154i	0.998613	+	0.0526599i	$0.0167699\pi$
$787$	43.3013	+	25.0000i	1.54352	+	0.891154i	0.544911	+	0.838494i	$0.316563\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	35.0908	+	41.0443i	1.23987	+	1.45023i
$802$		0			0
$803$	7.48361	−	4.32066i	0.264091	−	0.152473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.9898			1.82786			0.913932	−	0.405868i	$-0.133031\pi$
$809$	51.9898			1.82786			0.913932	+	0.405868i	$0.133031\pi$
$810$		0			0
$811$		−	55.7423i		−	1.95738i	−0.205347	−	0.978689i	$-0.565832\pi$
$811$		−	55.7423i		−	1.95738i	0.205347	−	0.978689i	$-0.434168\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−39.1969	−	67.8911i	−1.37133	−	2.37521i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$	−2.75255	−	3.89270i	−0.0958315	−	0.135526i
$826$		0			0
$827$			54.0000i			1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$			54.0000i			1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.6464	−	47.8850i	0.957892	−	1.65912i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$	2.86054	+	31.0454i	0.0985221	+	1.06926i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	22.0000	+	31.1127i	0.755038	+	1.06779i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.0000	+	46.7654i	0.922302	+	1.59747i	0.795843	+	0.605503i	$0.207028\pi$
$857$	27.0000	+	46.7654i	0.922302	+	1.59747i	0.126459	+	0.991972i	$0.459639\pi$
$858$		0			0
$859$	−31.4787	−	18.1742i	−1.07404	−	0.620097i	−0.144757	−	0.989467i	$-0.546240\pi$
$859$	−31.4787	−	18.1742i	−1.07404	−	0.620097i	−0.929282	+	0.369370i	$0.879573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−71.4106	−	32.8990i	−2.42523	−	1.11731i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	9.69694	−	27.4271i	0.328192	−	0.928266i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$		−	50.4393i		−	1.69742i	−0.528861	−	0.848709i	$-0.677381\pi$
$883$		−	50.4393i		−	1.69742i	0.528861	−	0.848709i	$-0.322619\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.78021	−	4.62372i	−0.0596394	−	0.154901i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	48.8779	−	28.2196i	1.62296	−	0.937018i	0.636841	−	0.770996i	$-0.280241\pi$
$907$	48.8779	−	28.2196i	1.62296	−	0.937018i	0.986122	−	0.166022i	$-0.0530924\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	4.95459	−	8.58161i	0.163973	−	0.284010i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	17.6464	−	38.3034i	0.581470	−	1.26214i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.0000	+	46.7654i	0.885841	+	1.53432i	0.844746	+	0.535167i	$0.179751\pi$
$929$	27.0000	+	46.7654i	0.885841	+	1.53432i	0.0410949	+	0.999155i	$0.486915\pi$
$930$		0			0
$931$	−38.4855	−	22.2196i	−1.26131	−	0.728219i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$	−5.46583	−	59.3207i	−0.178371	−	1.93586i
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−53.2955	+	30.7702i	−1.73187	+	0.999896i	−0.858710	+	0.512461i	$0.828734\pi$
$947$	−53.2955	+	30.7702i	−1.73187	+	0.999896i	−0.873160	+	0.487435i	$0.837933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.1918			0.589291			0.294646	−	0.955607i	$-0.404798\pi$
$953$	18.1918			0.589291			0.294646	+	0.955607i	$0.404798\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	−32.2572	+	27.5783i	−1.03947	+	0.888699i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$967$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$968$		0			0
$969$	−36.3434	+	78.8870i	−1.16752	+	2.53421i
$970$		0			0
$971$			54.0000i			1.73294i	0.499227	+	0.866471i	$0.333617\pi$
$971$			54.0000i			1.73294i	−0.499227	+	0.866471i	$0.666383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.4444	−	44.0710i	0.814038	−	1.40996i	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	25.4444	−	44.0710i	0.814038	−	1.40996i	0.910017	−	0.414572i	$-0.136069\pi$
$978$		0			0
$979$	−8.58161	+	4.95459i	−0.274269	+	0.158349i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	44.8434	−	4.13188i	1.42306	−	0.131121i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.r.c.481.1	yes	8
3.2	odd	2		1728.2.r.c.1441.3		8
4.3	odd	2	inner	576.2.r.c.481.4	yes	8
8.3	odd	2	CM	576.2.r.c.481.1	yes	8
8.5	even	2	inner	576.2.r.c.481.4	yes	8
9.2	odd	6		1728.2.r.c.289.2		8
9.4	even	3		5184.2.d.l.2593.2		4
9.5	odd	6		5184.2.d.e.2593.3		4
9.7	even	3	inner	576.2.r.c.97.4	yes	8
12.11	even	2		1728.2.r.c.1441.2		8
24.5	odd	2		1728.2.r.c.1441.2		8
24.11	even	2		1728.2.r.c.1441.3		8
36.7	odd	6	inner	576.2.r.c.97.1	✓	8
36.11	even	6		1728.2.r.c.289.3		8
36.23	even	6		5184.2.d.e.2593.2		4
36.31	odd	6		5184.2.d.l.2593.3		4
72.5	odd	6		5184.2.d.e.2593.2		4
72.11	even	6		1728.2.r.c.289.2		8
72.13	even	6		5184.2.d.l.2593.3		4
72.29	odd	6		1728.2.r.c.289.3		8
72.43	odd	6	inner	576.2.r.c.97.4	yes	8
72.59	even	6		5184.2.d.e.2593.3		4
72.61	even	6	inner	576.2.r.c.97.1	✓	8
72.67	odd	6		5184.2.d.l.2593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.c.97.1	✓	8	36.7	odd	6	inner
576.2.r.c.97.1	✓	8	72.61	even	6	inner
576.2.r.c.97.4	yes	8	9.7	even	3	inner
576.2.r.c.97.4	yes	8	72.43	odd	6	inner
576.2.r.c.481.1	yes	8	1.1	even	1	trivial
576.2.r.c.481.1	yes	8	8.3	odd	2	CM
576.2.r.c.481.4	yes	8	4.3	odd	2	inner
576.2.r.c.481.4	yes	8	8.5	even	2	inner
1728.2.r.c.289.2		8	9.2	odd	6
1728.2.r.c.289.2		8	72.11	even	6
1728.2.r.c.289.3		8	36.11	even	6
1728.2.r.c.289.3		8	72.29	odd	6
1728.2.r.c.1441.2		8	12.11	even	2
1728.2.r.c.1441.2		8	24.5	odd	2
1728.2.r.c.1441.3		8	3.2	odd	2
1728.2.r.c.1441.3		8	24.11	even	2
5184.2.d.e.2593.2		4	36.23	even	6
5184.2.d.e.2593.2		4	72.5	odd	6
5184.2.d.e.2593.3		4	9.5	odd	6
5184.2.d.e.2593.3		4	72.59	even	6
5184.2.d.l.2593.2		4	9.4	even	3
5184.2.d.l.2593.2		4	72.67	odd	6
5184.2.d.l.2593.3		4	36.31	odd	6
5184.2.d.l.2593.3		4	72.13	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.57313 - 0.724745i) q^{3} +(1.94949 + 2.28024i) q^{9} +(-0.476756 + 0.275255i) q^{11} +7.89898 q^{17} -6.34847i q^{19} +(-2.50000 - 4.33013i) q^{25} +(-1.41421 - 5.00000i) q^{27} +(0.949490 - 0.0874863i) q^{33} +(3.39898 - 5.88721i) q^{41} +(10.6941 - 6.17423i) q^{43} +(3.50000 - 6.06218i) q^{49} +(-12.4261 - 5.72474i) q^{51} +(-4.60102 + 9.98698i) q^{57} +(13.2047 + 7.62372i) q^{59} +(-0.301783 - 0.174235i) q^{67} -15.6969 q^{73} +(0.794593 + 8.62372i) q^{75} +(-1.39898 + 8.89060i) q^{81} +(-15.5885 + 9.00000i) q^{83} +18.0000 q^{89} +(-4.84847 - 8.39780i) q^{97} +(-1.55708 - 0.550510i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^9 + 24 * q^17 - 20 * q^25 - 12 * q^33 - 12 * q^41 + 28 * q^49 - 76 * q^57 - 8 * q^73 + 28 * q^81 + 144 * q^89 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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