LMFDB - Embedded newform 207.2.g.a.68.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,2,Mod(68,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("207.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.g (of order $6$, degree $2$, 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:	$1.65290332184$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.57352136505929721.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 $ x^12 - 3x^9 + x^6 - 24x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			68.2
Root			$-0.721506 + 1.21632i$ of defining polynomial
Character	$\chi$	$=$	207.68
Dual form			207.2.g.a.137.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.13562 + 1.23300i) q^{2} +(1.65147 - 0.522159i) q^{3} +(2.04058 - 3.53440i) q^{4} +(-2.88309 + 3.15140i) q^{6} +5.13217i q^{8} +(2.45470 - 1.72466i) q^{9} +O(q^{10})$	\(q+(-2.13562 + 1.23300i) q^{2} +(1.65147 - 0.522159i) q^{3} +(2.04058 - 3.53440i) q^{4} +(-2.88309 + 3.15140i) q^{6} +5.13217i q^{8} +(2.45470 - 1.72466i) q^{9} +(1.52444 - 6.90245i) q^{12} +(3.03669 - 5.25970i) q^{13} +(-2.24680 - 3.89157i) q^{16} +(-3.11580 + 6.70987i) q^{18} +(4.15331 + 2.39792i) q^{23} +(2.67981 + 8.47561i) q^{24} +(2.50000 + 4.33013i) q^{25} +14.9770i q^{26} +(3.15331 - 4.12997i) q^{27} +(1.86155 - 1.07477i) q^{29} +(-4.16396 + 7.21219i) q^{31} +(0.707446 + 0.408444i) q^{32} +(-1.08661 - 12.1952i) q^{36} +(2.26860 - 10.2719i) q^{39} +(-8.73043 - 5.04051i) q^{41} -11.8265 q^{46} +(8.90380 - 5.14061i) q^{47} +(-5.74254 - 5.25362i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-10.6781 - 6.16501i) q^{50} +(-12.3932 - 21.4657i) q^{52} +(-1.64202 + 12.7081i) q^{54} +(-2.65038 + 4.59059i) q^{58} +(-13.1533 - 7.59407i) q^{59} -20.5367i q^{62} +6.97274 q^{64} +(8.11116 + 1.79139i) q^{69} +16.7454i q^{71} +(8.85124 + 12.5979i) q^{72} -15.6654 q^{73} +(6.38969 + 5.84567i) q^{75} +(7.82036 + 24.7340i) q^{78} +(3.05110 - 8.46704i) q^{81} +24.8598 q^{82} +(2.51309 - 2.74697i) q^{87} +(16.9504 - 9.78630i) q^{92} +(-3.11074 + 14.0850i) q^{93} +(-12.6768 + 21.9568i) q^{94} +(1.38160 + 0.305133i) q^{96} -17.2620i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} - 3 q^{6}+O(q^{10}) $ 12 * q + 12 * q^4 - 3 * q^6	$ 12 q + 12 q^{4} - 3 q^{6} - 30 q^{12} - 24 q^{16} - 21 q^{18} - 12 q^{24} + 30 q^{25} - 12 q^{27} + 54 q^{32} + 33 q^{36} + 24 q^{39} + 48 q^{48} - 42 q^{49} - 3 q^{52} + 15 q^{58} - 108 q^{59} - 54 q^{64} + 42 q^{72} + 51 q^{78} - 66 q^{82} + 96 q^{87} + 6 q^{93} - 39 q^{94} - 69 q^{96}+O(q^{100}) $ 12 * q + 12 * q^4 - 3 * q^6 - 30 * q^12 - 24 * q^16 - 21 * q^18 - 12 * q^24 + 30 * q^25 - 12 * q^27 + 54 * q^32 + 33 * q^36 + 24 * q^39 + 48 * q^48 - 42 * q^49 - 3 * q^52 + 15 * q^58 - 108 * q^59 - 54 * q^64 + 42 * q^72 + 51 * q^78 - 66 * q^82 + 96 * q^87 + 6 * q^93 - 39 * q^94 - 69 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−2.13562	+	1.23300i	−1.51011	+	0.871864i	−0.510181	+	0.860067i	$0.670422\pi$
$2$	−2.13562	+	1.23300i	−1.51011	+	0.871864i	−0.999930	+	0.0117968i	$0.996245\pi$
$3$	1.65147	−	0.522159i	0.953476	−	0.301469i
$3$	1.65147	−	0.522159i	0.953476	−	0.301469i
$4$	2.04058	−	3.53440i	1.02029	−	1.76720i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$	−2.88309	+	3.15140i	−1.17702	+	1.28655i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$			5.13217i			1.81449i
$9$	2.45470	−	1.72466i	0.818233	−	0.574887i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	1.52444	−	6.90245i	0.440069	−	1.99257i
$13$	3.03669	−	5.25970i	0.842226	−	1.45878i	−0.0457831	−	0.998951i	$-0.514578\pi$
$13$	3.03669	−	5.25970i	0.842226	−	1.45878i	0.888009	−	0.459826i	$-0.152088\pi$
$14$		0			0
$15$		0			0
$16$	−2.24680	−	3.89157i	−0.561700	−	0.972892i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−3.11580	+	6.70987i	−0.734401	+	1.58153i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.15331	+	2.39792i	0.866025	+	0.500000i
$23$	4.15331	+	2.39792i	0.866025	+	0.500000i
$24$	2.67981	+	8.47561i	0.547014	+	1.73008i
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$			14.9770i			2.93722i
$27$	3.15331	−	4.12997i	0.606855	−	0.794812i
$28$		0			0
$29$	1.86155	−	1.07477i	0.345681	−	0.199579i	−0.317100	−	0.948392i	$-0.602709\pi$
$29$	1.86155	−	1.07477i	0.345681	−	0.199579i	0.662782	+	0.748813i	$0.269376\pi$
$30$		0			0
$31$	−4.16396	+	7.21219i	−0.747869	+	1.29535i	0.200973	+	0.979597i	$0.435590\pi$
$31$	−4.16396	+	7.21219i	−0.747869	+	1.29535i	−0.948842	+	0.315751i	$0.897744\pi$
$32$	0.707446	+	0.408444i	0.125060	+	0.0722034i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.08661	−	12.1952i	−0.181102	−	2.03253i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	2.26860	−	10.2719i	0.363266	−	1.64481i
$40$		0			0
$41$	−8.73043	−	5.04051i	−1.36346	−	0.787196i	−0.373381	−	0.927678i	$-0.621801\pi$
$41$	−8.73043	−	5.04051i	−1.36346	−	0.787196i	−0.990083	+	0.140482i	$0.955135\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$		0			0
$46$	−11.8265			−1.74373
$47$	8.90380	−	5.14061i	1.29875	−	0.749836i	0.318565	−	0.947901i	$-0.396799\pi$
$47$	8.90380	−	5.14061i	1.29875	−	0.749836i	0.980189	+	0.198066i	$0.0634659\pi$
$48$	−5.74254	−	5.25362i	−0.828864	−	0.758294i
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	−10.6781	−	6.16501i	−1.51011	−	0.871864i
$51$		0			0
$52$	−12.3932	−	21.4657i	−1.71863	−	2.97676i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−1.64202	+	12.7081i	−0.223451	+	1.72935i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−2.65038	+	4.59059i	−0.348012	+	0.602774i
$59$	−13.1533	−	7.59407i	−1.71242	−	0.988663i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−13.1533	−	7.59407i	−1.71242	−	0.988663i	−0.781133	−	0.624364i	$-0.785358\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		−	20.5367i		−	2.60816i
$63$		0			0
$64$	6.97274			0.871593
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	8.11116	+	1.79139i	0.976469	+	0.215658i
$70$		0			0
$71$			16.7454i			1.98731i	0.112482	+	0.993654i	$0.464120\pi$
$71$			16.7454i			1.98731i	−0.112482	+	0.993654i	$0.535880\pi$
$72$	8.85124	+	12.5979i	1.04313	+	1.48468i
$73$	−15.6654			−1.83350			−0.916748	−	0.399466i	$-0.869196\pi$
$73$	−15.6654			−1.83350			−0.916748	+	0.399466i	$0.869196\pi$
$74$		0			0
$75$	6.38969	+	5.84567i	0.737818	+	0.675000i
$76$		0			0
$77$		0			0
$78$	7.82036	+	24.7340i	0.885481	+	2.80057i
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	3.05110	−	8.46704i	0.339011	−	0.940783i
$82$	24.8598			2.74531
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.51309	−	2.74697i	0.269432	−	0.294506i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	16.9504	−	9.78630i	1.76720	−	1.02029i
$93$	−3.11074	+	14.0850i	−0.322568	+	1.46054i
$94$	−12.6768	+	21.9568i	−1.30751	+	2.26467i
$95$		0			0
$96$	1.38160	+	0.305133i	0.141009	+	0.0311425i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		−	17.2620i		−	1.74373i
$99$		0			0
$100$	20.4058			2.04058
$101$	−12.8066	+	7.39391i	−1.27431	+	0.735721i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−12.8066	+	7.39391i	−1.27431	+	0.735721i	−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	26.9936	+	15.5848i	2.64694	+	1.52821i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−8.16234	−	19.5726i	−0.785421	−	1.88337i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		−	8.77261i		−	0.814516i
$117$	−1.61704	−	18.1482i	−0.149495	−	1.67780i
$118$	37.4540			3.44792
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$	−17.0500	−	3.76558i	−1.53735	−	0.339531i
$124$	16.9938	+	29.4342i	1.52609	+	2.64327i
$125$		0			0
$126$		0			0
$127$	−15.0482			−1.33531			−0.667657	−	0.744469i	$-0.732703\pi$
$127$	−15.0482			−1.33531			−0.667657	+	0.744469i	$0.732703\pi$
$128$	−16.3060	+	9.41429i	−1.44126	+	0.832114i
$129$		0			0
$130$		0			0
$131$	10.7788	+	6.22314i	0.941748	+	0.543718i	0.890508	−	0.454968i	$-0.150349\pi$
$131$	10.7788	+	6.22314i	0.941748	+	0.543718i	0.0512401	+	0.998686i	$0.483683\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$	−19.5312	+	6.17533i	−1.66260	+	0.525679i
$139$	−11.6865	+	20.2415i	−0.991232	+	1.71686i	−0.381185	+	0.924499i	$0.624484\pi$
$139$	−11.6865	+	20.2415i	−0.991232	+	1.71686i	−0.610047	+	0.792365i	$0.708849\pi$
$140$		0			0
$141$	12.0201	−	13.1388i	1.01228	−	1.10648i
$142$	−20.6470	−	35.7617i	−1.73266	−	3.00106i
$143$		0			0
$144$	−12.2268	−	5.67767i	−1.01890	−	0.473139i
$145$		0			0
$146$	33.4554	−	19.3155i	2.76878	−	1.59856i
$147$	−2.61472	+	11.8391i	−0.215658	+	0.976469i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	−20.8537	−	4.60564i	−1.70270	−	0.376049i
$151$	8.78984	+	15.2245i	0.715307	+	1.23895i	0.962841	+	0.270069i	$0.0870466\pi$
$151$	8.78984	+	15.2245i	0.715307	+	1.23895i	−0.247533	+	0.968879i	$0.579620\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−31.6756	−	28.9787i	−2.53607	−	2.32015i
$157$		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
$162$	3.92389	+	21.8444i	0.308290	+	1.71626i
$163$	5.43294			0.425540			0.212770	−	0.977102i	$-0.431751\pi$
$163$	5.43294			0.425540			0.212770	+	0.977102i	$0.431751\pi$
$164$	−35.6303	+	20.5712i	−2.78226	+	1.60634i
$165$		0			0
$166$		0			0
$167$	13.8467	+	7.99439i	1.07149	+	0.618624i	0.928588	−	0.371113i	$-0.121024\pi$
$167$	13.8467	+	7.99439i	1.07149	+	0.618624i	0.142901	+	0.989737i	$0.454357\pi$
$168$		0			0
$169$	−11.9429	−	20.6858i	−0.918688	−	1.59121i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.6132	−	9.59166i	1.26308	−	0.729241i	0.289412	−	0.957205i	$-0.406540\pi$
$173$	16.6132	−	9.59166i	1.26308	−	0.729241i	0.973670	+	0.227964i	$0.0732068\pi$
$174$	−1.98000	+	8.96513i	−0.150103	+	0.679645i
$175$		0			0
$176$		0			0
$177$	−25.6876	−	5.67324i	−1.93080	−	0.426427i
$178$		0			0
$179$			9.88083i			0.738528i	0.929325	+	0.369264i	$0.120390\pi$
$179$			9.88083i			0.738528i	−0.929325	+	0.369264i	$0.879610\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−12.3065	+	21.3155i	−0.907247	+	1.57140i
$185$		0			0
$186$	−10.7234	−	33.9157i	−0.786279	−	2.48682i
$187$		0			0
$188$		−	41.9594i		−	3.06020i
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	11.5153	−	3.64088i	0.831043	−	0.262758i
$193$	13.7676	−	23.8462i	0.991015	−	1.71649i	0.379676	−	0.925120i	$-0.376035\pi$
$193$	13.7676	−	23.8462i	0.991015	−	1.71649i	0.611339	−	0.791369i	$-0.290631\pi$
$194$		0			0
$195$		0			0
$196$	14.2841	+	24.7408i	1.02029	+	1.76720i
$197$			10.4814i			0.746771i	0.927676	+	0.373385i	$0.121803\pi$
$197$			10.4814i			0.746771i	−0.927676	+	0.373385i	$0.878197\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−22.2229	+	12.8304i	−1.57140	+	0.907247i
$201$		0			0
$202$	18.2334	−	31.5812i	1.28290	−	2.22204i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	14.3307	−	1.27689i	0.996054	−	0.0887499i
$208$	−27.2913			−1.89231
$209$		0			0
$210$		0			0
$211$	11.4599	−	19.8492i	0.788935	−	1.36647i	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	11.4599	−	19.8492i	0.788935	−	1.36647i	0.926620	−	0.375999i	$-0.122700\pi$
$212$		0			0
$213$	8.74374	+	27.6544i	0.599111	+	1.89485i
$214$		0			0
$215$		0			0
$216$	21.1957	+	16.1833i	1.44218	+	1.10114i
$217$		0			0
$218$		0			0
$219$	−25.8709	+	8.17984i	−1.74819	+	0.552742i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	14.4599	+	25.0453i	0.968309	+	1.67716i	0.700449	+	0.713702i	$0.252983\pi$
$223$	14.4599	+	25.0453i	0.968309	+	1.67716i	0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	13.6047	+	6.31751i	0.906983	+	0.421167i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		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
$232$	5.51588	+	9.55379i	0.362135	+	0.627237i
$233$			14.3309i			0.938847i	0.882973	+	0.469423i	$0.155538\pi$
$233$			14.3309i			0.938847i	−0.882973	+	0.469423i	$0.844462\pi$
$234$	25.8302	+	36.7639i	1.68857	+	2.40333i
$235$		0			0
$236$	−53.6809	+	30.9927i	−3.49433	+	2.01745i
$237$		0			0
$238$		0			0
$239$	16.6821	+	9.63144i	1.07908	+	0.623006i	0.930648	−	0.365917i	$-0.119245\pi$
$239$	16.6821	+	9.63144i	1.07908	+	0.623006i	0.148431	+	0.988923i	$0.452578\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$			27.1260i			1.74373i
$243$	0.617644	−	15.5762i	0.0396219	−	0.999215i
$244$		0			0
$245$		0			0
$246$	41.0553	−	12.9808i	2.61759	−	0.827626i
$247$		0			0
$248$	−37.0142	−	21.3701i	−2.35040	−	1.35700i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	32.1373	−	18.5545i	2.01647	−	1.16421i
$255$		0			0
$256$	16.2429	−	28.1336i	1.01518	−	1.75835i
$257$	−27.1422	−	15.6706i	−1.69309	−	0.977504i	−0.952001	−	0.306093i	$-0.900978\pi$
$257$	−27.1422	−	15.6706i	−1.69309	−	0.977504i	−0.741085	−	0.671411i	$-0.765689\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.71594	−	5.84877i	0.168112	−	0.362030i
$262$	−30.6926			−1.89619
$263$		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
$267$		0			0
$268$		0			0
$269$		−	4.46686i		−	0.272350i	−0.990685	−	0.136175i	$-0.956519\pi$
$269$		−	4.46686i		−	0.272350i	0.990685	−	0.136175i	$-0.0434809\pi$
$270$		0			0
$271$	−16.9199			−1.02781			−0.513905	−	0.857847i	$-0.671801\pi$
$271$	−16.9199			−1.02781			−0.513905	+	0.857847i	$0.671801\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	22.8830	−	25.0126i	1.37739	−	1.50558i
$277$	−16.1606	−	27.9910i	−0.970998	−	1.68182i	−0.692556	−	0.721364i	$-0.743516\pi$
$277$	−16.1606	−	27.9910i	−0.970998	−	1.68182i	−0.278441	−	0.960453i	$-0.589818\pi$
$278$		−	57.6376i		−	3.45688i
$279$	2.21731	+	24.8852i	0.132747	+	1.48984i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$	−9.47033	+	42.8803i	−0.563950	+	2.55348i
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	59.1847	+	34.1703i	3.51197	+	2.02763i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.44099	−	0.217496i	0.143837	−	0.0128161i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−31.9666	+	55.3677i	−1.87070	+	3.24015i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$	−9.01352	−	28.5077i	−0.525679	−	1.66260i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	25.2246	−	14.5634i	1.45878	−	0.842226i
$300$	33.6996	−	10.6551i	1.94565	−	0.615173i
$301$		0			0
$302$	−37.5435	−	21.6758i	−2.16039	−	1.24730i
$303$	−17.2889	+	18.8979i	−0.993224	+	1.08566i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−34.9199			−1.99298			−0.996491	−	0.0836980i	$-0.973327\pi$
$307$	−34.9199			−1.99298			−0.996491	+	0.0836980i	$0.973327\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.402824	+	0.232570i	0.0228420	+	0.0131879i	0.511378	−	0.859356i	$-0.329135\pi$
$311$	0.402824	+	0.232570i	0.0228420	+	0.0131879i	−0.488535	+	0.872544i	$0.662469\pi$
$312$	52.7169	+	11.6428i	2.98451	+	0.659144i
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.1934	−	8.19455i	0.797179	−	0.460252i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	14.1934	−	8.19455i	0.797179	−	0.460252i	0.842484	+	0.538721i	$0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−23.6999	−	28.0615i	−1.31666	−	1.55897i
$325$	30.3669			1.68445
$326$	−11.6027	+	6.69882i	−0.642614	+	0.371013i
$327$		0			0
$328$	25.8688	−	44.8060i	1.42836	−	2.47400i
$329$		0			0
$330$		0			0
$331$	−3.17352	−	5.49670i	−0.174432	−	0.302126i	0.765532	−	0.643397i	$-0.222476\pi$
$331$	−3.17352	−	5.49670i	−0.174432	−	0.302126i	−0.939965	+	0.341272i	$0.889142\pi$
$332$		0			0
$333$		0			0
$334$	−39.4284			−2.15742
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	51.0112	+	29.4513i	2.77464	+	1.60194i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−23.6531	+	40.9683i	−1.27160	+	2.20247i
$347$	8.30662	+	4.79583i	0.445923	+	0.257454i	0.706107	−	0.708105i	$-0.250450\pi$
$347$	8.30662	+	4.79583i	0.445923	+	0.257454i	−0.260184	+	0.965559i	$0.583783\pi$
$348$	−4.58070	−	14.4877i	−0.245551	−	0.776622i
$349$	1.28061	+	2.21809i	0.0685496	+	0.118731i	0.898263	−	0.439458i	$-0.144830\pi$
$349$	1.28061	+	2.21809i	0.0685496	+	0.118731i	−0.829713	+	0.558190i	$0.811496\pi$
$350$		0			0
$351$	−12.1468	−	29.1269i	−0.648346	−	1.55468i
$352$		0			0
$353$	−29.4959	+	17.0294i	−1.56991	+	0.906386i	−0.573728	+	0.819046i	$0.694503\pi$
$353$	−29.4959	+	17.0294i	−1.56991	+	0.906386i	−0.996179	+	0.0873401i	$0.972163\pi$
$354$	61.8541	−	19.5569i	3.28751	−	1.03944i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−12.1831	−	21.1017i	−0.643896	−	1.11526i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	4.10884	−	18.6042i	0.215658	−	0.976469i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		−	21.5505i		−	1.12340i
$369$	−30.1237	+	2.68407i	−1.56818	+	0.139727i
$370$		0			0
$371$		0			0
$372$	43.4341	+	39.7361i	2.25195	+	2.06022i
$373$		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
$376$	26.3825	+	45.6958i	1.36057	+	2.35658i
$377$		−	13.0549i		−	0.672363i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−24.8517	+	7.85758i	−1.27319	+	0.402556i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	−22.0131	+	24.0618i	−1.12335	+	1.22790i
$385$		0			0
$386$			67.9020i			3.45612i
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$	−31.1121	−	17.9626i	−1.57140	−	0.907247i
$393$	21.0503	+	4.64907i	1.06185	+	0.234515i
$394$	−12.9236	−	22.3843i	−0.651082	−	1.12771i
$395$		0			0
$396$		0			0
$397$	35.8599			1.79976			0.899879	−	0.436140i	$-0.143655\pi$
$397$	35.8599			1.79976			0.899879	+	0.436140i	$0.143655\pi$
$398$		0			0
$399$		0			0
$400$	11.2340	−	19.4578i	0.561700	−	0.972892i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	25.2893	+	43.8023i	1.25975	+	2.18195i
$404$			60.3516i			3.00260i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8.82314	−	15.2821i	0.436276	−	0.755653i	−0.561123	−	0.827733i	$-0.689630\pi$
$409$	8.82314	−	15.2821i	0.436276	−	0.755653i	0.997399	+	0.0720801i	$0.0229637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−29.0306	+	20.3967i	−1.42678	+	1.00245i
$415$		0			0
$416$	4.29659	−	2.48064i	0.210657	−	0.121623i
$417$	−8.73051	+	39.5304i	−0.427535	+	1.93581i
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$			56.5205i			2.75137i
$423$	12.9903	−	27.9747i	0.631612	−	1.36018i
$424$		0			0
$425$		0			0
$426$	−52.7713	−	48.2783i	−2.55678	−	2.33909i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−23.1569	−	2.99213i	−1.11414	−	0.143959i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	45.1647	−	49.3679i	2.15805	−	2.35889i
$439$	−20.9366	−	36.2632i	−0.999249	−	1.73075i	−0.533177	−	0.846004i	$-0.679002\pi$
$439$	−20.9366	−	36.2632i	−0.999249	−	1.73075i	−0.466073	−	0.884747i	$-0.654331\pi$
$440$		0			0
$441$	1.86375	+	20.9171i	0.0887499	+	0.996054i
$442$		0			0
$443$	8.94531	−	5.16458i	0.425004	−	0.245376i	−0.272212	−	0.962237i	$-0.587755\pi$
$443$	8.94531	−	5.16458i	0.425004	−	0.245376i	0.697216	+	0.716861i	$0.254422\pi$
$444$		0			0
$445$		0			0
$446$	−61.7619	−	35.6582i	−2.92451	−	1.68847i
$447$		0			0
$448$		0			0
$449$		−	38.3667i		−	1.81063i	−0.424736	−	0.905317i	$-0.639633\pi$
$449$		−	38.3667i		−	1.81063i	0.424736	−	0.905317i	$-0.360367\pi$
$450$	−36.8441	+	3.28286i	−1.73685	+	0.154756i
$451$		0			0
$452$		0			0
$453$	22.4657	+	20.5530i	1.05553	+	0.965665i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.38368	−	4.84032i	0.390467	−	0.225436i	−0.291896	−	0.956450i	$-0.594286\pi$
$461$	8.38368	−	4.84032i	0.390467	−	0.225436i	0.682362	+	0.731014i	$0.260953\pi$
$462$		0			0
$463$	20.4599	−	35.4376i	0.950854	−	1.64693i	0.207271	−	0.978284i	$-0.433542\pi$
$463$	20.4599	−	35.4376i	0.950854	−	1.64693i	0.743583	−	0.668644i	$-0.233125\pi$
$464$	−8.36506	−	4.82957i	−0.388338	−	0.224207i
$465$		0			0
$466$	−17.6700	−	30.6053i	−0.818546	−	1.41776i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−67.4427	−	31.3177i	−3.11754	−	1.44766i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	38.9740	−	67.5050i	1.79392	−	3.10717i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−47.5023			−2.17270
$479$		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
$484$	−22.4464	−	38.8783i	−1.02029	−	1.76720i
$485$		0			0
$486$	17.8864	+	34.0264i	0.811345	+	1.54347i
$487$	−1.60111			−0.0725533			−0.0362766	−	0.999342i	$-0.511550\pi$
$487$	−1.60111			−0.0725533			−0.0362766	+	0.999342i	$0.511550\pi$
$488$		0			0
$489$	8.97233	−	2.83686i	0.405743	−	0.128287i
$490$		0			0
$491$	−9.25056	−	5.34081i	−0.417472	−	0.241027i	0.276523	−	0.961007i	$-0.410818\pi$
$491$	−9.25056	−	5.34081i	−0.417472	−	0.241027i	−0.693995	+	0.719980i	$0.744151\pi$
$492$	−48.1010	+	52.5774i	−2.16856	+	2.37037i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	37.4223			1.68031
$497$		0			0
$498$		0			0
$499$	−9.43028	+	16.3337i	−0.422157	+	0.731198i	−0.996150	−	0.0876621i	$-0.972060\pi$
$499$	−9.43028	+	16.3337i	−0.422157	+	0.731198i	0.573993	+	0.818860i	$0.305394\pi$
$500$		0			0
$501$	27.0417	+	5.97231i	1.20813	+	0.266823i
$502$		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
$507$	−30.5247	−	27.9258i	−1.35565	−	1.24023i
$508$	−30.7072	+	53.1864i	−1.36241	+	2.35976i
$509$	−38.0383	−	21.9615i	−1.68602	−	0.973424i	−0.957515	−	0.288384i	$-0.906882\pi$
$509$	−38.0383	−	21.9615i	−1.68602	−	0.973424i	−0.728505	−	0.685040i	$-0.759785\pi$
$510$		0			0
$511$		0			0
$512$			42.4530i			1.87617i
$513$		0			0
$514$	77.2874			3.40900
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	22.4279	−	24.5151i	0.984475	−	1.07609i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	1.41133	+	15.8395i	0.0617720	+	0.693277i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	43.9901	−	25.3977i	1.92172	−	1.10950i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	−45.3846	+	4.04384i	−1.96952	+	0.175488i
$532$		0			0
$533$	−53.0232	+	30.6129i	−2.29669	+	1.32599i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.15937	+	16.3179i	0.222643	+	0.704169i
$538$	5.50765	+	9.53953i	0.237452	+	0.411278i
$539$		0			0
$540$		0			0
$541$	30.3404			1.30443			0.652217	−	0.758032i	$-0.273839\pi$
$541$	30.3404			1.30443			0.652217	+	0.758032i	$0.273839\pi$
$542$	36.1344	−	20.8622i	1.55211	−	0.896109i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	20.0111	+	34.6603i	0.855614	+	1.48197i	0.876074	+	0.482176i	$0.160153\pi$
$547$	20.0111	+	34.6603i	0.855614	+	1.48197i	−0.0204604	+	0.999791i	$0.506513\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−9.19372	+	41.6278i	−0.391311	+	1.77180i
$553$		0			0
$554$	69.0259	+	39.8521i	2.93263	+	1.69315i
$555$		0			0
$556$	47.6944	+	82.6091i	2.02269	+	3.50341i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−35.4188	−	50.4114i	−1.49940	−	2.13408i
$559$		0			0
$560$		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	$0.166667\pi$
$564$	−21.9095	−	69.2947i	−0.922557	−	2.91783i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−85.9399			−3.60596
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			23.9792i			1.00000i
$576$	17.1160	−	12.0256i	0.713166	−	0.501067i
$577$	−29.0860			−1.21087			−0.605433	−	0.795896i	$-0.707000\pi$
$577$	−29.0860			−1.21087			−0.605433	+	0.795896i	$0.707000\pi$
$578$	36.3056	−	20.9610i	1.51011	−	0.871864i
$579$	10.2853	−	46.5702i	0.427441	−	1.93539i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	80.3974i		−	3.32687i
$585$		0			0
$586$		0			0
$587$	39.7826	−	22.9685i	1.64200	−	0.948011i	0.661883	−	0.749607i	$-0.269757\pi$
$587$	39.7826	−	22.9685i	1.64200	−	0.948011i	0.980120	−	0.198404i	$-0.0635758\pi$
$588$	36.5084	+	33.4000i	1.50558	+	1.37739i
$589$		0			0
$590$		0			0
$591$	5.47297	+	17.3097i	0.225128	+	0.712028i
$592$		0			0
$593$			45.1641i			1.85467i	0.374236	+	0.927333i	$0.377905\pi$
$593$			45.1641i			1.85467i	−0.374236	+	0.927333i	$0.622095\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−35.9135	+	62.2040i	−1.46861	+	2.54371i
$599$	−40.1533	−	23.1825i	−1.64062	−	0.947212i	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−40.1533	−	23.1825i	−1.64062	−	0.947212i	−0.660006	−	0.751260i	$-0.729446\pi$
$600$	−30.0010	+	32.7929i	−1.22478	+	1.33877i
$601$	−8.46962	−	14.6698i	−0.345483	−	0.598394i	0.639958	−	0.768410i	$-0.278952\pi$
$601$	−8.46962	−	14.6698i	−0.345483	−	0.598394i	−0.985441	+	0.170015i	$0.945618\pi$
$602$		0			0
$603$		0			0
$604$	71.7457			2.91929
$605$		0			0
$606$	13.6215	−	61.6761i	0.553335	−	2.50542i
$607$	2.45994	−	4.26073i	0.0998457	−	0.172938i	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	2.45994	−	4.26073i	0.0998457	−	0.172938i	0.911621	+	0.411033i	$0.134832\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	62.4418i		−	2.52612i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	74.5756	−	43.0562i	3.00963	−	1.73761i
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	23.0000	−	9.59166i	0.922958	−	0.384900i
$622$	−1.14704			−0.0459920
$623$		0			0
$624$	−45.0707	+	14.2504i	−1.80427	+	0.570473i
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	8.56129	−	38.7642i	0.340281	−	1.54074i
$634$	−20.2078	+	35.0009i	−0.802553	+	1.39006i
$635$		0			0
$636$		0			0
$637$	21.2568	+	36.8179i	0.842226	+	1.45878i
$638$		0			0
$639$	28.8800	+	41.1048i	1.14248	+	1.62608i
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.3435i		−	0.996355i	−0.867075	−	0.498178i	$-0.834003\pi$
$647$		−	25.3435i		−	0.996355i	0.867075	−	0.498178i	$-0.165997\pi$
$648$	43.4543	+	15.6587i	1.70704	+	0.615133i
$649$		0			0
$650$	−64.8521	+	37.4424i	−2.54371	+	1.46861i
$651$		0			0
$652$	11.0864	−	19.2021i	0.434175	−	0.752014i
$653$	43.9989	+	25.4028i	1.72181	+	0.994087i	0.915197	+	0.403007i	$0.132035\pi$
$653$	43.9989	+	25.4028i	1.72181	+	0.994087i	0.806613	+	0.591080i	$0.201298\pi$
$654$		0			0
$655$		0			0
$656$			45.3001i			1.76867i
$657$	−38.4538	+	27.0175i	−1.50023	+	1.05405i
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	13.5549	+	7.82591i	0.526825	+	0.304163i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.3088			0.399158
$668$	56.5107	−	32.6264i	2.18646	−	1.26236i
$669$	36.9578	+	33.8112i	1.42887	+	1.30722i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.84226	+	11.8511i	0.263750	+	0.456828i	0.967235	−	0.253882i	$-0.0817074\pi$
$673$	6.84226	+	11.8511i	0.263750	+	0.456828i	−0.703486	+	0.710710i	$0.748374\pi$
$674$		0			0
$675$	25.7666	+	3.32932i	0.991755	+	0.128146i
$676$	−97.4824			−3.74932
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	51.0057i		−	1.95168i	−0.218485	−	0.975840i	$-0.570112\pi$
$683$		−	51.0057i		−	1.95168i	0.218485	−	0.975840i	$-0.429888\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	23.4599	+	40.6338i	0.892458	+	1.54578i	0.836919	+	0.547326i	$0.184354\pi$
$691$	23.4599	+	40.6338i	0.892458	+	1.54578i	0.0555386	+	0.998457i	$0.482312\pi$
$692$		−	78.2904i		−	2.97615i
$693$		0			0
$694$	−23.6531			−0.897858
$695$		0			0
$696$	14.0979	+	12.8976i	0.534380	+	0.488883i
$697$		0			0
$698$	−5.46981	−	3.15799i	−0.207035	−	0.119532i
$699$	7.48300	+	23.6670i	0.283033	+	0.895168i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	61.8543	+	47.2270i	2.33454	+	1.78247i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	41.9947	−	72.7369i	1.58049	−	2.73749i
$707$		0			0
$708$	−72.4692	+	79.2134i	−2.72356	+	2.97702i
$709$		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
$713$	−34.5885	+	19.9697i	−1.29535	+	0.747869i
$714$		0			0
$715$		0			0
$716$	34.9228	+	20.1627i	1.30513	+	0.753514i
$717$	32.5792	+	7.19529i	1.21669	+	0.268713i
$718$		0			0
$719$		−	44.7638i		−	1.66941i	−0.550700	−	0.834703i	$-0.685639\pi$
$719$		−	44.7638i		−	1.66941i	0.550700	−	0.834703i	$-0.314361\pi$
$720$		0			0
$721$		0			0
$722$	−40.5768	+	23.4270i	−1.51011	+	0.871864i
$723$		0			0
$724$		0			0
$725$	9.30775	+	5.37383i	0.345681	+	0.199579i
$726$	14.1641	+	44.7978i	0.525679	+	1.66260i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−7.11325	−	26.0461i	−0.263454	−	0.964672i
$730$		0			0
$731$		0			0
$732$		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
$736$	1.95883	+	3.39279i	0.0722034	+	0.125060i
$737$		0			0
$738$	61.0234	−	42.8748i	2.24630	−	1.57824i
$739$	−54.0199			−1.98716			−0.993578	−	0.113150i	$-0.963906\pi$
$739$	−54.0199			−1.98716			−0.993578	+	0.113150i	$0.963906\pi$
$740$		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	$0.166667\pi$
$744$	−72.2863	−	15.9648i	−2.65015	−	0.585299i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−40.0101	−	23.0998i	−1.45902	−	0.842365i
$753$		0			0
$754$	16.0967	+	27.8804i	0.586209	+	1.01534i
$755$		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
$761$	−21.7590	−	12.5626i	−0.788764	−	0.455393i	0.0507631	−	0.998711i	$-0.483835\pi$
$761$	−21.7590	−	12.5626i	−0.788764	−	0.455393i	−0.839527	+	0.543317i	$0.817168\pi$
$762$	43.3854	−	47.4230i	1.57169	−	1.71795i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−79.8850	+	46.1216i	−2.88448	+	1.66536i
$768$	12.1345	−	54.9431i	0.437865	−	1.98259i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−53.0071	−	11.7069i	−1.90900	−	0.421614i
$772$	−56.1880	−	97.3204i	−2.02225	−	3.50264i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−41.6396			−1.49574
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.43130	−	11.0772i	0.0511505	−	0.395867i
$784$	31.4552			1.12340
$785$		0			0
$786$	−50.6878	+	16.0264i	−1.80797	+	0.571643i
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	37.0455	+	21.3882i	1.31969	+	0.761924i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−76.5832	+	44.2153i	−2.71784	+	1.56914i
$795$		0			0
$796$		0			0
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$			4.08444i			0.144407i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−108.017	−	62.3635i	−3.80473	−	2.19666i
$807$	−2.33242	−	7.37689i	−0.0821049	−	0.259679i
$808$	−37.9468	−	65.7257i	−1.33496	−	2.31222i
$809$		−	17.1897i		−	0.604359i	−0.953251	−	0.302180i	$-0.902286\pi$
$809$		−	17.1897i		−	0.604359i	0.953251	−	0.302180i	$-0.0977142\pi$
$810$		0			0
$811$	56.3146			1.97747			0.988736	−	0.149671i	$-0.0478215\pi$
$811$	56.3146			1.97747			0.988736	+	0.149671i	$0.0478215\pi$
$812$		0			0
$813$	−27.9426	+	8.83487i	−0.979991	+	0.309852i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			43.5158i			1.52149i
$819$		0			0
$820$		0			0
$821$	16.6132	−	9.59166i	0.579806	−	0.334751i	−0.181250	−	0.983437i	$-0.558014\pi$
$821$	16.6132	−	9.59166i	0.579806	−	0.334751i	0.761056	+	0.648686i	$0.224681\pi$
$822$		0			0
$823$	15.5037	−	26.8531i	0.540424	−	0.936041i	−0.458456	−	0.888717i	$-0.651597\pi$
$823$	15.5037	−	26.8531i	0.540424	−	0.936041i	0.998880	−	0.0473240i	$-0.0150693\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	24.7300	−	53.2560i	0.859427	−	1.85077i
$829$	48.8397			1.69627			0.848137	−	0.529777i	$-0.177724\pi$
$829$	48.8397			1.69627			0.848137	+	0.529777i	$0.177724\pi$
$830$		0			0
$831$	−41.3045	−	37.7879i	−1.43284	−	1.31085i
$832$	21.1740	−	36.6745i	0.734078	−	1.27146i
$833$		0			0
$834$	−30.0960	−	95.1868i	−1.04214	−	3.29605i
$835$		0			0
$836$		0			0
$837$	16.6558	+	39.9393i	0.575710	+	1.38050i
$838$		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
$841$	−12.1898	+	21.1133i	−0.420336	+	0.728044i
$842$		0			0
$843$		0			0
$844$	−46.7699	−	81.0079i	−1.60989	−	2.78841i
$845$		0			0
$846$	6.75037	+	75.7605i	0.232083	+	2.60470i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	115.584	+	25.5273i	3.95984	+	0.874552i
$853$	−27.4199	−	47.4926i	−0.938839	−	1.62612i	−0.767642	−	0.640879i	$-0.778570\pi$
$853$	−27.4199	−	47.4926i	−0.938839	−	1.62612i	−0.171197	−	0.985237i	$-0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−44.3456	+	25.6030i	−1.51482	+	0.874581i	−0.514969	+	0.857209i	$0.672197\pi$
$857$	−44.3456	+	25.6030i	−1.51482	+	0.874581i	−0.999849	+	0.0173721i	$0.994470\pi$
$858$		0			0
$859$	−4.96289	+	8.59598i	−0.169332	+	0.293291i	−0.938185	−	0.346134i	$-0.887494\pi$
$859$	−4.96289	+	8.59598i	−0.169332	+	0.293291i	0.768853	+	0.639425i	$0.220828\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			58.7189i			1.99882i	0.0344158	+	0.999408i	$0.489043\pi$
$863$			58.7189i			1.99882i	−0.0344158	+	0.999408i	$0.510957\pi$
$864$	3.91766	−	1.63378i	0.133281	−	0.0555822i
$865$		0			0
$866$		0			0
$867$	−28.0750	+	8.87671i	−0.953476	+	0.301469i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−23.8810	+	108.130i	−0.806865	+	3.65336i
$877$	−21.4199	+	37.1003i	−0.723298	+	1.25279i	0.236373	+	0.971662i	$0.424041\pi$
$877$	−21.4199	+	37.1003i	−0.723298	+	1.25279i	−0.959671	+	0.281126i	$0.909292\pi$
$878$	89.4253	+	51.6297i	3.01796	+	1.74242i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−29.7711	−	42.3731i	−1.00245	−	1.42678i
$883$	1.08013			0.0363492			0.0181746	−	0.999835i	$-0.494215\pi$
$883$	1.08013			0.0363492			0.0181746	+	0.999835i	$0.494215\pi$
$884$		0			0
$885$		0			0
$886$	−12.7359	+	22.0591i	−0.427869	+	0.741092i
$887$	−8.21030	−	4.74022i	−0.275675	−	0.159161i	0.355789	−	0.934566i	$-0.384212\pi$
$887$	−8.21030	−	4.74022i	−0.275675	−	0.159161i	−0.631464	+	0.775405i	$0.717546\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	118.027			3.95183
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	34.0532	−	37.2224i	1.13700	−	1.24282i
$898$	47.3061	+	81.9366i	1.57863	+	2.73426i
$899$			17.9011i			0.597037i
$900$	50.0902	−	35.1931i	1.66967	−	1.17310i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−73.3202	−	16.1932i	−2.43590	−	0.537982i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	−18.6844	+	40.2369i	−0.619723	+	1.33457i
$910$		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
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−57.6691	+	18.2337i	−1.90026	+	0.600822i
$922$	−11.9362	+	20.6742i	−0.393099	+	0.680867i
$923$	88.0755	+	50.8504i	2.89904	+	1.67376i
$924$		0			0
$925$		0			0
$926$			100.909i			3.31606i
$927$		0			0
$928$	1.75593			0.0576412
$929$	9.42393	−	5.44091i	0.309189	−	0.178510i	−0.337374	−	0.941371i	$-0.609539\pi$
$929$	9.42393	−	5.44091i	0.309189	−	0.178510i	0.646564	+	0.762860i	$0.276205\pi$
$930$		0			0
$931$		0			0
$932$	50.6510	+	29.2434i	1.65913	+	0.957898i
$933$	0.786690	+	0.173745i	0.0257551	+	0.00568814i
$934$		0			0
$935$		0			0
$936$	93.1397	−	8.29889i	3.04437	−	0.271258i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$	−24.1735	−	41.8697i	−0.787196	−	1.36346i
$944$			68.2494i			2.22133i
$945$		0			0
$946$		0			0
$947$	3.33259	−	1.92407i	0.108295	−	0.0625240i	−0.444874	−	0.895593i	$-0.646752\pi$
$947$	3.33259	−	1.92407i	0.108295	−	0.0625240i	0.553169	+	0.833069i	$0.313418\pi$
$948$		0			0
$949$	−47.5709	+	82.3953i	−1.54422	+	2.67466i
$950$		0			0
$951$	19.1611	−	20.9442i	0.621340	−	0.679164i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	68.0826	−	39.3075i	2.20195	−	1.27130i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−19.1771	−	33.2158i	−0.618617	−	1.07148i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	18.8389	−	32.6300i	0.605819	−	1.04931i	−0.386103	−	0.922456i	$-0.626179\pi$
$967$	18.8389	−	32.6300i	0.605819	−	1.04931i	0.991922	−	0.126853i	$-0.0404877\pi$
$968$	48.8904	+	28.2269i	1.57140	+	0.907247i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−53.7921	−	33.9676i	−1.72538	−	1.08951i
$973$		0			0
$974$	3.41937	−	1.97417i	0.109564	−	0.0632565i
$975$	50.1500	−	15.8564i	1.60608	−	0.507810i
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$	−15.6636	+	17.1213i	−0.500868	+	0.547480i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	26.3409			0.840572
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$	19.3256	−	87.5033i	0.616077	−	2.78950i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52.9199			−1.68106			−0.840528	−	0.541769i	$-0.817755\pi$
$991$	−52.9199			−1.68106			−0.840528	+	0.541769i	$0.817755\pi$
$992$	−5.89155	+	3.40149i	−0.187057	+	0.107997i
$993$	−8.11112	−	7.42054i	−0.257399	−	0.235484i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−18.4199	−	31.9042i	−0.583363	−	1.01041i	−0.995077	−	0.0991016i	$-0.968403\pi$
$997$	−18.4199	−	31.9042i	−0.583363	−	1.01041i	0.411714	−	0.911313i	$-0.364930\pi$
$998$		−	46.5102i		−	1.47225i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.2.g.a.68.2	✓	12
3.2	odd	2		621.2.g.a.206.5		12
9.2	odd	6	inner	207.2.g.a.137.2	yes	12
9.4	even	3		1863.2.c.a.1862.2		12
9.5	odd	6		1863.2.c.a.1862.11		12
9.7	even	3		621.2.g.a.413.5		12
23.22	odd	2	CM	207.2.g.a.68.2	✓	12
69.68	even	2		621.2.g.a.206.5		12
207.22	odd	6		1863.2.c.a.1862.2		12
207.68	even	6		1863.2.c.a.1862.11		12
207.137	even	6	inner	207.2.g.a.137.2	yes	12
207.160	odd	6		621.2.g.a.413.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.g.a.68.2	✓	12	1.1	even	1	trivial
207.2.g.a.68.2	✓	12	23.22	odd	2	CM
207.2.g.a.137.2	yes	12	9.2	odd	6	inner
207.2.g.a.137.2	yes	12	207.137	even	6	inner
621.2.g.a.206.5		12	3.2	odd	2
621.2.g.a.206.5		12	69.68	even	2
621.2.g.a.413.5		12	9.7	even	3
621.2.g.a.413.5		12	207.160	odd	6
1863.2.c.a.1862.2		12	9.4	even	3
1863.2.c.a.1862.2		12	207.22	odd	6
1863.2.c.a.1862.11		12	9.5	odd	6
1863.2.c.a.1862.11		12	207.68	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.13562 + 1.23300i) q^{2} +(1.65147 - 0.522159i) q^{3} +(2.04058 - 3.53440i) q^{4} +(-2.88309 + 3.15140i) q^{6} +5.13217i q^{8} +(2.45470 - 1.72466i) q^{9} +O(q^{10})\)	\(q+(-2.13562 + 1.23300i) q^{2} +(1.65147 - 0.522159i) q^{3} +(2.04058 - 3.53440i) q^{4} +(-2.88309 + 3.15140i) q^{6} +5.13217i q^{8} +(2.45470 - 1.72466i) q^{9} +(1.52444 - 6.90245i) q^{12} +(3.03669 - 5.25970i) q^{13} +(-2.24680 - 3.89157i) q^{16} +(-3.11580 + 6.70987i) q^{18} +(4.15331 + 2.39792i) q^{23} +(2.67981 + 8.47561i) q^{24} +(2.50000 + 4.33013i) q^{25} +14.9770i q^{26} +(3.15331 - 4.12997i) q^{27} +(1.86155 - 1.07477i) q^{29} +(-4.16396 + 7.21219i) q^{31} +(0.707446 + 0.408444i) q^{32} +(-1.08661 - 12.1952i) q^{36} +(2.26860 - 10.2719i) q^{39} +(-8.73043 - 5.04051i) q^{41} -11.8265 q^{46} +(8.90380 - 5.14061i) q^{47} +(-5.74254 - 5.25362i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-10.6781 - 6.16501i) q^{50} +(-12.3932 - 21.4657i) q^{52} +(-1.64202 + 12.7081i) q^{54} +(-2.65038 + 4.59059i) q^{58} +(-13.1533 - 7.59407i) q^{59} -20.5367i q^{62} +6.97274 q^{64} +(8.11116 + 1.79139i) q^{69} +16.7454i q^{71} +(8.85124 + 12.5979i) q^{72} -15.6654 q^{73} +(6.38969 + 5.84567i) q^{75} +(7.82036 + 24.7340i) q^{78} +(3.05110 - 8.46704i) q^{81} +24.8598 q^{82} +(2.51309 - 2.74697i) q^{87} +(16.9504 - 9.78630i) q^{92} +(-3.11074 + 14.0850i) q^{93} +(-12.6768 + 21.9568i) q^{94} +(1.38160 + 0.305133i) q^{96} -17.2620i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} - 3 q^{6}+O(q^{10}) \) 12 * q + 12 * q^4 - 3 * q^6	\( 12 q + 12 q^{4} - 3 q^{6} - 30 q^{12} - 24 q^{16} - 21 q^{18} - 12 q^{24} + 30 q^{25} - 12 q^{27} + 54 q^{32} + 33 q^{36} + 24 q^{39} + 48 q^{48} - 42 q^{49} - 3 q^{52} + 15 q^{58} - 108 q^{59} - 54 q^{64} + 42 q^{72} + 51 q^{78} - 66 q^{82} + 96 q^{87} + 6 q^{93} - 39 q^{94} - 69 q^{96}+O(q^{100}) \) 12 * q + 12 * q^4 - 3 * q^6 - 30 * q^12 - 24 * q^16 - 21 * q^18 - 12 * q^24 + 30 * q^25 - 12 * q^27 + 54 * q^32 + 33 * q^36 + 24 * q^39 + 48 * q^48 - 42 * q^49 - 3 * q^52 + 15 * q^58 - 108 * q^59 - 54 * q^64 + 42 * q^72 + 51 * q^78 - 66 * q^82 + 96 * q^87 + 6 * q^93 - 39 * q^94 - 69 * q^96

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