LMFDB - Embedded newform 364.2.x.a.103.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [364,2,Mod(103,364)] 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("364.103"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(364, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 364 = 2^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	364.x (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.90655463357$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.151613669376.6
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 95x^{4} - 588x^{2} + 2401 $ x^8 - 12x^6 + 95x^4 - 588*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			103.3
Root			$-1.85439 - 1.88713i$ of defining polynomial
Character	$\chi$	$=$	364.103
Dual form			364.2.x.a.311.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-1.85439 - 1.88713i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.14728 + 1.98715i) q^{11} -3.60555i q^{13} +(-3.62250 + 0.936750i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-0.122499 + 0.0707248i) q^{17} +(-2.12132 - 3.67423i) q^{18} +(-7.51124 - 4.33662i) q^{19} +3.24500 q^{22} +(2.50000 + 4.33013i) q^{25} +(-4.41588 - 2.54951i) q^{26} +(-1.41421 + 5.09902i) q^{28} +10.2450 q^{29} +(4.41588 - 2.54951i) q^{31} +(2.82843 + 4.89898i) q^{32} +0.200040i q^{34} -6.00000 q^{36} +(-10.6225 + 6.13290i) q^{38} +(2.29456 - 3.97429i) q^{44} +(7.33800 + 4.23660i) q^{47} +(-0.122499 + 6.99893i) q^{49} +7.07107 q^{50} +(-6.24500 + 3.60555i) q^{52} +(-6.74500 - 11.6827i) q^{53} +(5.24500 + 5.33760i) q^{56} +(7.24431 - 12.5475i) q^{58} +(-7.68448 + 4.43664i) q^{59} +(10.7450 + 6.20363i) q^{61} -7.21110i q^{62} +(-7.68448 + 1.98715i) q^{63} +8.00000 q^{64} +(6.09703 + 10.5604i) q^{67} +(0.244998 + 0.141450i) q^{68} +16.7832 q^{71} +(-4.24264 + 7.34847i) q^{72} +17.3465i q^{76} +(1.62250 - 5.85000i) q^{77} +(-4.50000 - 7.79423i) q^{81} -15.2971i q^{83} +(-3.24500 - 5.62050i) q^{88} +(-6.80413 + 6.68609i) q^{91} +(10.3775 - 5.99145i) q^{94} +(8.48528 + 5.09902i) q^{98} +6.88368 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 12 q^{9} - 4 q^{14} - 16 q^{16} + 24 q^{17} - 24 q^{22} + 20 q^{25} + 32 q^{29} - 48 q^{36} - 60 q^{38} + 24 q^{49} - 4 q^{53} - 8 q^{56} + 36 q^{61} + 64 q^{64} - 48 q^{68} - 12 q^{77}+ \cdots + 108 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 + 12 * q^9 - 4 * q^14 - 16 * q^16 + 24 * q^17 - 24 * q^22 + 20 * q^25 + 32 * q^29 - 48 * q^36 - 60 * q^38 + 24 * q^49 - 4 * q^53 - 8 * q^56 + 36 * q^61 + 64 * q^64 - 48 * q^68 - 12 * q^77 - 36 * q^81 + 24 * q^88 + 108 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/364\mathbb{Z}\right)^\times$.

$n$	$157$	$183$	$197$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

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 $

$2$	0.707107	−	1.22474i	0.500000	−	0.866025i
$2$	0.707107	−	1.22474i	0.500000	−	0.866025i
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−1.85439	−	1.88713i	−0.700892	−	0.713267i
$7$	−1.85439	−	1.88713i	−0.700892	−	0.713267i
$8$	−2.82843			−1.00000
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	1.14728	+	1.98715i	0.345918	+	0.599147i	0.985520	−	0.169559i	$-0.0542342\pi$
$11$	1.14728	+	1.98715i	0.345918	+	0.599147i	−0.639602	+	0.768706i	$0.720901\pi$
$12$		0			0
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$	−3.62250	+	0.936750i	−0.968154	+	0.250357i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−0.122499	+	0.0707248i	−0.0297104	+	0.0171533i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	−0.122499	+	0.0707248i	−0.0297104	+	0.0171533i	0.485071	+	0.874475i	$0.338794\pi$
$18$	−2.12132	−	3.67423i	−0.500000	−	0.866025i
$19$	−7.51124	−	4.33662i	−1.72320	−	0.994888i	−0.912090	−	0.409991i	$-0.865532\pi$
$19$	−7.51124	−	4.33662i	−1.72320	−	0.994888i	−0.811107	−	0.584898i	$-0.801135\pi$
$20$		0			0
$21$		0			0
$22$	3.24500			0.691836
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	−4.41588	−	2.54951i	−0.866025	−	0.500000i
$27$		0			0
$28$	−1.41421	+	5.09902i	−0.267261	+	0.963624i
$29$	10.2450			1.90245			0.951224	−	0.308500i	$-0.0998271\pi$
$29$	10.2450			1.90245			0.951224	+	0.308500i	$0.0998271\pi$
$30$		0			0
$31$	4.41588	−	2.54951i	0.793116	−	0.457905i	−0.0479427	−	0.998850i	$-0.515266\pi$
$31$	4.41588	−	2.54951i	0.793116	−	0.457905i	0.841058	+	0.540945i	$0.181933\pi$
$32$	2.82843	+	4.89898i	0.500000	+	0.866025i
$33$		0			0
$34$			0.200040i			0.0343066i
$35$		0			0
$36$	−6.00000			−1.00000
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	−10.6225	+	6.13290i	−1.72320	+	0.994888i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	2.29456	−	3.97429i	0.345918	−	0.599147i
$45$		0			0
$46$		0			0
$47$	7.33800	+	4.23660i	1.07036	+	0.617971i	0.928279	−	0.371884i	$-0.121288\pi$
$47$	7.33800	+	4.23660i	1.07036	+	0.617971i	0.142078	+	0.989855i	$0.454621\pi$
$48$		0			0
$49$	−0.122499	+	6.99893i	−0.0174999	+	0.999847i
$50$	7.07107			1.00000
$51$		0			0
$52$	−6.24500	+	3.60555i	−0.866025	+	0.500000i
$53$	−6.74500	−	11.6827i	−0.926497	−	1.60474i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	−6.74500	−	11.6827i	−0.926497	−	1.60474i	−0.137361	−	0.990521i	$-0.543862\pi$
$54$		0			0
$55$		0			0
$56$	5.24500	+	5.33760i	0.700892	+	0.713267i
$57$		0			0
$58$	7.24431	−	12.5475i	0.951224	−	1.64757i
$59$	−7.68448	+	4.43664i	−1.00043	+	0.577601i	−0.908377	−	0.418153i	$-0.862678\pi$
$59$	−7.68448	+	4.43664i	−1.00043	+	0.577601i	−0.0920575	+	0.995754i	$0.529344\pi$
$60$		0			0
$61$	10.7450	+	6.20363i	1.37576	+	0.794293i	0.991645	−	0.128994i	$-0.0411748\pi$
$61$	10.7450	+	6.20363i	1.37576	+	0.794293i	0.384111	+	0.923287i	$0.374508\pi$
$62$		−	7.21110i		−	0.915811i
$63$	−7.68448	+	1.98715i	−0.968154	+	0.250357i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	6.09703	+	10.5604i	0.744870	+	1.29015i	0.950255	+	0.311472i	$0.100822\pi$
$67$	6.09703	+	10.5604i	0.744870	+	1.29015i	−0.205385	+	0.978681i	$0.565845\pi$
$68$	0.244998	+	0.141450i	0.0297104	+	0.0171533i
$69$		0			0
$70$		0			0
$71$	16.7832			1.99180			0.995898	−	0.0904805i	$-0.0288403\pi$
$71$	16.7832			1.99180			0.995898	+	0.0904805i	$0.0288403\pi$
$72$	−4.24264	+	7.34847i	−0.500000	+	0.866025i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$		0			0
$75$		0			0
$76$			17.3465i			1.98978i
$77$	1.62250	−	5.85000i	0.184901	−	0.666670i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		−	15.2971i		−	1.67907i	−0.543305	−	0.839535i	$-0.682827\pi$
$83$		−	15.2971i		−	1.67907i	0.543305	−	0.839535i	$-0.317173\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−3.24500	−	5.62050i	−0.345918	−	0.599147i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−6.80413	+	6.68609i	−0.713267	+	0.700892i
$92$		0			0
$93$		0			0
$94$	10.3775	−	5.99145i	1.07036	−	0.617971i
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	8.48528	+	5.09902i	0.857143	+	0.515079i
$99$	6.88368			0.691836
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$	−12.4900	+	7.21110i	−1.24280	+	0.717532i	−0.969664	−	0.244443i	$-0.921395\pi$
$101$	−12.4900	+	7.21110i	−1.24280	+	0.717532i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$			10.1980i			1.00000i
$105$		0			0
$106$	−19.0777			−1.85299
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	10.2460	−	2.64953i	0.968154	−	0.250357i
$113$	−3.75500			−0.353241			−0.176620	−	0.984279i	$-0.556517\pi$
$113$	−3.75500			−0.353241			−0.176620	+	0.984279i	$0.556517\pi$
$114$		0			0
$115$		0			0
$116$	−10.2450	−	17.7449i	−0.951224	−	1.64757i
$117$	−9.36750	−	5.40833i	−0.866025	−	0.500000i
$118$			12.5487i			1.15520i
$119$	0.360627	+	0.100020i	0.0330586	+	0.00916882i
$120$		0			0
$121$	2.86750	−	4.96665i	0.260682	−	0.451514i
$122$	15.1957	−	8.77325i	1.37576	−	0.794293i
$123$		0			0
$124$	−8.83176	−	5.09902i	−0.793116	−	0.457905i
$125$		0			0
$126$	−3.00000	+	10.8167i	−0.267261	+	0.963624i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	5.65685	−	9.79796i	0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	5.74500	+	22.2164i	0.498155	+	1.92641i
$134$	17.2450			1.48974
$135$		0			0
$136$	0.346479	−	0.200040i	0.0297104	−	0.0171533i
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	11.8675	−	20.5551i	0.995898	−	1.72495i
$143$	7.16476	−	4.13658i	0.599147	−	0.345918i
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		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
$151$	−3.80247	−	6.58607i	−0.309440	−	0.535967i	0.668800	−	0.743443i	$-0.266808\pi$
$151$	−3.80247	−	6.58607i	−0.309440	−	0.535967i	−0.978240	+	0.207476i	$0.933475\pi$
$152$	21.2450	+	12.2658i	1.72320	+	0.994888i
$153$			0.424349i			0.0343066i
$154$	−6.01748	−	6.12372i	−0.484902	−	0.493464i
$155$		0			0
$156$		0			0
$157$	−21.1225	+	12.1951i	−1.68576	+	0.973273i	−0.728055	+	0.685519i	$0.759575\pi$
$157$	−21.1225	+	12.1951i	−1.68576	+	0.973273i	−0.957704	+	0.287754i	$0.907091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−12.7279			−1.00000
$163$	−10.6861	+	18.5090i	−0.837004	+	1.44973i	0.0553849	+	0.998465i	$0.482361\pi$
$163$	−10.6861	+	18.5090i	−0.837004	+	1.44973i	−0.892389	+	0.451268i	$0.850972\pi$
$164$		0			0
$165$		0			0
$166$	−18.7350	−	10.8167i	−1.45412	−	0.839535i
$167$		−	9.07331i		−	0.702114i	−0.936354	−	0.351057i	$-0.885822\pi$
$167$		−	9.07331i		−	0.702114i	0.936354	−	0.351057i	$-0.114178\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−22.5337	+	13.0099i	−1.72320	+	0.994888i
$172$		0			0
$173$	−10.2550	−	5.92073i	−0.779673	−	0.450145i	0.0566411	−	0.998395i	$-0.481961\pi$
$173$	−10.2550	−	5.92073i	−0.779673	−	0.450145i	−0.836315	+	0.548250i	$0.815294\pi$
$174$		0			0
$175$	3.53553	−	12.7475i	0.267261	−	0.963624i
$176$	−9.17824			−0.691836
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$			24.6731i			1.83393i	0.398963	+	0.916967i	$0.369370\pi$
$181$			24.6731i			1.83393i	−0.398963	+	0.916967i	$0.630630\pi$
$182$	3.37750	+	13.0611i	0.250357	+	0.968154i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.281081	−	0.162282i	−0.0205547	−	0.0118673i
$188$		−	16.9464i		−	1.23594i
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	12.2450	−	6.78675i	0.874643	−	0.484768i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	4.86750	−	8.43075i	0.345918	−	0.599147i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−7.07107	−	12.2474i	−0.500000	−	0.866025i
$201$		0			0
$202$			20.3961i			1.43506i
$203$	−18.9982	−	19.3336i	−1.33341	−	1.35695i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	12.4900	+	7.21110i	0.866025	+	0.500000i
$209$		−	19.9013i		−	1.37660i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−13.4900	+	23.3654i	−0.926497	+	1.60474i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.0000	−	3.60555i	−0.882498	−	0.244761i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.255002	+	0.441676i	0.0171533	+	0.0297104i
$222$		0			0
$223$		−	26.2197i		−	1.75580i	−0.478841	−	0.877902i	$-0.658943\pi$
$223$		−	26.2197i		−	1.75580i	0.478841	−	0.877902i	$-0.341057\pi$
$224$	4.00000	−	14.4222i	0.267261	−	0.963624i
$225$	15.0000			1.00000
$226$	−2.65519	+	4.59892i	−0.176620	+	0.305916i
$227$	4.41588	−	2.54951i	0.293092	−	0.169217i	−0.346243	−	0.938145i	$-0.612543\pi$
$227$	4.41588	−	2.54951i	0.293092	−	0.169217i	0.639336	+	0.768928i	$0.279209\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	−28.9772			−1.90245
$233$	−9.99000	+	17.3032i	−0.654466	+	1.13357i	0.327561	+	0.944830i	$0.393773\pi$
$233$	−9.99000	+	17.3032i	−0.654466	+	1.13357i	−0.982027	+	0.188739i	$0.939560\pi$
$234$	−13.2476	+	7.64853i	−0.866025	+	0.500000i
$235$		0			0
$236$	15.3690	+	8.87327i	1.00043	+	0.577601i
$237$		0			0
$238$	0.377501	−	0.370952i	0.0244698	−	0.0240452i
$239$	26.6827			1.72596			0.862979	−	0.505239i	$-0.168596\pi$
$239$	26.6827			1.72596			0.862979	+	0.505239i	$0.168596\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	−4.05525	−	7.02390i	−0.260682	−	0.451514i
$243$		0			0
$244$		−	24.8145i		−	1.58859i
$245$		0			0
$246$		0			0
$247$	−15.6359	+	27.0822i	−0.994888	+	1.72320i
$248$	−12.4900	+	7.21110i	−0.793116	+	0.457905i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	11.1263	+	11.3228i	0.700892	+	0.713267i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	24.9800	+	14.4222i	1.55821	+	0.899632i	0.997429	+	0.0716680i	$0.0228322\pi$
$257$	24.9800	+	14.4222i	1.55821	+	0.899632i	0.560781	+	0.827964i	$0.310501\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	15.3675	−	26.6173i	0.951224	−	1.64757i
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$	31.2718	+	8.67323i	1.91740	+	0.531790i
$267$		0			0
$268$	12.1941	−	21.1207i	0.744870	−	1.29015i
$269$	20.8775	−	12.0536i	1.27292	−	0.734923i	0.297386	−	0.954757i	$-0.403885\pi$
$269$	20.8775	−	12.0536i	1.27292	−	0.734923i	0.975537	+	0.219834i	$0.0705517\pi$
$270$		0			0
$271$	22.1872	+	12.8098i	1.34778	+	0.778141i	0.987935	−	0.154872i	$-0.0494964\pi$
$271$	22.1872	+	12.8098i	1.34778	+	0.778141i	0.359844	+	0.933012i	$0.382830\pi$
$272$		−	0.565799i		−	0.0343066i
$273$		0			0
$274$		0			0
$275$	−5.73640	+	9.93574i	−0.345918	+	0.599147i
$276$		0			0
$277$	−13.7450	−	23.8070i	−0.825857	−	1.43043i	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−13.7450	−	23.8070i	−0.825857	−	1.43043i	0.0754058	−	0.997153i	$-0.475975\pi$
$278$		0			0
$279$		−	15.2971i		−	0.915811i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$	−16.7832	−	29.0693i	−0.995898	−	1.72495i
$285$		0			0
$286$		−	11.7000i		−	0.691836i
$287$		0			0
$288$	16.9706			1.00000
$289$	−8.49000	+	14.7051i	−0.499412	+	0.865006i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−10.7550			−0.618881
$303$		0			0
$304$	30.0450	−	17.3465i	1.72320	−	0.994888i
$305$		0			0
$306$	0.519719	+	0.300060i	0.0297104	+	0.0171533i
$307$			8.07311i			0.460757i	0.973101	+	0.230378i	$0.0739964\pi$
$307$			8.07311i			0.460757i	−0.973101	+	0.230378i	$0.926004\pi$
$308$	−11.7550	+	3.03975i	−0.669803	+	0.173206i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−18.7350	−	10.8167i	−1.05896	−	0.611393i	−0.133819	−	0.991006i	$-0.542724\pi$
$313$	−18.7350	−	10.8167i	−1.05896	−	0.611393i	−0.925146	+	0.379612i	$0.876057\pi$
$314$			34.4929i			1.94655i
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	11.7539	+	20.3583i	0.658091	+	1.13985i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.22683			0.0682624
$324$	−9.00000	+	15.5885i	−0.500000	+	0.866025i
$325$	15.6125	−	9.01388i	0.866025	−	0.500000i
$326$	15.1125	+	26.1756i	0.837004	+	1.44973i
$327$		0			0
$328$		0			0
$329$	−5.61249	−	21.7040i	−0.309427	−	1.19658i
$330$		0			0
$331$	−3.53553	+	6.12372i	−0.194331	+	0.336590i	−0.946681	−	0.322173i	$-0.895587\pi$
$331$	−3.53553	+	6.12372i	−0.194331	+	0.336590i	0.752350	+	0.658763i	$0.228920\pi$
$332$	−26.4953	+	15.2971i	−1.45412	+	0.839535i
$333$		0			0
$334$	−11.1125	−	6.41580i	−0.608049	−	0.351057i
$335$		0			0
$336$		0			0
$337$	24.2450			1.32071			0.660355	−	0.750954i	$-0.270406\pi$
$337$	24.2450			1.32071			0.660355	+	0.750954i	$0.270406\pi$
$338$	−9.19239	+	15.9217i	−0.500000	+	0.866025i
$339$		0			0
$340$		0			0
$341$	10.1325	+	5.85000i	0.548706	+	0.316795i
$342$			36.7974i			1.98978i
$343$	13.4350	−	12.7475i	0.725423	−	0.688303i
$344$		0			0
$345$		0			0
$346$	−14.5028	+	8.37317i	−0.779673	+	0.450145i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−13.1125	−	13.3440i	−0.700892	−	0.713267i
$351$		0			0
$352$	−6.49000	+	11.2410i	−0.345918	+	0.599147i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.36396	−	11.0227i	0.335877	−	0.581756i	−0.647776	−	0.761831i	$-0.724301\pi$
$359$	6.36396	−	11.0227i	0.335877	−	0.581756i	0.983653	+	0.180075i	$0.0576340\pi$
$360$		0			0
$361$	28.1125	+	48.6923i	1.47960	+	2.56275i
$362$	30.2182	+	17.4465i	1.58823	+	0.916967i
$363$		0			0
$364$	18.3848	+	5.09902i	0.963624	+	0.267261i
$365$		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
$369$		0			0
$370$		0			0
$371$	−9.53887	+	34.3929i	−0.495233	+	1.78559i
$372$		0			0
$373$	1.36750	−	2.36857i	0.0708063	−	0.122640i	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	1.36750	−	2.36857i	0.0708063	−	0.122640i	0.899255	+	0.437425i	$0.144109\pi$
$374$	−0.397509	+	0.229502i	−0.0205547	+	0.0118673i
$375$		0			0
$376$	−20.7550	−	11.9829i	−1.07036	−	0.617971i
$377$		−	36.9389i		−	1.90245i
$378$		0			0
$379$	−15.5563			−0.799076			−0.399538	−	0.916717i	$-0.630829\pi$
$379$	−15.5563			−0.799076			−0.399538	+	0.916717i	$0.630829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	22.0794	+	12.7475i	1.12820	+	0.651369i	0.943483	−	0.331422i	$-0.107528\pi$
$383$	22.0794	+	12.7475i	1.12820	+	0.651369i	0.184722	+	0.982791i	$0.440862\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.6125	+	20.1134i	0.588777	+	1.01979i	0.994393	+	0.105748i	$0.0337237\pi$
$389$	11.6125	+	20.1134i	0.588777	+	1.01979i	−0.405616	+	0.914044i	$0.632943\pi$
$390$		0			0
$391$		0			0
$392$	0.346479	−	19.7960i	0.0174999	−	0.999847i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−6.88368	−	11.9229i	−0.345918	−	0.599147i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−9.19239	−	15.9217i	−0.457905	−	0.793116i
$404$	24.9800	+	14.4222i	1.24280	+	0.717532i
$405$		0			0
$406$	−37.1125	+	9.59700i	−1.84186	+	0.476291i
$407$		0			0
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.6225	+	6.27435i	1.11318	+	0.308741i
$414$		0			0
$415$		0			0
$416$	17.6635	−	10.1980i	0.866025	−	0.500000i
$417$		0			0
$418$	−24.3740	−	14.0723i	−1.19217	−	0.688299i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	22.0140	−	12.7098i	1.07036	−	0.617971i
$424$	19.0777	+	33.0436i	0.926497	+	1.60474i
$425$	−0.612495	−	0.353624i	−0.0297104	−	0.0171533i
$426$		0			0
$427$	−8.21835	−	31.7811i	−0.397714	−	1.53800i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.6066	+	18.3712i	0.510902	+	0.884908i	0.999920	+	0.0126347i	$0.00402184\pi$
$431$	10.6066	+	18.3712i	0.510902	+	0.884908i	−0.489018	+	0.872274i	$0.662645\pi$
$432$		0			0
$433$			11.5586i			0.555469i	0.960658	+	0.277734i	$0.0895835\pi$
$433$			11.5586i			0.555469i	−0.960658	+	0.277734i	$0.910416\pi$
$434$	−13.6083	+	13.3722i	−0.653218	+	0.641885i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	18.0000	+	10.8167i	0.857143	+	0.515079i
$442$	0.721255			0.0343066
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$	−32.1125	−	18.5402i	−1.52057	−	0.877902i
$447$		0			0
$448$	−14.8351	−	15.0970i	−0.700892	−	0.713267i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	10.6066	−	18.3712i	0.500000	−	0.866025i
$451$		0			0
$452$	3.75500	+	6.50385i	0.176620	+	0.305916i
$453$		0			0
$454$		−	7.21110i		−	0.338434i
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	24.0416			1.11731			0.558655	−	0.829400i	$-0.311318\pi$
$463$	24.0416			1.11731			0.558655	+	0.829400i	$0.311318\pi$
$464$	−20.4900	+	35.4897i	−0.951224	+	1.64757i
$465$		0			0
$466$	14.1280	+	24.4704i	0.654466	+	1.13357i
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$			21.6333i			1.00000i
$469$	8.62250	−	31.0889i	0.398150	−	1.43555i
$470$		0			0
$471$		0			0
$472$	21.7350	−	12.5487i	1.00043	−	0.577601i
$473$		0			0
$474$		0			0
$475$		−	43.3662i		−	1.98978i
$476$	−0.187388	−	0.724645i	−0.00858889	−	0.0332140i
$477$	−40.4700			−1.85299
$478$	18.8675	−	32.6795i	0.862979	−	1.49472i
$479$	−37.3830	+	21.5831i	−1.70807	+	0.986155i	−0.771124	+	0.636685i	$0.780305\pi$
$479$	−37.3830	+	21.5831i	−1.70807	+	0.986155i	−0.936947	+	0.349470i	$0.886362\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−11.4700			−0.521363
$485$		0			0
$486$		0			0
$487$	−8.75221	−	15.1593i	−0.396601	−	0.686933i	0.596703	−	0.802462i	$-0.296477\pi$
$487$	−8.75221	−	15.1593i	−0.396601	−	0.686933i	−0.993304	+	0.115529i	$0.963144\pi$
$488$	−30.3914	−	17.5465i	−1.37576	−	0.794293i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−1.25500	+	0.724576i	−0.0565224	+	0.0326333i
$494$	22.1125	+	38.3000i	0.994888	+	1.72320i
$495$		0			0
$496$			20.3961i			0.915811i
$497$	−31.1225	−	31.6720i	−1.39603	−	1.42068i
$498$		0			0
$499$	−13.4350	+	23.2702i	−0.601434	+	1.04172i	0.391170	+	0.920319i	$0.372071\pi$
$499$	−13.4350	+	23.2702i	−0.601434	+	1.04172i	−0.992604	+	0.121396i	$0.961263\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	21.7350	−	5.62050i	0.968154	−	0.250357i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$		0			0
$514$	35.3270	−	20.3961i	1.55821	−	0.899632i
$515$		0			0
$516$		0			0
$517$			19.4423i			0.855069i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	31.2250	−	18.0278i	1.36799	−	0.789810i	0.377320	−	0.926083i	$-0.376846\pi$
$521$	31.2250	−	18.0278i	1.36799	−	0.789810i	0.990671	+	0.136272i	$0.0435123\pi$
$522$	−21.7329	−	37.6425i	−0.951224	−	1.64757i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.360627	+	0.624625i	−0.0157092	+	0.0272091i
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$			26.6198i			1.15520i
$532$	32.7350	−	32.1671i	1.41924	−	1.39462i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−17.2450	−	29.8692i	−0.744870	−	1.29015i
$537$		0			0
$538$		−	34.0928i		−	1.46985i
$539$	−14.0484	+	7.78631i	−0.605109	+	0.335380i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	31.3775	−	18.1158i	1.34778	−	0.778141i
$543$		0			0
$544$	−0.692959	−	0.400080i	−0.0297104	−	0.0171533i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	32.2350	−	18.6109i	1.37576	−	0.794293i
$550$	8.11249	+	14.0513i	0.345918	+	0.599147i
$551$	−76.9526	−	44.4286i	−3.27829	−	1.89272i
$552$		0			0
$553$		0			0
$554$	−38.8767			−1.65171
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$	−18.7350	−	10.8167i	−0.793116	−	0.457905i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.36396	+	22.9456i	−0.267261	+	0.963624i
$568$	−47.4700			−1.99180
$569$	−2.99000	+	5.17882i	−0.125347	+	0.217108i	−0.921869	−	0.387503i	$-0.873338\pi$
$569$	−2.99000	+	5.17882i	−0.125347	+	0.217108i	0.796521	+	0.604610i	$0.206671\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$	−14.3295	−	8.27315i	−0.599147	−	0.345918i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	12.0067	+	20.7962i	0.499412	+	0.865006i
$579$		0			0
$580$		0			0
$581$	−28.8675	+	28.3667i	−1.19763	+	1.17685i
$582$		0			0
$583$	15.4768	−	26.8066i	0.640984	−	1.11022i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			9.47339i			0.391009i	0.980703	+	0.195504i	$0.0626344\pi$
$587$			9.47339i			0.391009i	−0.980703	+	0.195504i	$0.937366\pi$
$588$		0			0
$589$	−44.2250			−1.82226
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$			48.9218i			1.99556i	0.0665912	+	0.997780i	$0.478788\pi$
$601$			48.9218i			1.99556i	−0.0665912	+	0.997780i	$0.521212\pi$
$602$		0			0
$603$	36.5822			1.48974
$604$	−7.60493	+	13.1721i	−0.309440	+	0.535967i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		−	49.0632i		−	1.98978i
$609$		0			0
$610$		0			0
$611$	15.2753	−	26.4575i	0.617971	−	1.07036i
$612$	0.734994	−	0.424349i	0.0297104	−	0.0171533i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	9.88751	+	5.70855i	0.399027	+	0.230378i
$615$		0			0
$616$	−4.58912	+	16.5463i	−0.184901	+	0.666670i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	4.41588	−	2.54951i	0.177489	−	0.102473i	−0.408623	−	0.912703i	$-0.633991\pi$
$619$	4.41588	−	2.54951i	0.177489	−	0.102473i	0.586112	+	0.810230i	$0.300658\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	−26.4953	+	15.2971i	−1.05896	+	0.611393i
$627$		0			0
$628$	42.2450	+	24.3902i	1.68576	+	0.973273i
$629$		0			0
$630$		0			0
$631$	−35.3553			−1.40747			−0.703737	−	0.710461i	$-0.748487\pi$
$631$	−35.3553			−1.40747			−0.703737	+	0.710461i	$0.748487\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	25.2350	+	0.441676i	0.999847	+	0.0174999i
$638$	33.2450			1.31618
$639$	25.1748	−	43.6040i	0.995898	−	1.72495i
$640$		0			0
$641$	−2.00000	−	3.46410i	−0.0789953	−	0.136824i	0.823821	−	0.566849i	$-0.191838\pi$
$641$	−2.00000	−	3.46410i	−0.0789953	−	0.136824i	−0.902817	+	0.430026i	$0.858505\pi$
$642$		0			0
$643$			25.2195i			0.994562i	0.867590	+	0.497281i	$0.165668\pi$
$643$			25.2195i			0.994562i	−0.867590	+	0.497281i	$0.834332\pi$
$644$		0			0
$645$		0			0
$646$	0.867497	−	1.50255i	0.0341312	−	0.0591170i
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	12.7279	+	22.0454i	0.500000	+	0.866025i
$649$	−17.6325	−	10.1801i	−0.692136	−	0.399605i
$650$		−	25.4951i		−	1.00000i
$651$		0			0
$652$	42.7446			1.67401
$653$	−4.00000	+	6.92820i	−0.156532	+	0.271122i	−0.933616	−	0.358276i	$-0.883365\pi$
$653$	−4.00000	+	6.92820i	−0.156532	+	0.271122i	0.777084	+	0.629397i	$0.216698\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−30.5505	−	8.47319i	−1.19098	−	0.330319i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	5.00000	+	8.66025i	0.194331	+	0.336590i
$663$		0			0
$664$			43.2666i			1.67907i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−15.7154	+	9.07331i	−0.608049	+	0.351057i
$669$		0			0
$670$		0			0
$671$			28.4692i			1.09904i
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$	17.1438	−	29.6939i	0.660355	−	1.14377i
$675$		0			0
$676$	13.0000	+	22.5167i	0.500000	+	0.866025i
$677$	−31.2550	−	18.0451i	−1.20123	−	0.693529i	−0.240399	−	0.970674i	$-0.577278\pi$
$677$	−31.2550	−	18.0451i	−1.20123	−	0.693529i	−0.960828	+	0.277145i	$0.910612\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	14.3295	−	8.27315i	0.548706	−	0.316795i
$683$	−19.0919	−	33.0681i	−0.730531	−	1.26532i	−0.956657	−	0.291218i	$-0.905939\pi$
$683$	−19.0919	−	33.0681i	−0.730531	−	1.26532i	0.226126	−	0.974098i	$-0.427394\pi$
$684$	45.0674	+	26.0197i	1.72320	+	0.994888i
$685$		0			0
$686$	−6.11249	−	25.4684i	−0.233376	−	0.972387i
$687$		0			0
$688$		0			0
$689$	−42.1225	+	24.3194i	−1.60474	+	0.926497i
$690$		0			0
$691$	37.0365	+	21.3830i	1.40893	+	0.813449i	0.995286	−	0.0969879i	$-0.0309208\pi$
$691$	37.0365	+	21.3830i	1.40893	+	0.813449i	0.413649	+	0.910437i	$0.364254\pi$
$692$			23.6829i			0.900289i
$693$	−12.7650	−	12.9904i	−0.484902	−	0.493464i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−25.6149	+	6.62382i	−0.968154	+	0.250357i
$701$	16.0000			0.604312			0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000			0.604312			0.302156	+	0.953259i	$0.402294\pi$
$702$		0			0
$703$		0			0
$704$	9.17824	+	15.8972i	0.345918	+	0.599147i
$705$		0			0
$706$		0			0
$707$	36.7696	+	10.1980i	1.38286	+	0.383537i
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−9.00000	−	15.5885i	−0.335877	−	0.581756i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	79.5141			2.95921
$723$		0			0
$724$	42.7350	−	24.6731i	1.58823	−	0.916967i
$725$	25.6125	+	44.3621i	0.951224	+	1.64757i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	19.2450	−	18.9111i	0.713267	−	0.700892i
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.9900	+	24.2314i	−0.515328	+	0.892574i
$738$		0			0
$739$	20.5061	+	35.5176i	0.754329	+	1.30654i	0.945707	+	0.325020i	$0.105371\pi$
$739$	20.5061	+	35.5176i	0.754329	+	1.30654i	−0.191378	+	0.981516i	$0.561296\pi$
$740$		0			0
$741$		0			0
$742$	35.3775	+	36.0021i	1.29875	+	1.32168i
$743$	−12.9153			−0.473817			−0.236908	−	0.971532i	$-0.576134\pi$
$743$	−12.9153			−0.473817			−0.236908	+	0.971532i	$0.576134\pi$
$744$		0			0
$745$		0			0
$746$	−1.93393	−	3.34967i	−0.0708063	−	0.122640i
$747$	−39.7429	−	22.9456i	−1.45412	−	0.839535i
$748$			0.649129i			0.0237345i
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−29.3520	+	16.9464i	−1.07036	+	0.617971i
$753$		0			0
$754$	−45.2407	−	26.1197i	−1.64757	−	0.951224i
$755$		0			0
$756$		0			0
$757$	−54.4700			−1.97975			−0.989873	−	0.141958i	$-0.954660\pi$
$757$	−54.4700			−1.97975			−0.989873	+	0.141958i	$0.954660\pi$
$758$	−11.0000	+	19.0526i	−0.399538	+	0.692020i
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	31.2250	−	18.0278i	1.12820	−	0.651369i
$767$	15.9965	+	27.7068i	0.577601	+	1.00043i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	22.0794	+	12.7475i	0.793116	+	0.457905i
$776$		0			0
$777$		0			0
$778$	32.8451			1.17755
$779$		0			0
$780$		0			0
$781$	19.2550	+	33.3506i	0.688998	+	1.19338i
$782$		0			0
$783$		0			0
$784$	−24.0000	−	14.4222i	−0.857143	−	0.515079i
$785$		0			0
$786$		0			0
$787$	23.2267	−	13.4099i	0.827942	−	0.478012i	−0.0252056	−	0.999682i	$-0.508024\pi$
$787$	23.2267	−	13.4099i	0.827942	−	0.478012i	0.853147	+	0.521670i	$0.174691\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.96323	+	7.08617i	0.247584	+	0.251955i
$792$	−19.4700			−0.691836
$793$	22.3675	−	38.7416i	0.794293	−	1.37576i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	7.21110i		−	0.255430i	−0.991811	−	0.127715i	$-0.959236\pi$
$797$		−	7.21110i		−	0.255430i	0.991811	−	0.127715i	$-0.0407643\pi$
$798$		0			0
$799$	−1.19853			−0.0424010
$800$	−14.1421	+	24.4949i	−0.500000	+	0.866025i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−26.0000			−0.915811
$807$		0			0
$808$	35.3270	−	20.3961i	1.24280	−	0.717532i
$809$	4.61249	+	7.98908i	0.162167	+	0.280881i	0.935645	−	0.352941i	$-0.114818\pi$
$809$	4.61249	+	7.98908i	0.162167	+	0.280881i	−0.773479	+	0.633822i	$0.781485\pi$
$810$		0			0
$811$			56.0892i			1.96956i	0.173809	+	0.984779i	$0.444392\pi$
$811$			56.0892i			1.96956i	−0.173809	+	0.984779i	$0.555608\pi$
$812$	−14.4886	+	52.2394i	−0.508451	+	1.83325i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	7.16476	+	27.7068i	0.250357	+	0.968154i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$	23.6810	−	23.2702i	0.823967	−	0.809672i
$827$	−54.9386			−1.91040			−0.955202	−	0.295955i	$-0.904362\pi$
$827$	−54.9386			−1.91040			−0.955202	+	0.295955i	$0.904362\pi$
$828$		0			0
$829$	−9.76501	+	5.63783i	−0.339153	+	0.195810i	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−9.76501	+	5.63783i	−0.339153	+	0.195810i	0.320745	+	0.947166i	$0.396067\pi$
$830$		0			0
$831$		0			0
$832$		−	28.8444i		−	1.00000i
$833$	−0.479992	−	0.866025i	−0.0166307	−	0.0300060i
$834$		0			0
$835$		0			0
$836$	−34.4700	+	19.9013i	−1.19217	+	0.688299i
$837$		0			0
$838$		0			0
$839$		−	7.67303i		−	0.264903i	−0.991190	−	0.132451i	$-0.957715\pi$
$839$		−	7.67303i		−	0.264903i	0.991190	−	0.132451i	$-0.0422848\pi$
$840$		0			0
$841$	75.9600			2.61931
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	35.9487i		−	1.23594i
$847$	−14.6902	+	3.79876i	−0.504760	+	0.130527i
$848$	53.9600			1.85299
$849$		0			0
$850$	−0.866199	+	0.500100i	−0.0297104	+	0.0171533i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$	−44.7350	−	12.4073i	−1.53080	−	0.424568i
$855$		0			0
$856$		0			0
$857$	−32.4800	+	18.7523i	−1.10950	+	0.640567i	−0.938699	−	0.344739i	$-0.887967\pi$
$857$	−32.4800	+	18.7523i	−1.10950	+	0.640567i	−0.170797	+	0.985306i	$0.554634\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	30.0000			1.02180
$863$	−23.3345	+	40.4166i	−0.794316	+	1.37580i	0.128956	+	0.991650i	$0.458837\pi$
$863$	−23.3345	+	40.4166i	−0.794316	+	1.37580i	−0.923272	+	0.384146i	$0.874496\pi$
$864$		0			0
$865$		0			0
$866$	14.1563	+	8.17313i	0.481050	+	0.277734i
$867$		0			0
$868$	6.75500	+	26.1222i	0.229280	+	0.886646i
$869$		0			0
$870$		0			0
$871$	38.0759	−	21.9831i	1.29015	−	0.744870i
$872$		0			0
$873$		0			0
$874$		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
$881$		−	57.6888i		−	1.94359i	−0.235836	−	0.971793i	$-0.575783\pi$
$881$		−	57.6888i		−	1.94359i	0.235836	−	0.971793i	$-0.424217\pi$
$882$	25.9756	−	14.3969i	0.874643	−	0.484768i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	0.510004	−	0.883353i	0.0171533	−	0.0297104i
$885$		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
$891$	10.3255	−	17.8843i	0.345918	−	0.599147i
$892$	−45.4139	+	26.2197i	−1.52057	+	0.877902i
$893$	−36.7450	−	63.6442i	−1.22962	−	2.12977i
$894$		0			0
$895$		0			0
$896$	−28.9800	+	7.49400i	−0.968154	+	0.250357i
$897$		0			0
$898$		0			0
$899$	45.2407	−	26.1197i	1.50886	−	0.871142i
$900$	−15.0000	−	25.9808i	−0.500000	−	0.866025i
$901$	1.65251	+	0.954078i	0.0550531	+	0.0317849i
$902$		0			0
$903$		0			0
$904$	10.6207			0.353241
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	−8.83176	−	5.09902i	−0.293092	−	0.169217i
$909$			43.2666i			1.43506i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	30.3975	−	17.5500i	1.00601	−	0.580821i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	60.5126i		−	1.99180i
$924$		0			0
$925$		0			0
$926$	17.0000	−	29.4449i	0.558655	−	0.967618i
$927$		0			0
$928$	28.9772	+	50.1900i	0.951224	+	1.64757i
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	31.2718	−	52.0394i	1.02489	−	1.70552i
$932$	39.9600			1.30893
$933$		0			0
$934$		0			0
$935$		0			0
$936$	26.4953	+	15.2971i	0.866025	+	0.500000i
$937$		−	61.1876i		−	1.99891i	−0.0329954	−	0.999456i	$-0.510505\pi$
$937$		−	61.1876i		−	1.99891i	0.0329954	−	0.999456i	$-0.489495\pi$
$938$	−31.9789	−	32.5435i	−1.04415	−	1.06258i
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		−	35.4931i		−	1.15520i
$945$		0			0
$946$		0			0
$947$	30.1245	−	52.1772i	0.978915	−	1.69553i	0.312556	−	0.949899i	$-0.398815\pi$
$947$	30.1245	−	52.1772i	0.978915	−	1.69553i	0.666359	−	0.745631i	$-0.267852\pi$
$948$		0			0
$949$		0			0
$950$	−53.1125	−	30.6645i	−1.72320	−	0.994888i
$951$		0			0
$952$	−1.02001	−	0.282899i	−0.0330586	−	0.00916882i
$953$	60.9600			1.97469			0.987344	−	0.158595i	$-0.0506963\pi$
$953$	60.9600			1.97469			0.987344	+	0.158595i	$0.0506963\pi$
$954$	−28.6166	+	49.5654i	−0.926497	+	1.60474i
$955$		0			0
$956$	−26.6827	−	46.2157i	−0.862979	−	1.49472i
$957$		0			0
$958$			61.0461i			1.97231i
$959$		0			0
$960$		0			0
$961$	−2.50000	+	4.33013i	−0.0806452	+	0.139682i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−35.1397			−1.13002			−0.565008	−	0.825086i	$-0.691127\pi$
$967$	−35.1397			−1.13002			−0.565008	+	0.825086i	$0.691127\pi$
$968$	−8.11051	+	14.0478i	−0.260682	+	0.451514i
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$		0			0
$974$	−24.7550			−0.793201
$975$		0			0
$976$	−42.9800	+	24.8145i	−1.37576	+	0.794293i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−52.2322	+	30.1563i	−1.66595	+	0.961836i	−0.696161	+	0.717886i	$0.745110\pi$
$983$	−52.2322	+	30.1563i	−1.66595	+	0.961836i	−0.969788	+	0.243950i	$0.921557\pi$
$984$		0			0
$985$		0			0
$986$			2.04941i			0.0652665i
$987$		0			0
$988$	62.5436			1.98978
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$	24.9800	+	14.4222i	0.793116	+	0.457905i
$993$		0			0
$994$	−60.7970	+	15.7216i	−1.92836	+	0.498660i
$995$		0			0
$996$		0			0
$997$	53.2350	−	30.7352i	1.68597	−	0.973395i	0.728417	−	0.685134i	$-0.240256\pi$
$997$	53.2350	−	30.7352i	1.68597	−	0.973395i	0.957552	−	0.288261i	$-0.0930771\pi$
$998$	19.0000	+	32.9090i	0.601434	+	1.04172i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	364.2.x.a.103.3	yes	8
4.3	odd	2	inner	364.2.x.a.103.2	✓	8
7.3	odd	6	inner	364.2.x.a.311.3	yes	8
13.12	even	2	inner	364.2.x.a.103.2	✓	8
28.3	even	6	inner	364.2.x.a.311.2	yes	8
52.51	odd	2	CM	364.2.x.a.103.3	yes	8
91.38	odd	6	inner	364.2.x.a.311.2	yes	8
364.311	even	6	inner	364.2.x.a.311.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.x.a.103.2	✓	8	4.3	odd	2	inner
364.2.x.a.103.2	✓	8	13.12	even	2	inner
364.2.x.a.103.3	yes	8	1.1	even	1	trivial
364.2.x.a.103.3	yes	8	52.51	odd	2	CM
364.2.x.a.311.2	yes	8	28.3	even	6	inner
364.2.x.a.311.2	yes	8	91.38	odd	6	inner
364.2.x.a.311.3	yes	8	7.3	odd	6	inner
364.2.x.a.311.3	yes	8	364.311	even	6	inner

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:	\( N \)	\(=\)	\( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	364.x (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-1.85439 - 1.88713i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.14728 + 1.98715i) q^{11} -3.60555i q^{13} +(-3.62250 + 0.936750i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-0.122499 + 0.0707248i) q^{17} +(-2.12132 - 3.67423i) q^{18} +(-7.51124 - 4.33662i) q^{19} +3.24500 q^{22} +(2.50000 + 4.33013i) q^{25} +(-4.41588 - 2.54951i) q^{26} +(-1.41421 + 5.09902i) q^{28} +10.2450 q^{29} +(4.41588 - 2.54951i) q^{31} +(2.82843 + 4.89898i) q^{32} +0.200040i q^{34} -6.00000 q^{36} +(-10.6225 + 6.13290i) q^{38} +(2.29456 - 3.97429i) q^{44} +(7.33800 + 4.23660i) q^{47} +(-0.122499 + 6.99893i) q^{49} +7.07107 q^{50} +(-6.24500 + 3.60555i) q^{52} +(-6.74500 - 11.6827i) q^{53} +(5.24500 + 5.33760i) q^{56} +(7.24431 - 12.5475i) q^{58} +(-7.68448 + 4.43664i) q^{59} +(10.7450 + 6.20363i) q^{61} -7.21110i q^{62} +(-7.68448 + 1.98715i) q^{63} +8.00000 q^{64} +(6.09703 + 10.5604i) q^{67} +(0.244998 + 0.141450i) q^{68} +16.7832 q^{71} +(-4.24264 + 7.34847i) q^{72} +17.3465i q^{76} +(1.62250 - 5.85000i) q^{77} +(-4.50000 - 7.79423i) q^{81} -15.2971i q^{83} +(-3.24500 - 5.62050i) q^{88} +(-6.80413 + 6.68609i) q^{91} +(10.3775 - 5.99145i) q^{94} +(8.48528 + 5.09902i) q^{98} +6.88368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 12 q^{9} - 4 q^{14} - 16 q^{16} + 24 q^{17} - 24 q^{22} + 20 q^{25} + 32 q^{29} - 48 q^{36} - 60 q^{38} + 24 q^{49} - 4 q^{53} - 8 q^{56} + 36 q^{61} + 64 q^{64} - 48 q^{68} - 12 q^{77}+ \cdots + 108 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 + 12 * q^9 - 4 * q^14 - 16 * q^16 + 24 * q^17 - 24 * q^22 + 20 * q^25 + 32 * q^29 - 48 * q^36 - 60 * q^38 + 24 * q^49 - 4 * q^53 - 8 * q^56 + 36 * q^61 + 64 * q^64 - 48 * q^68 - 12 * q^77 - 36 * q^81 + 24 * q^88 + 108 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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