LMFDB - Embedded newform 364.2.x.a.311.1

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			311.1
Root			$-2.56149 + 0.662382i$ of defining polynomial
Character	$\chi$	$=$	364.311
Dual form			364.2.x.a.103.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.56149 + 0.662382i) q^{7} +2.82843 q^{8} +(1.50000 + 2.59808i) q^{9} +(3.26860 - 5.66138i) q^{11} -3.60555i q^{13} +(2.62250 + 2.66880i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(6.12250 + 3.53483i) q^{17} +(2.12132 - 3.67423i) q^{18} +(3.09536 - 1.78711i) q^{19} -9.24500 q^{22} +(2.50000 - 4.33013i) q^{25} +(-4.41588 + 2.54951i) q^{26} +(1.41421 - 5.09902i) q^{28} -2.24500 q^{29} +(4.41588 + 2.54951i) q^{31} +(-2.82843 + 4.89898i) q^{32} -9.99800i q^{34} -6.00000 q^{36} +(-4.37750 - 2.52735i) q^{38} +(6.53720 + 11.3228i) q^{44} +(-11.7539 + 6.78611i) q^{47} +(6.12250 - 3.39338i) q^{49} -7.07107 q^{50} +(6.24500 + 3.60555i) q^{52} +(5.74500 - 9.95063i) q^{53} +(-7.24500 + 1.87350i) q^{56} +(1.58745 + 2.74955i) q^{58} +(-5.56316 - 3.21189i) q^{59} +(-1.74500 + 1.00748i) q^{61} -7.21110i q^{62} +(-5.56316 - 5.66138i) q^{63} +8.00000 q^{64} +(-1.68115 + 2.91183i) q^{67} +(-12.2450 + 7.06965i) q^{68} +9.71211 q^{71} +(4.24264 + 7.34847i) q^{72} +7.14843i q^{76} +(-4.62250 + 16.6667i) q^{77} +(-4.50000 + 7.79423i) q^{81} +15.2971i q^{83} +(9.24500 - 16.0128i) q^{88} +(2.38825 + 9.23560i) q^{91} +(16.6225 + 9.59700i) q^{94} +(-8.48528 - 5.09902i) q^{98} +19.6116 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/364\mathbb{Z}\right)^\times$.

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

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	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−2.56149	+	0.662382i	−0.968154	+	0.250357i
$7$	−2.56149	+	0.662382i	−0.968154	+	0.250357i
$8$	2.82843			1.00000
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	3.26860	−	5.66138i	0.985520	−	1.70697i	0.345918	−	0.938265i	$-0.387568\pi$
$11$	3.26860	−	5.66138i	0.985520	−	1.70697i	0.639602	−	0.768706i	$-0.279099\pi$
$12$		0			0
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$	2.62250	+	2.66880i	0.700892	+	0.713267i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	6.12250	+	3.53483i	1.48492	+	0.857321i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	6.12250	+	3.53483i	1.48492	+	0.857321i	0.485071	+	0.874475i	$0.338794\pi$
$18$	2.12132	−	3.67423i	0.500000	−	0.866025i
$19$	3.09536	−	1.78711i	0.710124	−	0.409991i	−0.100983	−	0.994888i	$-0.532199\pi$
$19$	3.09536	−	1.78711i	0.710124	−	0.409991i	0.811107	+	0.584898i	$0.198865\pi$
$20$		0			0
$21$		0			0
$22$	−9.24500			−1.97104
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		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
$29$	−2.24500			−0.416886			−0.208443	−	0.978035i	$-0.566840\pi$
$29$	−2.24500			−0.416886			−0.208443	+	0.978035i	$0.566840\pi$
$30$		0			0
$31$	4.41588	+	2.54951i	0.793116	+	0.457905i	0.841058	−	0.540945i	$-0.181933\pi$
$31$	4.41588	+	2.54951i	0.793116	+	0.457905i	−0.0479427	+	0.998850i	$0.515266\pi$
$32$	−2.82843	+	4.89898i	−0.500000	+	0.866025i
$33$		0			0
$34$		−	9.99800i		−	1.71464i
$35$		0			0
$36$	−6.00000			−1.00000
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	−4.37750	−	2.52735i	−0.710124	−	0.409991i
$39$		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			$\pi$
$44$	6.53720	+	11.3228i	0.985520	+	1.70697i
$45$		0			0
$46$		0			0
$47$	−11.7539	+	6.78611i	−1.71448	+	0.989855i	−0.786201	+	0.617971i	$0.787955\pi$
$47$	−11.7539	+	6.78611i	−1.71448	+	0.989855i	−0.928279	+	0.371884i	$0.878712\pi$
$48$		0			0
$49$	6.12250	−	3.39338i	0.874643	−	0.484768i
$50$	−7.07107			−1.00000
$51$		0			0
$52$	6.24500	+	3.60555i	0.866025	+	0.500000i
$53$	5.74500	−	9.95063i	0.789136	−	1.36682i	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	5.74500	−	9.95063i	0.789136	−	1.36682i	0.926497	−	0.376303i	$-0.122805\pi$
$54$		0			0
$55$		0			0
$56$	−7.24500	+	1.87350i	−0.968154	+	0.250357i
$57$		0			0
$58$	1.58745	+	2.74955i	0.208443	+	0.361034i
$59$	−5.56316	−	3.21189i	−0.724262	−	0.418153i	0.0920575	−	0.995754i	$-0.470656\pi$
$59$	−5.56316	−	3.21189i	−0.724262	−	0.418153i	−0.816319	+	0.577601i	$0.803989\pi$
$60$		0			0
$61$	−1.74500	+	1.00748i	−0.223424	+	0.128994i	−0.607535	−	0.794293i	$-0.707841\pi$
$61$	−1.74500	+	1.00748i	−0.223424	+	0.128994i	0.384111	+	0.923287i	$0.374508\pi$
$62$		−	7.21110i		−	0.915811i
$63$	−5.56316	−	5.66138i	−0.700892	−	0.713267i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−1.68115	+	2.91183i	−0.205385	+	0.355737i	−0.950255	−	0.311472i	$-0.899178\pi$
$67$	−1.68115	+	2.91183i	−0.205385	+	0.355737i	0.744870	+	0.667209i	$0.232511\pi$
$68$	−12.2450	+	7.06965i	−1.48492	+	0.857321i
$69$		0			0
$70$		0			0
$71$	9.71211			1.15262			0.576308	−	0.817233i	$-0.304493\pi$
$71$	9.71211			1.15262			0.576308	+	0.817233i	$0.304493\pi$
$72$	4.24264	+	7.34847i	0.500000	+	0.866025i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		0			0
$76$			7.14843i			0.819981i
$77$	−4.62250	+	16.6667i	−0.526783	+	1.89934i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$			15.2971i			1.67907i	0.543305	+	0.839535i	$0.317173\pi$
$83$			15.2971i			1.67907i	−0.543305	+	0.839535i	$0.682827\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	9.24500	−	16.0128i	0.985520	−	1.70697i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	2.38825	+	9.23560i	0.250357	+	0.968154i
$92$		0			0
$93$		0			0
$94$	16.6225	+	9.59700i	1.71448	+	0.989855i
$95$		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
$99$	19.6116			1.97104
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$	12.4900	+	7.21110i	1.24280	+	0.717532i	0.969664	−	0.244443i	$-0.0786053\pi$
$101$	12.4900	+	7.21110i	1.24280	+	0.717532i	0.273138	+	0.961975i	$0.411939\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		−	10.1980i		−	1.00000i
$105$		0			0
$106$	−16.2493			−1.57827
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	7.41755	+	7.54851i	0.700892	+	0.713267i
$113$	−16.2450			−1.52820			−0.764100	−	0.645097i	$-0.776817\pi$
$113$	−16.2450			−1.52820			−0.764100	+	0.645097i	$0.776817\pi$
$114$		0			0
$115$		0			0
$116$	2.24500	−	3.88845i	0.208443	−	0.361034i
$117$	9.36750	−	5.40833i	0.866025	−	0.500000i
$118$			9.08460i			0.836305i
$119$	−18.0241	−	4.99900i	−1.65227	−	0.458258i
$120$		0			0
$121$	−15.8675	−	27.4833i	−1.44250	−	2.49848i
$122$	2.46780	+	1.42478i	0.223424	+	0.128994i
$123$		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	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−6.74500	+	6.62798i	−0.584866	+	0.574718i
$134$	4.75500			0.410770
$135$		0			0
$136$	17.3170	+	9.99800i	1.48492	+	0.857321i
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−6.86750	−	11.8949i	−0.576308	−	0.998194i
$143$	−20.4124	−	11.7851i	−1.70697	−	0.985520i
$144$	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	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	8.21835	−	14.2346i	0.668800	−	1.15839i	−0.309440	−	0.950919i	$-0.600142\pi$
$151$	8.21835	−	14.2346i	0.668800	−	1.15839i	0.978240	−	0.207476i	$-0.0665249\pi$
$152$	8.75500	−	5.05470i	0.710124	−	0.409991i
$153$			21.2090i			1.71464i
$154$	23.6810	−	6.12372i	1.90827	−	0.493464i
$155$		0			0
$156$		0			0
$157$	−14.8775	−	8.58953i	−1.18735	−	0.685519i	−0.229650	−	0.973273i	$-0.573758\pi$
$157$	−14.8775	−	8.58953i	−1.18735	−	0.685519i	−0.957704	+	0.287754i	$0.907091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	12.7279			1.00000
$163$	−11.3933	−	19.7337i	−0.892389	−	1.54566i	−0.837004	−	0.547197i	$-0.815695\pi$
$163$	−11.3933	−	19.7337i	−0.892389	−	1.54566i	−0.0553849	−	0.998465i	$-0.517639\pi$
$164$		0			0
$165$		0			0
$166$	18.7350	−	10.8167i	1.45412	−	0.839535i
$167$			16.4218i			1.27076i	0.772201	+	0.635378i	$0.219156\pi$
$167$			16.4218i			1.27076i	−0.772201	+	0.635378i	$0.780844\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	9.28608	+	5.36132i	0.710124	+	0.409991i
$172$		0			0
$173$	−22.7450	+	13.1318i	−1.72927	+	0.998395i	−0.836315	+	0.548250i	$0.815294\pi$
$173$	−22.7450	+	13.1318i	−1.72927	+	0.998395i	−0.892956	+	0.450145i	$0.851372\pi$
$174$		0			0
$175$	−3.53553	+	12.7475i	−0.267261	+	0.963624i
$176$	−26.1488			−1.97104
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	3.03975i		−	0.225943i	−0.993598	−	0.112972i	$-0.963963\pi$
$181$		−	3.03975i		−	0.225943i	0.993598	−	0.112972i	$-0.0360369\pi$
$182$	9.62250	−	9.45555i	0.713267	−	0.700892i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	40.0240	−	23.1079i	2.92685	−	1.68981i
$188$		−	27.1444i		−	1.97971i
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−0.244998	+	13.9979i	−0.0174999	+	0.999847i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−13.8675	−	24.0192i	−0.985520	−	1.70697i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	7.07107	−	12.2474i	0.500000	−	0.866025i
$201$		0			0
$202$		−	20.3961i		−	1.43506i
$203$	5.75055	−	1.48705i	0.403609	−	0.104370i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−12.4900	+	7.21110i	−0.866025	+	0.500000i
$209$		−	23.3654i		−	1.61622i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	11.4900	+	19.9013i	0.789136	+	1.36682i
$213$		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
$221$	12.7450	−	22.0750i	0.857321	−	1.48492i
$222$		0			0
$223$		−	0.724645i		−	0.0485258i	−0.999706	−	0.0242629i	$-0.992276\pi$
$223$		−	0.724645i		−	0.0485258i	0.999706	−	0.0242629i	$-0.00772388\pi$
$224$	4.00000	−	14.4222i	0.267261	−	0.963624i
$225$	15.0000			1.00000
$226$	11.4869	+	19.8960i	0.764100	+	1.32346i
$227$	4.41588	+	2.54951i	0.293092	+	0.169217i	0.639336	−	0.768928i	$-0.279209\pi$
$227$	4.41588	+	2.54951i	0.293092	+	0.169217i	−0.346243	+	0.938145i	$0.612543\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$	−6.34981			−0.416886
$233$	14.9900	+	25.9634i	0.982027	+	1.70092i	0.654466	+	0.756091i	$0.272893\pi$
$233$	14.9900	+	25.9634i	0.982027	+	1.70092i	0.327561	+	0.944830i	$0.393773\pi$
$234$	−13.2476	−	7.64853i	−0.866025	−	0.500000i
$235$		0			0
$236$	11.1263	−	6.42378i	0.724262	−	0.418153i
$237$		0			0
$238$	6.62250	+	25.6098i	0.429273	+	1.66004i
$239$	−0.187388			−0.0121211			−0.00606055	−	0.999982i	$-0.501929\pi$
$239$	−0.187388			−0.0121211			−0.00606055	+	0.999982i	$0.501929\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	−22.4400	+	38.8673i	−1.44250	+	2.49848i
$243$		0			0
$244$		−	4.02990i		−	0.257988i
$245$		0			0
$246$		0			0
$247$	−6.44351	−	11.1605i	−0.409991	−	0.710124i
$248$	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	$0.5\pi$
$252$	15.3690	−	3.97429i	0.968154	−	0.250357i
$253$		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	−0.560781	+	0.827964i	$0.689499\pi$
$257$	−24.9800	+	14.4222i	−1.55821	+	0.899632i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.36750	−	5.83268i	−0.208443	−	0.361034i
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$	12.8870	+	3.57421i	0.790154	+	0.219149i
$267$		0			0
$268$	−3.36229	−	5.82366i	−0.205385	−	0.355737i
$269$	27.1225	+	15.6592i	1.65369	+	0.954757i	0.975537	+	0.219834i	$0.0705517\pi$
$269$	27.1225	+	15.6592i	1.65369	+	0.954757i	0.678151	+	0.734923i	$0.262782\pi$
$270$		0			0
$271$	−26.6031	+	15.3593i	−1.61602	+	0.933012i	−0.628090	+	0.778141i	$0.716163\pi$
$271$	−26.6031	+	15.3593i	−1.61602	+	0.933012i	−0.987935	+	0.154872i	$0.950504\pi$
$272$		−	28.2786i		−	1.71464i
$273$		0			0
$274$		0			0
$275$	−16.3430	−	28.3069i	−0.985520	−	1.70697i
$276$		0			0
$277$	−1.25500	+	2.17373i	−0.0754058	+	0.130607i	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−1.25500	+	2.17373i	−0.0754058	+	0.130607i	0.825857	+	0.563880i	$0.190692\pi$
$278$		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	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	−9.71211	+	16.8219i	−0.576308	+	0.998194i
$285$		0			0
$286$			33.3333i			1.97104i
$287$		0			0
$288$	−16.9706			−1.00000
$289$	16.4900	+	28.5615i	0.970000	+	1.68009i
$290$		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
$302$	−23.2450			−1.33760
$303$		0			0
$304$	−12.3814	−	7.14843i	−0.710124	−	0.409991i
$305$		0			0
$306$	25.9756	−	14.9970i	1.48492	−	0.857321i
$307$			33.5682i			1.91584i	0.287037	+	0.957920i	$0.407330\pi$
$307$			33.5682i			1.91584i	−0.287037	+	0.957920i	$0.592670\pi$
$308$	−24.2450	−	24.6731i	−1.38149	−	1.40588i
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	18.7350	−	10.8167i	1.05896	−	0.611393i	0.133819	−	0.991006i	$-0.457276\pi$
$313$	18.7350	−	10.8167i	1.05896	−	0.611393i	0.925146	+	0.379612i	$0.123943\pi$
$314$			24.2949i			1.37104i
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	−7.33800	+	12.7098i	−0.410849	+	0.711612i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	25.2685			1.40597
$324$	−9.00000	−	15.5885i	−0.500000	−	0.866025i
$325$	−15.6125	−	9.01388i	−0.866025	−	0.500000i
$326$	−16.1125	+	27.9077i	−0.892389	+	1.54566i
$327$		0			0
$328$		0			0
$329$	25.6125	−	25.1681i	1.41206	−	1.38756i
$330$		0			0
$331$	3.53553	+	6.12372i	0.194331	+	0.336590i	0.946681	−	0.322173i	$-0.104413\pi$
$331$	3.53553	+	6.12372i	0.194331	+	0.336590i	−0.752350	+	0.658763i	$0.771080\pi$
$332$	−26.4953	−	15.2971i	−1.45412	−	0.839535i
$333$		0			0
$334$	20.1125	−	11.6120i	1.10051	−	0.635378i
$335$		0			0
$336$		0			0
$337$	11.7550			0.640336			0.320168	−	0.947361i	$-0.396261\pi$
$337$	11.7550			0.640336			0.320168	+	0.947361i	$0.396261\pi$
$338$	9.19239	+	15.9217i	0.500000	+	0.866025i
$339$		0			0
$340$		0			0
$341$	28.8675	−	16.6667i	1.56326	−	0.902550i
$342$		−	15.1641i		−	0.819981i
$343$	−13.4350	+	12.7475i	−0.725423	+	0.688303i
$344$		0			0
$345$		0			0
$346$	32.1663	+	18.5712i	1.72927	+	0.998395i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	18.1125	−	4.68375i	0.968154	−	0.250357i
$351$		0			0
$352$	18.4900	+	32.0256i	0.985520	+	1.70697i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		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	$-0.275699\pi$
$359$	−6.36396	−	11.0227i	−0.335877	−	0.581756i	−0.983653	+	0.180075i	$0.942366\pi$
$360$		0			0
$361$	−3.11249	+	5.39100i	−0.163816	+	0.283737i
$362$	−3.72292	+	2.14943i	−0.195672	+	0.112972i
$363$		0			0
$364$	−18.3848	−	5.09902i	−0.963624	−	0.267261i
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.12465	+	29.2939i	−0.421811	+	1.52086i
$372$		0			0
$373$	−17.3675	−	30.0814i	−0.899255	−	1.55756i	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−17.3675	−	30.0814i	−0.899255	−	1.55756i	−0.0708063	−	0.997490i	$-0.522557\pi$
$374$	−56.6025	−	32.6795i	−2.92685	−	1.68981i
$375$		0			0
$376$	−33.2450	+	19.1940i	−1.71448	+	0.989855i
$377$			8.09446i			0.416886i
$378$		0			0
$379$	15.5563			0.799076			0.399538	−	0.916717i	$-0.369171\pi$
$379$	15.5563			0.799076			0.399538	+	0.916717i	$0.369171\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	22.0794	−	12.7475i	1.12820	−	0.651369i	0.184722	−	0.982791i	$-0.440862\pi$
$383$	22.0794	−	12.7475i	1.12820	−	0.651369i	0.943483	+	0.331422i	$0.107528\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−19.6125	+	33.9698i	−0.994393	+	1.72234i	−0.405616	+	0.914044i	$0.632943\pi$
$389$	−19.6125	+	33.9698i	−0.994393	+	1.72234i	−0.588777	+	0.808296i	$0.700390\pi$
$390$		0			0
$391$		0			0
$392$	17.3170	−	9.59792i	0.874643	−	0.484768i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−19.6116	+	33.9683i	−0.985520	+	1.70697i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	9.19239	−	15.9217i	0.457905	−	0.793116i
$404$	−24.9800	+	14.4222i	−1.24280	+	0.717532i
$405$		0			0
$406$	−5.88751	−	5.99145i	−0.292192	−	0.297351i
$407$		0			0
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.3775	+	4.54230i	0.805884	+	0.223512i
$414$		0			0
$415$		0			0
$416$	17.6635	+	10.1980i	0.866025	+	0.500000i
$417$		0			0
$418$	−28.6166	+	16.5218i	−1.39968	+	0.808108i
$419$		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
$423$	−35.2616	−	20.3583i	−1.71448	−	0.989855i
$424$	16.2493	−	28.1446i	0.789136	−	1.36682i
$425$	30.6125	−	17.6741i	1.48492	−	0.857321i
$426$		0			0
$427$	3.80247	−	3.73650i	0.184014	−	0.180822i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.6066	+	18.3712i	−0.510902	+	0.884908i	0.489018	+	0.872274i	$0.337355\pi$
$431$	−10.6066	+	18.3712i	−0.510902	+	0.884908i	−0.999920	+	0.0126347i	$0.995978\pi$
$432$		0			0
$433$		−	40.4030i		−	1.94164i	−0.239804	−	0.970821i	$-0.577083\pi$
$433$		−	40.4030i		−	1.94164i	0.239804	−	0.970821i	$-0.422917\pi$
$434$	4.77651	+	18.4712i	0.229280	+	0.886646i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	18.0000	+	10.8167i	0.857143	+	0.515079i
$442$	−36.0483			−1.71464
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	−0.887505	+	0.512401i	−0.0420246	+	0.0242629i
$447$		0			0
$448$	−20.4919	+	5.29906i	−0.968154	+	0.250357i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−10.6066	−	18.3712i	−0.500000	−	0.866025i
$451$		0			0
$452$	16.2450	−	28.1372i	0.764100	−	1.32346i
$453$		0			0
$454$		−	7.21110i		−	0.338434i
$455$		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
$461$		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	$-0.688682\pi$
$463$	−24.0416			−1.11731			−0.558655	+	0.829400i	$0.688682\pi$
$464$	4.49000	+	7.77690i	0.208443	+	0.361034i
$465$		0			0
$466$	21.1991	−	36.7178i	0.982027	−	1.70092i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$			21.6333i			1.00000i
$469$	2.37750	−	8.57220i	0.109783	−	0.395827i
$470$		0			0
$471$		0			0
$472$	−15.7350	−	9.08460i	−0.724262	−	0.418153i
$473$		0			0
$474$		0			0
$475$		−	17.8711i		−	0.819981i
$476$	26.6827	−	26.2197i	1.22300	−	1.20178i
$477$	34.4700			1.57827
$478$	0.132503	+	0.229502i	0.00606055	+	0.0104972i
$479$	24.1353	+	13.9345i	1.10277	+	0.636685i	0.936947	−	0.349470i	$-0.113638\pi$
$479$	24.1353	+	13.9345i	1.10277	+	0.636685i	0.165824	+	0.986155i	$0.446972\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	63.4700			2.88500
$485$		0			0
$486$		0			0
$487$	13.1681	−	22.8078i	0.596703	−	1.03352i	−0.396601	−	0.917991i	$-0.629810\pi$
$487$	13.1681	−	22.8078i	0.596703	−	1.03352i	0.993304	−	0.115529i	$-0.0368564\pi$
$488$	−4.93560	+	2.84957i	−0.223424	+	0.128994i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−13.7450	−	7.93568i	−0.619044	−	0.357405i
$494$	−9.11249	+	15.7833i	−0.409991	+	0.710124i
$495$		0			0
$496$		−	20.3961i		−	0.915811i
$497$	−24.8775	+	6.43313i	−1.11591	+	0.288565i
$498$		0			0
$499$	13.4350	+	23.2702i	0.601434	+	1.04172i	0.992604	+	0.121396i	$0.0387372\pi$
$499$	13.4350	+	23.2702i	0.601434	+	1.04172i	−0.391170	+	0.920319i	$0.627929\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−15.7350	−	16.0128i	−0.700892	−	0.713267i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		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
$517$			88.7243i			3.90209i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.2250	−	18.0278i	−1.36799	−	0.789810i	−0.377320	−	0.926083i	$-0.623154\pi$
$521$	−31.2250	−	18.0278i	−1.36799	−	0.789810i	−0.990671	+	0.136272i	$0.956488\pi$
$522$	−4.76236	+	8.24865i	−0.208443	+	0.361034i
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.0241	+	31.2187i	0.785144	+	1.35991i
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		−	19.2714i		−	0.836305i
$532$	−4.73499	−	18.3107i	−0.205288	−	0.793868i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−4.75500	+	8.23591i	−0.205385	+	0.355737i
$537$		0			0
$538$		−	44.2909i		−	1.90951i
$539$	0.800801	−	45.7534i	0.0344929	−	1.97074i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$	37.6225	+	21.7214i	1.61602	+	0.933012i
$543$		0			0
$544$	−34.6341	+	19.9960i	−1.48492	+	0.857321i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−5.23499	−	3.02243i	−0.223424	−	0.128994i
$550$	−23.1125	+	40.0320i	−0.985520	+	1.70697i
$551$	−6.94908	+	4.01205i	−0.296041	+	0.170919i
$552$		0			0
$553$		0			0
$554$	3.54968			0.150812
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$	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	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.36396	−	22.9456i	0.267261	−	0.963624i
$568$	27.4700			1.15262
$569$	21.9900	+	38.0878i	0.921869	+	1.59672i	0.796521	+	0.604610i	$0.206671\pi$
$569$	21.9900	+	38.0878i	0.921869	+	1.59672i	0.125347	+	0.992113i	$0.459996\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$	40.8248	−	23.5702i	1.70697	−	0.985520i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	23.3204	−	40.3921i	0.970000	−	1.68009i
$579$		0			0
$580$		0			0
$581$	−10.1325	−	39.1833i	−0.420367	−	1.62560i
$582$		0			0
$583$	−37.5562	−	65.0493i	−1.55542	−	2.69406i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	36.4178i		−	1.50312i	−0.659663	−	0.751561i	$-0.729301\pi$
$587$		−	36.4178i		−	1.50312i	0.659663	−	0.751561i	$-0.270699\pi$
$588$		0			0
$589$	18.2250			0.750948
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		−	27.2885i		−	1.11312i	−0.830808	−	0.556560i	$-0.812121\pi$
$601$		−	27.2885i		−	1.11312i	0.830808	−	0.556560i	$-0.187879\pi$
$602$		0			0
$603$	−10.0869			−0.410770
$604$	16.4367	+	28.4692i	0.668800	+	1.15839i
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$			20.2188i			0.819981i
$609$		0			0
$610$		0			0
$611$	24.4677	+	42.3792i	0.989855	+	1.71448i
$612$	−36.7350	−	21.2090i	−1.48492	−	0.857321i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	41.1125	−	23.7363i	1.65917	−	0.957920i
$615$		0			0
$616$	−13.0744	+	47.1404i	−0.526783	+	1.89934i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	4.41588	+	2.54951i	0.177489	+	0.102473i	0.586112	−	0.810230i	$-0.300658\pi$
$619$	4.41588	+	2.54951i	0.177489	+	0.102473i	−0.408623	+	0.912703i	$0.633991\pi$
$620$		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
$628$	29.7550	−	17.1791i	1.18735	−	0.685519i
$629$		0			0
$630$		0			0
$631$	35.3553			1.40747			0.703737	−	0.710461i	$-0.251513\pi$
$631$	35.3553			1.40747			0.703737	+	0.710461i	$0.251513\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−12.2350	−	22.0750i	−0.484768	−	0.874643i
$638$	20.7550			0.821698
$639$	14.5682	+	25.2328i	0.576308	+	0.998194i
$640$		0			0
$641$	−2.00000	+	3.46410i	−0.0789953	+	0.136824i	−0.902817	−	0.430026i	$-0.858505\pi$
$641$	−2.00000	+	3.46410i	−0.0789953	+	0.136824i	0.823821	+	0.566849i	$0.191838\pi$
$642$		0			0
$643$			50.7146i			1.99999i	0.00313699	+	0.999995i	$0.499001\pi$
$643$			50.7146i			1.99999i	−0.00313699	+	0.999995i	$0.500999\pi$
$644$		0			0
$645$		0			0
$646$	−17.8675	−	30.9474i	−0.702987	−	1.21761i
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	−12.7279	+	22.0454i	−0.500000	+	0.866025i
$649$	−36.3675	+	20.9968i	−1.42755	+	0.824196i
$650$			25.4951i			1.00000i
$651$		0			0
$652$	45.5730			1.78478
$653$	−4.00000	−	6.92820i	−0.156532	−	0.271122i	0.777084	−	0.629397i	$-0.216698\pi$
$653$	−4.00000	−	6.92820i	−0.156532	−	0.271122i	−0.933616	+	0.358276i	$0.883365\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−48.9353	−	13.5722i	−1.90770	−	0.529100i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	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
$668$	−28.4434	−	16.4218i	−1.10051	−	0.635378i
$669$		0			0
$670$		0			0
$671$			13.1721i			0.508505i
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$	−8.31204	−	14.3969i	−0.320168	−	0.554547i
$675$		0			0
$676$	13.0000	−	22.5167i	0.500000	−	0.866025i
$677$	−43.7450	+	25.2562i	−1.68126	+	0.970674i	−0.720429	+	0.693529i	$0.756055\pi$
$677$	−43.7450	+	25.2562i	−1.68126	+	0.970674i	−0.960828	+	0.277145i	$0.910612\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−40.8248	−	23.5702i	−1.56326	−	0.902550i
$683$	19.0919	−	33.0681i	0.730531	−	1.26532i	−0.226126	−	0.974098i	$-0.572606\pi$
$683$	19.0919	−	33.0681i	0.730531	−	1.26532i	0.956657	−	0.291218i	$-0.0940605\pi$
$684$	−18.5722	+	10.7226i	−0.710124	+	0.409991i
$685$		0			0
$686$	25.1125	+	7.44060i	0.958800	+	0.284084i
$687$		0			0
$688$		0			0
$689$	−35.8775	−	20.7139i	−1.36682	−	0.789136i
$690$		0			0
$691$	−41.4524	+	23.9325i	−1.57692	+	0.910437i	−0.581637	+	0.813449i	$0.697587\pi$
$691$	−41.4524	+	23.9325i	−1.57692	+	0.910437i	−0.995286	+	0.0969879i	$0.969079\pi$
$692$		−	52.5273i		−	1.99679i
$693$	−50.2350	+	12.9904i	−1.90827	+	0.493464i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−18.5439	−	18.8713i	−0.700892	−	0.713267i
$701$	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
$704$	26.1488	−	45.2911i	0.985520	−	1.70697i
$705$		0			0
$706$		0			0
$707$	−36.7696	−	10.1980i	−1.38286	−	0.383537i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−9.00000	+	15.5885i	−0.335877	+	0.581756i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	8.80347			0.327631
$723$		0			0
$724$	5.26501	+	3.03975i	0.195672	+	0.112972i
$725$	−5.61249	+	9.72113i	−0.208443	+	0.361034i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	6.75500	+	26.1222i	0.250357	+	0.968154i
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.9900	+	19.0352i	0.404822	+	0.701172i
$738$		0			0
$739$	−20.5061	+	35.5176i	−0.754329	+	1.30654i	0.191378	+	0.981516i	$0.438704\pi$
$739$	−20.5061	+	35.5176i	−0.754329	+	1.30654i	−0.945707	+	0.325020i	$0.894629\pi$
$740$		0			0
$741$		0			0
$742$	41.6225	−	10.7633i	1.52801	−	0.395132i
$743$	39.4106			1.44583			0.722917	−	0.690935i	$-0.242801\pi$
$743$	39.4106			1.44583			0.722917	+	0.690935i	$0.242801\pi$
$744$		0			0
$745$		0			0
$746$	−24.5613	+	42.5415i	−0.899255	+	1.55756i
$747$	−39.7429	+	22.9456i	−1.45412	+	0.839535i
$748$			92.4315i			3.37963i
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	47.0155	+	27.1444i	1.71448	+	0.989855i
$753$		0			0
$754$	9.91364	−	5.72364i	0.361034	−	0.208443i
$755$		0			0
$756$		0			0
$757$	20.4700			0.743994			0.371997	−	0.928234i	$-0.378673\pi$
$757$	20.4700			0.743994			0.371997	+	0.928234i	$0.378673\pi$
$758$	−11.0000	−	19.0526i	−0.399538	−	0.692020i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−31.2250	−	18.0278i	−1.12820	−	0.651369i
$767$	−11.5806	+	20.0583i	−0.418153	+	0.724262i
$768$		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	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	22.0794	−	12.7475i	0.793116	−	0.457905i
$776$		0			0
$777$		0			0
$778$	55.4725			1.98879
$779$		0			0
$780$		0			0
$781$	31.7450	−	54.9839i	1.13593	−	1.96748i
$782$		0			0
$783$		0			0
$784$	−24.0000	−	14.4222i	−0.857143	−	0.515079i
$785$		0			0
$786$		0			0
$787$	25.3480	+	14.6347i	0.903559	+	0.521670i	0.878353	−	0.478012i	$-0.158643\pi$
$787$	25.3480	+	14.6347i	0.903559	+	0.521670i	0.0252056	+	0.999682i	$0.491976\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	41.6115	−	10.7604i	1.47953	−	0.382596i
$792$	55.4700			1.97104
$793$	3.63250	+	6.29168i	0.128994	+	0.223424i
$794$		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
$799$	−95.9508			−3.39450
$800$	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
$809$	−26.6125	+	46.0942i	−0.935645	+	1.62059i	−0.162167	+	0.986763i	$0.551848\pi$
$809$	−26.6125	+	46.0942i	−0.935645	+	1.62059i	−0.773479	+	0.633822i	$0.781485\pi$
$810$		0			0
$811$		−	56.0892i		−	1.96956i	−0.173809	−	0.984779i	$-0.555608\pi$
$811$		−	56.0892i		−	1.96956i	0.173809	−	0.984779i	$-0.444392\pi$
$812$	−3.17491	+	11.4473i	−0.111417	+	0.401721i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−20.4124	+	20.0583i	−0.713267	+	0.700892i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$	−6.01748	−	23.2702i	−0.209375	−	0.809672i
$827$	−42.2107			−1.46781			−0.733905	−	0.679252i	$-0.762304\pi$
$827$	−42.2107			−1.46781			−0.733905	+	0.679252i	$0.762304\pi$
$828$		0			0
$829$	−47.2350	−	27.2711i	−1.64054	−	0.947166i	−0.980642	−	0.195810i	$-0.937266\pi$
$829$	−47.2350	−	27.2711i	−1.64054	−	0.947166i	−0.659897	−	0.751356i	$-0.729400\pi$
$830$		0			0
$831$		0			0
$832$		−	28.8444i		−	1.00000i
$833$	49.4800	+	0.866025i	1.71438	+	0.0300060i
$834$		0			0
$835$		0			0
$836$	40.4700	+	23.3654i	1.39968	+	0.808108i
$837$		0			0
$838$		0			0
$839$		−	53.5642i		−	1.84924i	−0.380889	−	0.924621i	$-0.624382\pi$
$839$		−	53.5642i		−	1.84924i	0.380889	−	0.924621i	$-0.375618\pi$
$840$		0			0
$841$	−23.9600			−0.826206
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$			57.5820i			1.97971i
$847$	58.8490	+	59.8880i	2.02207	+	2.05778i
$848$	−45.9600			−1.57827
$849$		0			0
$850$	−43.2926	−	24.9950i	−1.48492	−	0.857321i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$	−7.26501	−	2.01495i	−0.248603	−	0.0689502i
$855$		0			0
$856$		0			0
$857$	17.4800	+	10.0921i	0.597105	+	0.344739i	0.767902	−	0.640567i	$-0.221301\pi$
$857$	17.4800	+	10.0921i	0.597105	+	0.344739i	−0.170797	+	0.985306i	$0.554634\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	30.0000			1.02180
$863$	23.3345	+	40.4166i	0.794316	+	1.37580i	0.923272	+	0.384146i	$0.125504\pi$
$863$	23.3345	+	40.4166i	0.794316	+	1.37580i	−0.128956	+	0.991650i	$0.541163\pi$
$864$		0			0
$865$		0			0
$866$	−49.4833	+	28.5692i	−1.68151	+	0.970821i
$867$		0			0
$868$	19.2450	−	18.9111i	0.653218	−	0.641885i
$869$		0			0
$870$		0			0
$871$	10.4988	+	6.06146i	0.355737	+	0.205385i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		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	$-0.424217\pi$
$882$	0.519719	−	29.6939i	0.0174999	−	0.999847i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	25.4900	+	44.1500i	0.857321	+	1.48492i
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	29.4174	+	50.9524i	0.985520	+	1.70697i
$892$	1.25512	+	0.724645i	0.0420246	+	0.0242629i
$893$	−24.2550	+	42.0109i	−0.811663	+	1.40584i
$894$		0			0
$895$		0			0
$896$	20.9800	+	21.3504i	0.700892	+	0.713267i
$897$		0			0
$898$		0			0
$899$	−9.91364	−	5.72364i	−0.330638	−	0.190894i
$900$	−15.0000	+	25.9808i	−0.500000	+	0.866025i
$901$	70.3475	−	40.6151i	2.34361	−	1.35309i
$902$		0			0
$903$		0			0
$904$	−45.9478			−1.52820
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−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
$913$	86.6025	+	50.0000i	2.86612	+	1.65476i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	35.0175i		−	1.15262i
$924$		0			0
$925$		0			0
$926$	17.0000	+	29.4449i	0.558655	+	0.967618i
$927$		0			0
$928$	6.34981	−	10.9982i	0.208443	−	0.361034i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	12.8870	−	21.4453i	0.422355	−	0.702841i
$932$	−59.9600			−1.96405
$933$		0			0
$934$		0			0
$935$		0			0
$936$	26.4953	−	15.2971i	0.866025	−	0.500000i
$937$			32.3432i			1.05661i	0.849056	+	0.528303i	$0.177171\pi$
$937$			32.3432i			1.05661i	−0.849056	+	0.528303i	$0.822829\pi$
$938$	−12.1799	+	3.14963i	−0.397688	+	0.102839i
$939$		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
$943$		0			0
$944$			25.6951i			0.836305i
$945$		0			0
$946$		0			0
$947$	9.61841	+	16.6596i	0.312556	+	0.541364i	0.978915	−	0.204268i	$-0.0654813\pi$
$947$	9.61841	+	16.6596i	0.312556	+	0.541364i	−0.666359	+	0.745631i	$0.732148\pi$
$948$		0			0
$949$		0			0
$950$	−21.8875	+	12.6368i	−0.710124	+	0.409991i
$951$		0			0
$952$	−50.9800	−	14.1393i	−1.65227	−	0.458258i
$953$	−38.9600			−1.26204			−0.631019	−	0.775768i	$-0.717363\pi$
$953$	−38.9600			−1.26204			−0.631019	+	0.775768i	$0.717363\pi$
$954$	−24.3740	−	42.2169i	−0.789136	−	1.36682i
$955$		0			0
$956$	0.187388	−	0.324565i	0.00606055	−	0.0104972i
$957$		0			0
$958$		−	39.4128i		−	1.27337i
$959$		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
$967$	−62.0097			−1.99410			−0.997049	−	0.0767681i	$-0.975540\pi$
$967$	−62.0097			−1.99410			−0.997049	+	0.0767681i	$0.975540\pi$
$968$	−44.8801	−	77.7345i	−1.44250	−	2.49848i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$	−37.2450			−1.19341
$975$		0			0
$976$	6.97999	+	4.02990i	0.223424	+	0.128994i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	38.9846	+	22.5078i	1.24341	+	0.717886i	0.969788	−	0.243950i	$-0.0784433\pi$
$983$	38.9846	+	22.5078i	1.24341	+	0.717886i	0.273627	+	0.961836i	$0.411777\pi$
$984$		0			0
$985$		0			0
$986$			22.4455i			0.714810i
$987$		0			0
$988$	25.7740			0.819981
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	−24.9800	+	14.4222i	−0.793116	+	0.457905i
$993$		0			0
$994$	25.4700	+	25.9197i	0.807859	+	0.822122i
$995$		0			0
$996$		0			0
$997$	15.7650	+	9.10193i	0.499283	+	0.288261i	0.728417	−	0.685134i	$-0.240256\pi$
$997$	15.7650	+	9.10193i	0.499283	+	0.288261i	−0.229135	+	0.973395i	$0.573590\pi$
$998$	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.311.1	yes	8
4.3	odd	2	inner	364.2.x.a.311.4	yes	8
7.5	odd	6	inner	364.2.x.a.103.1	✓	8
13.12	even	2	inner	364.2.x.a.311.4	yes	8
28.19	even	6	inner	364.2.x.a.103.4	yes	8
52.51	odd	2	CM	364.2.x.a.311.1	yes	8
91.12	odd	6	inner	364.2.x.a.103.4	yes	8
364.103	even	6	inner	364.2.x.a.103.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.x.a.103.1	✓	8	7.5	odd	6	inner
364.2.x.a.103.1	✓	8	364.103	even	6	inner
364.2.x.a.103.4	yes	8	28.19	even	6	inner
364.2.x.a.103.4	yes	8	91.12	odd	6	inner
364.2.x.a.311.1	yes	8	1.1	even	1	trivial
364.2.x.a.311.1	yes	8	52.51	odd	2	CM
364.2.x.a.311.4	yes	8	4.3	odd	2	inner
364.2.x.a.311.4	yes	8	13.12	even	2	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)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.56149 + 0.662382i) q^{7} +2.82843 q^{8} +(1.50000 + 2.59808i) q^{9} +(3.26860 - 5.66138i) q^{11} -3.60555i q^{13} +(2.62250 + 2.66880i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(6.12250 + 3.53483i) q^{17} +(2.12132 - 3.67423i) q^{18} +(3.09536 - 1.78711i) q^{19} -9.24500 q^{22} +(2.50000 - 4.33013i) q^{25} +(-4.41588 + 2.54951i) q^{26} +(1.41421 - 5.09902i) q^{28} -2.24500 q^{29} +(4.41588 + 2.54951i) q^{31} +(-2.82843 + 4.89898i) q^{32} -9.99800i q^{34} -6.00000 q^{36} +(-4.37750 - 2.52735i) q^{38} +(6.53720 + 11.3228i) q^{44} +(-11.7539 + 6.78611i) q^{47} +(6.12250 - 3.39338i) q^{49} -7.07107 q^{50} +(6.24500 + 3.60555i) q^{52} +(5.74500 - 9.95063i) q^{53} +(-7.24500 + 1.87350i) q^{56} +(1.58745 + 2.74955i) q^{58} +(-5.56316 - 3.21189i) q^{59} +(-1.74500 + 1.00748i) q^{61} -7.21110i q^{62} +(-5.56316 - 5.66138i) q^{63} +8.00000 q^{64} +(-1.68115 + 2.91183i) q^{67} +(-12.2450 + 7.06965i) q^{68} +9.71211 q^{71} +(4.24264 + 7.34847i) q^{72} +7.14843i q^{76} +(-4.62250 + 16.6667i) q^{77} +(-4.50000 + 7.79423i) q^{81} +15.2971i q^{83} +(9.24500 - 16.0128i) q^{88} +(2.38825 + 9.23560i) q^{91} +(16.6225 + 9.59700i) q^{94} +(-8.48528 - 5.09902i) q^{98} +19.6116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 12 q^{9} - 4 q^{14} - 16 q^{16} + 24 q^{17} - 24 q^{22} + 20 q^{25} + 32 q^{29} - 48 q^{36} - 60 q^{38} + 24 q^{49} - 4 q^{53} - 8 q^{56} + 36 q^{61} + 64 q^{64} - 48 q^{68} - 12 q^{77}+ \cdots + 108 q^{94}+O(q^{100}) \) 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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