LMFDB - Embedded newform 52.2.l.a.7.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(7,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 11]))

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

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

chi := DirichletCharacter("52.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$52 = 2^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	52.l (of order $12$ , degree $4$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.415222090511$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			7.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	52.7
Dual form			52.2.l.a.15.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.36603 + 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(2.36603 + 2.36603i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-4.09808 - 2.36603i) q^{10} +(-1.59808 - 3.23205i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-5.13397 + 2.96410i) q^{17} +(3.00000 + 3.00000i) q^{18} +(6.46410 + 1.73205i) q^{20} +6.19615i q^{25} +(3.36603 + 3.83013i) q^{26} +(5.33013 - 9.23205i) q^{29} +(-1.46410 + 5.46410i) q^{32} +(5.92820 - 5.92820i) q^{34} +(-5.19615 - 3.00000i) q^{36} +(1.30385 + 4.86603i) q^{37} -9.46410 q^{40} +(-11.3301 + 3.03590i) q^{41} +(2.59808 - 9.69615i) q^{45} +(6.06218 + 3.50000i) q^{49} +(-2.26795 - 8.46410i) q^{50} +(-6.00000 - 4.00000i) q^{52} +3.53590 q^{53} +(-3.90192 + 14.5622i) q^{58} +(7.69615 + 13.3301i) q^{61} -8.00000i q^{64} +(3.86603 - 11.4282i) q^{65} +(-5.92820 + 10.2679i) q^{68} +(8.19615 + 2.19615i) q^{72} +(1.16987 - 1.16987i) q^{73} +(-3.56218 - 6.16987i) q^{74} +(12.9282 - 3.46410i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(14.3660 - 8.29423i) q^{82} +(-19.1603 - 5.13397i) q^{85} +(1.09808 + 4.09808i) q^{89} +14.1962i q^{90} +(1.83013 - 6.83013i) q^{97} +(-9.56218 - 2.56218i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9} - 6 q^{10} + 4 q^{13} + 8 q^{16} - 24 q^{17} + 12 q^{18} + 12 q^{20} + 10 q^{26} + 4 q^{29} + 8 q^{32} - 4 q^{34} + 26 q^{37} - 24 q^{40} - 28 q^{41} - 16 q^{50}+ \cdots - 14 q^{98}+O(q^{100})$ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 - 6 * q^9 - 6 * q^10 + 4 * q^13 + 8 * q^16 - 24 * q^17 + 12 * q^18 + 12 * q^20 + 10 * q^26 + 4 * q^29 + 8 * q^32 - 4 * q^34 + 26 * q^37 - 24 * q^40 - 28 * q^41 - 16 * q^50 - 24 * q^52 + 28 * q^53 - 26 * q^58 + 10 * q^61 + 12 * q^65 + 4 * q^68 + 12 * q^72 + 22 * q^73 + 10 * q^74 + 24 * q^80 - 18 * q^81 + 54 * q^82 - 42 * q^85 - 6 * q^89 - 10 * q^97 - 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$ .

$n$	$27$	$41$
$\chi(n)$	$-1$	$e\left(\frac{11}{12}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−1.36603	+	0.366025i	−0.965926	+	0.258819i
$2$	−1.36603	+	0.366025i	−0.965926	+	0.258819i
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	2.36603	+	2.36603i	1.05812	+	1.05812i	0.998203	+	0.0599153i	$0.0190830\pi$
$5$	2.36603	+	2.36603i	1.05812	+	1.05812i	0.0599153	+	0.998203i	$0.480917\pi$
$6$		0			0
$7$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$	−4.09808	−	2.36603i	−1.29593	−	0.748203i
$11$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$11$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$12$		0			0
$13$	−1.59808	−	3.23205i	−0.443227	−	0.896410i
$13$	−1.59808	−	3.23205i	−0.443227	−	0.896410i
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	−5.13397	+	2.96410i	−1.24517	+	0.718900i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−5.13397	+	2.96410i	−1.24517	+	0.718900i	−0.275029	+	0.961436i	$0.588688\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$19$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$20$	6.46410	+	1.73205i	1.44542	+	0.387298i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$			6.19615i			1.23923i
$26$	3.36603	+	3.83013i	0.660132	+	0.751150i
$27$		0			0
$28$		0			0
$29$	5.33013	−	9.23205i	0.989780	−	1.71435i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	5.33013	−	9.23205i	0.989780	−	1.71435i	0.618389	−	0.785872i	$-0.287786\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$	−1.46410	+	5.46410i	−0.258819	+	0.965926i
$33$		0			0
$34$	5.92820	−	5.92820i	1.01668	−	1.01668i
$35$		0			0
$36$	−5.19615	−	3.00000i	−0.866025	−	0.500000i
$37$	1.30385	+	4.86603i	0.214351	+	0.799970i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	1.30385	+	4.86603i	0.214351	+	0.799970i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$	−9.46410			−1.49641
$41$	−11.3301	+	3.03590i	−1.76947	+	0.474128i	−0.988600	−	0.150567i	$-0.951890\pi$
$41$	−11.3301	+	3.03590i	−1.76947	+	0.474128i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	2.59808	−	9.69615i	0.387298	−	1.44542i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	6.06218	+	3.50000i	0.866025	+	0.500000i
$50$	−2.26795	−	8.46410i	−0.320736	−	1.19700i
$51$		0			0
$52$	−6.00000	−	4.00000i	−0.832050	−	0.554700i
$53$	3.53590			0.485693			0.242846	−	0.970065i	$-0.421919\pi$
$53$	3.53590			0.485693			0.242846	+	0.970065i	$0.421919\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.90192	+	14.5622i	−0.512348	+	1.91211i
$59$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$59$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	7.69615	+	13.3301i	0.985391	+	1.70675i	0.640184	+	0.768221i	$0.278858\pi$
$61$	7.69615	+	13.3301i	0.985391	+	1.70675i	0.345207	+	0.938527i	$0.387809\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	3.86603	−	11.4282i	0.479521	−	1.41749i
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	−5.92820	+	10.2679i	−0.718900	+	1.24517i
$69$		0			0
$70$		0			0
$71$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$71$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$72$	8.19615	+	2.19615i	0.965926	+	0.258819i
$73$	1.16987	−	1.16987i	0.136923	−	0.136923i	−0.635323	−	0.772246i	$-0.719133\pi$
$73$	1.16987	−	1.16987i	0.136923	−	0.136923i	0.772246	+	0.635323i	$0.219133\pi$
$74$	−3.56218	−	6.16987i	−0.414095	−	0.717233i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	12.9282	−	3.46410i	1.44542	−	0.387298i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$	14.3660	−	8.29423i	1.58646	−	0.915944i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−19.1603	−	5.13397i	−2.07822	−	0.556858i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.09808	+	4.09808i	0.116396	+	0.434395i	0.999388	−	0.0349934i	$-0.0111410\pi$
$89$	1.09808	+	4.09808i	0.116396	+	0.434395i	−0.882992	+	0.469389i	$0.844474\pi$
$90$			14.1962i			1.49641i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.83013	−	6.83013i	0.185821	−	0.693494i	−0.808632	−	0.588315i	$-0.799792\pi$
$97$	1.83013	−	6.83013i	0.185821	−	0.693494i	0.994453	−	0.105180i	$-0.0335417\pi$
$98$	−9.56218	−	2.56218i	−0.965926	−	0.258819i
$99$		0			0
$100$	6.19615	+	10.7321i	0.619615	+	1.07321i
$101$	7.16025	+	4.13397i	0.712472	+	0.411346i	0.811976	−	0.583691i	$-0.198392\pi$
$101$	7.16025	+	4.13397i	0.712472	+	0.411346i	−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	9.66025	+	3.26795i	0.947266	+	0.320449i
$105$		0			0
$106$	−4.83013	+	1.29423i	−0.469143	+	0.125707i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0622	−	17.4282i	−0.946570	−	1.63951i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−10.0622	−	17.4282i	−0.946570	−	1.63951i	−0.193993	−	0.981003i	$-0.562144\pi$
$114$		0			0
$115$		0			0
$116$		−	21.3205i		−	1.97956i
$117$	−6.00000	+	9.00000i	−0.554700	+	0.832050i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.52628	+	5.50000i	−0.866025	+	0.500000i
$122$	−15.3923	−	15.3923i	−1.39355	−	1.39355i
$123$		0			0
$124$		0			0
$125$	−2.83013	+	2.83013i	−0.253134	+	0.253134i
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$	2.92820	+	10.9282i	0.258819	+	0.965926i
$129$		0			0
$130$	−1.09808	+	17.0263i	−0.0963077	+	1.49330i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	4.33975	−	16.1962i	0.372130	−	1.38881i
$137$	12.9641	+	3.47372i	1.10760	+	0.296780i	0.765855	−	0.643013i	$-0.222316\pi$
$137$	12.9641	+	3.47372i	1.10760	+	0.296780i	0.341743	+	0.939793i	$0.388983\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$	34.4545	−	9.23205i	2.86129	−	0.766680i
$146$	−1.16987	+	2.02628i	−0.0968194	+	0.167696i
$147$		0			0
$148$	7.12436	+	7.12436i	0.585618	+	0.585618i
$149$	−2.16025	+	8.06218i	−0.176975	+	0.660479i	0.819232	+	0.573462i	$0.194400\pi$
$149$	−2.16025	+	8.06218i	−0.176975	+	0.660479i	−0.996207	+	0.0870170i	$0.972267\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	15.4019	+	8.89230i	1.24517	+	0.718900i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	13.0526			1.04171			0.520854	−	0.853646i	$-0.325614\pi$
$157$	13.0526			1.04171			0.520854	+	0.853646i	$0.325614\pi$
$158$		0			0
$159$		0			0
$160$	−16.3923	+	9.46410i	−1.29593	+	0.748203i
$161$		0			0
$162$	3.29423	−	12.2942i	0.258819	−	0.965926i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$	−16.5885	+	16.5885i	−1.29534	+	1.29534i
$165$		0			0
$166$		0			0
$167$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$167$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$	−7.89230	+	10.3301i	−0.607100	+	0.794625i
$170$	28.0526			2.15153
$171$		0			0
$172$		0			0
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	−0.362500	−	0.931984i	$-0.618077\pi$
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	0.625871	+	0.779926i	$0.284744\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−5.19615	−	19.3923i	−0.387298	−	1.44542i
$181$		−	8.32051i		−	0.618458i	−0.950988	−	0.309229i	$-0.899929\pi$
$181$		−	8.32051i		−	0.618458i	0.950988	−	0.309229i	$-0.100071\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.42820	+	14.5981i	−0.619654	+	1.07327i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		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
$193$	−6.93782	−	25.8923i	−0.499395	−	1.86377i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−6.93782	−	25.8923i	−0.499395	−	1.86377i	0.00447566	−	0.999990i	$-0.498575\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	14.0000			1.00000
$197$	−20.4904	+	5.49038i	−1.45988	+	0.391173i	−0.899448	−	0.437028i	$-0.856031\pi$
$197$	−20.4904	+	5.49038i	−1.45988	+	0.391173i	−0.560431	+	0.828201i	$0.689365\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−12.3923	−	12.3923i	−0.876268	−	0.876268i
$201$		0			0
$202$	−11.2942	−	3.02628i	−0.794659	−	0.212928i
$203$		0			0
$204$		0			0
$205$	−33.9904	−	19.6244i	−2.37399	−	1.37062i
$206$		0			0
$207$		0			0
$208$	−14.3923	−	0.928203i	−0.997927	−	0.0643593i
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	6.12436	−	3.53590i	0.420622	−	0.242846i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	12.1244	+	7.00000i	0.821165	+	0.474100i
$219$		0			0
$220$		0			0
$221$	17.7846	+	11.8564i	1.19632	+	0.797548i
$222$		0			0
$223$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$223$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$224$		0			0
$225$	16.0981	−	9.29423i	1.07321	−	0.619615i
$226$	20.1244	+	20.1244i	1.33865	+	1.33865i
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	−17.0000	+	17.0000i	−1.12339	+	1.12339i	−0.132164	+	0.991228i	$0.542192\pi$
$229$	−17.0000	+	17.0000i	−1.12339	+	1.12339i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$	7.80385	+	29.1244i	0.512348	+	1.91211i
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	4.90192	−	14.4904i	0.320449	−	0.947266i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−0.0358984	−	0.00961894i	−0.00231242	−	0.000619611i	0.257663	−	0.966235i	$-0.417048\pi$
$241$	−0.0358984	−	0.00961894i	−0.00231242	−	0.000619611i	−0.259975	+	0.965615i	$0.583714\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$	26.6603	+	15.3923i	1.70675	+	0.985391i
$245$	6.06218	+	22.6244i	0.387298	+	1.44542i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	2.83013	−	4.90192i	0.178993	−	0.310025i
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−15.3564	−	8.86603i	−0.957906	−	0.553047i	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−15.3564	−	8.86603i	−0.957906	−	0.553047i	−0.895528	+	0.445005i	$0.853202\pi$
$258$		0			0
$259$		0			0
$260$	−4.73205	−	23.6603i	−0.293469	−	1.46735i
$261$	−31.9808			−1.97956
$262$		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
$265$	8.36603	+	8.36603i	0.513921	+	0.513921i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.0000	+	17.3205i	0.609711	+	1.05605i	0.991288	+	0.131713i	$0.0420477\pi$
$269$	10.0000	+	17.3205i	0.609711	+	1.05605i	−0.381577	+	0.924337i	$0.624619\pi$
$270$		0			0
$271$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$271$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$272$			23.7128i			1.43780i
$273$		0			0
$274$	−18.9808			−1.14667
$275$		0			0
$276$		0			0
$277$	25.6244	−	14.7942i	1.53962	−	0.888899i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	25.6244	−	14.7942i	1.53962	−	0.888899i	0.998861	−	0.0477206i	$-0.0151957\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.6865	−	23.6865i	1.41302	−	1.41302i	0.677466	−	0.735554i	$-0.263078\pi$
$281$	23.6865	−	23.6865i	1.41302	−	1.41302i	0.735554	−	0.677466i	$-0.236922\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.3923	−	4.39230i	0.965926	−	0.258819i
$289$	9.07180	−	15.7128i	0.533635	−	0.924283i
$290$	−43.6865	+	25.2224i	−2.56536	+	1.48111i
$291$		0			0
$292$	0.856406	−	3.19615i	0.0501174	−	0.187041i
$293$	30.7224	+	8.23205i	1.79482	+	0.480922i	0.993151	−	0.116841i	$-0.0372769\pi$
$293$	30.7224	+	8.23205i	1.79482	+	0.480922i	0.801673	+	0.597763i	$0.203944\pi$
$294$		0			0
$295$		0			0
$296$	−12.3397	−	7.12436i	−0.717233	−	0.414095i
$297$		0			0
$298$		−	11.8038i		−	0.683779i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.3301	+	49.7487i	−0.763281	+	2.84860i
$306$	−24.2942	−	6.50962i	−1.38881	−	0.372130i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−24.0000			−1.35656			−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000			−1.35656			−0.678280	+	0.734803i	$0.737274\pi$
$314$	−17.8301	+	4.77757i	−1.00621	+	0.269614i
$315$		0			0
$316$		0			0
$317$	−20.1506	−	20.1506i	−1.13177	−	1.13177i	−0.989882	−	0.141890i	$-0.954682\pi$
$317$	−20.1506	−	20.1506i	−1.13177	−	1.13177i	−0.141890	−	0.989882i	$-0.545318\pi$
$318$		0			0
$319$		0			0
$320$	18.9282	−	18.9282i	1.05812	−	1.05812i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	20.0263	−	9.90192i	1.11086	−	0.549260i
$326$		0			0
$327$		0			0
$328$	16.5885	−	28.7321i	0.915944	−	1.58646i
$329$		0			0
$330$		0			0
$331$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$331$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$332$		0			0
$333$	10.6865	−	10.6865i	0.585618	−	0.585618i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			36.7128i			1.99987i	0.0112091	+	0.999937i	$0.496432\pi$
$337$			36.7128i			1.99987i	−0.0112091	+	0.999937i	$0.503568\pi$
$338$	7.00000	−	17.0000i	0.380750	−	0.924678i
$339$		0			0
$340$	−38.3205	+	10.2679i	−2.07822	+	0.556858i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−4.00000	+	4.00000i	−0.215041	+	0.215041i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−8.41858	−	31.4186i	−0.450636	−	1.68180i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	−8.41858	−	31.4186i	−0.450636	−	1.68180i	0.249973	−	0.968253i	$-0.419578\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.42820	−	1.72243i	0.342139	−	0.0916758i	−0.0836583	−	0.996495i	$-0.526660\pi$
$353$	6.42820	−	1.72243i	0.342139	−	0.0916758i	0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$	6.00000	+	6.00000i	0.317999	+	0.317999i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	14.1962	+	24.5885i	0.748203	+	1.29593i
$361$	−16.4545	−	9.50000i	−0.866025	−	0.500000i
$362$	3.04552	+	11.3660i	0.160069	+	0.597385i
$363$		0			0
$364$		0			0
$365$	5.53590			0.289762
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	24.8827	+	24.8827i	1.29534	+	1.29534i
$370$	6.16987	−	23.0263i	0.320756	−	1.19708i
$371$		0			0
$372$		0			0
$373$	2.93782	+	5.08846i	0.152115	+	0.263470i	0.932005	−	0.362446i	$-0.118058\pi$
$373$	2.93782	+	5.08846i	0.152115	+	0.263470i	−0.779890	+	0.625917i	$0.784725\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−38.3564	−	2.47372i	−1.97546	−	0.127403i
$378$		0			0
$379$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$379$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$383$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$384$		0			0
$385$		0			0
$386$	18.9545	+	32.8301i	0.964758	+	1.67101i
$387$		0			0
$388$	−3.66025	−	13.6603i	−0.185821	−	0.693494i
$389$		−	34.3205i		−	1.74012i	−0.492947	−	0.870059i	$-0.664080\pi$
$389$		−	34.3205i		−	1.74012i	0.492947	−	0.870059i	$-0.335920\pi$
$390$		0			0
$391$		0			0
$392$	−19.1244	+	5.12436i	−0.965926	+	0.258819i
$393$		0			0
$394$	25.9808	−	15.0000i	1.30889	−	0.755689i
$395$		0			0
$396$		0			0
$397$	34.1506	+	9.15064i	1.71397	+	0.459257i	0.976392	−	0.216004i	$-0.0693024\pi$
$397$	34.1506	+	9.15064i	1.71397	+	0.459257i	0.737579	+	0.675261i	$0.235969\pi$
$398$		0			0
$399$		0			0
$400$	21.4641	+	12.3923i	1.07321	+	0.619615i
$401$	−2.17949	−	8.13397i	−0.108839	−	0.406191i	0.889914	−	0.456129i	$-0.150764\pi$
$401$	−2.17949	−	8.13397i	−0.108839	−	0.406191i	−0.998752	+	0.0499376i	$0.984098\pi$
$402$		0			0
$403$		0			0
$404$	16.5359			0.822692
$405$	−29.0885	+	7.79423i	−1.44542	+	0.387298i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.4019	+	38.8205i	−0.514342	+	1.91955i	−0.148340	+	0.988936i	$0.547393\pi$
$409$	−10.4019	+	38.8205i	−0.514342	+	1.91955i	−0.366002	+	0.930614i	$0.619274\pi$
$410$	53.6147	+	14.3660i	2.64784	+	0.709487i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	20.0000	−	4.00000i	0.980581	−	0.196116i
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	15.3660	+	15.3660i	0.748894	+	0.748894i	0.974272	−	0.225377i	$-0.0723615\pi$
$421$	15.3660	+	15.3660i	0.748894	+	0.748894i	−0.225377	+	0.974272i	$0.572361\pi$
$422$		0			0
$423$		0			0
$424$	−7.07180	+	7.07180i	−0.343437	+	0.343437i
$425$	−18.3660	−	31.8109i	−0.890883	−	1.54305i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$		0			0
$433$	−35.8923	+	20.7224i	−1.72487	+	0.995857i	−0.816968	+	0.576683i	$0.804347\pi$
$433$	−35.8923	+	20.7224i	−1.72487	+	0.995857i	−0.907906	+	0.419173i	$0.862320\pi$
$434$		0			0
$435$		0			0
$436$	−19.1244	−	5.12436i	−0.915891	−	0.245412i
$437$		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
$441$		−	21.0000i		−	1.00000i
$442$	−28.6340	−	9.68653i	−1.36198	−	0.460741i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−7.09808	+	12.2942i	−0.336481	+	0.582802i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−36.8827	−	9.88269i	−1.74060	−	0.466393i	−0.758021	−	0.652230i	$-0.773834\pi$
$449$	−36.8827	−	9.88269i	−1.74060	−	0.466393i	−0.982581	+	0.185837i	$0.940500\pi$
$450$	−18.5885	+	18.5885i	−0.876268	+	0.876268i
$451$		0			0
$452$	−34.8564	−	20.1244i	−1.63951	−	0.946570i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	37.1865	−	9.96410i	1.73951	−	0.466101i	0.757174	−	0.653213i	$-0.226579\pi$
$457$	37.1865	−	9.96410i	1.73951	−	0.466101i	0.982339	+	0.187112i	$0.0599128\pi$
$458$	17.0000	−	29.4449i	0.794358	−	1.37587i
$459$		0			0
$460$		0			0
$461$	10.8397	−	40.4545i	0.504857	−	1.88415i	0.0391109	−	0.999235i	$-0.487547\pi$
$461$	10.8397	−	40.4545i	0.504857	−	1.88415i	0.465746	−	0.884918i	$-0.345786\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−21.3205	−	36.9282i	−0.989780	−	1.71435i
$465$		0			0
$466$	5.85641	+	21.8564i	0.271293	+	1.01248i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−1.39230	+	21.5885i	−0.0643593	+	0.997927i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.30385	−	9.18653i	−0.242846	−	0.420622i
$478$		0			0
$479$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$479$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$480$		0			0
$481$	13.6436	−	11.9904i	0.622094	−	0.546714i
$482$	0.0525589			0.00239399
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$	20.4904	−	11.8301i	0.930420	−	0.537178i
$486$		0			0
$487$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$487$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$488$	−42.0526	−	11.2679i	−1.90363	−	0.510076i
$489$		0			0
$490$	−16.5622	−	28.6865i	−0.748203	−	1.29593i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$			63.1962i			2.84621i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	−2.07180	+	7.73205i	−0.0926536	+	0.345788i
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	7.16025	+	26.7224i	0.318627	+	1.18913i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.66987	−	0.447441i	0.0740158	−	0.0198325i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	1.66987	−	0.447441i	0.0740158	−	0.0198325i	0.295637	+	0.955300i	$0.404468\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	24.2224	+	6.49038i	1.06841	+	0.286278i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	15.1244	+	30.5885i	0.663247	+	1.34139i
$521$	39.0526			1.71092			0.855462	−	0.517866i	$-0.173273\pi$
$521$	39.0526			1.71092			0.855462	+	0.517866i	$0.173273\pi$
$522$	43.6865	−	11.7058i	1.91211	−	0.512348i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$	−14.4904	−	8.36603i	−0.629422	−	0.363397i
$531$		0			0
$532$		0			0
$533$	27.9186	+	31.7679i	1.20929	+	1.37602i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−20.0000	−	20.0000i	−0.862261	−	0.862261i
$539$		0			0
$540$		0			0
$541$	−21.3468	+	21.3468i	−0.917770	+	0.917770i	−0.996867	−	0.0790969i	$-0.974796\pi$
$541$	−21.3468	+	21.3468i	−0.917770	+	0.917770i	0.0790969	+	0.996867i	$0.474796\pi$
$542$		0			0
$543$		0			0
$544$	−8.67949	−	32.3923i	−0.372130	−	1.38881i
$545$		−	33.1244i		−	1.41889i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	25.9282	−	6.94744i	1.10760	−	0.296780i
$549$	23.0885	−	39.9904i	0.985391	−	1.70675i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−29.5885	+	29.5885i	−1.25709	+	1.25709i
$555$		0			0
$556$		0			0
$557$	9.54552	+	35.6244i	0.404457	+	1.50945i	0.805056	+	0.593199i	$0.202135\pi$
$557$	9.54552	+	35.6244i	0.404457	+	1.50945i	−0.400599	+	0.916253i	$0.631198\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−23.6865	+	41.0263i	−0.999156	+	1.73059i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	17.4282	−	65.0429i	0.733210	−	2.73638i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.998608	−	0.0527519i	$-0.0167993\pi$
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.453619	+	0.891196i	$0.350133\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−20.7846	+	12.0000i	−0.866025	+	0.500000i
$577$	−33.1506	−	33.1506i	−1.38008	−	1.38008i	−0.844459	−	0.535620i	$-0.820078\pi$
$577$	−33.1506	−	33.1506i	−1.38008	−	1.38008i	−0.535620	−	0.844459i	$-0.679922\pi$
$578$	−6.64102	+	24.7846i	−0.276230	+	1.03090i
$579$		0			0
$580$	50.4449	−	50.4449i	2.09461	−	2.09461i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			4.67949i			0.193639i
$585$	−35.4904	+	7.09808i	−1.46735	+	0.293469i
$586$	−44.9808			−1.85814
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	19.4641	+	5.21539i	0.799970	+	0.214351i
$593$	−34.3468	+	34.3468i	−1.41045	+	1.41045i	−0.653698	+	0.756756i	$0.726783\pi$
$593$	−34.3468	+	34.3468i	−1.41045	+	1.41045i	−0.756756	+	0.653698i	$0.773217\pi$
$594$		0			0
$595$		0			0
$596$	4.32051	+	16.1244i	0.176975	+	0.660479i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−7.66987	+	13.2846i	−0.312861	+	0.541891i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−7.66987	+	13.2846i	−0.312861	+	0.541891i	0.666120	+	0.745845i	$0.267954\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−35.5526	−	9.52628i	−1.44542	−	0.387298i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		−	72.8372i		−	2.94909i
$611$		0			0
$612$	35.5692			1.43780
$613$	−42.0885	+	11.2776i	−1.69994	+	0.455497i	−0.972924	−	0.231127i	$-0.925759\pi$
$613$	−42.0885	+	11.2776i	−1.69994	+	0.455497i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.35641	−	27.4545i	0.296158	−	1.10528i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	7.35641	−	27.4545i	0.296158	−	1.10528i	0.940294	−	0.340365i	$-0.110551\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.5885			0.703538
$626$	32.7846	−	8.78461i	1.31034	−	0.351104i
$627$		0			0
$628$	22.6077	−	13.0526i	0.902145	−	0.520854i
$629$	−21.1173	−	21.1173i	−0.842002	−	0.842002i
$630$		0			0
$631$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$631$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$632$		0			0
$633$		0			0
$634$	34.9019	+	20.1506i	1.38613	+	0.800284i
$635$		0			0
$636$		0			0
$637$	1.62436	−	25.1865i	0.0643593	−	0.997927i
$638$		0			0
$639$		0			0
$640$	−18.9282	+	32.7846i	−0.748203	+	1.29593i
$641$	−27.6506	+	15.9641i	−1.09213	+	0.630544i	−0.934144	−	0.356897i	$-0.883835\pi$
$641$	−27.6506	+	15.9641i	−1.09213	+	0.630544i	−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$643$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	−6.58846	−	24.5885i	−0.258819	−	0.965926i
$649$		0			0
$650$	−23.7321	+	20.8564i	−0.930848	+	0.818056i
$651$		0			0
$652$		0			0
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	0.0101092	+	0.999949i	$0.496782\pi$
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	−0.871036	+	0.491220i	$0.836551\pi$
$654$		0			0
$655$		0			0
$656$	−12.1436	+	45.3205i	−0.474128	+	1.76947i
$657$	−4.79423	−	1.28461i	−0.187041	−	0.0501174i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−11.6962	−	43.6506i	−0.454928	−	1.69781i	−0.688301	−	0.725426i	$-0.741643\pi$
$661$	−11.6962	−	43.6506i	−0.454928	−	1.69781i	0.233373	−	0.972387i	$-0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−10.6865	+	18.5096i	−0.414095	+	0.717233i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	37.9186	+	21.8923i	1.46165	+	0.843886i	0.999088	−	0.0426985i	$-0.0135955\pi$
$673$	37.9186	+	21.8923i	1.46165	+	0.843886i	0.462566	+	0.886585i	$0.346929\pi$
$674$	−13.4378	−	50.1506i	−0.517606	−	1.93173i
$675$		0			0
$676$	−3.33975	+	25.7846i	−0.128452	+	0.991716i
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$	48.5885	−	28.0526i	1.86328	−	1.07577i
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$	22.4545	+	38.8923i	0.857942	+	1.48600i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.65064	−	11.4282i	−0.215272	−	0.435380i
$690$		0			0
$691$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$691$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$692$	4.00000	−	6.92820i	0.152057	−	0.263371i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	49.1699	−	49.1699i	1.86244	−	1.86244i
$698$	23.0000	+	39.8372i	0.870563	+	1.50786i
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−8.15064	+	4.70577i	−0.306753	+	0.177104i
$707$		0			0
$708$		0			0
$709$	−48.5526	−	13.0096i	−1.82343	−	0.488586i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−48.5526	−	13.0096i	−1.82343	−	0.488586i	−0.997202	+	0.0747503i	$0.976184\pi$
$710$		0			0
$711$		0			0
$712$	−10.3923	−	6.00000i	−0.389468	−	0.224860i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	−28.3923	−	28.3923i	−1.05812	−	1.05812i
$721$		0			0
$722$	25.9545	+	6.95448i	0.965926	+	0.258819i
$723$		0			0
$724$	−8.32051	−	14.4115i	−0.309229	−	0.535601i
$725$	57.2032	+	33.0263i	2.12447	+	1.22657i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	−7.56218	+	2.02628i	−0.279889	+	0.0749960i
$731$		0			0
$732$		0			0
$733$	−7.15064	−	7.15064i	−0.264115	−	0.264115i	0.562609	−	0.826723i	$-0.309798\pi$
$733$	−7.15064	−	7.15064i	−0.264115	−	0.264115i	−0.826723	+	0.562609i	$0.809798\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−43.0981	−	24.8827i	−1.58646	−	0.915944i
$739$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$739$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$740$			33.7128i			1.23931i
$741$		0			0
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$	−24.1865	+	13.9641i	−0.886126	+	0.511605i
$746$	−5.87564	−	5.87564i	−0.215123	−	0.215123i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$	53.3013	−	10.6603i	1.94112	−	0.388224i
$755$		0			0
$756$		0			0
$757$	−9.00000	+	15.5885i	−0.327111	+	0.566572i	−0.981937	−	0.189207i	$-0.939408\pi$
$757$	−9.00000	+	15.5885i	−0.327111	+	0.566572i	0.654827	+	0.755779i	$0.272742\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.36603	−	0.366025i	−0.0495184	−	0.0132684i	0.233975	−	0.972243i	$-0.424827\pi$
$761$	−1.36603	−	0.366025i	−0.0495184	−	0.0132684i	−0.283493	+	0.958974i	$0.591493\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	15.4019	+	57.4808i	0.556858	+	2.07822i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	50.5429	−	13.5429i	1.82263	−	0.488371i	0.825518	−	0.564376i	$-0.190883\pi$
$769$	50.5429	−	13.5429i	1.82263	−	0.488371i	0.997107	+	0.0760054i	$0.0242166\pi$
$770$		0			0
$771$		0			0
$772$	−37.9090	−	37.9090i	−1.36437	−	1.36437i
$773$	1.83013	−	6.83013i	0.0658251	−	0.245663i	−0.925172	−	0.379549i	$-0.876079\pi$
$773$	1.83013	−	6.83013i	0.0658251	−	0.245663i	0.990997	+	0.133887i	$0.0427458\pi$
$774$		0			0
$775$		0			0
$776$	10.0000	+	17.3205i	0.358979	+	0.621770i
$777$		0			0
$778$	12.5622	+	46.8827i	0.450376	+	1.68083i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	24.2487	−	14.0000i	0.866025	−	0.500000i
$785$	30.8827	+	30.8827i	1.10225	+	1.10225i
$786$		0			0
$787$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$787$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$788$	−30.0000	+	30.0000i	−1.06871	+	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	30.7846	−	46.1769i	1.09319	−	1.63979i
$794$	−50.0000			−1.77443
$795$		0			0
$796$		0			0
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	−0.797922	−	0.602761i	$-0.794067\pi$
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	0.123045	+	0.992401i	$0.460734\pi$
$798$		0			0
$799$		0			0
$800$	−33.8564	−	9.07180i	−1.19700	−	0.320736i
$801$	9.00000	−	9.00000i	0.317999	−	0.317999i
$802$	5.95448	+	10.3135i	0.210260	+	0.364181i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−22.5885	+	6.05256i	−0.794659	+	0.212928i
$809$	18.3301	−	31.7487i	0.644453	−	1.11623i	−0.339975	−	0.940435i	$-0.610418\pi$
$809$	18.3301	−	31.7487i	0.644453	−	1.11623i	0.984428	−	0.175791i	$-0.0562482\pi$
$810$	36.8827	−	21.2942i	1.29593	−	0.748203i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	56.8372i		−	1.98726i
$819$		0			0
$820$	−78.4974			−2.74125
$821$	15.0263	−	4.02628i	0.524421	−	0.140518i	0.0131101	−	0.999914i	$-0.495827\pi$
$821$	15.0263	−	4.02628i	0.524421	−	0.140518i	0.511311	+	0.859396i	$0.329160\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	−31.8397	−	18.3827i	−1.10584	−	0.638457i	−0.168091	−	0.985771i	$-0.553760\pi$
$829$	−31.8397	−	18.3827i	−1.10584	−	0.638457i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$	−25.8564	+	12.7846i	−0.896410	+	0.443227i
$833$	−41.4974			−1.43780
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$839$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$840$		0			0
$841$	−42.3205	−	73.3013i	−1.45933	−	2.52763i
$842$	−26.6147	−	15.3660i	−0.917204	−	0.529548i
$843$		0			0
$844$		0			0
$845$	−43.1147	+	5.76795i	−1.48319	+	0.198423i
$846$		0			0
$847$		0			0
$848$	7.07180	−	12.2487i	0.242846	−	0.420622i
$849$		0			0
$850$	36.7321	+	36.7321i	1.25990	+	1.25990i
$851$		0			0
$852$		0			0
$853$	−24.8301	+	24.8301i	−0.850167	+	0.850167i	−0.990153	−	0.139986i	$-0.955294\pi$
$853$	−24.8301	+	24.8301i	−0.850167	+	0.850167i	0.139986	+	0.990153i	$0.455294\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			46.2295i			1.57917i	0.613642	+	0.789584i	$0.289704\pi$
$857$			46.2295i			1.57917i	−0.613642	+	0.789584i	$0.710296\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	12.9282	+	3.46410i	0.439572	+	0.117783i
$866$	41.4449	−	41.4449i	1.40835	−	1.40835i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	28.0000			0.948200
$873$	−20.4904	+	5.49038i	−0.693494	+	0.185821i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.885263	+	3.30385i	−0.0298932	+	0.111563i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−0.885263	+	3.30385i	−0.0298932	+	0.111563i	0.949367	+	0.314169i	$0.101726\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.6436	+	13.6506i	0.796573	+	0.459902i	0.842271	−	0.539054i	$-0.181218\pi$
$881$	23.6436	+	13.6506i	0.796573	+	0.459902i	−0.0456985	+	0.998955i	$0.514551\pi$
$882$	7.68653	+	28.6865i	0.258819	+	0.965926i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	42.6603	+	2.75129i	1.43482	+	0.0925359i
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	5.19615	−	19.3923i	0.174175	−	0.650032i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	54.0000			1.80200
$899$		0			0
$900$	18.5885	−	32.1962i	0.619615	−	1.07321i
$901$	−18.1532	+	10.4808i	−0.604771	+	0.349165i
$902$		0			0
$903$		0			0
$904$	54.9808	+	14.7321i	1.82863	+	0.489981i
$905$	19.6865	−	19.6865i	0.654403	−	0.654403i
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$		−	24.8038i		−	0.822692i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−47.1506	+	27.2224i	−1.55960	+	0.900438i
$915$		0			0
$916$	−12.4449	+	46.4449i	−0.411190	+	1.53458i
$917$		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
$922$			59.2295i			1.95062i
$923$		0			0
$924$		0			0
$925$	−30.1506	+	8.07884i	−0.991347	+	0.265631i
$926$		0			0
$927$		0			0
$928$	42.6410	+	42.6410i	1.39976	+	1.39976i
$929$	−6.91858	+	25.8205i	−0.226991	+	0.847143i	0.754606	+	0.656179i	$0.227828\pi$
$929$	−6.91858	+	25.8205i	−0.226991	+	0.847143i	−0.981597	+	0.190965i	$0.938838\pi$
$930$		0			0
$931$		0			0
$932$	−16.0000	−	27.7128i	−0.524097	−	0.907763i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−6.00000	−	30.0000i	−0.196116	−	0.980581i
$937$	22.5692			0.737304			0.368652	−	0.929567i	$-0.379819\pi$
$937$	22.5692			0.737304			0.368652	+	0.929567i	$0.379819\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	+	19.0000i	0.619382	+	0.619382i	0.945373	−	0.325991i	$-0.105698\pi$
$941$	19.0000	+	19.0000i	0.619382	+	0.619382i	−0.325991	+	0.945373i	$0.605698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$947$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$948$		0			0
$949$	−5.65064	−	1.91154i	−0.183427	−	0.0620513i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.574937	−	0.818198i	$-0.305026\pi$
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.996048	−	0.0888114i	$-0.0283068\pi$
$954$	10.6077	+	10.6077i	0.343437	+	0.343437i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			31.0000i			1.00000i
$962$	−14.2487	+	21.3731i	−0.459397	+	0.689095i
$963$		0			0
$964$	−0.0717968	+	0.0192379i	−0.00231242	+	0.000619611i
$965$	44.8468	−	77.6769i	1.44367	−	2.50051i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	8.05256	−	30.0526i	0.258819	−	0.965926i
$969$		0			0
$970$	−23.6603	+	23.6603i	−0.759685	+	0.759685i
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	61.5692			1.97078
$977$	−59.8468	+	16.0359i	−1.91467	+	0.513034i	−0.922890	+	0.385063i	$0.874180\pi$
$977$	−59.8468	+	16.0359i	−1.91467	+	0.513034i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$	33.1244	+	33.1244i	1.05812	+	1.05812i
$981$	−7.68653	+	28.6865i	−0.245412	+	0.915891i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−61.4711	−	35.4904i	−1.95863	−	1.13082i
$986$	−23.1314	−	86.3275i	−0.736654	−	2.74923i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20.6962	+	35.8468i	0.655454	+	1.13528i	0.981780	+	0.190022i	$0.0608559\pi$
$997$	20.6962	+	35.8468i	0.655454	+	1.13528i	−0.326326	+	0.945257i	$0.605811\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.2.l.a.7.1	✓	4
3.2	odd	2		468.2.cb.d.163.1		4
4.3	odd	2	CM	52.2.l.a.7.1	✓	4
8.3	odd	2		832.2.bu.d.319.1		4
8.5	even	2		832.2.bu.d.319.1		4
12.11	even	2		468.2.cb.d.163.1		4
13.2	odd	12	inner	52.2.l.a.15.1	yes	4
13.3	even	3		676.2.l.c.427.1		4
13.4	even	6		676.2.f.d.99.1		4
13.5	odd	4		676.2.l.c.19.1		4
13.6	odd	12		676.2.f.e.239.2		4
13.7	odd	12		676.2.f.d.239.1		4
13.8	odd	4		676.2.l.e.19.1		4
13.9	even	3		676.2.f.e.99.2		4
13.10	even	6		676.2.l.e.427.1		4
13.11	odd	12		676.2.l.d.587.1		4
13.12	even	2		676.2.l.d.319.1		4
39.2	even	12		468.2.cb.d.379.1		4
52.3	odd	6		676.2.l.c.427.1		4
52.7	even	12		676.2.f.d.239.1		4
52.11	even	12		676.2.l.d.587.1		4
52.15	even	12	inner	52.2.l.a.15.1	yes	4
52.19	even	12		676.2.f.e.239.2		4
52.23	odd	6		676.2.l.e.427.1		4
52.31	even	4		676.2.l.c.19.1		4
52.35	odd	6		676.2.f.e.99.2		4
52.43	odd	6		676.2.f.d.99.1		4
52.47	even	4		676.2.l.e.19.1		4
52.51	odd	2		676.2.l.d.319.1		4
104.67	even	12		832.2.bu.d.639.1		4
104.93	odd	12		832.2.bu.d.639.1		4
156.119	odd	12		468.2.cb.d.379.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.l.a.7.1	✓	4	1.1	even	1	trivial
52.2.l.a.7.1	✓	4	4.3	odd	2	CM
52.2.l.a.15.1	yes	4	13.2	odd	12	inner
52.2.l.a.15.1	yes	4	52.15	even	12	inner
468.2.cb.d.163.1		4	3.2	odd	2
468.2.cb.d.163.1		4	12.11	even	2
468.2.cb.d.379.1		4	39.2	even	12
468.2.cb.d.379.1		4	156.119	odd	12
676.2.f.d.99.1		4	13.4	even	6
676.2.f.d.99.1		4	52.43	odd	6
676.2.f.d.239.1		4	13.7	odd	12
676.2.f.d.239.1		4	52.7	even	12
676.2.f.e.99.2		4	13.9	even	3
676.2.f.e.99.2		4	52.35	odd	6
676.2.f.e.239.2		4	13.6	odd	12
676.2.f.e.239.2		4	52.19	even	12
676.2.l.c.19.1		4	13.5	odd	4
676.2.l.c.19.1		4	52.31	even	4
676.2.l.c.427.1		4	13.3	even	3
676.2.l.c.427.1		4	52.3	odd	6
676.2.l.d.319.1		4	13.12	even	2
676.2.l.d.319.1		4	52.51	odd	2
676.2.l.d.587.1		4	13.11	odd	12
676.2.l.d.587.1		4	52.11	even	12
676.2.l.e.19.1		4	13.8	odd	4
676.2.l.e.19.1		4	52.47	even	4
676.2.l.e.427.1		4	13.10	even	6
676.2.l.e.427.1		4	52.23	odd	6
832.2.bu.d.319.1		4	8.3	odd	2
832.2.bu.d.319.1		4	8.5	even	2
832.2.bu.d.639.1		4	104.67	even	12
832.2.bu.d.639.1		4	104.93	odd	12

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	52.2.l.a.7.1

Level	$52$
Weight	$2$
Character	52.7
Analytic conductor	$0.415$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-4
Inner twists	$4$