LMFDB - Embedded newform 20.10.e.a.7.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,10,Mod(3,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	20.e (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.3007167233$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			7.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	20.7
Dual form			20.10.e.a.3.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-16.0000 - 16.0000i) q^{2} +512.000i q^{4} +(718.000 - 1199.00i) q^{5} +(8192.00 - 8192.00i) q^{8} -19683.0i q^{9} +O(q^{10})$	\(q+(-16.0000 - 16.0000i) q^{2} +512.000i q^{4} +(718.000 - 1199.00i) q^{5} +(8192.00 - 8192.00i) q^{8} -19683.0i q^{9} +(-30672.0 + 7696.00i) q^{10} +(-142561. + 142561. i) q^{13} -262144. q^{16} +(-481437. - 481437. i) q^{17} +(-314928. + 314928. i) q^{18} +(613888. + 367616. i) q^{20} +(-922077. - 1.72176e6i) q^{25} +4.56195e6 q^{26} -2.12688e6i q^{29} +(4.19430e6 + 4.19430e6i) q^{32} +1.54060e7i q^{34} +1.00777e7 q^{36} +(-1.23211e7 - 1.23211e7i) q^{37} +(-3.94035e6 - 1.57041e7i) q^{40} +7.56191e6 q^{41} +(-2.35999e7 - 1.41324e7i) q^{45} +4.03536e7i q^{49} +(-1.27950e7 + 4.23015e7i) q^{50} +(-7.29912e7 - 7.29912e7i) q^{52} +(-1.20197e7 + 1.20197e7i) q^{53} +(-3.40300e7 + 3.40300e7i) q^{58} +2.16178e8 q^{61} -1.34218e8i q^{64} +(6.85718e7 + 2.73289e8i) q^{65} +(2.46496e8 - 2.46496e8i) q^{68} +(-1.61243e8 - 1.61243e8i) q^{72} +(2.62875e8 - 2.62875e8i) q^{73} +3.94276e8i q^{74} +(-1.88219e8 + 3.14311e8i) q^{80} -3.87420e8 q^{81} +(-1.20991e8 - 1.20991e8i) q^{82} +(-9.22915e8 + 2.31571e8i) q^{85} -3.66772e8i q^{89} +(1.51480e8 + 6.03717e8i) q^{90} +(-1.21673e9 - 1.21673e9i) q^{97} +(6.45658e8 - 6.45658e8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{2} + 1436 q^{5} + 16384 q^{8}+O(q^{10}) $ 2 * q - 32 * q^2 + 1436 * q^5 + 16384 * q^8	$ 2 q - 32 q^{2} + 1436 q^{5} + 16384 q^{8} - 61344 q^{10} - 285122 q^{13} - 524288 q^{16} - 962874 q^{17} - 629856 q^{18} + 1227776 q^{20} - 1844154 q^{25} + 9123904 q^{26} + 8388608 q^{32} + 20155392 q^{36} - 24642254 q^{37} - 7880704 q^{40} + 15123824 q^{41} - 47199834 q^{45} - 25589984 q^{50} - 145982464 q^{52} - 24039342 q^{53} - 68060032 q^{58} + 432356184 q^{61} + 137143682 q^{65} + 492991488 q^{68} - 322486272 q^{72} + 525750698 q^{73} - 376438784 q^{80} - 774840978 q^{81} - 241981184 q^{82} - 1845829458 q^{85} + 302960736 q^{90} - 2433462694 q^{97} + 1291315424 q^{98}+O(q^{100}) $ 2 * q - 32 * q^2 + 1436 * q^5 + 16384 * q^8 - 61344 * q^10 - 285122 * q^13 - 524288 * q^16 - 962874 * q^17 - 629856 * q^18 + 1227776 * q^20 - 1844154 * q^25 + 9123904 * q^26 + 8388608 * q^32 + 20155392 * q^36 - 24642254 * q^37 - 7880704 * q^40 + 15123824 * q^41 - 47199834 * q^45 - 25589984 * q^50 - 145982464 * q^52 - 24039342 * q^53 - 68060032 * q^58 + 432356184 * q^61 + 137143682 * q^65 + 492991488 * q^68 - 322486272 * q^72 + 525750698 * q^73 - 376438784 * q^80 - 774840978 * q^81 - 241981184 * q^82 - 1845829458 * q^85 + 302960736 * q^90 - 2433462694 * q^97 + 1291315424 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$2$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$			512.000i			1.00000i
$5$	718.000	−	1199.00i	0.513759	−	0.857935i
$5$	718.000	−	1199.00i	0.513759	−	0.857935i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	8192.00	−	8192.00i	0.707107	−	0.707107i
$9$		−	19683.0i		−	1.00000i
$10$	−30672.0	+	7696.00i	−0.969934	+	0.243369i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−142561.	+	142561.i	−1.38438	+	1.38438i	−0.547718	+	0.836663i	$0.684503\pi$
$13$	−142561.	+	142561.i	−1.38438	+	1.38438i	−0.836663	+	0.547718i	$0.815497\pi$
$14$		0			0
$15$		0			0
$16$	−262144.			−1.00000
$17$	−481437.	−	481437.i	−1.39804	−	1.39804i	−0.805658	−	0.592382i	$-0.798188\pi$
$17$	−481437.	−	481437.i	−1.39804	−	1.39804i	−0.592382	−	0.805658i	$-0.701812\pi$
$18$	−314928.	+	314928.i	−0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	613888.	+	367616.i	0.857935	+	0.513759i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	−922077.	−	1.72176e6i	−0.472103	−	0.881543i
$26$	4.56195e6			1.95781
$27$		0			0
$28$		0			0
$29$		−	2.12688e6i		−	0.558407i	−0.960232	−	0.279204i	$-0.909930\pi$
$29$		−	2.12688e6i		−	0.558407i	0.960232	−	0.279204i	$-0.0900705\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	4.19430e6	+	4.19430e6i	0.707107	+	0.707107i
$33$		0			0
$34$			1.54060e7i			1.97713i
$35$		0			0
$36$	1.00777e7			1.00000
$37$	−1.23211e7	−	1.23211e7i	−1.08079	−	1.08079i	−0.996436	−	0.0843579i	$-0.973116\pi$
$37$	−1.23211e7	−	1.23211e7i	−1.08079	−	1.08079i	−0.0843579	−	0.996436i	$-0.526884\pi$
$38$		0			0
$39$		0			0
$40$	−3.94035e6	−	1.57041e7i	−0.243369	−	0.969934i
$41$	7.56191e6			0.417931			0.208965	−	0.977923i	$-0.432990\pi$
$41$	7.56191e6			0.417931			0.208965	+	0.977923i	$0.432990\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−2.35999e7	−	1.41324e7i	−0.857935	−	0.513759i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			4.03536e7i			1.00000i
$50$	−1.27950e7	+	4.23015e7i	−0.289518	+	0.957173i
$51$		0			0
$52$	−7.29912e7	−	7.29912e7i	−1.38438	−	1.38438i
$53$	−1.20197e7	+	1.20197e7i	−0.209243	+	0.209243i	−0.803946	−	0.594703i	$-0.797270\pi$
$53$	−1.20197e7	+	1.20197e7i	−0.209243	+	0.209243i	0.594703	+	0.803946i	$0.297270\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.40300e7	+	3.40300e7i	−0.394854	+	0.394854i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.16178e8			1.99907			0.999534	−	0.0305361i	$-0.00972145\pi$
$61$	2.16178e8			1.99907			0.999534	+	0.0305361i	$0.00972145\pi$
$62$		0			0
$63$		0			0
$64$		−	1.34218e8i		−	1.00000i
$65$	6.85718e7	+	2.73289e8i	0.476470	+	1.89895i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	2.46496e8	−	2.46496e8i	1.39804	−	1.39804i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−1.61243e8	−	1.61243e8i	−0.707107	−	0.707107i
$73$	2.62875e8	−	2.62875e8i	1.08342	−	1.08342i	0.0872324	−	0.996188i	$-0.472198\pi$
$73$	2.62875e8	−	2.62875e8i	1.08342	−	1.08342i	0.996188	−	0.0872324i	$-0.0278023\pi$
$74$			3.94276e8i			1.52847i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.88219e8	+	3.14311e8i	−0.513759	+	0.857935i
$81$	−3.87420e8			−1.00000
$82$	−1.20991e8	−	1.20991e8i	−0.295522	−	0.295522i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−9.22915e8	+	2.31571e8i	−1.91768	+	0.481171i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	3.66772e8i		−	0.619642i	−0.950795	−	0.309821i	$-0.899731\pi$
$89$		−	3.66772e8i		−	0.619642i	0.950795	−	0.309821i	$-0.100269\pi$
$90$	1.51480e8	+	6.03717e8i	0.243369	+	0.969934i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.21673e9	−	1.21673e9i	−1.39547	−	1.39547i	−0.812466	−	0.583008i	$-0.801876\pi$
$97$	−1.21673e9	−	1.21673e9i	−1.39547	−	1.39547i	−0.583008	−	0.812466i	$-0.698124\pi$
$98$	6.45658e8	−	6.45658e8i	0.707107	−	0.707107i
$99$		0			0
$100$	8.81543e8	−	4.72103e8i	0.881543	−	0.472103i
$101$	1.63451e9			1.56294			0.781470	−	0.623943i	$-0.214470\pi$
$101$	1.63451e9			1.56294			0.781470	+	0.623943i	$0.214470\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$			2.33572e9i			1.95781i
$105$		0			0
$106$	3.84629e8			0.295914
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		−	1.46065e9i		−	0.991124i	−0.868573	−	0.495562i	$-0.834962\pi$
$109$		−	1.46065e9i		−	0.991124i	0.868573	−	0.495562i	$-0.165038\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.38222e9	+	1.38222e9i	−0.797490	+	0.797490i	−0.982699	−	0.185209i	$-0.940704\pi$
$113$	−1.38222e9	+	1.38222e9i	−0.797490	+	0.797490i	0.185209	+	0.982699i	$0.440704\pi$
$114$		0			0
$115$		0			0
$116$	1.08896e9			0.558407
$117$	2.80603e9	+	2.80603e9i	1.38438	+	1.38438i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.35795e9			1.00000
$122$	−3.45885e9	−	3.45885e9i	−1.41355	−	1.41355i
$123$		0			0
$124$		0			0
$125$	−2.72645e9	−	1.30656e8i	−0.998854	−	0.0478669i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−2.14748e9	+	2.14748e9i	−0.707107	+	0.707107i
$129$		0			0
$130$	3.27548e9	−	5.46978e9i	1.00584	−	1.67967i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−7.88786e9			−1.97713
$137$	4.13420e9	+	4.13420e9i	1.00265	+	1.00265i	0.999996	+	0.00265362i	$0.000844674\pi$
$137$	4.13420e9	+	4.13420e9i	1.00265	+	1.00265i	0.00265362	+	0.999996i	$0.499155\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			5.15978e9i			1.00000i
$145$	−2.55012e9	−	1.52710e9i	−0.479077	−	0.286887i
$146$	−8.41201e9			−1.53219
$147$		0			0
$148$	6.30842e9	−	6.30842e9i	1.08079	−	1.08079i
$149$		−	8.51916e9i		−	1.41599i	−0.706220	−	0.707993i	$-0.749601\pi$
$149$		−	8.51916e9i		−	1.41599i	0.706220	−	0.707993i	$-0.250399\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−9.47612e9	+	9.47612e9i	−1.39804	+	1.39804i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.08073e9	+	9.08073e9i	1.19281	+	1.19281i	0.976274	+	0.216540i	$0.0694770\pi$
$157$	9.08073e9	+	9.08073e9i	1.19281	+	1.19281i	0.216540	+	0.976274i	$0.430523\pi$
$158$		0			0
$159$		0			0
$160$	8.04048e9	−	2.01746e9i	0.969934	−	0.243369i
$161$		0			0
$162$	6.19873e9	+	6.19873e9i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$			3.87170e9i			0.417931i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	3.00428e10i		−	2.83302i
$170$	1.84718e10	+	1.10615e10i	1.69624	+	1.01577i
$171$		0			0
$172$		0			0
$173$	9.24846e9	−	9.24846e9i	0.784986	−	0.784986i	−0.195682	−	0.980667i	$-0.562692\pi$
$173$	9.24846e9	−	9.24846e9i	0.784986	−	0.784986i	0.980667	+	0.195682i	$0.0626919\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−5.86835e9	+	5.86835e9i	−0.438153	+	0.438153i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	7.23579e9	−	1.20832e10i	0.513759	−	0.857935i
$181$	−8.86894e9			−0.614212			−0.307106	−	0.951675i	$-0.599361\pi$
$181$	−8.86894e9			−0.614212			−0.307106	+	0.951675i	$0.599361\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.36196e10	+	5.92646e9i	−1.48252	+	0.371983i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−1.84837e10	+	1.84837e10i	−0.958919	+	0.958919i	−0.999189	−	0.0402702i	$-0.987178\pi$
$193$	−1.84837e10	+	1.84837e10i	−0.958919	+	0.958919i	0.0402702	+	0.999189i	$0.487178\pi$
$194$			3.89354e10i			1.97350i
$195$		0			0
$196$	−2.06610e10			−1.00000
$197$	4.27924e9	+	4.27924e9i	0.202427	+	0.202427i	0.801039	−	0.598612i	$-0.204281\pi$
$197$	4.27924e9	+	4.27924e9i	0.202427	+	0.202427i	−0.598612	+	0.801039i	$0.704281\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−2.16583e10	−	6.55104e9i	−0.957173	−	0.289518i
$201$		0			0
$202$	−2.61522e10	−	2.61522e10i	−1.10517	−	1.10517i
$203$		0			0
$204$		0			0
$205$	5.42945e9	−	9.06673e9i	0.214716	−	0.358557i
$206$		0			0
$207$		0			0
$208$	3.73715e10	−	3.73715e10i	1.38438	−	1.38438i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−6.15407e9	−	6.15407e9i	−0.209243	−	0.209243i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.33704e10	+	2.33704e10i	−0.700830	+	0.700830i
$219$		0			0
$220$		0			0
$221$	1.37268e11			3.87084
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	−3.38895e10	+	1.81492e10i	−0.881543	+	0.472103i
$226$	4.42311e10			1.12782
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			7.73612e10i			1.85893i	0.368907	+	0.929466i	$0.379732\pi$
$229$			7.73612e10i			1.85893i	−0.368907	+	0.929466i	$0.620268\pi$
$230$		0			0
$231$		0			0
$232$	−1.74234e10	−	1.74234e10i	−0.394854	−	0.394854i
$233$	3.23224e10	−	3.23224e10i	0.718458	−	0.718458i	−0.249831	−	0.968289i	$-0.580375\pi$
$233$	3.23224e10	−	3.23224e10i	0.718458	−	0.718458i	0.968289	+	0.249831i	$0.0803751\pi$
$234$		−	8.97929e10i		−	1.95781i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−7.48549e10			−1.42937			−0.714683	−	0.699448i	$-0.753429\pi$
$241$	−7.48549e10			−1.42937			−0.714683	+	0.699448i	$0.753429\pi$
$242$	−3.77272e10	−	3.77272e10i	−0.707107	−	0.707107i
$243$		0			0
$244$			1.10683e11i			1.99907i
$245$	4.83840e10	+	2.89739e10i	0.857935	+	0.513759i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	4.15326e10	+	4.57136e10i	0.672449	+	0.740143i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	6.87195e10			1.00000
$257$	−9.64412e10	−	9.64412e10i	−1.37900	−	1.37900i	−0.846314	−	0.532684i	$-0.821183\pi$
$257$	−9.64412e10	−	9.64412e10i	−1.37900	−	1.37900i	−0.532684	−	0.846314i	$-0.678817\pi$
$258$		0			0
$259$		0			0
$260$	−1.39924e11	+	3.51088e10i	−1.89895	+	0.476470i
$261$	−4.18633e10			−0.558407
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	5.78146e9	+	2.30417e10i	0.0720164	+	0.287017i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.59382e11i		−	1.85590i	−0.372707	−	0.927949i	$-0.621570\pi$
$269$		−	1.59382e11i		−	1.85590i	0.372707	−	0.927949i	$-0.378430\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	1.26206e11	+	1.26206e11i	1.39804	+	1.39804i
$273$		0			0
$274$		−	1.32295e11i		−	1.41796i
$275$		0			0
$276$		0			0
$277$	−4.82635e10	−	4.82635e10i	−0.492561	−	0.492561i	0.416551	−	0.909112i	$-0.363239\pi$
$277$	−4.82635e10	−	4.82635e10i	−0.492561	−	0.492561i	−0.909112	+	0.416551i	$0.863239\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.91234e11			−1.82973			−0.914863	−	0.403765i	$-0.867701\pi$
$281$	−1.91234e11			−1.82973			−0.914863	+	0.403765i	$0.867701\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	8.25565e10	−	8.25565e10i	0.707107	−	0.707107i
$289$			3.44975e11i			2.90903i
$290$	1.63684e10	+	6.52355e10i	0.135899	+	0.541618i
$291$		0			0
$292$	1.34592e11	+	1.34592e11i	1.08342	+	1.08342i
$293$	1.72014e11	−	1.72014e11i	1.36351	−	1.36351i	0.494116	−	0.869396i	$-0.335492\pi$
$293$	1.72014e11	−	1.72014e11i	1.36351	−	1.36351i	0.869396	−	0.494116i	$-0.164508\pi$
$294$		0			0
$295$		0			0
$296$	−2.01869e11			−1.52847
$297$		0			0
$298$	−1.36307e11	+	1.36307e11i	−1.00125	+	1.00125i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.55216e11	−	2.59198e11i	1.02704	−	1.71507i
$306$	3.03236e11			1.97713
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−8.45520e10	+	8.45520e10i	−0.497937	+	0.497937i	−0.910795	−	0.412858i	$-0.864531\pi$
$313$	−8.45520e10	+	8.45520e10i	−0.497937	+	0.497937i	0.412858	+	0.910795i	$0.364531\pi$
$314$		−	2.90583e11i		−	1.68689i
$315$		0			0
$316$		0			0
$317$	2.23630e11	+	2.23630e11i	1.24383	+	1.24383i	0.958398	+	0.285437i	$0.0921387\pi$
$317$	2.23630e11	+	2.23630e11i	1.24383	+	1.24383i	0.285437	+	0.958398i	$0.407861\pi$
$318$		0			0
$319$		0			0
$320$	−1.60927e11	−	9.63683e10i	−0.857935	−	0.513759i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	1.98359e11i		−	1.00000i
$325$	3.76909e11	+	1.14004e11i	1.87396	+	0.566821i
$326$		0			0
$327$		0			0
$328$	6.19472e10	−	6.19472e10i	0.295522	−	0.295522i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−2.42517e11	+	2.42517e11i	−1.08079	+	1.08079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.11724e11	−	2.11724e11i	−0.894203	−	0.894203i	0.100713	−	0.994916i	$-0.467888\pi$
$337$	−2.11724e11	−	2.11724e11i	−0.894203	−	0.894203i	−0.994916	+	0.100713i	$0.967888\pi$
$338$	−4.80684e11	+	4.80684e11i	−2.00325	+	2.00325i
$339$		0			0
$340$	−1.18564e11	−	4.72532e11i	−0.481171	−	1.91768i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.95951e11			−1.11014
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			4.22853e11i			1.52572i	0.646563	+	0.762860i	$0.276206\pi$
$349$			4.22853e11i			1.52572i	−0.646563	+	0.762860i	$0.723794\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.30320e10	−	1.30320e10i	0.0446710	−	0.0446710i	−0.684418	−	0.729089i	$-0.739944\pi$
$353$	1.30320e10	−	1.30320e10i	0.0446710	−	0.0446710i	0.729089	+	0.684418i	$0.239944\pi$
$354$		0			0
$355$		0			0
$356$	1.87787e11			0.619642
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−3.09103e11	+	7.75579e10i	−0.969934	+	0.243369i
$361$	−3.22688e11			−1.00000
$362$	1.41903e11	+	1.41903e11i	0.434314	+	0.434314i
$363$		0			0
$364$		0			0
$365$	−1.26443e11	−	5.03932e11i	−0.372887	−	1.48612i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		−	1.48841e11i		−	0.417931i
$370$	4.72737e11	+	2.83090e11i	1.31133	+	0.785267i
$371$		0			0
$372$		0			0
$373$	2.93692e11	−	2.93692e11i	0.785602	−	0.785602i	−0.195168	−	0.980770i	$-0.562525\pi$
$373$	2.93692e11	−	2.93692e11i	0.785602	−	0.785602i	0.980770	+	0.195168i	$0.0625253\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.03210e11	+	3.03210e11i	0.773049	+	0.773049i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	5.91479e11			1.35612
$387$		0			0
$388$	6.22966e11	−	6.22966e11i	1.39547	−	1.39547i
$389$			6.59929e10i			0.146125i	0.997327	+	0.0730625i	$0.0232772\pi$
$389$			6.59929e10i			0.146125i	−0.997327	+	0.0730625i	$0.976723\pi$
$390$		0			0
$391$		0			0
$392$	3.30577e11	+	3.30577e11i	0.707107	+	0.707107i
$393$		0			0
$394$		−	1.36936e11i		−	0.286275i
$395$		0			0
$396$		0			0
$397$	−6.45596e11	−	6.45596e11i	−1.30438	−	1.30438i	−0.925409	−	0.378970i	$-0.876278\pi$
$397$	−6.45596e11	−	6.45596e11i	−1.30438	−	1.30438i	−0.378970	−	0.925409i	$-0.623722\pi$
$398$		0			0
$399$		0			0
$400$	2.41717e11	+	4.51350e11i	0.472103	+	0.881543i
$401$	4.50088e11			0.869257			0.434628	−	0.900610i	$-0.356880\pi$
$401$	4.50088e11			0.869257			0.434628	+	0.900610i	$0.356880\pi$
$402$		0			0
$403$		0			0
$404$			8.36871e11i			1.56294i
$405$	−2.78168e11	+	4.64517e11i	−0.513759	+	0.857935i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	1.10183e12i		−	1.94697i	−0.228744	−	0.973487i	$-0.573462\pi$
$409$		−	1.10183e12i		−	1.94697i	0.228744	−	0.973487i	$-0.426538\pi$
$410$	−2.31939e11	+	5.81965e10i	−0.405365	+	0.101711i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.19589e12			−1.95781
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−5.89774e11			−0.914990			−0.457495	−	0.889212i	$-0.651253\pi$
$421$	−5.89774e11			−0.914990			−0.457495	+	0.889212i	$0.651253\pi$
$422$		0			0
$423$		0			0
$424$			1.96930e11i			0.295914i
$425$	−3.84999e11	+	1.27284e12i	−0.572413	+	1.89245i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	3.53501e10	−	3.53501e10i	0.0483276	−	0.0483276i	−0.682530	−	0.730858i	$-0.739120\pi$
$433$	3.53501e10	−	3.53501e10i	0.0483276	−	0.0483276i	0.730858	+	0.682530i	$0.239120\pi$
$434$		0			0
$435$		0			0
$436$	7.47854e11			0.991124
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	7.94280e11			1.00000
$442$	−2.19629e12	−	2.19629e12i	−2.73710	−	2.73710i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	−4.39759e11	−	2.63342e11i	−0.531612	−	0.318347i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.91685e11i			0.222577i	0.993788	+	0.111288i	$0.0354977\pi$
$449$			1.91685e11i			0.222577i	−0.993788	+	0.111288i	$0.964502\pi$
$450$	8.32620e11	+	2.51844e11i	0.957173	+	0.289518i
$451$		0			0
$452$	−7.07698e11	−	7.07698e11i	−0.797490	−	0.797490i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.03996e12	+	1.03996e12i	1.11531	+	1.11531i	0.992421	+	0.122886i	$0.0392149\pi$
$457$	1.03996e12	+	1.03996e12i	1.11531	+	1.11531i	0.122886	+	0.992421i	$0.460785\pi$
$458$	1.23778e12	−	1.23778e12i	1.31446	−	1.31446i
$459$		0			0
$460$		0			0
$461$	6.68844e11			0.689716			0.344858	−	0.938655i	$-0.387927\pi$
$461$	6.68844e11			0.689716			0.344858	+	0.938655i	$0.387927\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$			5.57548e11i			0.558407i
$465$		0			0
$466$	−1.03432e12			−1.01605
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−1.43669e12	+	1.43669e12i	−1.38438	+	1.38438i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.36583e11	+	2.36583e11i	0.209243	+	0.209243i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	3.51302e12			2.99246
$482$	1.19768e12	+	1.19768e12i	1.01071	+	1.01071i
$483$		0			0
$484$			1.20727e12i			1.00000i
$485$	−2.33247e12	+	5.85248e11i	−1.91416	+	0.480288i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	1.77093e12	−	1.77093e12i	1.41355	−	1.41355i
$489$		0			0
$490$	−3.10561e11	−	1.23773e12i	−0.243369	−	0.969934i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−1.02396e12	+	1.02396e12i	−0.780675	+	0.780675i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	6.68960e10	−	1.39594e12i	0.0478669	−	0.998854i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	1.17358e12	−	1.95978e12i	0.802974	−	1.34090i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	1.29141e12i		−	0.852771i	−0.904542	−	0.426386i	$-0.859787\pi$
$509$		−	1.29141e12i		−	0.852771i	0.904542	−	0.426386i	$-0.140213\pi$
$510$		0			0
$511$		0			0
$512$	−1.09951e12	−	1.09951e12i	−0.707107	−	0.707107i
$513$		0			0
$514$			3.08612e12i			1.95020i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	2.80053e12	+	1.67705e12i	1.67967	+	1.00584i
$521$	−3.30505e12			−1.96521			−0.982604	−	0.185716i	$-0.940540\pi$
$521$	−3.30505e12			−1.96521			−0.982604	+	0.185716i	$0.940540\pi$
$522$	6.69813e11	+	6.69813e11i	0.394854	+	0.394854i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	1.80115e12i		−	1.00000i
$530$	2.76164e11	−	4.61171e11i	0.152029	−	0.253875i
$531$		0			0
$532$		0			0
$533$	−1.07803e12	+	1.07803e12i	−0.578575	+	0.578575i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−2.55011e12	+	2.55011e12i	−1.31232	+	1.31232i
$539$		0			0
$540$		0			0
$541$	−2.60539e12			−1.30763			−0.653815	−	0.756654i	$-0.726833\pi$
$541$	−2.60539e12			−1.30763			−0.653815	+	0.756654i	$0.726833\pi$
$542$		0			0
$543$		0			0
$544$		−	4.03859e12i		−	1.97713i
$545$	−1.75132e12	−	1.04875e12i	−0.850319	−	0.509199i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−2.11671e12	+	2.11671e12i	−1.00265	+	1.00265i
$549$		−	4.25503e12i		−	1.99907i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			1.54443e12i			0.696587i
$555$		0			0
$556$		0			0
$557$	6.93109e11	+	6.93109e11i	0.305108	+	0.305108i	0.843008	−	0.537901i	$-0.180782\pi$
$557$	6.93109e11	+	6.93109e11i	0.305108	+	0.305108i	−0.537901	+	0.843008i	$0.680782\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.05974e12	+	3.05974e12i	1.29381	+	1.29381i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	6.64849e11	+	2.64972e12i	0.274477	+	1.09391i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.44716e12i			1.77860i	0.457328	+	0.889298i	$0.348807\pi$
$569$			4.44716e12i			1.77860i	−0.457328	+	0.889298i	$0.651193\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.64181e12			−1.00000
$577$	1.50303e12	+	1.50303e12i	0.564516	+	0.564516i	0.930587	−	0.366071i	$-0.119297\pi$
$577$	1.50303e12	+	1.50303e12i	0.564516	+	0.564516i	−0.366071	+	0.930587i	$0.619297\pi$
$578$	5.51960e12	−	5.51960e12i	2.05699	−	2.05699i
$579$		0			0
$580$	7.81874e11	−	1.30566e12i	0.286887	−	0.479077i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	4.30695e12i		−	1.53219i
$585$	5.37916e12	−	1.34970e12i	1.89895	−	0.476470i
$586$	−5.50444e12			−1.92830
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	3.22991e12	+	3.22991e12i	1.08079	+	1.08079i
$593$	2.59898e12	−	2.59898e12i	0.863092	−	0.863092i	−0.128604	−	0.991696i	$-0.541050\pi$
$593$	2.59898e12	−	2.59898e12i	0.863092	−	0.863092i	0.991696	+	0.128604i	$0.0410496\pi$
$594$		0			0
$595$		0			0
$596$	4.36181e12			1.41599
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	1.75404e12			0.548409			0.274204	−	0.961671i	$-0.411586\pi$
$601$	1.75404e12			0.548409			0.274204	+	0.961671i	$0.411586\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.69301e12	−	2.82718e12i	0.513759	−	0.857935i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	−6.63061e12	+	1.66371e12i	−1.93896	+	0.486511i
$611$		0			0
$612$	−4.85178e12	−	4.85178e12i	−1.39804	−	1.39804i
$613$	−1.25704e12	+	1.25704e12i	−0.359565	+	0.359565i	−0.863653	−	0.504088i	$-0.831829\pi$
$613$	−1.25704e12	+	1.25704e12i	−0.359565	+	0.359565i	0.504088	+	0.863653i	$0.331829\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.54233e12	+	3.54233e12i	0.984025	+	0.984025i	0.999874	−	0.0158499i	$-0.00504538\pi$
$617$	3.54233e12	+	3.54233e12i	0.984025	+	0.984025i	−0.0158499	+	0.999874i	$0.505045\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.11425e12	+	3.17520e12i	−0.554237	+	0.832359i
$626$	2.70566e12			0.704189
$627$		0			0
$628$	−4.64934e12	+	4.64934e12i	−1.19281	+	1.19281i
$629$			1.18637e13i			3.02198i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	7.15615e12i		−	1.75905i
$635$		0			0
$636$		0			0
$637$	−5.75285e12	−	5.75285e12i	−1.38438	−	1.38438i
$638$		0			0
$639$		0			0
$640$	1.03294e12	+	4.11673e12i	0.243369	+	0.969934i
$641$	−8.46139e12			−1.97961			−0.989807	−	0.142413i	$-0.954514\pi$
$641$	−8.46139e12			−1.97961			−0.989807	+	0.142413i	$0.954514\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−3.17375e12	+	3.17375e12i	−0.707107	+	0.707107i
$649$		0			0
$650$	−4.20647e12	−	7.85460e12i	−0.924289	−	1.72589i
$651$		0			0
$652$		0			0
$653$	5.04927e12	−	5.04927e12i	1.08672	−	1.08672i	0.0908593	−	0.995864i	$-0.471039\pi$
$653$	5.04927e12	−	5.04927e12i	1.08672	−	1.08672i	0.995864	−	0.0908593i	$-0.0289613\pi$
$654$		0			0
$655$		0			0
$656$	−1.98231e12			−0.417931
$657$	−5.17418e12	−	5.17418e12i	−1.08342	−	1.08342i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	8.37299e12			1.70598			0.852990	−	0.521927i	$-0.174787\pi$
$661$	8.37299e12			1.70598			0.852990	+	0.521927i	$0.174787\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	7.76054e12			1.52847
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.93778e12	+	2.93778e12i	−0.552016	+	0.552016i	−0.927022	−	0.375006i	$-0.877641\pi$
$673$	−2.93778e12	+	2.93778e12i	−0.552016	+	0.552016i	0.375006	+	0.927022i	$0.377641\pi$
$674$			6.77518e12i			1.26459i
$675$		0			0
$676$	1.53819e13			2.83302
$677$	−6.99541e12	−	6.99541e12i	−1.27986	−	1.27986i	−0.940742	−	0.339122i	$-0.889870\pi$
$677$	−6.99541e12	−	6.99541e12i	−1.27986	−	1.27986i	−0.339122	−	0.940742i	$-0.610130\pi$
$678$		0			0
$679$		0			0
$680$	−5.66349e12	+	9.45755e12i	−1.01577	+	1.69624i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	7.92527e12	−	1.98855e12i	1.37533	−	0.345088i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	3.42707e12i		−	0.579344i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	4.73521e12	+	4.73521e12i	0.784986	+	0.784986i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.64058e12	−	3.64058e12i	−0.584283	−	0.584283i
$698$	6.76565e12	−	6.76565e12i	1.07885	−	1.07885i
$699$		0			0
$700$		0			0
$701$	−1.77289e12			−0.277300			−0.138650	−	0.990341i	$-0.544276\pi$
$701$	−1.77289e12			−0.277300			−0.138650	+	0.990341i	$0.544276\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−4.17025e11			−0.0631743
$707$		0			0
$708$		0			0
$709$			8.39203e12i			1.24727i	0.781718	+	0.623633i	$0.214344\pi$
$709$			8.39203e12i			1.24727i	−0.781718	+	0.623633i	$0.785656\pi$
$710$		0			0
$711$		0			0
$712$	−3.00460e12	−	3.00460e12i	−0.438153	−	0.438153i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	6.18658e12	+	3.70472e12i	0.857935	+	0.513759i
$721$		0			0
$722$	5.16300e12	+	5.16300e12i	0.707107	+	0.707107i
$723$		0			0
$724$		−	4.54090e12i		−	0.614212i
$725$	−3.66198e12	+	1.96114e12i	−0.492260	+	0.263626i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			7.62560e12i			1.00000i
$730$	−6.03982e12	+	1.00860e13i	−0.787175	+	1.31452i
$731$		0			0
$732$		0			0
$733$	1.09737e13	−	1.09737e13i	1.40405	−	1.40405i	0.617414	−	0.786638i	$-0.288180\pi$
$733$	1.09737e13	−	1.09737e13i	1.40405	−	1.40405i	0.786638	−	0.617414i	$-0.211820\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−2.38146e12	+	2.38146e12i	−0.295522	+	0.295522i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−3.03435e12	−	1.20932e13i	−0.371983	−	1.48252i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−1.02145e13	−	6.11676e12i	−1.21482	−	0.727475i
$746$	−9.39814e12			−1.11101
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		−	9.70271e12i		−	1.09326i
$755$		0			0
$756$		0			0
$757$	1.02258e13	+	1.02258e13i	1.13179	+	1.13179i	0.989879	+	0.141913i	$0.0453254\pi$
$757$	1.02258e13	+	1.02258e13i	1.13179	+	1.13179i	0.141913	+	0.989879i	$0.454675\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.74541e12			−1.05334			−0.526671	−	0.850069i	$-0.676560\pi$
$761$	−9.74541e12			−1.05334			−0.526671	+	0.850069i	$0.676560\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	4.55802e12	+	1.81657e13i	0.481171	+	1.91768i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	1.50237e13i		−	1.54921i	−0.632448	−	0.774603i	$-0.717950\pi$
$769$		−	1.50237e13i		−	1.54921i	0.632448	−	0.774603i	$-0.282050\pi$
$770$		0			0
$771$		0			0
$772$	−9.46367e12	−	9.46367e12i	−0.958919	−	0.958919i
$773$	5.76549e12	−	5.76549e12i	0.580803	−	0.580803i	−0.354321	−	0.935124i	$-0.615288\pi$
$773$	5.76549e12	−	5.76549e12i	0.580803	−	0.580803i	0.935124	+	0.354321i	$0.115288\pi$
$774$		0			0
$775$		0			0
$776$	−1.99349e13			−1.97350
$777$		0			0
$778$	1.05589e12	−	1.05589e12i	0.103326	−	0.103326i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	1.05785e13i		−	1.00000i
$785$	1.74078e13	−	4.36783e12i	1.63617	−	0.410537i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−2.19097e12	+	2.19097e12i	−0.202427	+	0.202427i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.08186e13	+	3.08186e13i	−2.76747	+	2.76747i
$794$			2.06591e13i			1.84467i
$795$		0			0
$796$		0			0
$797$	1.52669e13	+	1.52669e13i	1.34026	+	1.34026i	0.895798	+	0.444462i	$0.146605\pi$
$797$	1.52669e13	+	1.52669e13i	1.34026	+	1.34026i	0.444462	+	0.895798i	$0.353395\pi$
$798$		0			0
$799$		0			0
$800$	3.35413e12	−	1.10891e13i	0.289518	−	0.957173i
$801$	−7.21917e12			−0.619642
$802$	−7.20141e12	−	7.20141e12i	−0.614657	−	0.614657i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.33899e13	−	1.33899e13i	1.10517	−	1.10517i
$809$			4.77154e11i			0.0391643i	0.999808	+	0.0195821i	$0.00623358\pi$
$809$			4.77154e11i			0.0391643i	−0.999808	+	0.0195821i	$0.993766\pi$
$810$	1.18830e13	−	2.98159e12i	0.969934	−	0.243369i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.76293e13	+	1.76293e13i	−1.37672	+	1.37672i
$819$		0			0
$820$	4.64217e12	+	2.77988e12i	0.358557	+	0.214716i
$821$	2.58550e13			1.98609			0.993047	−	0.117720i	$-0.0375587\pi$
$821$	2.58550e13			1.98609			0.993047	+	0.117720i	$0.0375587\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		−	2.71622e13i		−	1.99742i	−0.0507364	−	0.998712i	$-0.516157\pi$
$829$		−	2.71622e13i		−	1.99742i	0.0507364	−	0.998712i	$-0.483843\pi$
$830$		0			0
$831$		0			0
$832$	1.91342e13	+	1.91342e13i	1.38438	+	1.38438i
$833$	1.94277e13	−	1.94277e13i	1.39804	−	1.39804i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	9.98354e12			0.688181
$842$	9.43639e12	+	9.43639e12i	0.646996	+	0.646996i
$843$		0			0
$844$		0			0
$845$	−3.60213e13	−	2.15707e13i	−2.43055	−	1.45549i
$846$		0			0
$847$		0			0
$848$	3.15088e12	−	3.15088e12i	0.209243	−	0.209243i
$849$		0			0
$850$	2.65255e13	−	1.42055e13i	1.74292	−	0.933408i
$851$		0			0
$852$		0			0
$853$	−1.00710e13	+	1.00710e13i	−0.651333	+	0.651333i	−0.953314	−	0.301981i	$-0.902352\pi$
$853$	−1.00710e13	+	1.00710e13i	−0.651333	+	0.651333i	0.301981	+	0.953314i	$0.402352\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.01263e13	+	2.01263e13i	1.27453	+	1.27453i	0.943684	+	0.330847i	$0.107334\pi$
$857$	2.01263e13	+	2.01263e13i	1.27453	+	1.27453i	0.330847	+	0.943684i	$0.392666\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	−4.44851e12	−	1.77293e13i	−0.270173	−	1.07676i
$866$	−1.13120e12			−0.0683455
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.19657e13	−	1.19657e13i	−0.700830	−	0.700830i
$873$	−2.39489e13	+	2.39489e13i	−1.39547	+	1.39547i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.14998e13	−	2.14998e13i	−1.22726	−	1.22726i	−0.964997	−	0.262262i	$-0.915532\pi$
$877$	−2.14998e13	−	2.14998e13i	−1.22726	−	1.22726i	−0.262262	−	0.964997i	$-0.584468\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.27776e13			−1.83310			−0.916550	−	0.399920i	$-0.869038\pi$
$881$	−3.27776e13			−1.83310			−0.916550	+	0.399920i	$0.869038\pi$
$882$	−1.27085e13	−	1.27085e13i	−0.707107	−	0.707107i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$			7.02814e13i			3.87084i
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$	2.82268e12	+	1.12496e13i	0.150802	+	0.601012i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	3.06696e12	−	3.06696e12i	0.157385	−	0.157385i
$899$		0			0
$900$	−9.29241e12	−	1.73514e13i	−0.472103	−	0.881543i
$901$	1.15734e13			0.585060
$902$		0			0
$903$		0			0
$904$			2.26463e13i			1.12782i
$905$	−6.36790e12	+	1.06339e13i	−0.315557	+	0.526954i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	3.21721e13i		−	1.56294i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	3.32788e13i		−	1.57728i
$915$		0			0
$916$	−3.96089e13			−1.85893
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	−1.07015e13	−	1.07015e13i	−0.487703	−	0.487703i
$923$		0			0
$924$		0			0
$925$	−9.85305e12	+	3.25751e13i	−0.442520	+	1.46301i
$926$		0			0
$927$		0			0
$928$	8.92076e12	−	8.92076e12i	0.394854	−	0.394854i
$929$		−	4.48355e13i		−	1.97493i	−0.157844	−	0.987464i	$-0.550454\pi$
$929$		−	4.48355e13i		−	1.97493i	0.157844	−	0.987464i	$-0.449546\pi$
$930$		0			0
$931$		0			0
$932$	1.65491e13	+	1.65491e13i	0.718458	+	0.718458i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	4.59740e13			1.95781
$937$	1.68393e13	+	1.68393e13i	0.713667	+	0.713667i	0.967300	−	0.253634i	$-0.0816257\pi$
$937$	1.68393e13	+	1.68393e13i	0.713667	+	0.713667i	−0.253634	+	0.967300i	$0.581626\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.75419e13			1.97662			0.988310	−	0.152461i	$-0.0487197\pi$
$941$	4.75419e13			1.97662			0.988310	+	0.152461i	$0.0487197\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$			7.49515e13i			2.99973i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.60068e13	−	3.60068e13i	1.41406	−	1.41406i	0.696451	−	0.717605i	$-0.254762\pi$
$953$	3.60068e13	−	3.60068e13i	1.41406	−	1.41406i	0.717605	−	0.696451i	$-0.245238\pi$
$954$		−	7.57066e12i		−	0.295914i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.64396e13			1.00000
$962$	−5.62084e13	−	5.62084e13i	−2.11599	−	2.11599i
$963$		0			0
$964$		−	3.83257e13i		−	1.42937i
$965$	8.89068e12	+	3.54333e13i	0.330036	+	1.31534i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	1.93163e13	−	1.93163e13i	0.707107	−	0.707107i
$969$		0			0
$970$	4.66835e13	+	2.79556e13i	1.69313	+	1.01390i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−5.66698e13			−1.99907
$977$	1.45459e13	+	1.45459e13i	0.510756	+	0.510756i	0.914758	−	0.404002i	$-0.132381\pi$
$977$	1.45459e13	+	1.45459e13i	0.510756	+	0.510756i	−0.404002	+	0.914758i	$0.632381\pi$
$978$		0			0
$979$		0			0
$980$	−1.48346e13	+	2.47726e13i	−0.513759	+	0.857935i
$981$	−2.87500e13			−0.991124
$982$		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
$985$	8.20331e12	−	2.05832e12i	0.277668	−	0.0696705i
$986$	3.27666e13			1.10404
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.61904e13	−	2.61904e13i	−0.839488	−	0.839488i	0.149303	−	0.988791i	$-0.452297\pi$
$997$	−2.61904e13	−	2.61904e13i	−0.839488	−	0.839488i	−0.988791	+	0.149303i	$0.952297\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.a.7.1	yes	2
4.3	odd	2	CM	20.10.e.a.7.1	yes	2
5.2	odd	4		100.10.e.c.43.1		2
5.3	odd	4	inner	20.10.e.a.3.1	✓	2
5.4	even	2		100.10.e.c.7.1		2
20.3	even	4	inner	20.10.e.a.3.1	✓	2
20.7	even	4		100.10.e.c.43.1		2
20.19	odd	2		100.10.e.c.7.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.a.3.1	✓	2	5.3	odd	4	inner
20.10.e.a.3.1	✓	2	20.3	even	4	inner
20.10.e.a.7.1	yes	2	1.1	even	1	trivial
20.10.e.a.7.1	yes	2	4.3	odd	2	CM
100.10.e.c.7.1		2	5.4	even	2
100.10.e.c.7.1		2	20.19	odd	2
100.10.e.c.43.1		2	5.2	odd	4
100.10.e.c.43.1		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-16.0000 - 16.0000i) q^{2} +512.000i q^{4} +(718.000 - 1199.00i) q^{5} +(8192.00 - 8192.00i) q^{8} -19683.0i q^{9} +O(q^{10})\)	\(q+(-16.0000 - 16.0000i) q^{2} +512.000i q^{4} +(718.000 - 1199.00i) q^{5} +(8192.00 - 8192.00i) q^{8} -19683.0i q^{9} +(-30672.0 + 7696.00i) q^{10} +(-142561. + 142561. i) q^{13} -262144. q^{16} +(-481437. - 481437. i) q^{17} +(-314928. + 314928. i) q^{18} +(613888. + 367616. i) q^{20} +(-922077. - 1.72176e6i) q^{25} +4.56195e6 q^{26} -2.12688e6i q^{29} +(4.19430e6 + 4.19430e6i) q^{32} +1.54060e7i q^{34} +1.00777e7 q^{36} +(-1.23211e7 - 1.23211e7i) q^{37} +(-3.94035e6 - 1.57041e7i) q^{40} +7.56191e6 q^{41} +(-2.35999e7 - 1.41324e7i) q^{45} +4.03536e7i q^{49} +(-1.27950e7 + 4.23015e7i) q^{50} +(-7.29912e7 - 7.29912e7i) q^{52} +(-1.20197e7 + 1.20197e7i) q^{53} +(-3.40300e7 + 3.40300e7i) q^{58} +2.16178e8 q^{61} -1.34218e8i q^{64} +(6.85718e7 + 2.73289e8i) q^{65} +(2.46496e8 - 2.46496e8i) q^{68} +(-1.61243e8 - 1.61243e8i) q^{72} +(2.62875e8 - 2.62875e8i) q^{73} +3.94276e8i q^{74} +(-1.88219e8 + 3.14311e8i) q^{80} -3.87420e8 q^{81} +(-1.20991e8 - 1.20991e8i) q^{82} +(-9.22915e8 + 2.31571e8i) q^{85} -3.66772e8i q^{89} +(1.51480e8 + 6.03717e8i) q^{90} +(-1.21673e9 - 1.21673e9i) q^{97} +(6.45658e8 - 6.45658e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{2} + 1436 q^{5} + 16384 q^{8}+O(q^{10}) \) 2 * q - 32 * q^2 + 1436 * q^5 + 16384 * q^8	\( 2 q - 32 q^{2} + 1436 q^{5} + 16384 q^{8} - 61344 q^{10} - 285122 q^{13} - 524288 q^{16} - 962874 q^{17} - 629856 q^{18} + 1227776 q^{20} - 1844154 q^{25} + 9123904 q^{26} + 8388608 q^{32} + 20155392 q^{36} - 24642254 q^{37} - 7880704 q^{40} + 15123824 q^{41} - 47199834 q^{45} - 25589984 q^{50} - 145982464 q^{52} - 24039342 q^{53} - 68060032 q^{58} + 432356184 q^{61} + 137143682 q^{65} + 492991488 q^{68} - 322486272 q^{72} + 525750698 q^{73} - 376438784 q^{80} - 774840978 q^{81} - 241981184 q^{82} - 1845829458 q^{85} + 302960736 q^{90} - 2433462694 q^{97} + 1291315424 q^{98}+O(q^{100}) \) 2 * q - 32 * q^2 + 1436 * q^5 + 16384 * q^8 - 61344 * q^10 - 285122 * q^13 - 524288 * q^16 - 962874 * q^17 - 629856 * q^18 + 1227776 * q^20 - 1844154 * q^25 + 9123904 * q^26 + 8388608 * q^32 + 20155392 * q^36 - 24642254 * q^37 - 7880704 * q^40 + 15123824 * q^41 - 47199834 * q^45 - 25589984 * q^50 - 145982464 * q^52 - 24039342 * q^53 - 68060032 * q^58 + 432356184 * q^61 + 137143682 * q^65 + 492991488 * q^68 - 322486272 * q^72 + 525750698 * q^73 - 376438784 * q^80 - 774840978 * q^81 - 241981184 * q^82 - 1845829458 * q^85 + 302960736 * q^90 - 2433462694 * q^97 + 1291315424 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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