LMFDB - Embedded newform 20.40.e.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,40,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, 40, names="a")

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 40);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 40 $
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:	$192.679102779$
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			3.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	20.3
Dual form			20.40.e.a.7.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-524288. + 524288. i) q^{2} -5.49756e11i q^{4} +(-3.06987e13 - 2.96071e13i) q^{5} +(2.88230e17 + 2.88230e17i) q^{8} +4.05256e18i q^{9} +O(q^{10})$	\(q+(-524288. + 524288. i) q^{2} -5.49756e11i q^{4} +(-3.06987e13 - 2.96071e13i) q^{5} +(2.88230e17 + 2.88230e17i) q^{8} +4.05256e18i q^{9} +(3.16176e19 - 5.72309e17i) q^{10} +(7.36435e21 + 7.36435e21i) q^{13} -3.02231e23 q^{16} +(-1.10467e24 + 1.10467e24i) q^{17} +(-2.12471e24 - 2.12471e24i) q^{18} +(-1.62767e25 + 1.68768e25i) q^{20} +(6.58294e25 + 1.81780e27i) q^{25} -7.72208e27 q^{26} +5.01667e28i q^{29} +(1.58456e29 - 1.58456e29i) q^{32} -1.15833e30i q^{34} +2.22792e30 q^{36} +(-3.15754e30 + 3.15754e30i) q^{37} +(-3.14630e29 - 1.73820e31i) q^{40} +5.20783e31 q^{41} +(1.19984e32 - 1.24408e32i) q^{45} -9.09544e32i q^{49} +(-9.87563e32 - 9.18536e32i) q^{50} +(4.04860e33 - 4.04860e33i) q^{52} +(3.56493e33 + 3.56493e33i) q^{53} +(-2.63018e34 - 2.63018e34i) q^{58} -4.96918e34 q^{61} +1.66153e35i q^{64} +(-8.03888e33 - 4.44113e35i) q^{65} +(6.07298e35 + 6.07298e35i) q^{68} +(-1.16807e36 + 1.16807e36i) q^{72} +(1.82640e36 + 1.82640e36i) q^{73} -3.31092e36i q^{74} +(9.27811e36 + 8.94820e36i) q^{80} -1.64232e37 q^{81} +(-2.73040e37 + 2.73040e37i) q^{82} +(6.66179e37 - 1.20585e36i) q^{85} -2.01778e38i q^{89} +(2.31931e36 + 1.28132e38i) q^{90} +(-1.41801e38 + 1.41801e38i) q^{97} +(4.76863e38 + 4.76863e38i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1048576 q^{2} - 61397374177364 q^{5} + 57\!\cdots\!88 q^{8}+O(q^{10}) $ 2 * q - 1048576 * q^2 - 61397374177364 * q^5 + 576460752303423488 * q^8	$ 2 q - 1048576 q^{2} - 61397374177364 q^{5} + 57\!\cdots\!88 q^{8}+ \cdots + 95\!\cdots\!68 q^{98}+O(q^{100}) $ 2 * q - 1048576 * q^2 - 61397374177364 * q^5 + 576460752303423488 * q^8 + 63235194701145440256 * q^10 + 14728708243289840454646 * q^13 - 604462909807314587353088 * q^16 - 2209335626245349613005082 * q^17 - 4249412072132026058145792 * q^18 - 32553344107485460875444224 * q^20 + 131658748783530764069832246 * q^25 - 15444169974915887744570884096 * q^26 + 316912650057057350374175801344 * q^32 + 4455831512947911355946281992192 * q^36 - 6315070876415894212107982079222 * q^37 - 629260578443681810867556450304 * q^40 + 104156506162896529409552831978224 * q^41 + 239968762636254052800519097107966 * q^45 - 1975126277432191896766247609565184 * q^50 + 8097192987808700953825579680923648 * q^52 + 7129862979151660093343761890759306 * q^53 - 52603641793583427999156653076250624 * q^58 - 99383542409702374129137867076998216 * q^61 - 16077756259844071201512336322846718 * q^65 + 1214595105358266350072714684390178816 * q^68 - 2336138992260434548986364293122359296 * q^72 + 3652790957208407144721971672057732186 * q^73 + 18556217724888960595218879231102550016 * q^80 - 32846406536521316292462935601418510578 * q^81 - 54608006303132695611075635172199104512 * q^82 + 133235710607339644158613592089648668342 * q^85 + 4638628888212487180995247479047847936 * q^90 - 283601189444481118373198525559476429542 * q^97 + 953725673935849275596687298935278010368 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$11$	$17$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−524288.	+	524288.i	−0.707107	+	0.707107i
$2$	−524288.	+	524288.i	−0.707107	+	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	5.49756e11i		−	1.00000i
$5$	−3.06987e13	−	2.96071e13i	−0.719788	−	0.694194i
$5$	−3.06987e13	−	2.96071e13i	−0.719788	−	0.694194i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	2.88230e17	+	2.88230e17i	0.707107	+	0.707107i
$9$			4.05256e18i			1.00000i
$10$	3.16176e19	−	5.72309e17i	0.999836	−	0.0180980i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	7.36435e21	+	7.36435e21i	1.39714	+	1.39714i	0.808120	+	0.589018i	$0.200485\pi$
$13$	7.36435e21	+	7.36435e21i	1.39714	+	1.39714i	0.589018	+	0.808120i	$0.299515\pi$
$14$		0			0
$15$		0			0
$16$	−3.02231e23			−1.00000
$17$	−1.10467e24	+	1.10467e24i	−1.12067	+	1.12067i	−0.129029	+	0.991641i	$0.541186\pi$
$17$	−1.10467e24	+	1.10467e24i	−1.12067	+	1.12067i	−0.991641	+	0.129029i	$0.958814\pi$
$18$	−2.12471e24	−	2.12471e24i	−0.707107	−	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.62767e25	+	1.68768e25i	−0.694194	+	0.719788i
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	6.58294e25	+	1.81780e27i	0.0361901	+	0.999345i
$26$	−7.72208e27			−1.97585
$27$		0			0
$28$		0			0
$29$			5.01667e28i			1.52635i	0.646190	+	0.763176i	$0.276361\pi$
$29$			5.01667e28i			1.52635i	−0.646190	+	0.763176i	$0.723639\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	1.58456e29	−	1.58456e29i	0.707107	−	0.707107i
$33$		0			0
$34$		−	1.15833e30i		−	1.58487i
$35$		0			0
$36$	2.22792e30			1.00000
$37$	−3.15754e30	+	3.15754e30i	−0.830643	+	0.830643i	−0.987605	−	0.156961i	$-0.949830\pi$
$37$	−3.15754e30	+	3.15754e30i	−0.830643	+	0.830643i	0.156961	+	0.987605i	$0.449830\pi$
$38$		0			0
$39$		0			0
$40$	−3.14630e29	−	1.73820e31i	−0.0180980	−	0.999836i
$41$	5.20783e31			1.85085			0.925425	−	0.378931i	$-0.123708\pi$
$41$	5.20783e31			1.85085			0.925425	+	0.378931i	$0.123708\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	1.19984e32	−	1.24408e32i	0.694194	−	0.719788i
$46$		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
$49$		−	9.09544e32i		−	1.00000i
$50$	−9.87563e32	−	9.18536e32i	−0.732234	−	0.681053i
$51$		0			0
$52$	4.04860e33	−	4.04860e33i	1.39714	−	1.39714i
$53$	3.56493e33	+	3.56493e33i	0.848539	+	0.848539i	0.989951	−	0.141412i	$-0.0451642\pi$
$53$	3.56493e33	+	3.56493e33i	0.848539	+	0.848539i	−0.141412	+	0.989951i	$0.545164\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−2.63018e34	−	2.63018e34i	−1.07929	−	1.07929i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−4.96918e34			−0.762698			−0.381349	−	0.924431i	$-0.624540\pi$
$61$	−4.96918e34			−0.762698			−0.381349	+	0.924431i	$0.624540\pi$
$62$		0			0
$63$		0			0
$64$			1.66153e35i			1.00000i
$65$	−8.03888e33	−	4.44113e35i	−0.0357590	−	1.97553i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	6.07298e35	+	6.07298e35i	1.12067	+	1.12067i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−1.16807e36	+	1.16807e36i	−0.707107	+	0.707107i
$73$	1.82640e36	+	1.82640e36i	0.844888	+	0.844888i	0.989490	−	0.144602i	$-0.0461902\pi$
$73$	1.82640e36	+	1.82640e36i	0.844888	+	0.844888i	−0.144602	+	0.989490i	$0.546190\pi$
$74$		−	3.31092e36i		−	1.17471i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	9.27811e36	+	8.94820e36i	0.719788	+	0.694194i
$81$	−1.64232e37			−1.00000
$82$	−2.73040e37	+	2.73040e37i	−1.30875	+	1.30875i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	6.66179e37	−	1.20585e36i	1.58461	−	0.0286829i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.01778e38i		−	1.95781i	−0.204326	−	0.978903i	$-0.565500\pi$
$89$		−	2.01778e38i		−	1.95781i	0.204326	−	0.978903i	$-0.434500\pi$
$90$	2.31931e36	+	1.28132e38i	0.0180980	+	0.999836i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.41801e38	+	1.41801e38i	−0.256820	+	0.256820i	−0.823760	−	0.566939i	$-0.808127\pi$
$97$	−1.41801e38	+	1.41801e38i	−0.256820	+	0.256820i	0.566939	+	0.823760i	$0.308127\pi$
$98$	4.76863e38	+	4.76863e38i	0.707107	+	0.707107i
$99$		0			0
$100$	9.99345e38	−	3.61901e37i	0.999345	−	0.0361901i
$101$	1.64716e39			1.35666			0.678328	−	0.734759i	$-0.262705\pi$
$101$	1.64716e39			1.35666			0.678328	+	0.734759i	$0.262705\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			4.24526e39i			1.97585i
$105$		0			0
$106$	−3.73810e39			−1.20002
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$			1.00047e40i			1.86376i	0.362772	+	0.931878i	$0.381830\pi$
$109$			1.00047e40i			1.86376i	−0.362772	+	0.931878i	$0.618170\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.95309e39	+	7.95309e39i	0.733679	+	0.733679i	0.971347	−	0.237668i	$-0.0763829\pi$
$113$	7.95309e39	+	7.95309e39i	0.733679	+	0.733679i	−0.237668	+	0.971347i	$0.576383\pi$
$114$		0			0
$115$		0			0
$116$	2.75795e40			1.52635
$117$	−2.98445e40	+	2.98445e40i	−1.39714	+	1.39714i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.11448e40			1.00000
$122$	2.60528e40	−	2.60528e40i	0.539309	−	0.539309i
$123$		0			0
$124$		0			0
$125$	5.17988e40	−	5.77530e40i	0.667690	−	0.744440i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−8.71123e40	−	8.71123e40i	−0.707107	−	0.707107i
$129$		0			0
$130$	2.37058e41	+	2.28628e41i	1.42219	+	1.37162i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−6.36798e41			−1.58487
$137$	−6.35064e41	+	6.35064e41i	−1.37014	+	1.37014i	−0.509925	+	0.860219i	$0.670327\pi$
$137$	−6.35064e41	+	6.35064e41i	−1.37014	+	1.37014i	−0.860219	+	0.509925i	$0.829673\pi$
$138$		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
$144$		−	1.22481e42i		−	1.00000i
$145$	1.48529e42	−	1.54005e42i	1.05958	−	1.09865i
$146$	−1.91511e42			−1.19485
$147$		0			0
$148$	1.73587e42	+	1.73587e42i	0.830643	+	0.830643i
$149$		−	4.61032e42i		−	1.93463i	−0.253569	−	0.967317i	$-0.581604\pi$
$149$		−	4.61032e42i		−	1.93463i	0.253569	−	0.967317i	$-0.418396\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−4.47673e42	−	4.47673e42i	−1.12067	−	1.12067i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.49274e42	+	1.49274e42i	−0.225913	+	0.225913i	−0.810983	−	0.585070i	$-0.801067\pi$
$157$	−1.49274e42	+	1.49274e42i	−0.225913	+	0.225913i	0.585070	+	0.810983i	$0.301067\pi$
$158$		0			0
$159$		0			0
$160$	−9.55583e42	+	1.72970e41i	−0.999836	+	0.0180980i
$161$		0			0
$162$	8.61049e42	−	8.61049e42i	0.707107	−	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		−	2.86303e43i		−	1.85085i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			8.06837e43i			2.90399i
$170$	−3.42947e43	+	3.55592e43i	−1.10020	+	1.14077i
$171$		0			0
$172$		0			0
$173$	−2.72779e43	−	2.72779e43i	−0.622175	−	0.622175i	0.323912	−	0.946087i	$-0.395002\pi$
$173$	−2.72779e43	−	2.72779e43i	−0.622175	−	0.622175i	−0.946087	+	0.323912i	$0.895002\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.05790e44	+	1.05790e44i	1.38438	+	1.38438i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−6.83941e43	−	6.59621e43i	−0.719788	−	0.694194i
$181$	−6.36700e43			−0.601454			−0.300727	−	0.953710i	$-0.597229\pi$
$181$	−6.36700e43			−0.601454			−0.300727	+	0.953710i	$0.597229\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.90418e44	−	3.44674e42i	1.17451	−	0.0212599i
$186$		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
$193$	−4.28798e44	−	4.28798e44i	−1.15847	−	1.15847i	−0.984805	−	0.173666i	$-0.944439\pi$
$193$	−4.28798e44	−	4.28798e44i	−1.15847	−	1.15847i	−0.173666	−	0.984805i	$-0.555561\pi$
$194$		−	1.48689e44i		−	0.363199i
$195$		0			0
$196$	−5.00027e44			−1.00000
$197$	−3.21845e44	+	3.21845e44i	−0.582848	+	0.582848i	−0.935685	−	0.352837i	$-0.885217\pi$
$197$	−3.21845e44	+	3.21845e44i	−0.582848	+	0.582848i	0.352837	+	0.935685i	$0.385217\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−5.04971e44	+	5.42919e44i	−0.681053	+	0.732234i
$201$		0			0
$202$	−8.63588e44	+	8.63588e44i	−0.959301	+	0.959301i
$203$		0			0
$204$		0			0
$205$	−1.59873e45	−	1.54189e45i	−1.33222	−	1.28485i
$206$		0			0
$207$		0			0
$208$	−2.22574e45	−	2.22574e45i	−1.39714	−	1.39714i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	1.95984e45	−	1.95984e45i	0.848539	−	0.848539i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−5.24536e45	−	5.24536e45i	−1.31787	−	1.31787i
$219$		0			0
$220$		0			0
$221$	−1.62703e46			−3.13146
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	−7.36673e45	+	2.66777e44i	−0.999345	+	0.0361901i
$226$	−8.33942e45			−1.03758
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	1.86540e46i		−	1.79465i	−0.441366	−	0.897327i	$-0.645506\pi$
$229$		−	1.86540e46i		−	1.79465i	0.441366	−	0.897327i	$-0.354494\pi$
$230$		0			0
$231$		0			0
$232$	−1.44596e46	+	1.44596e46i	−1.07929	+	1.07929i
$233$	1.96296e46	+	1.96296e46i	1.34732	+	1.34732i	0.888558	+	0.458765i	$0.151708\pi$
$233$	1.96296e46	+	1.96296e46i	1.34732	+	1.34732i	0.458765	+	0.888558i	$0.348292\pi$
$234$		−	3.12942e46i		−	1.97585i
$235$		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
$241$	−3.78962e46			−1.34668			−0.673339	−	0.739334i	$-0.735140\pi$
$241$	−3.78962e46			−1.34668			−0.673339	+	0.739334i	$0.735140\pi$
$242$	−2.15717e46	+	2.15717e46i	−0.707107	+	0.707107i
$243$		0			0
$244$			2.73183e46i			0.762698i
$245$	−2.69289e46	+	2.79218e46i	−0.694194	+	0.719788i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	3.12171e45	+	5.74367e46i	0.0542703	+	0.998526i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	9.13439e46			1.00000
$257$	1.38958e47	−	1.38958e47i	1.40990	−	1.40990i	0.649753	−	0.760146i	$-0.274872\pi$
$257$	1.38958e47	−	1.38958e47i	1.40990	−	1.40990i	0.760146	−	0.649753i	$-0.225128\pi$
$258$		0			0
$259$		0			0
$260$	−2.44154e47	+	4.41942e45i	−1.97553	+	0.0357590i
$261$	−2.03303e47			−1.52635
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	−3.89145e45	−	2.14986e47i	−0.0217179	−	1.19982i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.34364e47i			1.81002i	0.425386	+	0.905012i	$0.360138\pi$
$269$			4.34364e47i			1.81002i	−0.425386	+	0.905012i	$0.639862\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	3.33865e47	−	3.33865e47i	1.12067	−	1.12067i
$273$		0			0
$274$		−	6.65912e47i		−	1.93768i
$275$		0			0
$276$		0			0
$277$	−2.85166e47	+	2.85166e47i	−0.671031	+	0.671031i	−0.957954	−	0.286923i	$-0.907368\pi$
$277$	−2.85166e47	+	2.85166e47i	−0.671031	+	0.671031i	0.286923	+	0.957954i	$0.407368\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.19524e47			1.45810			0.729052	−	0.684459i	$-0.239961\pi$
$281$	8.19524e47			1.45810			0.729052	+	0.684459i	$0.239961\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	6.42153e47	+	6.42153e47i	0.707107	+	0.707107i
$289$		−	1.46894e48i		−	1.51180i
$290$	2.87109e46	+	1.58615e48i	0.0276239	+	1.52610i
$291$		0			0
$292$	1.00407e48	−	1.00407e48i	0.844888	−	0.844888i
$293$	1.07324e48	+	1.07324e48i	0.844846	+	0.844846i	0.989485	−	0.144639i	$-0.0462020\pi$
$293$	1.07324e48	+	1.07324e48i	0.844846	+	0.844846i	−0.144639	+	0.989485i	$0.546202\pi$
$294$		0			0
$295$		0			0
$296$	−1.82020e48			−1.17471
$297$		0			0
$298$	2.41714e48	+	2.41714e48i	1.36799	+	1.36799i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.52547e48	+	1.47123e48i	0.548981	+	0.529460i
$306$	4.69419e48			1.58487
$307$		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		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	6.51029e48	+	6.51029e48i	1.41412	+	1.41412i	0.715281	+	0.698837i	$0.246299\pi$
$313$	6.51029e48	+	6.51029e48i	1.41412	+	1.41412i	0.698837	+	0.715281i	$0.253701\pi$
$314$		−	1.56525e48i		−	0.319489i
$315$		0			0
$316$		0			0
$317$	8.32763e48	−	8.32763e48i	1.41210	−	1.41210i	0.667419	−	0.744683i	$-0.267399\pi$
$317$	8.32763e48	−	8.32763e48i	1.41210	−	1.41210i	0.744683	−	0.667419i	$-0.232601\pi$
$318$		0			0
$319$		0			0
$320$	4.91932e48	−	5.10069e48i	0.694194	−	0.719788i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			9.02875e48i			1.00000i
$325$	−1.29021e49	+	1.38717e49i	−1.34566	+	1.44679i
$326$		0			0
$327$		0			0
$328$	1.50105e49	+	1.50105e49i	1.30875	+	1.30875i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−1.27961e49	−	1.27961e49i	−0.830643	−	0.830643i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.58473e49	+	2.58473e49i	−1.32933	+	1.32933i	−0.423371	+	0.905956i	$0.639153\pi$
$337$	−2.58473e49	+	2.58473e49i	−1.32933	+	1.32933i	−0.905956	+	0.423371i	$0.860847\pi$
$338$	−4.23015e49	−	4.23015e49i	−2.05343	−	2.05343i
$339$		0			0
$340$	−6.62922e47	−	3.66236e49i	−0.0286829	−	1.58461i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	2.86030e49			0.879889
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	3.19775e49i		−	0.831281i	−0.909529	−	0.415641i	$-0.863557\pi$
$349$		−	3.19775e49i		−	0.831281i	0.909529	−	0.415641i	$-0.136443\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.36508e49	−	5.36508e49i	−1.11678	−	1.11678i	−0.992210	−	0.124574i	$-0.960243\pi$
$353$	−5.36508e49	−	5.36508e49i	−1.11678	−	1.11678i	−0.124574	−	0.992210i	$-0.539757\pi$
$354$		0			0
$355$		0			0
$356$	−1.10929e50			−1.95781
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	7.04413e49	−	1.27506e48i	0.999836	−	0.0180980i
$361$	−7.43687e49			−1.00000
$362$	3.33814e49	−	3.33814e49i	0.425293	−	0.425293i
$363$		0			0
$364$		0			0
$365$	−1.99368e48	−	1.10142e50i	−0.0216244	−	1.19466i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			2.11050e50i			1.85085i
$370$	−9.80266e49	+	1.01641e50i	−0.815474	+	0.845540i
$371$		0			0
$372$		0			0
$373$	−8.79508e49	−	8.79508e49i	−0.625054	−	0.625054i	0.321765	−	0.946819i	$-0.395724\pi$
$373$	−8.79508e49	−	8.79508e49i	−0.625054	−	0.625054i	−0.946819	+	0.321765i	$0.895724\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.69446e50	+	3.69446e50i	−2.13253	+	2.13253i
$378$		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	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	4.49627e50			1.63833
$387$		0			0
$388$	7.79557e49	+	7.79557e49i	0.256820	+	0.256820i
$389$			1.98911e50i			0.623219i	0.950210	+	0.311609i	$0.100868\pi$
$389$			1.98911e50i			0.623219i	−0.950210	+	0.311609i	$0.899132\pi$
$390$		0			0
$391$		0			0
$392$	2.62158e50	−	2.62158e50i	0.707107	−	0.707107i
$393$		0			0
$394$		−	3.37479e50i		−	0.824271i
$395$		0			0
$396$		0			0
$397$	9.91946e49	−	9.91946e49i	0.208965	−	0.208965i	−0.594863	−	0.803827i	$-0.702794\pi$
$397$	9.91946e49	−	9.91946e49i	0.208965	−	0.208965i	0.803827	+	0.594863i	$0.202794\pi$
$398$		0			0
$399$		0			0
$400$	−1.98957e49	−	5.49396e50i	−0.0361901	−	0.999345i
$401$	−1.07306e51			−1.85912			−0.929559	−	0.368673i	$-0.879812\pi$
$401$	−1.07306e51			−1.85912			−0.929559	+	0.368673i	$0.879812\pi$
$402$		0			0
$403$		0			0
$404$		−	9.05537e50i		−	1.35666i
$405$	5.04171e50	+	4.86243e50i	0.719788	+	0.694194i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			7.78247e50i			0.917303i	0.888616	+	0.458652i	$0.151667\pi$
$409$			7.78247e50i			0.917303i	−0.888616	+	0.458652i	$0.848333\pi$
$410$	1.64659e51	−	2.98049e49i	1.85055	−	0.0334967i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	2.33386e51			1.97585
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.52840e51			−1.69568			−0.847841	−	0.530251i	$-0.822098\pi$
$421$	−2.52840e51			−1.69568			−0.847841	+	0.530251i	$0.822098\pi$
$422$		0			0
$423$		0			0
$424$			2.05504e51i			1.20002i
$425$	−2.08078e51	−	1.93534e51i	−1.16049	−	1.07938i
$426$		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
$433$	3.39364e51	+	3.39364e51i	1.31568	+	1.31568i	0.917157	+	0.398527i	$0.130478\pi$
$433$	3.39364e51	+	3.39364e51i	1.31568	+	1.31568i	0.398527	+	0.917157i	$0.369522\pi$
$434$		0			0
$435$		0			0
$436$	5.50016e51			1.86376
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	3.68598e51			1.00000
$442$	8.53034e51	−	8.53034e51i	2.21428	−	2.21428i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−5.97407e51	+	6.19433e51i	−1.35910	+	1.40921i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.59504e51i			1.06905i	0.845152	+	0.534526i	$0.179510\pi$
$449$			5.59504e51i			1.06905i	−0.845152	+	0.534526i	$0.820490\pi$
$450$	3.72242e51	−	4.00215e51i	0.681053	−	0.732234i
$451$		0			0
$452$	4.37226e51	−	4.37226e51i	0.733679	−	0.733679i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00602e52	+	1.00602e52i	−1.36220	+	1.36220i	−0.491092	+	0.871108i	$0.663402\pi$
$457$	−1.00602e52	+	1.00602e52i	−1.36220	+	1.36220i	−0.871108	+	0.491092i	$0.836598\pi$
$458$	9.78008e51	+	9.78008e51i	1.26901	+	1.26901i
$459$		0			0
$460$		0			0
$461$	−1.74887e52			−1.99797			−0.998984	−	0.0450680i	$-0.985650\pi$
$461$	−1.74887e52			−1.99797			−0.998984	+	0.0450680i	$0.985650\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	1.51620e52i		−	1.52635i
$465$		0			0
$466$	−2.05832e52			−1.90540
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	1.64072e52	+	1.64072e52i	1.39714	+	1.39714i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.44471e52	+	1.44471e52i	−0.848539	+	0.848539i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−4.65064e52			−2.32105
$482$	1.98685e52	−	1.98685e52i	0.952245	−	0.952245i
$483$		0			0
$484$		−	2.26196e52i		−	1.00000i
$485$	8.55140e51	−	1.54789e50i	0.363139	−	0.00657317i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−1.43227e52	−	1.43227e52i	−0.539309	−	0.539309i
$489$		0			0
$490$	−5.20540e50	−	2.87576e52i	−0.0180980	−	0.999836i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−5.54176e52	−	5.54176e52i	−1.71054	−	1.71054i
$494$		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	$0.5\pi$
$500$	−3.17501e52	−	2.84767e52i	−0.744440	−	0.667690i
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	−5.05657e52	−	4.87677e52i	−0.976505	−	0.941782i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			9.16653e52i			1.51778i	0.651220	+	0.758889i	$0.274258\pi$
$509$			9.16653e52i			1.51778i	−0.651220	+	0.758889i	$0.725742\pi$
$510$		0			0
$511$		0			0
$512$	−4.78905e52	+	4.78905e52i	−0.707107	+	0.707107i
$513$		0			0
$514$			1.45708e53i			1.99390i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.25690e53	−	1.30324e53i	1.37162	−	1.42219i
$521$	−1.31263e53			−1.37977			−0.689884	−	0.723920i	$-0.742339\pi$
$521$	−1.31263e53			−1.37977			−0.689884	+	0.723920i	$0.742339\pi$
$522$	1.06590e53	−	1.06590e53i	1.07929	−	1.07929i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			1.28052e53i			1.00000i
$530$	1.14755e53	+	1.10674e53i	0.863757	+	0.833043i
$531$		0			0
$532$		0			0
$533$	3.83523e53	+	3.83523e53i	2.58589	+	2.58589i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−2.27732e53	−	2.27732e53i	−1.27988	−	1.27988i
$539$		0			0
$540$		0			0
$541$	2.12446e52			0.107128			0.0535639	−	0.998564i	$-0.482942\pi$
$541$	2.12446e52			0.107128			0.0535639	+	0.998564i	$0.482942\pi$
$542$		0			0
$543$		0			0
$544$			3.50083e53i			1.58487i
$545$	2.96211e53	−	3.07132e53i	1.29381	−	1.34151i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	3.49130e53	+	3.49130e53i	1.37014	+	1.37014i
$549$		−	2.01379e53i		−	0.762698i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	2.99018e53i		−	0.948981i
$555$		0			0
$556$		0			0
$557$	4.52331e53	−	4.52331e53i	1.29205	−	1.29205i	0.358535	−	0.933516i	$-0.383276\pi$
$557$	4.52331e53	−	4.52331e53i	1.29205	−	1.29205i	0.933516	−	0.358535i	$-0.116724\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−4.29667e53	+	4.29667e53i	−1.03103	+	1.03103i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−8.68153e51	−	4.79617e53i	−0.0187781	−	1.03741i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.96982e53i			0.936811i	0.883514	+	0.468405i	$0.155171\pi$
$569$			4.96982e53i			0.936811i	−0.883514	+	0.468405i	$0.844829\pi$
$570$		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
$576$	−6.73346e53			−1.00000
$577$	6.58442e53	−	6.58442e53i	0.945343	−	0.945343i	−0.0532391	−	0.998582i	$-0.516955\pi$
$577$	6.58442e53	−	6.58442e53i	0.945343	−	0.945343i	0.998582	+	0.0532391i	$0.0169545\pi$
$578$	7.70146e53	+	7.70146e53i	1.06901	+	1.06901i
$579$		0			0
$580$	−8.46653e53	−	8.16548e53i	−1.09865	−	1.05958i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			1.05285e54i			1.19485i
$585$	1.79979e54	−	3.25780e52i	1.97553	−	0.0357590i
$586$	−1.12537e54			−1.19479
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	9.54307e53	−	9.54307e53i	0.830643	−	0.830643i
$593$	−4.91451e53	−	4.91451e53i	−0.413917	−	0.413917i	0.469184	−	0.883101i	$-0.344548\pi$
$593$	−4.91451e53	−	4.91451e53i	−0.413917	−	0.413917i	−0.883101	+	0.469184i	$0.844548\pi$
$594$		0			0
$595$		0			0
$596$	−2.53455e54			−1.93463
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	4.80455e53			0.311602			0.155801	−	0.987788i	$-0.450204\pi$
$601$	4.80455e53			0.311602			0.155801	+	0.987788i	$0.450204\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.26309e54	−	1.21818e54i	−0.719788	−	0.694194i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$	−1.57113e54	+	2.84391e52i	−0.762573	+	0.0138033i
$611$		0			0
$612$	−2.46111e54	+	2.46111e54i	−1.12067	+	1.12067i
$613$	2.87747e54	+	2.87747e54i	1.26920	+	1.26920i	0.946500	+	0.322704i	$0.104592\pi$
$613$	2.87747e54	+	2.87747e54i	1.26920	+	1.26920i	0.322704	+	0.946500i	$0.395408\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.98532e53	−	6.98532e53i	0.271410	−	0.271410i	−0.558258	−	0.829668i	$-0.688530\pi$
$617$	6.98532e53	−	6.98532e53i	0.271410	−	0.271410i	0.829668	+	0.558258i	$0.188530\pi$
$618$		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
$625$	−3.30006e54	+	2.39329e53i	−0.997381	+	0.0723327i
$626$	−6.82653e54			−1.99986
$627$		0			0
$628$	8.20643e53	+	8.20643e53i	0.225913	+	0.225913i
$629$		−	6.97606e54i		−	1.86175i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			8.73215e54i			1.99701i
$635$		0			0
$636$		0			0
$637$	6.69820e54	−	6.69820e54i	1.39714	−	1.39714i
$638$		0			0
$639$		0			0
$640$	9.50912e52	+	5.25337e54i	0.0180980	+	0.999836i
$641$	−1.03441e54			−0.190969			−0.0954844	−	0.995431i	$-0.530440\pi$
$641$	−1.03441e54			−0.190969			−0.0954844	+	0.995431i	$0.530440\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−4.73367e54	−	4.73367e54i	−0.707107	−	0.707107i
$649$		0			0
$650$	−5.08340e53	−	1.40372e55i	−0.0715062	−	1.97456i
$651$		0			0
$652$		0			0
$653$	4.19289e54	+	4.19289e54i	0.539147	+	0.539147i	0.923278	−	0.384132i	$-0.125499\pi$
$653$	4.19289e54	+	4.19289e54i	0.539147	+	0.539147i	−0.384132	+	0.923278i	$0.625499\pi$
$654$		0			0
$655$		0			0
$656$	−1.57397e55			−1.85085
$657$	−7.40157e54	+	7.40157e54i	−0.844888	+	0.844888i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−4.65977e54			−0.472537			−0.236268	−	0.971688i	$-0.575924\pi$
$661$	−4.65977e54			−0.472537			−0.236268	+	0.971688i	$0.575924\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.34177e55			1.17471
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.26144e55	−	1.26144e55i	−0.900686	−	0.900686i	0.0948094	−	0.995495i	$-0.469776\pi$
$673$	−1.26144e55	−	1.26144e55i	−0.900686	−	0.900686i	−0.995495	+	0.0948094i	$0.969776\pi$
$674$		−	2.71029e55i		−	1.87995i
$675$		0			0
$676$	4.43563e55			2.90399
$677$	1.45571e55	−	1.45571e55i	0.925968	−	0.925968i	−0.0714743	−	0.997442i	$-0.522770\pi$
$677$	1.45571e55	−	1.45571e55i	0.925968	−	0.925968i	0.997442	+	0.0714743i	$0.0227704\pi$
$678$		0			0
$679$		0			0
$680$	1.95489e55	+	1.88537e55i	1.14077	+	1.10020i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	3.82980e55	−	6.93231e53i	1.93736	−	0.0350681i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			5.25068e55i			2.37105i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−1.49962e55	+	1.49962e55i	−0.622175	+	0.622175i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5.75292e55	+	5.75292e55i	−2.07419	+	2.07419i
$698$	1.67654e55	+	1.67654e55i	0.587805	+	0.587805i
$699$		0			0
$700$		0			0
$701$	2.66655e55			0.859902			0.429951	−	0.902852i	$-0.358531\pi$
$701$	2.66655e55			0.859902			0.429951	+	0.902852i	$0.358531\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	5.62569e55			1.57937
$707$		0			0
$708$		0			0
$709$			1.71591e55i			0.443499i	0.975104	+	0.221749i	$0.0711767\pi$
$709$			1.71591e55i			0.443499i	−0.975104	+	0.221749i	$0.928823\pi$
$710$		0			0
$711$		0			0
$712$	5.81587e55	−	5.81587e55i	1.38438	−	1.38438i
$713$		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	$0.5\pi$
$720$	−3.62631e55	+	3.76000e55i	−0.694194	+	0.719788i
$721$		0			0
$722$	3.89906e55	−	3.89906e55i	0.707107	−	0.707107i
$723$		0			0
$724$			3.50030e55i			0.601454i
$725$	−9.11930e55	+	3.30245e54i	−1.52535	+	0.0552388i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	6.65559e55i		−	1.00000i
$730$	5.87915e55	+	5.67010e55i	0.860040	+	0.829459i
$731$		0			0
$732$		0			0
$733$	−9.04917e55	−	9.04917e55i	−1.22203	−	1.22203i	−0.966910	−	0.255118i	$-0.917886\pi$
$733$	−9.04917e55	−	9.04917e55i	−1.22203	−	1.22203i	−0.255118	−	0.966910i	$-0.582114\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.10651e56	−	1.10651e56i	−1.30875	−	1.30875i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−1.89487e54	−	1.04683e56i	−0.0212599	−	1.17451i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	−1.36498e56	+	1.41531e56i	−1.34301	+	1.39253i
$746$	9.22231e55			0.883960
$747$		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
$754$		−	3.87392e56i		−	3.01585i
$755$		0			0
$756$		0			0
$757$	−1.84028e56	+	1.84028e56i	−1.32591	+	1.32591i	−0.417014	+	0.908900i	$0.636923\pi$
$757$	−1.84028e56	+	1.84028e56i	−1.32591	+	1.32591i	−0.908900	+	0.417014i	$0.863077\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00589e56			−1.95422			−0.977108	−	0.212744i	$-0.931760\pi$
$761$	−3.00589e56			−1.95422			−0.977108	+	0.212744i	$0.931760\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	4.88676e54	+	2.69973e56i	0.0286829	+	1.58461i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	3.71489e56i		−	1.96962i	−0.173635	−	0.984810i	$-0.555551\pi$
$769$		−	3.71489e56i		−	1.96962i	0.173635	−	0.984810i	$-0.444449\pi$
$770$		0			0
$771$		0			0
$772$	−2.35734e56	+	2.35734e56i	−1.15847	+	1.15847i
$773$	7.60105e55	+	7.60105e55i	0.364228	+	0.364228i	0.865367	−	0.501139i	$-0.167085\pi$
$773$	7.60105e55	+	7.60105e55i	0.364228	+	0.364228i	−0.501139	+	0.865367i	$0.667085\pi$
$774$		0			0
$775$		0			0
$776$	−8.17425e55			−0.363199
$777$		0			0
$778$	−1.04286e56	−	1.04286e56i	−0.440682	−	0.440682i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			2.74893e56i			1.00000i
$785$	9.00209e55	−	1.62947e54i	0.319437	−	0.00578212i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	1.76936e56	+	1.76936e56i	0.582848	+	0.582848i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.65948e56	−	3.65948e56i	−1.06559	−	1.06559i
$794$			1.04013e56i			0.295521i
$795$		0			0
$796$		0			0
$797$	4.14268e56	−	4.14268e56i	1.09356	−	1.09356i	0.0984181	−	0.995145i	$-0.468622\pi$
$797$	4.14268e56	−	4.14268e56i	1.09356	−	1.09356i	0.995145	−	0.0984181i	$-0.0313782\pi$
$798$		0			0
$799$		0			0
$800$	2.98473e56	+	2.77610e56i	0.732234	+	0.681053i
$801$	8.17718e56			1.95781
$802$	5.62590e56	−	5.62590e56i	1.31460	−	1.31460i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	4.74762e56	+	4.74762e56i	0.959301	+	0.959301i
$809$			8.31994e56i			1.64106i	0.571605	+	0.820529i	$0.306321\pi$
$809$			8.31994e56i			1.64106i	−0.571605	+	0.820529i	$0.693679\pi$
$810$	−5.19262e56	+	9.39915e54i	−0.999836	+	0.0180980i
$811$		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
$818$	−4.08026e56	−	4.08026e56i	−0.648632	−	0.648632i
$819$		0			0
$820$	−8.47661e56	+	8.78913e56i	−1.28485	+	1.33222i
$821$	1.17839e57			1.74421			0.872105	−	0.489318i	$-0.162754\pi$
$821$	1.17839e57			1.74421			0.872105	+	0.489318i	$0.162754\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		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
$829$		−	4.88110e56i		−	0.598004i	−0.954253	−	0.299002i	$-0.903346\pi$
$829$		−	4.88110e56i		−	0.598004i	0.954253	−	0.299002i	$-0.0966537\pi$
$830$		0			0
$831$		0			0
$832$	−1.22361e57	+	1.22361e57i	−1.39714	+	1.39714i
$833$	1.00474e57	+	1.00474e57i	1.12067	+	1.12067i
$834$		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
$841$	−1.43646e57			−1.32975
$842$	1.32561e57	−	1.32561e57i	1.19903	−	1.19903i
$843$		0			0
$844$		0			0
$845$	2.38881e57	−	2.47688e57i	2.01593	−	2.09026i
$846$		0			0
$847$		0			0
$848$	−1.07743e57	−	1.07743e57i	−0.848539	−	0.848539i
$849$		0			0
$850$	2.10561e57	−	7.62520e55i	1.58383	−	0.0573565i
$851$		0			0
$852$		0			0
$853$	−4.09564e55	−	4.09564e55i	−0.0287618	−	0.0287618i	0.692580	−	0.721341i	$-0.256474\pi$
$853$	−4.09564e55	−	4.09564e55i	−0.0287618	−	0.0287618i	−0.721341	+	0.692580i	$0.756474\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.18067e57	−	2.18067e57i	1.39786	−	1.39786i	0.591721	−	0.806143i	$-0.298449\pi$
$857$	2.18067e57	−	2.18067e57i	1.39786	−	1.39786i	0.806143	−	0.591721i	$-0.201551\pi$
$858$		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	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	2.97764e55	+	1.64502e57i	0.0159242	+	0.879745i
$866$	−3.55849e57			−1.86066
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−2.88367e57	+	2.88367e57i	−1.31787	+	1.31787i
$873$	−5.74655e56	−	5.74655e56i	−0.256820	−	0.256820i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.18248e57	−	2.18248e57i	0.892190	−	0.892190i	−0.102539	−	0.994729i	$-0.532697\pi$
$877$	2.18248e57	−	2.18248e57i	0.892190	−	0.892190i	0.994729	+	0.102539i	$0.0326967\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.12531e57			0.420960			0.210480	−	0.977598i	$-0.432497\pi$
$881$	1.12531e57			0.420960			0.210480	+	0.977598i	$0.432497\pi$
$882$	−1.93251e57	+	1.93251e57i	−0.707107	+	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$			8.94471e57i			3.13146i
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	−1.15480e56	−	6.37975e57i	−0.0354324	−	1.95749i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−2.93341e57	−	2.93341e57i	−0.755935	−	0.755935i
$899$		0			0
$900$	1.46662e56	+	4.04990e57i	0.0361901	+	0.999345i
$901$	−7.87613e57			−1.90186
$902$		0			0
$903$		0			0
$904$			4.58464e57i			1.03758i
$905$	1.95459e57	+	1.88508e57i	0.432920	+	0.417526i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			6.67522e57i			1.35666i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	1.05489e58i		−	1.92644i
$915$		0			0
$916$	−1.02552e58			−1.79465
$917$		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
$922$	9.16909e57	−	9.16909e57i	1.41278	−	1.41278i
$923$		0			0
$924$		0			0
$925$	−5.94762e57	−	5.53190e57i	−0.860160	−	0.800038i
$926$		0			0
$927$		0			0
$928$	7.94924e57	+	7.94924e57i	1.07929	+	1.07929i
$929$		−	1.40770e58i		−	1.87156i	−0.352587	−	0.935779i	$-0.614698\pi$
$929$		−	1.40770e58i		−	1.87156i	0.352587	−	0.935779i	$-0.385302\pi$
$930$		0			0
$931$		0			0
$932$	1.07915e58	−	1.07915e58i	1.34732	−	1.34732i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.72042e58			−1.97585
$937$	−2.47049e57	+	2.47049e57i	−0.277882	+	0.277882i	−0.832263	−	0.554381i	$-0.812955\pi$
$937$	−2.47049e57	+	2.47049e57i	−0.277882	+	0.277882i	0.554381	+	0.832263i	$0.312955\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.79142e58			−1.85439			−0.927193	−	0.374585i	$-0.877785\pi$
$941$	−1.79142e58			−1.85439			−0.927193	+	0.374585i	$0.877785\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$			2.69004e58i			2.36085i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.74613e57	−	7.74613e57i	−0.626287	−	0.626287i	0.320845	−	0.947132i	$-0.396033\pi$
$953$	−7.74613e57	−	7.74613e57i	−0.626287	−	0.626287i	−0.947132	+	0.320845i	$0.896033\pi$
$954$		−	1.51489e58i		−	1.20002i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.45581e58			1.00000
$962$	2.43828e58	−	2.43828e58i	1.64123	−	1.64123i
$963$		0			0
$964$			2.08337e58i			1.34668i
$965$	4.68073e56	+	2.58590e58i	0.0296504	+	1.63806i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	1.18592e58	+	1.18592e58i	0.707107	+	0.707107i
$969$		0			0
$970$	−4.40224e57	+	4.56455e57i	−0.252130	+	0.261426i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.50184e58			0.762698
$977$	1.34498e58	−	1.34498e58i	0.669532	−	0.669532i	−0.288076	−	0.957608i	$-0.593016\pi$
$977$	1.34498e58	−	1.34498e58i	0.669532	−	0.669532i	0.957608	+	0.288076i	$0.0930155\pi$
$978$		0			0
$979$		0			0
$980$	1.53502e58	+	1.48043e58i	0.719788	+	0.694194i
$981$	−4.05448e58			−1.86376
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	1.94091e58	−	3.51324e56i	0.824136	−	0.0149177i
$986$	5.81095e58			2.41907
$987$		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
$997$	1.59418e58	−	1.59418e58i	0.534541	−	0.534541i	−0.387380	−	0.921920i	$-0.626620\pi$
$997$	1.59418e58	−	1.59418e58i	0.534541	−	0.534541i	0.921920	+	0.387380i	$0.126620\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.40.e.a.3.1	✓	2
4.3	odd	2	CM	20.40.e.a.3.1	✓	2
5.2	odd	4	inner	20.40.e.a.7.1	yes	2
20.7	even	4	inner	20.40.e.a.7.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.40.e.a.3.1	✓	2	1.1	even	1	trivial
20.40.e.a.3.1	✓	2	4.3	odd	2	CM
20.40.e.a.7.1	yes	2	5.2	odd	4	inner
20.40.e.a.7.1	yes	2	20.7	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 40 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-524288. + 524288. i) q^{2} -5.49756e11i q^{4} +(-3.06987e13 - 2.96071e13i) q^{5} +(2.88230e17 + 2.88230e17i) q^{8} +4.05256e18i q^{9} +O(q^{10})\)	\(q+(-524288. + 524288. i) q^{2} -5.49756e11i q^{4} +(-3.06987e13 - 2.96071e13i) q^{5} +(2.88230e17 + 2.88230e17i) q^{8} +4.05256e18i q^{9} +(3.16176e19 - 5.72309e17i) q^{10} +(7.36435e21 + 7.36435e21i) q^{13} -3.02231e23 q^{16} +(-1.10467e24 + 1.10467e24i) q^{17} +(-2.12471e24 - 2.12471e24i) q^{18} +(-1.62767e25 + 1.68768e25i) q^{20} +(6.58294e25 + 1.81780e27i) q^{25} -7.72208e27 q^{26} +5.01667e28i q^{29} +(1.58456e29 - 1.58456e29i) q^{32} -1.15833e30i q^{34} +2.22792e30 q^{36} +(-3.15754e30 + 3.15754e30i) q^{37} +(-3.14630e29 - 1.73820e31i) q^{40} +5.20783e31 q^{41} +(1.19984e32 - 1.24408e32i) q^{45} -9.09544e32i q^{49} +(-9.87563e32 - 9.18536e32i) q^{50} +(4.04860e33 - 4.04860e33i) q^{52} +(3.56493e33 + 3.56493e33i) q^{53} +(-2.63018e34 - 2.63018e34i) q^{58} -4.96918e34 q^{61} +1.66153e35i q^{64} +(-8.03888e33 - 4.44113e35i) q^{65} +(6.07298e35 + 6.07298e35i) q^{68} +(-1.16807e36 + 1.16807e36i) q^{72} +(1.82640e36 + 1.82640e36i) q^{73} -3.31092e36i q^{74} +(9.27811e36 + 8.94820e36i) q^{80} -1.64232e37 q^{81} +(-2.73040e37 + 2.73040e37i) q^{82} +(6.66179e37 - 1.20585e36i) q^{85} -2.01778e38i q^{89} +(2.31931e36 + 1.28132e38i) q^{90} +(-1.41801e38 + 1.41801e38i) q^{97} +(4.76863e38 + 4.76863e38i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1048576 q^{2} - 61397374177364 q^{5} + 57\!\cdots\!88 q^{8}+O(q^{10}) \) 2 * q - 1048576 * q^2 - 61397374177364 * q^5 + 576460752303423488 * q^8	\( 2 q - 1048576 q^{2} - 61397374177364 q^{5} + 57\!\cdots\!88 q^{8}+ \cdots + 95\!\cdots\!68 q^{98}+O(q^{100}) \) 2 * q - 1048576 * q^2 - 61397374177364 * q^5 + 576460752303423488 * q^8 + 63235194701145440256 * q^10 + 14728708243289840454646 * q^13 - 604462909807314587353088 * q^16 - 2209335626245349613005082 * q^17 - 4249412072132026058145792 * q^18 - 32553344107485460875444224 * q^20 + 131658748783530764069832246 * q^25 - 15444169974915887744570884096 * q^26 + 316912650057057350374175801344 * q^32 + 4455831512947911355946281992192 * q^36 - 6315070876415894212107982079222 * q^37 - 629260578443681810867556450304 * q^40 + 104156506162896529409552831978224 * q^41 + 239968762636254052800519097107966 * q^45 - 1975126277432191896766247609565184 * q^50 + 8097192987808700953825579680923648 * q^52 + 7129862979151660093343761890759306 * q^53 - 52603641793583427999156653076250624 * q^58 - 99383542409702374129137867076998216 * q^61 - 16077756259844071201512336322846718 * q^65 + 1214595105358266350072714684390178816 * q^68 - 2336138992260434548986364293122359296 * q^72 + 3652790957208407144721971672057732186 * q^73 + 18556217724888960595218879231102550016 * q^80 - 32846406536521316292462935601418510578 * q^81 - 54608006303132695611075635172199104512 * q^82 + 133235710607339644158613592089648668342 * q^85 + 4638628888212487180995247479047847936 * q^90 - 283601189444481118373198525559476429542 * q^97 + 953725673935849275596687298935278010368 * q^98

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