LMFDB - Embedded newform 20.8.e.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,8,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, 8, names="a")

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
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:	$6.24770050968$
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			20.8.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)

$f(q)$	$=$	$q+(-8.00000 + 8.00000i) q^{2} -128.000i q^{4} +(278.000 - 29.0000i) q^{5} +(1024.00 + 1024.00i) q^{8} +2187.00i q^{9} +O(q^{10})$	\(q+(-8.00000 + 8.00000i) q^{2} -128.000i q^{4} +(278.000 - 29.0000i) q^{5} +(1024.00 + 1024.00i) q^{8} +2187.00i q^{9} +(-1992.00 + 2456.00i) q^{10} +(11003.0 + 11003.0i) q^{13} -16384.0 q^{16} +(17139.0 - 17139.0i) q^{17} +(-17496.0 - 17496.0i) q^{18} +(-3712.00 - 35584.0i) q^{20} +(76443.0 - 16124.0i) q^{25} -176048. q^{26} +120844. i q^{29} +(131072. - 131072. i) q^{32} +274224. i q^{34} +279936. q^{36} +(157829. - 157829. i) q^{37} +(314368. + 254976. i) q^{40} -882568. q^{41} +(63423.0 + 607986. i) q^{45} -823543. i q^{49} +(-482552. + 740536. i) q^{50} +(1.40838e6 - 1.40838e6i) q^{52} +(-1.40691e6 - 1.40691e6i) q^{53} +(-966752. - 966752. i) q^{58} +532572. q^{61} +2.09715e6i q^{64} +(3.37792e6 + 2.73975e6i) q^{65} +(-2.19379e6 - 2.19379e6i) q^{68} +(-2.23949e6 + 2.23949e6i) q^{72} +(4.64401e6 + 4.64401e6i) q^{73} +2.52526e6i q^{74} +(-4.55475e6 + 475136. i) q^{80} -4.78297e6 q^{81} +(7.06054e6 - 7.06054e6i) q^{82} +(4.26761e6 - 5.26167e6i) q^{85} -9.56210e6i q^{89} +(-5.37127e6 - 4.35650e6i) q^{90} +(-1.06929e7 + 1.06929e7i) q^{97} +(6.58834e6 + 6.58834e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} + 556 q^{5} + 2048 q^{8}+O(q^{10}) $ 2 * q - 16 * q^2 + 556 * q^5 + 2048 * q^8	$ 2 q - 16 q^{2} + 556 q^{5} + 2048 q^{8} - 3984 q^{10} + 22006 q^{13} - 32768 q^{16} + 34278 q^{17} - 34992 q^{18} - 7424 q^{20} + 152886 q^{25} - 352096 q^{26} + 262144 q^{32} + 559872 q^{36} + 315658 q^{37} + 628736 q^{40} - 1765136 q^{41} + 126846 q^{45} - 965104 q^{50} + 2816768 q^{52} - 2813814 q^{53} - 1933504 q^{58} + 1065144 q^{61} + 6755842 q^{65} - 4387584 q^{68} - 4478976 q^{72} + 9288026 q^{73} - 9109504 q^{80} - 9565938 q^{81} + 14121088 q^{82} + 8535222 q^{85} - 10742544 q^{90} - 21385702 q^{97} + 13176688 q^{98}+O(q^{100}) $ 2 * q - 16 * q^2 + 556 * q^5 + 2048 * q^8 - 3984 * q^10 + 22006 * q^13 - 32768 * q^16 + 34278 * q^17 - 34992 * q^18 - 7424 * q^20 + 152886 * q^25 - 352096 * q^26 + 262144 * q^32 + 559872 * q^36 + 315658 * q^37 + 628736 * q^40 - 1765136 * q^41 + 126846 * q^45 - 965104 * q^50 + 2816768 * q^52 - 2813814 * q^53 - 1933504 * q^58 + 1065144 * q^61 + 6755842 * q^65 - 4387584 * q^68 - 4478976 * q^72 + 9288026 * q^73 - 9109504 * q^80 - 9565938 * q^81 + 14121088 * q^82 + 8535222 * q^85 - 10742544 * q^90 - 21385702 * q^97 + 13176688 * 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 $

$2$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$2$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	128.000i		−	1.00000i
$5$	278.000	−	29.0000i	0.994603	−	0.103754i
$5$	278.000	−	29.0000i	0.994603	−	0.103754i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	1024.00	+	1024.00i	0.707107	+	0.707107i
$9$			2187.00i			1.00000i
$10$	−1992.00	+	2456.00i	−0.629926	+	0.776655i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	11003.0	+	11003.0i	1.38902	+	1.38902i	0.827379	+	0.561643i	$0.189831\pi$
$13$	11003.0	+	11003.0i	1.38902	+	1.38902i	0.561643	+	0.827379i	$0.310169\pi$
$14$		0			0
$15$		0			0
$16$	−16384.0			−1.00000
$17$	17139.0	−	17139.0i	0.846086	−	0.846086i	−0.143557	−	0.989642i	$-0.545854\pi$
$17$	17139.0	−	17139.0i	0.846086	−	0.846086i	0.989642	+	0.143557i	$0.0458539\pi$
$18$	−17496.0	−	17496.0i	−0.707107	−	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−3712.00	−	35584.0i	−0.103754	−	0.994603i
$21$		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
$25$	76443.0	−	16124.0i	0.978470	−	0.206387i
$26$	−176048.			−1.96437
$27$		0			0
$28$		0			0
$29$			120844.i			0.920094i	0.887895	+	0.460047i	$0.152167\pi$
$29$			120844.i			0.920094i	−0.887895	+	0.460047i	$0.847833\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	131072.	−	131072.i	0.707107	−	0.707107i
$33$		0			0
$34$			274224.i			1.19655i
$35$		0			0
$36$	279936.			1.00000
$37$	157829.	−	157829.i	0.512249	−	0.512249i	−0.402966	−	0.915215i	$-0.632021\pi$
$37$	157829.	−	157829.i	0.512249	−	0.512249i	0.915215	+	0.402966i	$0.132021\pi$
$38$		0			0
$39$		0			0
$40$	314368.	+	254976.i	0.776655	+	0.629926i
$41$	−882568.			−1.99988			−0.999942	−	0.0107974i	$-0.996563\pi$
$41$	−882568.			−1.99988			−0.999942	+	0.0107974i	$0.996563\pi$
$42$		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
$45$	63423.0	+	607986.i	0.103754	+	0.994603i
$46$		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
$49$		−	823543.i		−	1.00000i
$50$	−482552.	+	740536.i	−0.545945	+	0.837821i
$51$		0			0
$52$	1.40838e6	−	1.40838e6i	1.38902	−	1.38902i
$53$	−1.40691e6	−	1.40691e6i	−1.29808	−	1.29808i	−0.929662	−	0.368413i	$-0.879901\pi$
$53$	−1.40691e6	−	1.40691e6i	−1.29808	−	1.29808i	−0.368413	−	0.929662i	$-0.620099\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−966752.	−	966752.i	−0.650605	−	0.650605i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	532572.			0.300417			0.150208	−	0.988654i	$-0.452006\pi$
$61$	532572.			0.300417			0.150208	+	0.988654i	$0.452006\pi$
$62$		0			0
$63$		0			0
$64$			2.09715e6i			1.00000i
$65$	3.37792e6	+	2.73975e6i	1.52564	+	1.23741i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−2.19379e6	−	2.19379e6i	−0.846086	−	0.846086i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−2.23949e6	+	2.23949e6i	−0.707107	+	0.707107i
$73$	4.64401e6	+	4.64401e6i	1.39722	+	1.39722i	0.807911	+	0.589305i	$0.200598\pi$
$73$	4.64401e6	+	4.64401e6i	1.39722	+	1.39722i	0.589305	+	0.807911i	$0.299402\pi$
$74$			2.52526e6i			0.724429i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−4.55475e6	+	475136.i	−0.994603	+	0.103754i
$81$	−4.78297e6			−1.00000
$82$	7.06054e6	−	7.06054e6i	1.41413	−	1.41413i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	4.26761e6	−	5.26167e6i	0.753735	−	0.929304i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.56210e6i		−	1.43777i	−0.695131	−	0.718883i	$-0.744654\pi$
$89$		−	9.56210e6i		−	1.43777i	0.695131	−	0.718883i	$-0.255346\pi$
$90$	−5.37127e6	−	4.35650e6i	−0.776655	−	0.629926i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.06929e7	+	1.06929e7i	−1.18958	+	1.18958i	−0.212392	+	0.977185i	$0.568125\pi$
$97$	−1.06929e7	+	1.06929e7i	−1.18958	+	1.18958i	−0.977185	+	0.212392i	$0.931875\pi$
$98$	6.58834e6	+	6.58834e6i	0.707107	+	0.707107i
$99$		0			0
$100$	−2.06387e6	−	9.78470e6i	−0.206387	−	0.978470i
$101$	−1.33042e7			−1.28488			−0.642442	−	0.766334i	$-0.722079\pi$
$101$	−1.33042e7			−1.28488			−0.642442	+	0.766334i	$0.722079\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			2.25341e7i			1.96437i
$105$		0			0
$106$	2.25105e7			1.83576
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	2.41162e7i		−	1.78368i	−0.452352	−	0.891839i	$-0.649415\pi$
$109$		−	2.41162e7i		−	1.78368i	0.452352	−	0.891839i	$-0.350585\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.74590e6	+	9.74590e6i	0.635400	+	0.635400i	0.949417	−	0.314017i	$-0.101675\pi$
$113$	9.74590e6	+	9.74590e6i	0.635400	+	0.635400i	−0.314017	+	0.949417i	$0.601675\pi$
$114$		0			0
$115$		0			0
$116$	1.54680e7			0.920094
$117$	−2.40636e7	+	2.40636e7i	−1.38902	+	1.38902i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.94872e7			1.00000
$122$	−4.26058e6	+	4.26058e6i	−0.212427	+	0.212427i
$123$		0			0
$124$		0			0
$125$	2.07836e7	−	6.69932e6i	0.951776	−	0.306793i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−1.67772e7	−	1.67772e7i	−0.707107	−	0.707107i
$129$		0			0
$130$	−4.89413e7	+	5.10539e6i	−1.95377	+	0.203811i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	3.51007e7			1.19655
$137$	3.62213e6	−	3.62213e6i	0.120349	−	0.120349i	−0.644367	−	0.764716i	$-0.722879\pi$
$137$	3.62213e6	−	3.62213e6i	0.120349	−	0.120349i	0.764716	+	0.644367i	$0.222879\pi$
$138$		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
$144$		−	3.58318e7i		−	1.00000i
$145$	3.50448e6	+	3.35946e7i	0.0954630	+	0.915128i
$146$	−7.43042e7			−1.97596
$147$		0			0
$148$	−2.02021e7	−	2.02021e7i	−0.512249	−	0.512249i
$149$		−	7.31577e7i		−	1.81179i	−0.423502	−	0.905895i	$-0.639199\pi$
$149$		−	7.31577e7i		−	1.81179i	0.423502	−	0.905895i	$-0.360801\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	3.74830e7	+	3.74830e7i	0.846086	+	0.846086i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.86842e7	−	2.86842e7i	0.591554	−	0.591554i	−0.346497	−	0.938051i	$-0.612629\pi$
$157$	2.86842e7	−	2.86842e7i	0.591554	−	0.591554i	0.938051	+	0.346497i	$0.112629\pi$
$158$		0			0
$159$		0			0
$160$	3.26369e7	−	4.02391e7i	0.629926	−	0.776655i
$161$		0			0
$162$	3.82638e7	−	3.82638e7i	0.707107	−	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$			1.12969e8i			1.99988i
$165$		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
$169$			1.79384e8i			2.85877i
$170$	7.95250e6	+	7.62343e7i	0.124146	+	1.19009i
$171$		0			0
$172$		0			0
$173$	−9.24761e7	−	9.24761e7i	−1.35790	−	1.35790i	−0.876501	−	0.481400i	$-0.840129\pi$
$173$	−9.24761e7	−	9.24761e7i	−1.35790	−	1.35790i	−0.481400	−	0.876501i	$-0.659871\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	7.64968e7	+	7.64968e7i	1.01665	+	1.01665i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	7.78222e7	−	8.11814e6i	0.994603	−	0.103754i
$181$	−1.45860e8			−1.82836			−0.914180	−	0.405309i	$-0.867164\pi$
$181$	−1.45860e8			−1.82836			−0.914180	+	0.405309i	$0.867164\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.92994e7	−	4.84535e7i	0.456336	−	0.562632i
$186$		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
$193$	−3.22647e7	−	3.22647e7i	−0.323056	−	0.323056i	0.526883	−	0.849938i	$-0.323361\pi$
$193$	−3.22647e7	−	3.22647e7i	−0.323056	−	0.323056i	−0.849938	+	0.526883i	$0.823361\pi$
$194$		−	1.71086e8i		−	1.68232i
$195$		0			0
$196$	−1.05414e8			−1.00000
$197$	1.45581e8	−	1.45581e8i	1.35667	−	1.35667i	0.478681	−	0.877989i	$-0.341115\pi$
$197$	1.45581e8	−	1.45581e8i	1.35667	−	1.35667i	0.877989	−	0.478681i	$-0.158885\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	9.47886e7	+	6.17667e7i	0.837821	+	0.545945i
$201$		0			0
$202$	1.06434e8	−	1.06434e8i	0.908550	−	0.908550i
$203$		0			0
$204$		0			0
$205$	−2.45354e8	+	2.55945e7i	−1.98909	+	0.207495i
$206$		0			0
$207$		0			0
$208$	−1.80273e8	−	1.80273e8i	−1.38902	−	1.38902i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−1.80084e8	+	1.80084e8i	−1.29808	+	1.29808i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.92930e8	+	1.92930e8i	1.26125	+	1.26125i
$219$		0			0
$220$		0			0
$221$	3.77161e8			2.35046
$222$		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
$225$	3.52632e7	+	1.67181e8i	0.206387	+	0.978470i
$226$	−1.55934e8			−0.898592
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			2.90890e8i			1.60068i	0.599547	+	0.800339i	$0.295347\pi$
$229$			2.90890e8i			1.60068i	−0.599547	+	0.800339i	$0.704653\pi$
$230$		0			0
$231$		0			0
$232$	−1.23744e8	+	1.23744e8i	−0.650605	+	0.650605i
$233$	1.78471e7	+	1.78471e7i	0.0924320	+	0.0924320i	0.751811	−	0.659379i	$-0.229181\pi$
$233$	1.78471e7	+	1.78471e7i	0.0924320	+	0.0924320i	−0.659379	+	0.751811i	$0.729181\pi$
$234$		−	3.85017e8i		−	1.96437i
$235$		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
$241$	4.20719e8			1.93612			0.968061	−	0.250717i	$-0.0806662\pi$
$241$	4.20719e8			1.93612			0.968061	+	0.250717i	$0.0806662\pi$
$242$	−1.55897e8	+	1.55897e8i	−0.707107	+	0.707107i
$243$		0			0
$244$		−	6.81692e7i		−	0.300417i
$245$	−2.38827e7	−	2.28945e8i	−0.103754	−	0.994603i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−1.12674e8	+	2.19863e8i	−0.456072	+	0.889943i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	2.68435e8			1.00000
$257$	−1.31406e8	+	1.31406e8i	−0.482893	+	0.482893i	−0.906054	−	0.423161i	$-0.860920\pi$
$257$	−1.31406e8	+	1.31406e8i	−0.482893	+	0.482893i	0.423161	+	0.906054i	$0.360920\pi$
$258$		0			0
$259$		0			0
$260$	3.50688e8	−	4.32374e8i	1.23741	−	1.52564i
$261$	−2.64286e8			−0.920094
$262$		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
$265$	−4.31920e8	−	3.50320e8i	−1.42575	−	1.15639i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	7.80336e7i		−	0.244427i	−0.992504	−	0.122213i	$-0.961001\pi$
$269$		−	7.80336e7i		−	0.244427i	0.992504	−	0.122213i	$-0.0389992\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−2.80805e8	+	2.80805e8i	−0.846086	+	0.846086i
$273$		0			0
$274$			5.79541e7i			0.170199i
$275$		0			0
$276$		0			0
$277$	3.61729e7	−	3.61729e7i	0.102259	−	0.102259i	−0.654126	−	0.756386i	$-0.726963\pi$
$277$	3.61729e7	−	3.61729e7i	0.102259	−	0.102259i	0.756386	+	0.654126i	$0.226963\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.88434e8			−1.04435			−0.522174	−	0.852839i	$-0.674879\pi$
$281$	−3.88434e8			−1.04435			−0.522174	+	0.852839i	$0.674879\pi$
$282$		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
$288$	2.86654e8	+	2.86654e8i	0.707107	+	0.707107i
$289$		−	1.77152e8i		−	0.431721i
$290$	−2.96793e8	−	2.40721e8i	−0.714596	−	0.579591i
$291$		0			0
$292$	5.94434e8	−	5.94434e8i	1.39722	−	1.39722i
$293$	−2.09198e7	−	2.09198e7i	−0.0485871	−	0.0485871i	0.682396	−	0.730983i	$-0.260938\pi$
$293$	−2.09198e7	−	2.09198e7i	−0.0485871	−	0.0485871i	−0.730983	+	0.682396i	$0.760938\pi$
$294$		0			0
$295$		0			0
$296$	3.23234e8			0.724429
$297$		0			0
$298$	5.85261e8	+	5.85261e8i	1.28113	+	1.28113i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.48055e8	−	1.54446e7i	0.298795	−	0.0311693i
$306$	−5.99728e8			−1.19655
$307$		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
$313$	2.11916e8	+	2.11916e8i	0.390623	+	0.390623i	0.874910	−	0.484286i	$-0.160921\pi$
$313$	2.11916e8	+	2.11916e8i	0.390623	+	0.390623i	−0.484286	+	0.874910i	$0.660921\pi$
$314$			4.58948e8i			0.836583i
$315$		0			0
$316$		0			0
$317$	−5.95125e8	+	5.95125e8i	−1.04930	+	1.04930i	−0.0505827	+	0.998720i	$0.516108\pi$
$317$	−5.95125e8	+	5.95125e8i	−1.04930	+	1.04930i	−0.998720	+	0.0505827i	$0.983892\pi$
$318$		0			0
$319$		0			0
$320$	6.08174e7	+	5.83008e8i	0.103754	+	0.994603i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			6.12220e8i			1.00000i
$325$	1.01851e9	+	6.63690e8i	1.64579	+	1.07244i
$326$		0			0
$327$		0			0
$328$	−9.03750e8	−	9.03750e8i	−1.41413	−	1.41413i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	3.45172e8	+	3.45172e8i	0.512249	+	0.512249i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.36651e8	−	9.36651e8i	1.33313	−	1.33313i	0.430578	−	0.902553i	$-0.358310\pi$
$337$	9.36651e8	−	9.36651e8i	1.33313	−	1.33313i	0.902553	−	0.430578i	$-0.141690\pi$
$338$	−1.43507e9	−	1.43507e9i	−2.02145	−	2.02145i
$339$		0			0
$340$	−6.73494e8	−	5.46254e8i	−0.929304	−	0.753735i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.47962e9			1.92036
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	5.08394e8i		−	0.640194i	−0.947385	−	0.320097i	$-0.896285\pi$
$349$		−	5.08394e8i		−	0.640194i	0.947385	−	0.320097i	$-0.103715\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.73037e8	−	7.73037e8i	−0.935382	−	0.935382i	0.0626535	−	0.998035i	$-0.480044\pi$
$353$	−7.73037e8	−	7.73037e8i	−0.935382	−	0.935382i	−0.998035	+	0.0626535i	$0.980044\pi$
$354$		0			0
$355$		0			0
$356$	−1.22395e9			−1.43777
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−5.57633e8	+	6.87523e8i	−0.629926	+	0.776655i
$361$	−8.93872e8			−1.00000
$362$	1.16688e9	−	1.16688e9i	1.29285	−	1.29285i
$363$		0			0
$364$		0			0
$365$	1.42571e9	+	1.15636e9i	1.53464	+	1.24471i
$366$		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
$369$		−	1.93018e9i		−	1.99988i
$370$	7.32327e7	+	7.02023e8i	0.0751621	+	0.720519i
$371$		0			0
$372$		0			0
$373$	3.36984e8	+	3.36984e8i	0.336223	+	0.336223i	0.854944	−	0.518720i	$-0.173591\pi$
$373$	3.36984e8	+	3.36984e8i	0.336223	+	0.336223i	−0.518720	+	0.854944i	$0.673591\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.32965e9	+	1.32965e9i	−1.27803	+	1.27803i
$378$		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
$386$	5.16235e8			0.456870
$387$		0			0
$388$	1.36868e9	+	1.36868e9i	1.18958	+	1.18958i
$389$			1.94162e9i			1.67240i	0.548422	+	0.836202i	$0.315229\pi$
$389$			1.94162e9i			1.67240i	−0.548422	+	0.836202i	$0.684771\pi$
$390$		0			0
$391$		0			0
$392$	8.43308e8	−	8.43308e8i	0.707107	−	0.707107i
$393$		0			0
$394$			2.32930e9i			1.91862i
$395$		0			0
$396$		0			0
$397$	3.76230e8	−	3.76230e8i	0.301777	−	0.301777i	−0.539932	−	0.841709i	$-0.681550\pi$
$397$	3.76230e8	−	3.76230e8i	0.301777	−	0.301777i	0.841709	+	0.539932i	$0.181550\pi$
$398$		0			0
$399$		0			0
$400$	−1.25244e9	+	2.64176e8i	−0.978470	+	0.206387i
$401$	−8.84817e8			−0.685248			−0.342624	−	0.939473i	$-0.611316\pi$
$401$	−8.84817e8			−0.685248			−0.342624	+	0.939473i	$0.611316\pi$
$402$		0			0
$403$		0			0
$404$			1.70294e9i			1.28488i
$405$	−1.32967e9	+	1.38706e8i	−0.994603	+	0.103754i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	2.38968e9i		−	1.72706i	−0.504296	−	0.863531i	$-0.668248\pi$
$409$		−	2.38968e9i		−	1.72706i	0.504296	−	0.863531i	$-0.331752\pi$
$410$	1.75808e9	−	2.16759e9i	1.25978	−	1.55322i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	2.88437e9			1.96437
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.61988e9			−1.71117			−0.855586	−	0.517661i	$-0.826803\pi$
$421$	−2.61988e9			−1.71117			−0.855586	+	0.517661i	$0.826803\pi$
$422$		0			0
$423$		0			0
$424$		−	2.88135e9i		−	1.83576i
$425$	1.03381e9	−	1.58651e9i	0.653248	−	1.00249i
$426$		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
$433$	8.76483e8	+	8.76483e8i	0.518843	+	0.518843i	0.917221	−	0.398378i	$-0.130427\pi$
$433$	8.76483e8	+	8.76483e8i	0.518843	+	0.518843i	−0.398378	+	0.917221i	$0.630427\pi$
$434$		0			0
$435$		0			0
$436$	−3.08688e9			−1.78368
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	1.80109e9			1.00000
$442$	−3.01729e9	+	3.01729e9i	−1.66203	+	1.66203i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−2.77301e8	−	2.65826e9i	−0.149173	−	1.43001i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.71111e9i			1.41346i	0.707481	+	0.706732i	$0.249832\pi$
$449$			2.71111e9i			1.41346i	−0.707481	+	0.706732i	$0.750168\pi$
$450$	−1.61955e9	−	1.05534e9i	−0.837821	−	0.545945i
$451$		0			0
$452$	1.24748e9	−	1.24748e9i	0.635400	−	0.635400i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.48649e9	+	2.48649e9i	−1.21865	+	1.21865i	−0.250548	+	0.968104i	$0.580611\pi$
$457$	−2.48649e9	+	2.48649e9i	−1.21865	+	1.21865i	−0.968104	+	0.250548i	$0.919389\pi$
$458$	−2.32712e9	−	2.32712e9i	−1.13185	−	1.13185i
$459$		0			0
$460$		0			0
$461$	4.07657e9			1.93795			0.968974	−	0.247164i	$-0.0794988\pi$
$461$	4.07657e9			1.93795			0.968974	+	0.247164i	$0.0794988\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	1.97991e9i		−	0.920094i
$465$		0			0
$466$	−2.85554e8			−0.130719
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	3.08014e9	+	3.08014e9i	1.38902	+	1.38902i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.07691e9	−	3.07691e9i	1.29808	−	1.29808i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	3.47318e9			1.42305
$482$	−3.36575e9	+	3.36575e9i	−1.36904	+	1.36904i
$483$		0			0
$484$		−	2.49436e9i		−	1.00000i
$485$	−2.66252e9	+	3.28271e9i	−1.05973	+	1.30658i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	5.45354e8	+	5.45354e8i	0.212427	+	0.212427i
$489$		0			0
$490$	2.02262e9	+	1.64050e9i	0.776655	+	0.629926i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	2.07115e9	+	2.07115e9i	0.778478	+	0.778478i
$494$		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	$0.5\pi$
$500$	−8.57513e8	−	2.66030e9i	−0.306793	−	0.951776i
$501$		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
$505$	−3.69857e9	+	3.85822e8i	−1.27795	+	0.133311i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	3.84397e7i		−	0.0129201i	−0.999979	−	0.00646007i	$-0.997944\pi$
$509$		−	3.84397e7i		−	0.0129201i	0.999979	−	0.00646007i	$-0.00205632\pi$
$510$		0			0
$511$		0			0
$512$	−2.14748e9	+	2.14748e9i	−0.707107	+	0.707107i
$513$		0			0
$514$		−	2.10250e9i		−	0.682914i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	6.53490e8	+	6.26449e9i	0.203811	+	1.95377i
$521$	2.38452e9			0.738701			0.369350	−	0.929290i	$-0.379580\pi$
$521$	2.38452e9			0.738701			0.369350	+	0.929290i	$0.379580\pi$
$522$	2.11429e9	−	2.11429e9i	0.650605	−	0.650605i
$523$		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
$529$			3.40483e9i			1.00000i
$530$	6.25792e9	−	6.52805e8i	1.82585	−	0.190466i
$531$		0			0
$532$		0			0
$533$	−9.71090e9	−	9.71090e9i	−2.77788	−	2.77788i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	6.24269e8	+	6.24269e8i	0.172836	+	0.172836i
$539$		0			0
$540$		0			0
$541$	−7.36235e9			−1.99906			−0.999530	−	0.0306534i	$-0.990241\pi$
$541$	−7.36235e9			−1.99906			−0.999530	+	0.0306534i	$0.990241\pi$
$542$		0			0
$543$		0			0
$544$		−	4.49289e9i		−	1.19655i
$545$	−6.99371e8	−	6.70431e9i	−0.185063	−	1.77405i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−4.63633e8	−	4.63633e8i	−0.120349	−	0.120349i
$549$			1.16473e9i			0.300417i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			5.78766e8i			0.144617i
$555$		0			0
$556$		0			0
$557$	2.85686e9	−	2.85686e9i	0.700480	−	0.700480i	−0.264034	−	0.964513i	$-0.585053\pi$
$557$	2.85686e9	−	2.85686e9i	0.700480	−	0.700480i	0.964513	+	0.264034i	$0.0850530\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.10747e9	−	3.10747e9i	0.738465	−	0.738465i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	2.99199e9	+	2.42673e9i	0.697896	+	0.566046i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			6.84617e9i			1.55796i	0.627052	+	0.778978i	$0.284262\pi$
$569$			6.84617e9i			1.55796i	−0.627052	+	0.778978i	$0.715738\pi$
$570$		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
$576$	−4.58647e9			−1.00000
$577$	5.74586e9	−	5.74586e9i	1.24520	−	1.24520i	0.287387	−	0.957814i	$-0.407213\pi$
$577$	5.74586e9	−	5.74586e9i	1.24520	−	1.24520i	0.957814	−	0.287387i	$-0.0927867\pi$
$578$	1.41722e9	+	1.41722e9i	0.305273	+	0.305273i
$579$		0			0
$580$	4.30011e9	−	4.48573e8i	0.915128	−	0.0954630i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			9.51094e9i			1.97596i
$585$	−5.99183e9	+	7.38751e9i	−1.23741	+	1.52564i
$586$	3.34717e8			0.0687126
$587$		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
$592$	−2.58587e9	+	2.58587e9i	−0.512249	+	0.512249i
$593$	7.18075e9	+	7.18075e9i	1.41409	+	1.41409i	0.716265	+	0.697829i	$0.245850\pi$
$593$	7.18075e9	+	7.18075e9i	1.41409	+	1.41409i	0.697829	+	0.716265i	$0.254150\pi$
$594$		0			0
$595$		0			0
$596$	−9.36418e9			−1.81179
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−1.41174e9			−0.265273			−0.132637	−	0.991165i	$-0.542344\pi$
$601$	−1.41174e9			−0.265273			−0.132637	+	0.991165i	$0.542344\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.41743e9	−	5.65128e8i	0.994603	−	0.103754i
$606$		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
$610$	−1.06088e9	+	1.30800e9i	−0.189240	+	0.233320i
$611$		0			0
$612$	4.79782e9	−	4.79782e9i	0.846086	−	0.846086i
$613$	1.60192e9	+	1.60192e9i	0.280885	+	0.280885i	0.833462	−	0.552577i	$-0.186355\pi$
$613$	1.60192e9	+	1.60192e9i	0.280885	+	0.280885i	−0.552577	+	0.833462i	$0.686355\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.64897e9	+	4.64897e9i	−0.796817	+	0.796817i	−0.982592	−	0.185776i	$-0.940520\pi$
$617$	−4.64897e9	+	4.64897e9i	−0.796817	+	0.796817i	0.185776	+	0.982592i	$0.440520\pi$
$618$		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
$625$	5.58355e9	−	2.46513e9i	0.914809	−	0.403888i
$626$	−3.39065e9			−0.552425
$627$		0			0
$628$	−3.67158e9	−	3.67158e9i	−0.591554	−	0.591554i
$629$		−	5.41006e9i		−	0.866812i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	9.52200e9i		−	1.48394i
$635$		0			0
$636$		0			0
$637$	9.06144e9	−	9.06144e9i	1.38902	−	1.38902i
$638$		0			0
$639$		0			0
$640$	−5.15061e9	−	4.17753e9i	−0.776655	−	0.629926i
$641$	1.19487e10			1.79192			0.895959	−	0.444136i	$-0.146489\pi$
$641$	1.19487e10			1.79192			0.895959	+	0.444136i	$0.146489\pi$
$642$		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	$0.750000\pi$
$648$	−4.89776e9	−	4.89776e9i	−0.707107	−	0.707107i
$649$		0			0
$650$	−1.34576e10	+	2.83860e9i	−1.92208	+	0.405422i
$651$		0			0
$652$		0			0
$653$	−9.87957e9	−	9.87957e9i	−1.38849	−	1.38849i	−0.828504	−	0.559983i	$-0.810808\pi$
$653$	−9.87957e9	−	9.87957e9i	−1.38849	−	1.38849i	−0.559983	−	0.828504i	$-0.689192\pi$
$654$		0			0
$655$		0			0
$656$	1.44600e10			1.99988
$657$	−1.01565e10	+	1.01565e10i	−1.39722	+	1.39722i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−1.48051e10			−1.99392			−0.996958	−	0.0779394i	$-0.975166\pi$
$661$	−1.48051e10			−1.99392			−0.996958	+	0.0779394i	$0.975166\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−5.52275e9			−0.724429
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9.47042e9	+	9.47042e9i	1.19761	+	1.19761i	0.974879	+	0.222734i	$0.0714982\pi$
$673$	9.47042e9	+	9.47042e9i	1.19761	+	1.19761i	0.222734	+	0.974879i	$0.428502\pi$
$674$			1.49864e10i			1.88533i
$675$		0			0
$676$	2.29611e10			2.85877
$677$	−5.63649e9	+	5.63649e9i	−0.698149	+	0.698149i	−0.964011	−	0.265862i	$-0.914343\pi$
$677$	−5.63649e9	+	5.63649e9i	−0.698149	+	0.698149i	0.265862	+	0.964011i	$0.414343\pi$
$678$		0			0
$679$		0			0
$680$	9.75799e9	−	1.01792e9i	1.19009	−	0.124146i
$681$		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
$685$	9.01910e8	−	1.11199e9i	0.107213	−	0.132186i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	3.09604e10i		−	3.60611i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−1.18369e10	+	1.18369e10i	−1.35790	+	1.35790i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.51263e10	+	1.51263e10i	−1.69207	+	1.69207i
$698$	4.06715e9	+	4.06715e9i	0.452685	+	0.452685i
$699$		0			0
$700$		0			0
$701$	4.35144e9			0.477111			0.238556	−	0.971129i	$-0.423326\pi$
$701$	4.35144e9			0.477111			0.238556	+	0.971129i	$0.423326\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.23686e10			1.32283
$707$		0			0
$708$		0			0
$709$			9.48757e9i			0.999754i	0.866096	+	0.499877i	$0.166621\pi$
$709$			9.48757e9i			0.999754i	−0.866096	+	0.499877i	$0.833379\pi$
$710$		0			0
$711$		0			0
$712$	9.79159e9	−	9.79159e9i	1.01665	−	1.01665i
$713$		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	$0.5\pi$
$720$	−1.03912e9	−	9.96124e9i	−0.103754	−	0.994603i
$721$		0			0
$722$	7.15097e9	−	7.15097e9i	0.707107	−	0.707107i
$723$		0			0
$724$			1.86701e10i			1.82836i
$725$	1.94849e9	+	9.23768e9i	0.189896	+	0.900285i
$726$		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
$729$		−	1.04604e10i		−	1.00000i
$730$	−2.06566e10	+	2.15482e9i	−1.96530	+	0.205013i
$731$		0			0
$732$		0			0
$733$	3.99048e9	+	3.99048e9i	0.374250	+	0.374250i	0.869022	−	0.494773i	$-0.164749\pi$
$733$	3.99048e9	+	3.99048e9i	0.374250	+	0.374250i	−0.494773	+	0.869022i	$0.664749\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.54414e10	+	1.54414e10i	1.41413	+	1.41413i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−6.20205e9	−	5.03033e9i	−0.562632	−	0.456336i
$741$		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
$745$	−2.12157e9	−	2.03378e10i	−0.187980	−	1.80201i
$746$	−5.39174e9			−0.475492
$747$		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
$754$		−	2.12743e10i		−	1.80741i
$755$		0			0
$756$		0			0
$757$	−3.96360e8	+	3.96360e8i	−0.0332089	+	0.0332089i	−0.723516	−	0.690307i	$-0.757475\pi$
$757$	−3.96360e8	+	3.96360e8i	−0.0332089	+	0.0332089i	0.690307	+	0.723516i	$0.257475\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.99861e10			−1.64392			−0.821961	−	0.569543i	$-0.807120\pi$
$761$	−1.99861e10			−1.64392			−0.821961	+	0.569543i	$0.807120\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.15073e10	+	9.33327e9i	0.929304	+	0.753735i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			2.25801e8i			0.0179054i	0.999960	+	0.00895271i	$0.00284977\pi$
$769$			2.25801e8i			0.0179054i	−0.999960	+	0.00895271i	$0.997150\pi$
$770$		0			0
$771$		0			0
$772$	−4.12988e9	+	4.12988e9i	−0.323056	+	0.323056i
$773$	1.13946e10	+	1.13946e10i	0.887299	+	0.887299i	0.994263	−	0.106964i	$-0.0341129\pi$
$773$	1.13946e10	+	1.13946e10i	0.887299	+	0.887299i	−0.106964	+	0.994263i	$0.534113\pi$
$774$		0			0
$775$		0			0
$776$	−2.18990e10			−1.68232
$777$		0			0
$778$	−1.55330e10	−	1.55330e10i	−1.18257	−	1.18257i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			1.34929e10i			1.00000i
$785$	7.14237e9	−	8.80606e9i	0.526985	−	0.649737i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−1.86344e10	−	1.86344e10i	−1.35667	−	1.35667i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.85989e9	+	5.85989e9i	0.417285	+	0.417285i
$794$			6.01968e9i			0.426778i
$795$		0			0
$796$		0			0
$797$	1.82397e10	−	1.82397e10i	1.27619	−	1.27619i	0.333400	−	0.942785i	$-0.391804\pi$
$797$	1.82397e10	−	1.82397e10i	1.27619	−	1.27619i	0.942785	−	0.333400i	$-0.108196\pi$
$798$		0			0
$799$		0			0
$800$	7.90613e9	−	1.21329e10i	0.545945	−	0.837821i
$801$	2.09123e10			1.43777
$802$	7.07853e9	−	7.07853e9i	0.484544	−	0.484544i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.36235e10	−	1.36235e10i	−0.908550	−	0.908550i
$809$			9.86925e9i			0.655336i	0.944793	+	0.327668i	$0.106263\pi$
$809$			9.86925e9i			0.655336i	−0.944793	+	0.327668i	$0.893737\pi$
$810$	9.52767e9	−	1.17470e10i	0.629926	−	0.776655i
$811$		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
$818$	1.91174e10	+	1.91174e10i	1.22122	+	1.22122i
$819$		0			0
$820$	3.27609e9	+	3.14053e10i	0.207495	+	1.98909i
$821$	−1.32726e10			−0.837059			−0.418530	−	0.908203i	$-0.637454\pi$
$821$	−1.32726e10			−0.837059			−0.418530	+	0.908203i	$0.637454\pi$
$822$		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
$829$			2.59446e10i			1.58163i	0.612052	+	0.790817i	$0.290344\pi$
$829$			2.59446e10i			1.58163i	−0.612052	+	0.790817i	$0.709656\pi$
$830$		0			0
$831$		0			0
$832$	−2.30750e10	+	2.30750e10i	−1.38902	+	1.38902i
$833$	−1.41147e10	−	1.41147e10i	−0.846086	−	0.846086i
$834$		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
$841$	2.64660e9			0.153427
$842$	2.09590e10	−	2.09590e10i	1.20998	−	1.20998i
$843$		0			0
$844$		0			0
$845$	5.20212e9	+	4.98686e10i	0.296607	+	2.84334i
$846$		0			0
$847$		0			0
$848$	2.30508e10	+	2.30508e10i	1.29808	+	1.29808i
$849$		0			0
$850$	4.42159e9	+	2.09625e10i	0.246952	+	1.17078i
$851$		0			0
$852$		0			0
$853$	−1.69291e10	−	1.69291e10i	−0.933926	−	0.933926i	0.0640220	−	0.997948i	$-0.479607\pi$
$853$	−1.69291e10	−	1.69291e10i	−0.933926	−	0.933926i	−0.997948	+	0.0640220i	$0.979607\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.51299e9	+	4.51299e9i	−0.244924	+	0.244924i	−0.818884	−	0.573960i	$-0.805407\pi$
$857$	−4.51299e9	+	4.51299e9i	−0.244924	+	0.244924i	0.573960	+	0.818884i	$0.305407\pi$
$858$		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
$865$	−2.83902e10	−	2.30265e10i	−1.49146	−	1.20969i
$866$	−1.40237e10			−0.733754
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	2.46950e10	−	2.46950e10i	1.26125	−	1.26125i
$873$	−2.33853e10	−	2.33853e10i	−1.18958	−	1.18958i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.26128e10	−	2.26128e10i	1.13203	−	1.13203i	0.142187	−	0.989840i	$-0.454587\pi$
$877$	2.26128e10	−	2.26128e10i	1.13203	−	1.13203i	0.989840	−	0.142187i	$-0.0454134\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.03243e10			1.49408			0.747041	−	0.664778i	$-0.231474\pi$
$881$	3.03243e10			1.49408			0.747041	+	0.664778i	$0.231474\pi$
$882$	−1.44087e10	+	1.44087e10i	−0.707107	+	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		−	4.82766e10i		−	2.35046i
$885$		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
$890$	2.34845e10	+	1.90477e10i	1.11665	+	0.905685i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−2.16889e10	−	2.16889e10i	−0.999471	−	0.999471i
$899$		0			0
$900$	2.13991e10	−	4.51369e9i	0.978470	−	0.206387i
$901$	−4.82260e10			−2.19657
$902$		0			0
$903$		0			0
$904$			1.99596e10i			0.898592i
$905$	−4.05491e10	+	4.22994e9i	−1.81849	+	0.189699i
$906$		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
$909$		−	2.90963e10i		−	1.28488i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	3.97838e10i		−	1.72343i
$915$		0			0
$916$	3.72339e10			1.60068
$917$		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
$922$	−3.26126e10	+	3.26126e10i	−1.37034	+	1.37034i
$923$		0			0
$924$		0			0
$925$	9.52009e9	−	1.46098e10i	0.395499	−	0.606942i
$926$		0			0
$927$		0			0
$928$	1.58393e10	+	1.58393e10i	0.650605	+	0.650605i
$929$			1.43415e10i			0.586866i	0.955980	+	0.293433i	$0.0947978\pi$
$929$			1.43415e10i			0.586866i	−0.955980	+	0.293433i	$0.905202\pi$
$930$		0			0
$931$		0			0
$932$	2.28443e9	−	2.28443e9i	0.0924320	−	0.0924320i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−4.92822e10			−1.96437
$937$	2.45467e10	−	2.45467e10i	0.974776	−	0.974776i	−0.0249138	−	0.999690i	$-0.507931\pi$
$937$	2.45467e10	−	2.45467e10i	0.974776	−	0.974776i	0.999690	+	0.0249138i	$0.00793112\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.49808e10			1.36857			0.684283	−	0.729217i	$-0.260115\pi$
$941$	3.49808e10			1.36857			0.684283	+	0.729217i	$0.260115\pi$
$942$		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
$949$			1.02196e11i			3.88153i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.92270e10	+	2.92270e10i	1.09385	+	1.09385i	0.995113	+	0.0987400i	$0.0314812\pi$
$953$	2.92270e10	+	2.92270e10i	1.09385	+	1.09385i	0.0987400	+	0.995113i	$0.468519\pi$
$954$			4.92305e10i			1.83576i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.75126e10			1.00000
$962$	−2.77855e10	+	2.77855e10i	−1.00625	+	1.00625i
$963$		0			0
$964$		−	5.38520e10i		−	1.93612i
$965$	−9.90527e9	−	8.03391e9i	−0.354830	−	0.287794i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	1.99549e10	+	1.99549e10i	0.707107	+	0.707107i
$969$		0			0
$970$	−4.96148e9	−	4.75618e10i	−0.174546	−	1.67324i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−8.72566e9			−0.300417
$977$	−4.09613e10	+	4.09613e10i	−1.40522	+	1.40522i	−0.622970	+	0.782246i	$0.714074\pi$
$977$	−4.09613e10	+	4.09613e10i	−1.40522	+	1.40522i	−0.782246	+	0.622970i	$0.785926\pi$
$978$		0			0
$979$		0			0
$980$	−2.93050e10	+	3.05699e9i	−0.994603	+	0.103754i
$981$	5.27422e10			1.78368
$982$		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
$985$	3.62498e10	−	4.46935e10i	1.20859	−	1.49011i
$986$	−3.31383e10			−1.10093
$987$		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
$997$	2.32398e10	−	2.32398e10i	0.742676	−	0.742676i	−0.230416	−	0.973092i	$-0.574009\pi$
$997$	2.32398e10	−	2.32398e10i	0.742676	−	0.742676i	0.973092	+	0.230416i	$0.0740087\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.a.3.1	✓	2
4.3	odd	2	CM	20.8.e.a.3.1	✓	2
5.2	odd	4	inner	20.8.e.a.7.1	yes	2
5.3	odd	4		100.8.e.c.7.1		2
5.4	even	2		100.8.e.c.43.1		2
20.3	even	4		100.8.e.c.7.1		2
20.7	even	4	inner	20.8.e.a.7.1	yes	2
20.19	odd	2		100.8.e.c.43.1		2

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

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.00000 + 8.00000i) q^{2} -128.000i q^{4} +(278.000 - 29.0000i) q^{5} +(1024.00 + 1024.00i) q^{8} +2187.00i q^{9} +O(q^{10})\)	\(q+(-8.00000 + 8.00000i) q^{2} -128.000i q^{4} +(278.000 - 29.0000i) q^{5} +(1024.00 + 1024.00i) q^{8} +2187.00i q^{9} +(-1992.00 + 2456.00i) q^{10} +(11003.0 + 11003.0i) q^{13} -16384.0 q^{16} +(17139.0 - 17139.0i) q^{17} +(-17496.0 - 17496.0i) q^{18} +(-3712.00 - 35584.0i) q^{20} +(76443.0 - 16124.0i) q^{25} -176048. q^{26} +120844. i q^{29} +(131072. - 131072. i) q^{32} +274224. i q^{34} +279936. q^{36} +(157829. - 157829. i) q^{37} +(314368. + 254976. i) q^{40} -882568. q^{41} +(63423.0 + 607986. i) q^{45} -823543. i q^{49} +(-482552. + 740536. i) q^{50} +(1.40838e6 - 1.40838e6i) q^{52} +(-1.40691e6 - 1.40691e6i) q^{53} +(-966752. - 966752. i) q^{58} +532572. q^{61} +2.09715e6i q^{64} +(3.37792e6 + 2.73975e6i) q^{65} +(-2.19379e6 - 2.19379e6i) q^{68} +(-2.23949e6 + 2.23949e6i) q^{72} +(4.64401e6 + 4.64401e6i) q^{73} +2.52526e6i q^{74} +(-4.55475e6 + 475136. i) q^{80} -4.78297e6 q^{81} +(7.06054e6 - 7.06054e6i) q^{82} +(4.26761e6 - 5.26167e6i) q^{85} -9.56210e6i q^{89} +(-5.37127e6 - 4.35650e6i) q^{90} +(-1.06929e7 + 1.06929e7i) q^{97} +(6.58834e6 + 6.58834e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} + 556 q^{5} + 2048 q^{8}+O(q^{10}) \) 2 * q - 16 * q^2 + 556 * q^5 + 2048 * q^8	\( 2 q - 16 q^{2} + 556 q^{5} + 2048 q^{8} - 3984 q^{10} + 22006 q^{13} - 32768 q^{16} + 34278 q^{17} - 34992 q^{18} - 7424 q^{20} + 152886 q^{25} - 352096 q^{26} + 262144 q^{32} + 559872 q^{36} + 315658 q^{37} + 628736 q^{40} - 1765136 q^{41} + 126846 q^{45} - 965104 q^{50} + 2816768 q^{52} - 2813814 q^{53} - 1933504 q^{58} + 1065144 q^{61} + 6755842 q^{65} - 4387584 q^{68} - 4478976 q^{72} + 9288026 q^{73} - 9109504 q^{80} - 9565938 q^{81} + 14121088 q^{82} + 8535222 q^{85} - 10742544 q^{90} - 21385702 q^{97} + 13176688 q^{98}+O(q^{100}) \) 2 * q - 16 * q^2 + 556 * q^5 + 2048 * q^8 - 3984 * q^10 + 22006 * q^13 - 32768 * q^16 + 34278 * q^17 - 34992 * q^18 - 7424 * q^20 + 152886 * q^25 - 352096 * q^26 + 262144 * q^32 + 559872 * q^36 + 315658 * q^37 + 628736 * q^40 - 1765136 * q^41 + 126846 * q^45 - 965104 * q^50 + 2816768 * q^52 - 2813814 * q^53 - 1933504 * q^58 + 1065144 * q^61 + 6755842 * q^65 - 4387584 * q^68 - 4478976 * q^72 + 9288026 * q^73 - 9109504 * q^80 - 9565938 * q^81 + 14121088 * q^82 + 8535222 * q^85 - 10742544 * q^90 - 21385702 * q^97 + 13176688 * q^98

Introduction

L-functions

Modular forms

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Database

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