LMFDB - Embedded newform 21.8.g.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,8,Mod(5,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	21.g (of order $6$, 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:	$6.56008553517$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			5.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.5
Dual form			21.8.g.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.5000 - 23.3827i) q^{3} +(-64.0000 - 110.851i) q^{4} +(-881.500 - 215.640i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	$q+(40.5000 - 23.3827i) q^{3} +(-64.0000 - 110.851i) q^{4} +(-881.500 - 215.640i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-5184.00 - 2992.98i) q^{12} -15714.9i q^{13} +(-8192.00 + 14189.0i) q^{16} +(50269.5 + 29023.1i) q^{19} +(-40743.0 + 11878.4i) q^{21} +(39062.5 + 67658.2i) q^{25} -102276. i q^{27} +(32512.0 + 111516. i) q^{28} +(-13222.5 + 7634.01i) q^{31} -279936. q^{36} +(167831. - 290693. i) q^{37} +(-367456. - 636453. i) q^{39} +409495. q^{43} +766204. i q^{48} +(730542. + 380174. i) q^{49} +(-1.74202e6 + 1.00575e6i) q^{52} +2.71455e6 q^{57} +(-230574. - 133122. i) q^{61} +(-1.37234e6 + 1.43376e6i) q^{63} +2.09715e6 q^{64} +(-2.22176e6 - 3.84821e6i) q^{67} +(-5.65641e6 + 3.26573e6i) q^{73} +(3.16406e6 + 1.82677e6i) q^{75} -7.42992e6i q^{76} +(2.25881e6 - 3.91237e6i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +(3.92429e6 + 3.75619e6i) q^{84} +(-3.38877e6 + 1.38527e7i) q^{91} +(-357008. + 618355. i) q^{93} +1.31624e7i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 81 q^{3} - 128 q^{4} - 1763 q^{7} + 2187 q^{9}+O(q^{10}) $ 2 * q + 81 * q^3 - 128 * q^4 - 1763 * q^7 + 2187 * q^9	$ 2 q + 81 q^{3} - 128 q^{4} - 1763 q^{7} + 2187 q^{9} - 10368 q^{12} - 16384 q^{16} + 100539 q^{19} - 81486 q^{21} + 78125 q^{25} + 65024 q^{28} - 26445 q^{31} - 559872 q^{36} + 335663 q^{37} - 734913 q^{39} + 818990 q^{43} + 1461083 q^{49} - 3484032 q^{52} + 5429106 q^{57} - 461148 q^{61} - 2744685 q^{63} + 4194304 q^{64} - 4443527 q^{67} - 11312811 q^{73} + 6328125 q^{75} + 4517617 q^{79} - 4782969 q^{81} + 7848576 q^{84} - 6777531 q^{91} - 714015 q^{93}+O(q^{100}) $ 2 * q + 81 * q^3 - 128 * q^4 - 1763 * q^7 + 2187 * q^9 - 10368 * q^12 - 16384 * q^16 + 100539 * q^19 - 81486 * q^21 + 78125 * q^25 + 65024 * q^28 - 26445 * q^31 - 559872 * q^36 + 335663 * q^37 - 734913 * q^39 + 818990 * q^43 + 1461083 * q^49 - 3484032 * q^52 + 5429106 * q^57 - 461148 * q^61 - 2744685 * q^63 + 4194304 * q^64 - 4443527 * q^67 - 11312811 * q^73 + 6328125 * q^75 + 4517617 * q^79 - 4782969 * q^81 + 7848576 * q^84 - 6777531 * q^91 - 714015 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	40.5000	−	23.3827i	0.866025	−	0.500000i
$3$	40.5000	−	23.3827i	0.866025	−	0.500000i
$4$	−64.0000	−	110.851i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−881.500	−	215.640i	−0.971358	−	0.237622i
$7$	−881.500	−	215.640i	−0.971358	−	0.237622i
$8$		0			0
$9$	1093.50	−	1894.00i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−5184.00	−	2992.98i	−0.866025	−	0.500000i
$13$		−	15714.9i		−	1.98385i	−0.126808	−	0.991927i	$-0.540473\pi$
$13$		−	15714.9i		−	1.98385i	0.126808	−	0.991927i	$-0.459527\pi$
$14$		0			0
$15$		0			0
$16$	−8192.00	+	14189.0i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	50269.5	+	29023.1i	1.68138	+	0.970748i	0.960743	+	0.277439i	$0.0894857\pi$
$19$	50269.5	+	29023.1i	1.68138	+	0.970748i	0.720641	+	0.693308i	$0.243848\pi$
$20$		0			0
$21$	−40743.0	+	11878.4i	−0.960031	+	0.279892i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	39062.5	+	67658.2i	0.500000	+	0.866025i
$26$		0			0
$27$		−	102276.i		−	1.00000i
$28$	32512.0	+	111516.i	0.279892	+	0.960031i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−13222.5	+	7634.01i	−0.0797164	+	0.0460243i	−0.539328	−	0.842096i	$-0.681322\pi$
$31$	−13222.5	+	7634.01i	−0.0797164	+	0.0460243i	0.459612	+	0.888120i	$0.347988\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−279936.			−1.00000
$37$	167831.	−	290693.i	0.544713	−	0.943470i	−0.453912	−	0.891046i	$-0.649972\pi$
$37$	167831.	−	290693.i	0.544713	−	0.943470i	0.998625	−	0.0524236i	$-0.0166946\pi$
$38$		0			0
$39$	−367456.	−	636453.i	−0.991927	−	1.71807i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	409495.			0.785433			0.392716	−	0.919660i	$-0.371535\pi$
$43$	409495.			0.785433			0.392716	+	0.919660i	$0.371535\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$			766204.i			1.00000i
$49$	730542.	+	380174.i	0.887071	+	0.461632i
$50$		0			0
$51$		0			0
$52$	−1.74202e6	+	1.00575e6i	−1.71807	+	0.991927i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.71455e6			1.94150
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−230574.	−	133122.i	−0.130064	−	0.0750923i	0.433556	−	0.901126i	$-0.357258\pi$
$61$	−230574.	−	133122.i	−0.130064	−	0.0750923i	−0.563620	+	0.826034i	$0.690592\pi$
$62$		0			0
$63$	−1.37234e6	+	1.43376e6i	−0.691466	+	0.722409i
$64$	2.09715e6			1.00000
$65$		0			0
$66$		0			0
$67$	−2.22176e6	−	3.84821e6i	−0.902477	−	1.56314i	−0.824269	−	0.566198i	$-0.808414\pi$
$67$	−2.22176e6	−	3.84821e6i	−0.902477	−	1.56314i	−0.0782078	−	0.996937i	$-0.524920\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−5.65641e6	+	3.26573e6i	−1.70181	+	0.982539i	−0.757876	+	0.652398i	$0.773763\pi$
$73$	−5.65641e6	+	3.26573e6i	−1.70181	+	0.982539i	−0.943932	+	0.330141i	$0.892904\pi$
$74$		0			0
$75$	3.16406e6	+	1.82677e6i	0.866025	+	0.500000i
$76$		−	7.42992e6i		−	1.94150i
$77$		0			0
$78$		0			0
$79$	2.25881e6	−	3.91237e6i	0.515448	−	0.892782i	−0.484392	−	0.874851i	$-0.660959\pi$
$79$	2.25881e6	−	3.91237e6i	0.515448	−	0.892782i	0.999839	−	0.0179303i	$-0.00570769\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	3.92429e6	+	3.75619e6i	0.722409	+	0.691466i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−3.38877e6	+	1.38527e7i	−0.471408	+	1.92703i
$92$		0			0
$93$	−357008.	+	618355.i	−0.0460243	+	0.0797164i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.31624e7i			1.46431i	0.681137	+	0.732156i	$0.261486\pi$
$97$			1.31624e7i			1.46431i	−0.681137	+	0.732156i	$0.738514\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e6	−	8.66025e6i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	1.49730e7	+	8.64465e6i	1.35014	+	0.779502i	0.988268	−	0.152727i	$-0.0488056\pi$
$103$	1.49730e7	+	8.64465e6i	1.35014	+	0.779502i	0.361868	+	0.932229i	$0.382139\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−1.13374e7	+	6.54566e6i	−0.866025	+	0.500000i
$109$	−4.96086e6	−	8.59247e6i	−0.366914	−	0.635514i	0.622167	−	0.782884i	$-0.286252\pi$
$109$	−4.96086e6	−	8.59247e6i	−0.366914	−	0.635514i	−0.989081	+	0.147370i	$0.952919\pi$
$110$		0			0
$111$		−	1.56974e7i		−	1.08943i
$112$	1.02810e7	−	1.07410e7i	0.691466	−	0.722409i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.97640e7	−	1.71842e7i	−1.71807	−	0.991927i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.74359e6	+	1.68764e7i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	1.69248e6	+	977154.i	0.0797164	+	0.0460243i
$125$		0			0
$126$		0			0
$127$	3.10119e7			1.34343			0.671715	−	0.740810i	$-0.265558\pi$
$127$	3.10119e7			1.34343			0.671715	+	0.740810i	$0.265558\pi$
$128$		0			0
$129$	1.65845e7	−	9.57509e6i	0.680205	−	0.392716i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−3.80540e7	−	3.64240e7i	−1.40255	−	1.34248i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			3.25075e7i			1.02667i	0.858188	+	0.513336i	$0.171591\pi$
$139$			3.25075e7i			1.02667i	−0.858188	+	0.513336i	$0.828409\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.79159e7	+	3.10313e7i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	3.84764e7	−	1.68498e6i	0.999042	−	0.0437506i
$148$	−4.29649e7			−1.08943
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	3.85195e7	+	6.67177e7i	0.910460	+	1.57696i	0.813416	+	0.581683i	$0.197606\pi$
$151$	3.85195e7	+	6.67177e7i	0.910460	+	1.57696i	0.0970443	+	0.995280i	$0.469061\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.70344e7	+	8.14660e7i	−0.991927	+	1.71807i
$157$	4.48255e7	−	2.58800e7i	0.924435	−	0.533723i	0.0393881	−	0.999224i	$-0.487459\pi$
$157$	4.48255e7	−	2.58800e7i	0.924435	−	0.533723i	0.885047	+	0.465501i	$0.154126\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.36832e7	−	7.56615e7i	0.790056	−	1.36842i	−0.135876	−	0.990726i	$-0.543385\pi$
$163$	4.36832e7	−	7.56615e7i	0.790056	−	1.36842i	0.925932	−	0.377691i	$-0.123282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.84209e8			−2.93568
$170$		0			0
$171$	1.09939e8	−	6.34735e7i	1.68138	−	0.970748i
$172$	−2.62077e7	−	4.53930e7i	−0.392716	−	0.680205i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−1.98438e7	−	6.80642e7i	−0.279892	−	0.960031i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	4.27835e7i		−	0.536292i	−0.963378	−	0.268146i	$-0.913589\pi$
$181$		−	4.27835e7i		−	0.536292i	0.963378	−	0.268146i	$-0.0864109\pi$
$182$		0			0
$183$	−1.24510e7			−0.150185
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.20548e7	+	9.01562e7i	−0.237622	+	0.971358i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	8.49347e7	−	4.90370e7i	0.866025	−	0.500000i
$193$	9.98550e7	+	1.72954e8i	0.999814	+	1.73173i	0.516613	+	0.856219i	$0.327193\pi$
$193$	9.98550e7	+	1.72954e8i	0.999814	+	1.73173i	0.483201	+	0.875509i	$0.339474\pi$
$194$		0			0
$195$		0			0
$196$	−4.61190e6	−	1.05313e8i	−0.0437506	−	0.999042i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.46130e8	+	8.43683e7i	−1.31448	+	0.758916i	−0.982835	−	0.184488i	$-0.940937\pi$
$199$	−1.46130e8	+	8.43683e7i	−1.31448	+	0.758916i	−0.331646	+	0.943404i	$0.607604\pi$
$200$		0			0
$201$	−1.79963e8	−	1.03902e8i	−1.56314	−	0.902477i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2.22978e8	+	1.28736e8i	1.71807	+	0.991927i
$209$		0			0
$210$		0			0
$211$	2.20581e8			1.61651			0.808256	−	0.588832i	$-0.200412\pi$
$211$	2.20581e8			1.61651			0.808256	+	0.588832i	$0.200412\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.33018e7	−	3.87808e6i	0.0883695	−	0.0257637i
$218$		0			0
$219$	−1.52723e8	+	2.64524e8i	−0.982539	+	1.70181i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			2.01450e8i			1.21647i	0.793759	+	0.608233i	$0.208121\pi$
$223$			2.01450e8i			1.21647i	−0.793759	+	0.608233i	$0.791879\pi$
$224$		0			0
$225$	1.70859e8			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	−1.73731e8	−	3.00912e8i	−0.970748	−	1.68138i
$229$	−1.98178e7	−	1.14418e7i	−0.109051	−	0.0629609i	0.444482	−	0.895788i	$-0.353388\pi$
$229$	−1.98178e7	−	1.14418e7i	−0.109051	−	0.0629609i	−0.553534	+	0.832827i	$0.686721\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	2.11268e8i		−	1.03090i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−3.73056e8	+	2.15384e8i	−1.71678	+	0.991182i	−0.792136	+	0.610345i	$0.791031\pi$
$241$	−3.73056e8	+	2.15384e8i	−1.71678	+	0.991182i	−0.924642	+	0.380838i	$0.875636\pi$
$242$		0			0
$243$	−1.93710e8	−	1.11839e8i	−0.866025	−	0.500000i
$244$			3.40792e7i			0.150185i
$245$		0			0
$246$		0			0
$247$	4.56095e8	−	7.89980e8i	1.92582	−	3.33562i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2.46764e8	+	6.03655e7i	0.971358	+	0.237622i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.34218e8	−	2.32472e8i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−2.10629e8	+	2.20054e8i	−0.753300	+	0.787011i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.84386e8	+	4.92571e8i	−0.902477	+	1.56314i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	3.81069e8	+	2.20010e8i	1.16309	+	0.671508i	0.952041	−	0.305969i	$-0.0989804\pi$
$271$	3.81069e8	+	2.20010e8i	1.16309	+	0.671508i	0.211044	+	0.977477i	$0.432314\pi$
$272$		0			0
$273$	1.86668e8	+	6.40272e8i	0.555265	+	1.90456i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.04871e8	−	5.28052e8i	−0.861861	−	1.49279i	−0.870131	−	0.492821i	$-0.835966\pi$
$277$	−3.04871e8	−	5.28052e8i	−0.861861	−	1.49279i	0.00827024	−	0.999966i	$-0.497367\pi$
$278$		0			0
$279$			3.33912e7i			0.0920486i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−6.83654e7	+	3.94708e7i	−0.179302	+	0.103520i	−0.586964	−	0.809613i	$-0.699677\pi$
$283$	−6.83654e7	+	3.94708e7i	−0.179302	+	0.103520i	0.407663	+	0.913132i	$0.366344\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.05169e8	−	3.55364e8i	0.500000	−	0.866025i
$290$		0			0
$291$	3.07772e8	+	5.33076e8i	0.732156	+	1.26813i
$292$	7.24020e8	+	4.18013e8i	1.70181	+	0.982539i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	4.67654e8i		−	1.00000i
$301$	−3.60970e8	−	8.83036e7i	−0.762936	−	0.186636i
$302$		0			0
$303$		0			0
$304$	−8.23615e8	+	4.75515e8i	−1.68138	+	0.970748i
$305$		0			0
$306$		0			0
$307$		−	3.43989e8i		−	0.678516i	−0.940693	−	0.339258i	$-0.889824\pi$
$307$		−	3.43989e8i		−	0.678516i	0.940693	−	0.339258i	$-0.110176\pi$
$308$		0			0
$309$	8.08541e8			1.55900
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−8.06379e8	−	4.65563e8i	−1.48639	−	0.858170i	−0.486515	−	0.873672i	$-0.661732\pi$
$313$	−8.06379e8	−	4.65563e8i	−1.48639	−	0.858170i	−0.999880	+	0.0155022i	$0.995065\pi$
$314$		0			0
$315$		0			0
$316$	−5.78255e8			−1.03090
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−3.06110e8	+	5.30198e8i	−0.500000	+	0.866025i
$325$	1.06324e9	−	6.13863e8i	1.71807	−	0.991927i
$326$		0			0
$327$	−4.01830e8	−	2.31997e8i	−0.635514	−	0.366914i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.26161e8	−	2.18517e8i	0.191217	−	0.331197i	−0.754437	−	0.656372i	$-0.772090\pi$
$331$	1.26161e8	−	2.18517e8i	0.191217	−	0.331197i	0.945654	+	0.325175i	$0.105423\pi$
$332$		0			0
$333$	−3.67047e8	−	6.35745e8i	−0.544713	−	0.943470i
$334$		0			0
$335$		0			0
$336$	1.65224e8	−	6.75409e8i	0.237622	−	0.971358i
$337$	1.14809e9			1.63408			0.817039	−	0.576582i	$-0.195614\pi$
$337$	1.14809e9			1.63408			0.817039	+	0.576582i	$0.195614\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−5.61992e8	−	4.92657e8i	−0.751970	−	0.659198i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$			1.42801e9i			1.79822i	0.437728	+	0.899108i	$0.355783\pi$
$349$			1.42801e9i			1.79822i	−0.437728	+	0.899108i	$0.644217\pi$
$350$		0			0
$351$	−1.60725e9			−1.98385
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	1.23775e9	+	2.14384e9i	1.38470	+	2.39837i
$362$		0			0
$363$			9.11325e8i			1.00000i
$364$	1.75247e9	−	5.10923e8i	1.90456	−	0.555265i
$365$		0			0
$366$		0			0
$367$	−3.39552e8	+	1.96041e8i	−0.358571	+	0.207021i	−0.668454	−	0.743754i	$-0.733044\pi$
$367$	−3.39552e8	+	1.96041e8i	−0.358571	+	0.207021i	0.309882	+	0.950775i	$0.399710\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	9.13939e7			0.0920486
$373$	−6.74912e8	+	1.16898e9i	−0.673389	+	1.16634i	0.303548	+	0.952816i	$0.401829\pi$
$373$	−6.74912e8	+	1.16898e9i	−0.673389	+	1.16634i	−0.976937	+	0.213528i	$0.931505\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.27572e9			−1.20370			−0.601849	−	0.798610i	$-0.705569\pi$
$379$	−1.27572e9			−1.20370			−0.601849	+	0.798610i	$0.705569\pi$
$380$		0			0
$381$	1.25598e9	−	7.25141e8i	1.16344	−	0.671715i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.47783e8	−	7.75583e8i	0.392716	−	0.680205i
$388$	1.45907e9	−	8.42392e8i	1.26813	−	0.732156i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.36224e8	−	4.25059e8i	−0.590531	−	0.340944i	0.174776	−	0.984608i	$-0.444080\pi$
$397$	−7.36224e8	−	4.25059e8i	−0.590531	−	0.340944i	−0.765308	+	0.643665i	$0.777413\pi$
$398$		0			0
$399$	−2.39288e9	−	5.85367e8i	−1.88589	−	0.461342i
$400$	−1.28000e9			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	1.19968e8	+	2.07790e8i	0.0913055	+	0.158146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8.29025e8	−	4.78638e8i	0.599151	−	0.345920i	−0.169557	−	0.985520i	$-0.554234\pi$
$409$	8.29025e8	−	4.78638e8i	0.599151	−	0.345920i	0.768707	+	0.639601i	$0.220900\pi$
$410$		0			0
$411$		0			0
$412$		−	2.21303e9i		−	1.55900i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.60113e8	+	1.31655e9i	0.513336	+	0.889124i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6.92297e8			−0.452173			−0.226087	−	0.974107i	$-0.572593\pi$
$421$	−6.92297e8			−0.452173			−0.226087	+	0.974107i	$0.572593\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.74545e8	+	1.67068e8i	0.108495	+	0.103847i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	1.45119e9	+	8.37844e8i	0.866025	+	0.500000i
$433$		−	2.56073e9i		−	1.51585i	−0.652341	−	0.757925i	$-0.726213\pi$
$433$		−	2.56073e9i		−	1.51585i	0.652341	−	0.757925i	$-0.273787\pi$
$434$		0			0
$435$		0			0
$436$	−6.34991e8	+	1.09984e9i	−0.366914	+	0.635514i
$437$		0			0
$438$		0			0
$439$	−1.22376e9	−	7.06540e8i	−0.690353	−	0.398576i	0.113391	−	0.993550i	$-0.463829\pi$
$439$	−1.22376e9	−	7.06540e8i	−0.690353	−	0.398576i	−0.803744	+	0.594975i	$0.797162\pi$
$440$		0			0
$441$	1.51890e9	−	9.67924e8i	0.843321	−	0.537410i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.74008e9	+	1.00463e9i	−0.943470	+	0.544713i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.84864e9	−	4.52231e8i	−0.971358	−	0.237622i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3.12008e9	+	1.80138e9i	1.57696	+	0.910460i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.72496e8	+	1.51121e9i	−0.427619	+	0.740657i	−0.996661	−	0.0816509i	$-0.973981\pi$
$457$	−8.72496e8	+	1.51121e9i	−0.427619	+	0.740657i	0.569042	+	0.822308i	$0.307314\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	3.08012e9			1.44223			0.721114	−	0.692817i	$-0.243631\pi$
$463$	3.08012e9			1.44223			0.721114	+	0.692817i	$0.243631\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$			4.39917e9i			1.98385i
$469$	1.12866e9	+	3.87130e9i	0.505192	+	1.73281i
$470$		0			0
$471$	1.21029e9	−	2.09628e9i	0.533723	−	0.924435i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			4.53486e9i			1.94150i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−4.56821e9	−	2.63745e9i	−1.87171	−	1.08063i
$482$		0			0
$483$		0			0
$484$	2.49436e9			1.00000
$485$		0			0
$486$		0			0
$487$	−2.24025e9	−	3.88022e9i	−0.878910	−	1.52232i	−0.852539	−	0.522664i	$-0.824938\pi$
$487$	−2.24025e9	−	3.88022e9i	−0.878910	−	1.52232i	−0.0263708	−	0.999652i	$-0.508395\pi$
$488$		0			0
$489$		−	4.08572e9i		−	1.58011i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	2.50151e8i		−	0.0920486i
$497$		0			0
$498$		0			0
$499$	9.59533e8	−	1.66196e9i	0.345707	−	0.598782i	−0.639775	−	0.768562i	$-0.720972\pi$
$499$	9.59533e8	−	1.66196e9i	0.345707	−	0.598782i	0.985482	+	0.169780i	$0.0543058\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.46048e9	+	4.30731e9i	−2.54237	+	1.46784i
$508$	−1.98476e9	−	3.43770e9i	−0.671715	−	1.16344i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	5.69034e9	−	1.65899e9i	1.88654	−	0.550010i
$512$		0			0
$513$	2.96836e9	−	5.14136e9i	0.970748	−	1.68138i
$514$		0			0
$515$		0			0
$516$	−2.12282e9	−	1.22561e9i	−0.680205	−	0.392716i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	5.57140e9	+	3.21665e9i	1.70298	+	0.983213i	0.942717	+	0.333592i	$0.108261\pi$
$523$	5.57140e9	+	3.21665e9i	1.70298	+	0.983213i	0.760258	+	0.649621i	$0.225072\pi$
$524$		0			0
$525$	−2.39520e9	−	2.29260e9i	−0.722409	−	0.691466i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.70241e9	−	2.94867e9i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−1.60219e9	+	6.54947e9i	−0.461342	+	1.88589i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.68290e9	+	6.37898e9i	−1.00000	+	1.73205i	−0.499332	+	0.866411i	$0.666421\pi$
$541$	−3.68290e9	+	6.37898e9i	−1.00000	+	1.73205i	−0.500668	+	0.865640i	$0.666912\pi$
$542$		0			0
$543$	−1.00039e9	−	1.73273e9i	−0.268146	−	0.464442i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.98519e9			1.56359			0.781793	−	0.623537i	$-0.214305\pi$
$547$	5.98519e9			1.56359			0.781793	+	0.623537i	$0.214305\pi$
$548$		0			0
$549$	−5.04265e8	+	2.91138e8i	−0.130064	+	0.0750923i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.83480e9	+	2.96166e9i	−0.712829	+	0.744728i
$554$		0			0
$555$		0			0
$556$	3.60350e9	−	2.08048e9i	0.889124	−	0.513336i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	6.43517e9i		−	1.55818i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.21487e9	+	4.16703e9i	0.279892	+	0.960031i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	4.33318e9	+	7.50529e9i	0.974048	+	1.68710i	0.683042	+	0.730379i	$0.260657\pi$
$571$	4.33318e9	+	7.50529e9i	0.974048	+	1.68710i	0.291006	+	0.956721i	$0.406010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	2.29324e9	−	3.97200e9i	0.500000	−	0.866025i
$577$	−6.72912e9	+	3.88506e9i	−1.45829	+	0.841942i	−0.998927	−	0.0463088i	$-0.985254\pi$
$577$	−6.72912e9	+	3.88506e9i	−1.45829	+	0.841942i	−0.459359	+	0.888251i	$0.651921\pi$
$578$		0			0
$579$	8.08825e9	+	4.66976e9i	1.73173	+	0.999814i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−2.64927e9	−	4.15732e9i	−0.537410	−	0.843321i
$589$	−8.86251e8			−0.178712
$590$		0			0
$591$		0			0
$592$	2.74975e9	+	4.76271e9i	0.544713	+	0.943470i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.94552e9	+	6.83384e9i	−0.758916	+	1.31448i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		−	2.60577e9i		−	0.489638i	−0.969569	−	0.244819i	$-0.921272\pi$
$601$		−	2.60577e9i		−	0.489638i	0.969569	−	0.244819i	$-0.0787285\pi$
$602$		0			0
$603$	−9.71799e9			−1.80495
$604$	4.93049e9	−	8.53986e9i	0.910460	−	1.57696i
$605$		0			0
$606$		0			0
$607$	−6.43205e9	−	3.71354e9i	−1.16732	−	0.673951i	−0.214270	−	0.976774i	$-0.568737\pi$
$607$	−6.43205e9	−	3.71354e9i	−1.16732	−	0.673951i	−0.953047	+	0.302824i	$0.902071\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−5.64129e9	−	9.77100e9i	−0.989161	−	1.71328i	−0.621745	−	0.783220i	$-0.713576\pi$
$613$	−5.64129e9	−	9.77100e9i	−0.989161	−	1.71328i	−0.367415	−	0.930057i	$-0.619757\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	9.61402e9	−	5.55065e9i	1.62925	−	0.940648i	0.644932	−	0.764240i	$-0.276886\pi$
$619$	9.61402e9	−	5.55065e9i	1.62925	−	0.940648i	0.984317	−	0.176408i	$-0.0564478\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.20408e10			1.98385
$625$	−3.05176e9	+	5.28580e9i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−5.73767e9	−	3.31264e9i	−0.924435	−	0.533723i
$629$		0			0
$630$		0			0
$631$	−1.17017e10			−1.85415			−0.927077	−	0.374870i	$-0.877687\pi$
$631$	−1.17017e10			−1.85415			−0.927077	+	0.374870i	$0.877687\pi$
$632$		0			0
$633$	8.93352e9	−	5.15777e9i	1.39994	−	0.808256i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.97439e9	−	1.14804e10i	0.915811	−	1.75982i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$			7.62962e9i			1.13179i	0.824479	+	0.565893i	$0.191469\pi$
$643$			7.62962e9i			1.13179i	−0.824479	+	0.565893i	$0.808531\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	4.48044e8	−	4.68095e8i	0.0636484	−	0.0664967i
$652$	−1.11829e10			−1.58011
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1.42843e10i			1.96508i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.07203e10	+	6.18939e9i	−1.44379	+	0.833571i	−0.998100	−	0.0616172i	$-0.980374\pi$
$661$	−1.07203e10	+	6.18939e9i	−1.44379	+	0.833571i	−0.445688	+	0.895188i	$0.647041\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.71044e9	+	8.15872e9i	0.608233	+	1.05349i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.04948e10			1.32715			0.663576	−	0.748109i	$-0.269038\pi$
$673$	1.04948e10			1.32715			0.663576	+	0.748109i	$0.269038\pi$
$674$		0			0
$675$	6.91980e9	−	3.99515e9i	0.866025	−	0.500000i
$676$	1.17894e10	+	2.04199e10i	1.46784	+	2.54237i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	2.83834e9	−	1.16026e10i	0.347953	−	1.42237i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−1.40722e10	−	8.12461e9i	−1.68138	−	0.970748i
$685$		0			0
$686$		0			0
$687$	−1.07016e9			−0.125922
$688$	−3.35458e9	+	5.81031e9i	−0.392716	+	0.680205i
$689$		0			0
$690$		0			0
$691$	7.79710e9	+	4.50166e9i	0.899001	+	0.519038i	0.876876	−	0.480717i	$-0.159624\pi$
$691$	7.79710e9	+	4.50166e9i	0.899001	+	0.519038i	0.0221249	+	0.999755i	$0.492957\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−6.27500e9	+	6.55581e9i	−0.691466	+	0.722409i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.68736e10	−	9.74198e9i	1.83174	−	1.05756i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.51659e9	−	2.62681e9i	0.159811	−	0.276801i	−0.774989	−	0.631974i	$-0.782245\pi$
$709$	1.51659e9	−	2.62681e9i	0.159811	−	0.276801i	0.934800	+	0.355173i	$0.115578\pi$
$710$		0			0
$711$	−4.94001e9	−	8.55636e9i	−0.515448	−	0.892782i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−1.13345e10	−	1.08490e10i	−1.12624	−	1.07800i
$722$		0			0
$723$	−1.00725e10	+	1.74461e10i	−0.991182	+	1.71678i
$724$	−4.74260e9	+	2.73814e9i	−0.464442	+	0.268146i
$725$		0			0
$726$		0			0
$727$			2.61157e9i			0.252076i	0.992025	+	0.126038i	$0.0402261\pi$
$727$			2.61157e9i			0.252076i	−0.992025	+	0.126038i	$0.959774\pi$
$728$		0			0
$729$	−1.04604e10			−1.00000
$730$		0			0
$731$		0			0
$732$	7.96864e8	+	1.38021e9i	0.0750923	+	0.130064i
$733$	1.13717e10	+	6.56543e9i	1.06650	+	0.615743i	0.927223	−	0.374511i	$-0.122189\pi$
$733$	1.13717e10	+	6.56543e9i	1.06650	+	0.615743i	0.139275	+	0.990254i	$0.455523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	3.78041e7	+	6.54787e7i	0.00344575	+	0.00596821i	0.867743	−	0.497013i	$-0.165570\pi$
$739$	3.78041e7	+	6.54787e7i	0.00344575	+	0.00596821i	−0.864297	+	0.502981i	$0.832237\pi$
$740$		0			0
$741$		−	4.26589e10i		−	3.85164i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.16816e9	−	7.21947e9i	0.359091	−	0.621964i	−0.628718	−	0.777633i	$-0.716420\pi$
$751$	4.16816e9	−	7.21947e9i	0.359091	−	0.621964i	0.987809	+	0.155669i	$0.0497534\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.14054e10	−	3.32519e9i	0.960031	−	0.279892i
$757$	−7.12787e9			−0.597206			−0.298603	−	0.954377i	$-0.596521\pi$
$757$	−7.12787e9			−0.597206			−0.298603	+	0.954377i	$0.596521\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	2.52012e9	+	8.64402e9i	0.205393	+	0.704498i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.08716e10	−	6.27674e9i	−0.866025	−	0.500000i
$769$			6.75455e9i			0.535617i	0.963472	+	0.267808i	$0.0862994\pi$
$769$			6.75455e9i			0.535617i	−0.963472	+	0.267808i	$0.913701\pi$
$770$		0			0
$771$		0			0
$772$	1.27814e10	−	2.21381e10i	0.999814	−	1.73173i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−1.03301e9	−	5.96407e8i	−0.0797164	−	0.0460243i
$776$		0			0
$777$	−3.38499e9	+	1.38373e10i	−0.258872	+	1.05822i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.13789e10	+	7.25124e9i	−0.843321	+	0.537410i
$785$		0			0
$786$		0			0
$787$	−9.92652e9	+	5.73108e9i	−0.725915	+	0.419107i	−0.816926	−	0.576743i	$-0.804323\pi$
$787$	−9.92652e9	+	5.73108e9i	−0.725915	+	0.419107i	0.0910110	+	0.995850i	$0.470990\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.09200e9	+	3.62345e9i	−0.148972	+	0.258027i
$794$		0			0
$795$		0			0
$796$	1.87047e10	+	1.07991e10i	1.31448	+	0.758916i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			2.65988e10i			1.80495i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$			2.91438e10i			1.91855i	0.282472	+	0.959276i	$0.408846\pi$
$811$			2.91438e10i			1.91855i	−0.282472	+	0.959276i	$0.591154\pi$
$812$		0			0
$813$	2.05777e10			1.34302
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.05851e10	+	1.18848e10i	1.32061	+	0.762457i
$818$		0			0
$819$	2.25313e10	+	2.15662e10i	1.43316	+	1.37177i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−1.59311e10	−	2.75935e10i	−0.996202	−	1.72547i	−0.573512	−	0.819197i	$-0.694419\pi$
$823$	−1.59311e10	−	2.75935e10i	−0.996202	−	1.72547i	−0.422690	−	0.906274i	$-0.638914\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−1.86867e10	+	1.07888e10i	−1.13918	+	0.657704i	−0.946227	−	0.323505i	$-0.895139\pi$
$829$	−1.86867e10	+	1.07888e10i	−1.13918	+	0.657704i	−0.192950	+	0.981209i	$0.561805\pi$
$830$		0			0
$831$	−2.46946e10	−	1.42574e10i	−1.49279	−	0.861861i
$832$		−	3.29565e10i		−	1.98385i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.80775e8	+	1.35234e9i	0.0460243	+	0.0797164i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.72499e10			1.00000
$842$		0			0
$843$		0			0
$844$	−1.41172e10	−	2.44516e10i	−0.808256	−	1.39994i
$845$		0			0
$846$		0			0
$847$	1.22282e10	−	1.27754e10i	0.691466	−	0.722409i
$848$		0			0
$849$	−1.84587e9	+	3.19713e9i	−0.103520	+	0.179302i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			7.73385e9i			0.426653i	0.976981	+	0.213326i	$0.0684297\pi$
$853$			7.73385e9i			0.426653i	−0.976981	+	0.213326i	$0.931570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−2.77792e10	−	1.60383e10i	−1.49535	−	0.863343i	−0.495368	−	0.868683i	$-0.664967\pi$
$859$	−2.77792e10	−	1.60383e10i	−1.49535	−	0.863343i	−0.999986	+	0.00533998i	$0.998300\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.91896e10i		−	1.00000i
$868$	−1.28121e9	−	1.22633e9i	−0.0664967	−	0.0636484i
$869$		0			0
$870$		0			0
$871$	−6.04742e10	+	3.49148e10i	−3.10103	+	1.79038i
$872$		0			0
$873$	2.49295e10	+	1.43931e10i	1.26813	+	0.732156i
$874$		0			0
$875$		0			0
$876$	3.90971e10			1.96508
$877$	1.81380e10	−	3.14159e10i	0.908008	−	1.57272i	0.0911805	−	0.995834i	$-0.470936\pi$
$877$	1.81380e10	−	3.14159e10i	0.908008	−	1.57272i	0.816828	−	0.576882i	$-0.195731\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	4.16837e9			0.203753			0.101876	−	0.994797i	$-0.467515\pi$
$883$	4.16837e9			0.203753			0.101876	+	0.994797i	$0.467515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−2.73370e10	−	6.68741e9i	−1.30495	−	0.319228i
$890$		0			0
$891$		0			0
$892$	2.23310e10	−	1.28928e10i	1.05349	−	0.608233i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.09350e10	−	1.89400e10i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−1.66841e10	+	4.86415e9i	−0.754040	+	0.219836i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.40390e9	+	5.89573e9i	0.151479	+	0.262369i	0.931771	−	0.363046i	$-0.118263\pi$
$907$	3.40390e9	+	5.89573e9i	0.151479	+	0.262369i	−0.780293	+	0.625415i	$0.784930\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−2.22376e10	+	3.85167e10i	−0.970748	+	1.68138i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			2.92911e9i			0.125922i
$917$		0			0
$918$		0			0
$919$	2.71062e9	−	4.69493e9i	0.115203	−	0.199538i	−0.802658	−	0.596440i	$-0.796581\pi$
$919$	2.71062e9	−	4.69493e9i	0.115203	−	0.199538i	0.917861	+	0.396902i	$0.129915\pi$
$920$		0			0
$921$	−8.04338e9	−	1.39315e10i	−0.339258	−	0.587612i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.62237e10			1.08943
$926$		0			0
$927$	3.27459e10	−	1.89059e10i	1.35014	−	0.779502i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	2.56901e10	+	4.03137e10i	1.04338	+	1.63730i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	2.20132e10i		−	0.874167i	−0.899421	−	0.437084i	$-0.856011\pi$
$937$		−	2.20132e10i		−	0.874167i	0.899421	−	0.437084i	$-0.143989\pi$
$938$		0			0
$939$	−4.35445e10			−1.71634
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−2.34193e10	+	1.35212e10i	−0.892782	+	0.515448i
$949$	5.13206e10	+	8.88898e10i	1.94921	+	3.37614i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.36398e10	+	2.36247e10i	−0.495764	+	0.858688i
$962$		0			0
$963$		0			0
$964$	4.77511e10	+	2.75691e10i	1.71678	+	0.991182i
$965$		0			0
$966$		0			0
$967$	−1.10279e10			−0.392194			−0.196097	−	0.980585i	$-0.562827\pi$
$967$	−1.10279e10			−0.392194			−0.196097	+	0.980585i	$0.562827\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			2.86307e10i			1.00000i
$973$	7.00993e9	−	2.86554e10i	0.243960	−	0.997266i
$974$		0			0
$975$	2.87075e10	−	4.97229e10i	0.991927	−	1.71807i
$976$	3.77772e9	−	2.18107e9i	0.130064	−	0.0750923i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−2.16988e10			−0.733828
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.16760e11			−3.85164
$989$		0			0
$990$		0			0
$991$	−2.88528e10	−	4.99745e10i	−0.941737	−	1.63114i	−0.762156	−	0.647394i	$-0.775859\pi$
$991$	−2.88528e10	−	4.99745e10i	−0.941737	−	1.63114i	−0.179581	−	0.983743i	$-0.557474\pi$
$992$		0			0
$993$		−	1.17999e10i		−	0.382434i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.40815e10	−	3.12240e10i	1.72829	−	0.997826i	0.831192	−	0.555985i	$-0.187659\pi$
$997$	5.40815e10	−	3.12240e10i	1.72829	−	0.997826i	0.897093	−	0.441841i	$-0.145674\pi$
$998$		0			0
$999$	−2.97308e10	−	1.71651e10i	−0.943470	−	0.544713i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.8.g.a.5.1	✓	2
3.2	odd	2	CM	21.8.g.a.5.1	✓	2
7.2	even	3		147.8.c.a.146.2		2
7.3	odd	6	inner	21.8.g.a.17.1	yes	2
7.5	odd	6		147.8.c.a.146.1		2
21.2	odd	6		147.8.c.a.146.2		2
21.5	even	6		147.8.c.a.146.1		2
21.17	even	6	inner	21.8.g.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.g.a.5.1	✓	2	1.1	even	1	trivial
21.8.g.a.5.1	✓	2	3.2	odd	2	CM
21.8.g.a.17.1	yes	2	7.3	odd	6	inner
21.8.g.a.17.1	yes	2	21.17	even	6	inner
147.8.c.a.146.1		2	7.5	odd	6
147.8.c.a.146.1		2	21.5	even	6
147.8.c.a.146.2		2	7.2	even	3
147.8.c.a.146.2		2	21.2	odd	6

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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(40.5000 - 23.3827i) q^{3} +(-64.0000 - 110.851i) q^{4} +(-881.500 - 215.640i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(40.5000 - 23.3827i) q^{3} +(-64.0000 - 110.851i) q^{4} +(-881.500 - 215.640i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-5184.00 - 2992.98i) q^{12} -15714.9i q^{13} +(-8192.00 + 14189.0i) q^{16} +(50269.5 + 29023.1i) q^{19} +(-40743.0 + 11878.4i) q^{21} +(39062.5 + 67658.2i) q^{25} -102276. i q^{27} +(32512.0 + 111516. i) q^{28} +(-13222.5 + 7634.01i) q^{31} -279936. q^{36} +(167831. - 290693. i) q^{37} +(-367456. - 636453. i) q^{39} +409495. q^{43} +766204. i q^{48} +(730542. + 380174. i) q^{49} +(-1.74202e6 + 1.00575e6i) q^{52} +2.71455e6 q^{57} +(-230574. - 133122. i) q^{61} +(-1.37234e6 + 1.43376e6i) q^{63} +2.09715e6 q^{64} +(-2.22176e6 - 3.84821e6i) q^{67} +(-5.65641e6 + 3.26573e6i) q^{73} +(3.16406e6 + 1.82677e6i) q^{75} -7.42992e6i q^{76} +(2.25881e6 - 3.91237e6i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +(3.92429e6 + 3.75619e6i) q^{84} +(-3.38877e6 + 1.38527e7i) q^{91} +(-357008. + 618355. i) q^{93} +1.31624e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 81 q^{3} - 128 q^{4} - 1763 q^{7} + 2187 q^{9}+O(q^{10}) \) 2 * q + 81 * q^3 - 128 * q^4 - 1763 * q^7 + 2187 * q^9	\( 2 q + 81 q^{3} - 128 q^{4} - 1763 q^{7} + 2187 q^{9} - 10368 q^{12} - 16384 q^{16} + 100539 q^{19} - 81486 q^{21} + 78125 q^{25} + 65024 q^{28} - 26445 q^{31} - 559872 q^{36} + 335663 q^{37} - 734913 q^{39} + 818990 q^{43} + 1461083 q^{49} - 3484032 q^{52} + 5429106 q^{57} - 461148 q^{61} - 2744685 q^{63} + 4194304 q^{64} - 4443527 q^{67} - 11312811 q^{73} + 6328125 q^{75} + 4517617 q^{79} - 4782969 q^{81} + 7848576 q^{84} - 6777531 q^{91} - 714015 q^{93}+O(q^{100}) \) 2 * q + 81 * q^3 - 128 * q^4 - 1763 * q^7 + 2187 * q^9 - 10368 * q^12 - 16384 * q^16 + 100539 * q^19 - 81486 * q^21 + 78125 * q^25 + 65024 * q^28 - 26445 * q^31 - 559872 * q^36 + 335663 * q^37 - 734913 * q^39 + 818990 * q^43 + 1461083 * q^49 - 3484032 * q^52 + 5429106 * q^57 - 461148 * q^61 - 2744685 * q^63 + 4194304 * q^64 - 4443527 * q^67 - 11312811 * q^73 + 6328125 * q^75 + 4517617 * q^79 - 4782969 * q^81 + 7848576 * q^84 - 6777531 * q^91 - 714015 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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