LMFDB - Embedded newform 84.12.k.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(5,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.5");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.k (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:	$64.5408271670$
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:	$\Z[a_1, \ldots, a_{7}]$
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	$\chi$	$=$	84.5
Dual form			84.12.k.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)

$f(q)$	$=$	$q+(364.500 - 210.444i) q^{3} +(38442.5 + 22349.5i) q^{7} +(88573.5 - 153414. i) q^{9} +O(q^{10})$	$q+(364.500 - 210.444i) q^{3} +(38442.5 + 22349.5i) q^{7} +(88573.5 - 153414. i) q^{9} -1.45340e6i q^{13} +(-5.63558e6 - 3.25371e6i) q^{19} +(1.87156e7 + 56399.0i) q^{21} +(2.44141e7 + 4.22864e7i) q^{25} -7.45591e7i q^{27} +(2.73106e8 - 1.57678e8i) q^{31} +(5.96873e7 - 1.03381e8i) q^{37} +(-3.05860e8 - 5.29764e8i) q^{39} +2.18925e8 q^{43} +(9.78325e8 + 1.71834e9i) q^{49} -2.73889e9 q^{57} +(-1.07618e10 - 6.21334e9i) q^{61} +(6.83371e9 - 3.91804e9i) q^{63} +(-2.97565e9 - 5.15397e9i) q^{67} +(2.75721e10 - 1.59188e10i) q^{73} +(1.77979e10 + 1.02756e10i) q^{75} +(2.71481e10 - 4.70219e10i) q^{79} +(-1.56905e10 - 2.71768e10i) q^{81} +(3.24828e10 - 5.58723e10i) q^{91} +(6.63647e10 - 1.14947e11i) q^{93} -1.26194e11i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9}+O(q^{10}) $ 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9	$ 2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9} - 11271165 q^{19} + 37431234 q^{21} + 48828125 q^{25} + 546211707 q^{31} + 119374607 q^{37} - 611719209 q^{39} + 437849438 q^{43} + 1956649739 q^{49} - 5477786190 q^{57} - 21523645452 q^{61} + 13667422491 q^{63} - 5951291615 q^{67} + 55144201461 q^{73} + 35595703125 q^{75} + 54296224537 q^{79} - 31381059609 q^{81} + 64965586941 q^{91} + 132729444801 q^{93}+O(q^{100}) $ 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9 - 11271165 * q^19 + 37431234 * q^21 + 48828125 * q^25 + 546211707 * q^31 + 119374607 * q^37 - 611719209 * q^39 + 437849438 * q^43 + 1956649739 * q^49 - 5477786190 * q^57 - 21523645452 * q^61 + 13667422491 * q^63 - 5951291615 * q^67 + 55144201461 * q^73 + 35595703125 * q^75 + 54296224537 * q^79 - 31381059609 * q^81 + 64965586941 * q^91 + 132729444801 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$2$		0			0
$3$	364.500	−	210.444i	0.866025	−	0.500000i
$3$	364.500	−	210.444i	0.866025	−	0.500000i
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	38442.5	+	22349.5i	0.864515	+	0.502607i
$7$	38442.5	+	22349.5i	0.864515	+	0.502607i
$8$		0			0
$9$	88573.5	−	153414.i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$		−	1.45340e6i		−	1.08567i	−0.839840	−	0.542834i	$-0.817351\pi$
$13$		−	1.45340e6i		−	1.08567i	0.839840	−	0.542834i	$-0.182649\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−5.63558e6	−	3.25371e6i	−0.522148	−	0.301463i	0.215665	−	0.976467i	$-0.430808\pi$
$19$	−5.63558e6	−	3.25371e6i	−0.522148	−	0.301463i	−0.737813	+	0.675005i	$0.764141\pi$
$20$		0			0
$21$	1.87156e7	+	56399.0i	0.999995	+	0.00301346i
$22$		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
$25$	2.44141e7	+	4.22864e7i	0.500000	+	0.866025i
$26$		0			0
$27$		−	7.45591e7i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	2.73106e8	−	1.57678e8i	1.71333	−	0.989193i	0.783357	−	0.621572i	$-0.213506\pi$
$31$	2.73106e8	−	1.57678e8i	1.71333	−	0.989193i	0.929976	−	0.367621i	$-0.119828\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.96873e7	−	1.03381e8i	0.141505	−	0.245094i	−0.786558	−	0.617516i	$-0.788139\pi$
$37$	5.96873e7	−	1.03381e8i	0.141505	−	0.245094i	0.928064	+	0.372422i	$0.121472\pi$
$38$		0			0
$39$	−3.05860e8	−	5.29764e8i	−0.542834	−	0.940216i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	2.18925e8			0.227101			0.113550	−	0.993532i	$-0.463778\pi$
$43$	2.18925e8			0.227101			0.113550	+	0.993532i	$0.463778\pi$
$44$		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	$0.166667\pi$
$48$		0			0
$49$	9.78325e8	+	1.71834e9i	0.494771	+	0.869023i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		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
$57$	−2.73889e9			−0.602925
$58$		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
$61$	−1.07618e10	−	6.21334e9i	−1.63144	−	0.941914i	−0.983649	−	0.180098i	$-0.942359\pi$
$61$	−1.07618e10	−	6.21334e9i	−1.63144	−	0.941914i	−0.647794	−	0.761816i	$-0.724308\pi$
$62$		0			0
$63$	6.83371e9	−	3.91804e9i	0.867528	−	0.497388i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.97565e9	−	5.15397e9i	−0.269259	−	0.466370i	0.699412	−	0.714719i	$-0.253445\pi$
$67$	−2.97565e9	−	5.15397e9i	−0.269259	−	0.466370i	−0.968671	+	0.248349i	$0.920112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	2.75721e10	−	1.59188e10i	1.55666	−	0.898739i	0.559088	−	0.829108i	$-0.311151\pi$
$73$	2.75721e10	−	1.59188e10i	1.55666	−	0.898739i	0.997573	−	0.0696308i	$-0.0221821\pi$
$74$		0			0
$75$	1.77979e10	+	1.02756e10i	0.866025	+	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.71481e10	−	4.70219e10i	0.992637	−	1.71930i	0.391422	−	0.920211i	$-0.371983\pi$
$79$	2.71481e10	−	4.70219e10i	0.992637	−	1.71930i	0.601215	−	0.799087i	$-0.294684\pi$
$80$		0			0
$81$	−1.56905e10	−	2.71768e10i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		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
$91$	3.24828e10	−	5.58723e10i	0.545665	−	0.938575i
$92$		0			0
$93$	6.63647e10	−	1.14947e11i	0.989193	−	1.71333i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.26194e11i		−	1.49209i	−0.665897	−	0.746043i	$-0.731951\pi$
$97$		−	1.26194e11i		−	1.49209i	0.665897	−	0.746043i	$-0.268049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	4.20529e10	+	2.42793e10i	0.357430	+	0.206363i	0.667953	−	0.744203i	$-0.267171\pi$
$103$	4.20529e10	+	2.42793e10i	0.357430	+	0.206363i	−0.310523	+	0.950566i	$0.600504\pi$
$104$		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	$0.666667\pi$
$108$		0			0
$109$	1.28975e9	+	2.23392e9i	0.00802899	+	0.0139066i	0.870012	−	0.493031i	$-0.164111\pi$
$109$	1.28975e9	+	2.23392e9i	0.00802899	+	0.0139066i	−0.861983	+	0.506937i	$0.830778\pi$
$110$		0			0
$111$		−	5.02434e10i		−	0.283010i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.22972e11	−	1.28733e11i	−0.940216	−	0.542834i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.42656e11	+	2.47087e11i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.49271e11			0.938085			0.469043	−	0.883176i	$-0.344599\pi$
$127$	3.49271e11			0.938085			0.469043	+	0.883176i	$0.344599\pi$
$128$		0			0
$129$	7.97981e10	−	4.60714e10i	0.196675	−	0.113550i
$130$		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
$133$	−1.43927e11	−	2.51033e11i	−0.299888	−	0.523055i
$134$		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
$139$		−	7.85878e11i		−	1.28462i	−0.766446	−	0.642309i	$-0.777977\pi$
$139$		−	7.85878e11i		−	1.28462i	0.766446	−	0.642309i	$-0.222023\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.18215e11	+	4.20453e11i	0.862996	+	0.505210i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	9.39793e11	+	1.62777e12i	0.974225	+	1.68741i	0.682470	+	0.730913i	$0.260906\pi$
$151$	9.39793e11	+	1.62777e12i	0.974225	+	1.68741i	0.291754	+	0.956493i	$0.405761\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.01217e12	+	1.16173e12i	−1.68352	+	0.971979i	−0.724225	+	0.689564i	$0.757802\pi$
$157$	−2.01217e12	+	1.16173e12i	−1.68352	+	0.971979i	−0.959292	+	0.282415i	$0.908864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.22698e11	+	9.05340e11i	−0.355811	+	0.616283i	−0.987256	−	0.159138i	$-0.949129\pi$
$163$	−5.22698e11	+	9.05340e11i	−0.355811	+	0.616283i	0.631445	+	0.775420i	$0.282462\pi$
$164$		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
$169$	−3.20212e11			−0.178674
$170$		0			0
$171$	−9.98327e11	+	5.76384e11i	−0.522148	+	0.301463i
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−6.54297e9	+	2.17124e12i	−0.00301346	+	0.999995i
$176$		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
$181$			3.63955e12i			1.39256i	0.717768	+	0.696282i	$0.245164\pi$
$181$			3.63955e12i			1.39256i	−0.717768	+	0.696282i	$0.754836\pi$
$182$		0			0
$183$	−5.23025e12			−1.88383
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.66636e12	−	2.86624e12i	0.502607	−	0.864515i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	3.58652e12	+	6.21203e12i	0.964068	+	1.66981i	0.712099	+	0.702080i	$0.247745\pi$
$193$	3.58652e12	+	6.21203e12i	0.964068	+	1.66981i	0.251969	+	0.967735i	$0.418922\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−7.44676e12	+	4.29939e12i	−1.69151	+	0.976595i	−0.738213	+	0.674567i	$0.764330\pi$
$199$	−7.44676e12	+	4.29939e12i	−1.69151	+	0.976595i	−0.953299	+	0.302028i	$0.902336\pi$
$200$		0			0
$201$	−2.16925e12	−	1.25241e12i	−0.466370	−	0.269259i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.60648e12			−0.264436			−0.132218	−	0.991221i	$-0.542210\pi$
$211$	−1.60648e12			−0.264436			−0.132218	+	0.991221i	$0.542210\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.40229e13	+	4.22576e10i	1.97838	+	0.00596179i
$218$		0			0
$219$	6.70002e12	−	1.16048e13i	0.898739	−	1.55666i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.14390e13i		−	1.38903i	−0.719478	−	0.694515i	$-0.755619\pi$
$223$		−	1.14390e13i		−	1.38903i	0.719478	−	0.694515i	$-0.244381\pi$
$224$		0			0
$225$	8.64976e12			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−8.34802e11	−	4.81973e11i	−0.0875968	−	0.0505740i	0.455562	−	0.890204i	$-0.349438\pi$
$229$	−8.34802e11	−	4.81973e11i	−0.0875968	−	0.0505740i	−0.543159	+	0.839630i	$0.682772\pi$
$230$		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
$237$		−	2.28526e13i		−	1.98527i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	9.17716e12	−	5.29844e12i	0.727134	−	0.419811i	−0.0902386	−	0.995920i	$-0.528763\pi$
$241$	9.17716e12	−	5.29844e12i	0.727134	−	0.419811i	0.817373	+	0.576109i	$0.195430\pi$
$242$		0			0
$243$	−1.14384e13	−	6.60396e12i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.72894e12	+	8.19076e12i	−0.327288	+	0.566880i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	4.60505e12	−	2.64026e12i	0.245520	−	0.140766i
$260$		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
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	7.84832e12	+	4.53123e12i	0.326171	+	0.188315i	0.654140	−	0.756374i	$-0.273031\pi$
$271$	7.84832e12	+	4.53123e12i	0.326171	+	0.188315i	−0.327969	+	0.944689i	$0.606364\pi$
$272$		0			0
$273$	8.19704e10	−	2.72013e13i	0.00327162	−	1.08566i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.70373e13	+	4.68299e13i	0.996148	+	1.72538i	0.574010	+	0.818848i	$0.305387\pi$
$277$	2.70373e13	+	4.68299e13i	0.996148	+	1.72538i	0.422138	+	0.906531i	$0.361280\pi$
$278$		0			0
$279$		−	5.58643e13i		−	1.97839i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−5.17181e13	+	2.98595e13i	−1.69363	+	0.977815i	−0.742077	+	0.670314i	$0.766159\pi$
$283$	−5.17181e13	+	2.98595e13i	−1.69363	+	0.977815i	−0.951548	+	0.307501i	$0.900507\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.71359e13	−	2.96803e13i	0.500000	−	0.866025i
$290$		0			0
$291$	−2.65568e13	−	4.59977e13i	−0.746043	−	1.29218i
$292$		0			0
$293$		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
$300$		0			0
$301$	8.41601e12	+	4.89286e12i	0.196332	+	0.114142i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			9.55436e13i			1.99959i	0.0203149	+	0.999794i	$0.493533\pi$
$307$			9.55436e13i			1.99959i	−0.0203149	+	0.999794i	$0.506467\pi$
$308$		0			0
$309$	2.04377e13			0.412725
$310$		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
$313$	4.99891e13	+	2.88612e13i	0.940549	+	0.543026i	0.890132	−	0.455702i	$-0.150612\pi$
$313$	4.99891e13	+	2.88612e13i	0.940549	+	0.543026i	0.0504169	+	0.998728i	$0.483945\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		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
$324$		0			0
$325$	6.14591e13	−	3.54834e13i	0.940216	−	0.542834i
$326$		0			0
$327$	9.40230e11	+	5.42842e11i	0.0139066	+	0.00802899i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.05239e13	+	1.22151e14i	−0.975623	+	1.68983i	−0.297760	+	0.954641i	$0.596240\pi$
$331$	−7.05239e13	+	1.22151e14i	−0.975623	+	1.68983i	−0.677863	+	0.735188i	$0.737094\pi$
$332$		0			0
$333$	−1.05734e13	−	1.83137e13i	−0.141505	−	0.245094i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.90134e13			−1.11555			−0.557777	−	0.829991i	$-0.688346\pi$
$337$	−8.90134e13			−1.11555			−0.557777	+	0.829991i	$0.688346\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.94876e11	+	8.79225e13i	−0.00904027	+	0.999959i
$344$		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
$349$			9.06273e13i			0.936956i	0.883475	+	0.468478i	$0.155197\pi$
$349$			9.06273e13i			0.936956i	−0.883475	+	0.468478i	$0.844803\pi$
$350$		0			0
$351$	−1.08364e14			−1.08567
$352$		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
$361$	−3.70719e13	−	6.42105e13i	−0.318241	−	0.551209i
$362$		0			0
$363$			1.20084e14i			1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.29594e14	+	7.48214e13i	−1.01607	+	0.586628i	−0.912963	−	0.408043i	$-0.866211\pi$
$367$	−1.29594e14	+	7.48214e13i	−1.01607	+	0.586628i	−0.103106	+	0.994670i	$0.532878\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.38591e14	+	2.40046e14i	−0.993884	+	1.72146i	−0.401311	+	0.915942i	$0.631445\pi$
$373$	−1.38591e14	+	2.40046e14i	−0.993884	+	1.72146i	−0.592574	+	0.805516i	$0.701888\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	5.50919e13			0.361886			0.180943	−	0.983494i	$-0.442085\pi$
$379$	5.50919e13			0.361886			0.180943	+	0.983494i	$0.442085\pi$
$380$		0			0
$381$	1.27309e14	−	7.35021e13i	0.812405	−	0.469043i
$382$		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
$387$	1.93909e13	−	3.35861e13i	0.113550	−	0.196675i
$388$		0			0
$389$		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
$397$	3.27450e14	+	1.89053e14i	1.66647	+	0.962135i	0.969520	+	0.245013i	$0.0787921\pi$
$397$	3.27450e14	+	1.89053e14i	1.66647	+	0.962135i	0.696947	+	0.717123i	$0.254541\pi$
$398$		0			0
$399$	−1.05290e14	−	6.12129e13i	−0.521238	−	0.303035i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−2.29169e14	−	3.96932e14i	−1.07393	−	1.86011i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.94349e14	−	2.27678e14i	1.70374	−	0.983654i	0.761833	−	0.647773i	$-0.224300\pi$
$409$	3.94349e14	−	2.27678e14i	1.70374	−	0.983654i	0.941905	−	0.335880i	$-0.109034\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.65383e14	−	2.86453e14i	−0.642309	−	1.11251i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	4.85242e14			1.78816			0.894081	−	0.447905i	$-0.147830\pi$
$421$	4.85242e14			1.78816			0.894081	+	0.447905i	$0.147830\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.74846e14	−	4.79378e14i	−0.936993	−	1.63427i
$428$		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	$0.333333\pi$
$432$		0			0
$433$		−	5.50234e14i		−	1.73726i	−0.495463	−	0.868629i	$-0.665002\pi$
$433$		−	5.50234e14i		−	1.73726i	0.495463	−	0.868629i	$-0.334998\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.14904e14	+	6.63401e13i	0.336343	+	0.194188i	0.658654	−	0.752446i	$-0.271126\pi$
$439$	1.14904e14	+	6.63401e13i	0.336343	+	0.194188i	−0.322311	+	0.946634i	$0.604460\pi$
$440$		0			0
$441$	3.50271e14	+	2.11109e12i	0.999982	+	0.00602689i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.85109e14	+	3.95548e14i	1.68741	+	0.974225i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.05212e14	−	3.55438e14i	0.481575	−	0.834112i	−0.518202	−	0.855258i	$-0.673398\pi$
$457$	2.05212e14	−	3.55438e14i	0.481575	−	0.834112i	0.999776	+	0.0211466i	$0.00673167\pi$
$458$		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
$463$	−8.27316e14			−1.80707			−0.903537	−	0.428510i	$-0.859039\pi$
$463$	−8.27316e14			−1.80707			−0.903537	+	0.428510i	$0.859039\pi$
$464$		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	$0.166667\pi$
$468$		0			0
$469$	7.97473e11	−	2.64636e14i	0.00162280	−	0.538515i
$470$		0			0
$471$	−4.88958e14	+	8.46901e14i	−0.971979	+	1.68352i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	3.17745e14i		−	0.602925i
$476$		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
$481$	−1.50255e14	−	8.67495e13i	−0.266091	−	0.153628i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−5.94429e14	−	1.02958e15i	−0.983310	−	1.70314i	−0.649216	−	0.760604i	$-0.724903\pi$
$487$	−5.94429e14	−	1.02958e15i	−0.983310	−	1.70314i	−0.334094	−	0.942540i	$-0.608430\pi$
$488$		0			0
$489$			4.39995e14i			0.711622i
$490$		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
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.72379e14	+	4.71774e14i	−0.394113	+	0.682623i	−0.992987	−	0.118219i	$-0.962281\pi$
$499$	−2.72379e14	+	4.71774e14i	−0.394113	+	0.682623i	0.598875	+	0.800843i	$0.295615\pi$
$500$		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
$507$	−1.16717e14	+	6.73867e13i	−0.154736	+	0.0893368i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	1.41572e15	+	4.26623e12i	1.79747	+	0.00541663i
$512$		0			0
$513$	−2.42593e14	+	4.20184e14i	−0.301463	+	0.522148i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		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
$523$	9.79489e14	+	5.65508e14i	1.09456	+	0.631945i	0.934787	−	0.355208i	$-0.115590\pi$
$523$	9.79489e14	+	5.65508e14i	1.09456	+	0.631945i	0.159774	+	0.987154i	$0.448923\pi$
$524$		0			0
$525$	4.54539e14	+	7.92793e14i	0.497388	+	0.867528i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−4.76405e14	−	8.25157e14i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.91351e14	+	1.54387e15i	−0.826921	+	1.43227i	0.0735219	+	0.997294i	$0.476576\pi$
$541$	−8.91351e14	+	1.54387e15i	−0.826921	+	1.43227i	−0.900443	+	0.434975i	$0.856757\pi$
$542$		0			0
$543$	7.65922e14	+	1.32662e15i	0.696282	+	1.20600i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.19604e15			−1.91739			−0.958695	−	0.284436i	$-0.908194\pi$
$547$	−2.19604e15			−1.91739			−0.958695	+	0.284436i	$0.908194\pi$
$548$		0			0
$549$	−1.90642e15	+	1.10067e15i	−1.63144	+	0.941914i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.09456e15	−	1.20089e15i	1.72228	−	0.987452i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	3.18185e14i		−	0.246556i
$560$		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
$567$	4.20506e12	−	1.39542e15i	0.00301346	−	0.999995i
$568$		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
$571$	6.09372e14	+	1.05546e15i	0.420131	+	0.727687i	0.995952	−	0.0898881i	$-0.0286510\pi$
$571$	6.09372e14	+	1.05546e15i	0.420131	+	0.727687i	−0.575821	+	0.817576i	$0.695318\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−2.19553e15	+	1.26759e15i	−1.42913	+	0.825109i	−0.997052	−	0.0767272i	$-0.975553\pi$
$577$	−2.19553e15	+	1.26759e15i	−1.42913	+	0.825109i	−0.432078	+	0.901836i	$0.642220\pi$
$578$		0			0
$579$	2.61457e15	+	1.50952e15i	1.66981	+	0.964068i
$580$		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	$0.5\pi$
$588$		0			0
$589$	−2.05215e15			−1.19282
$590$		0			0
$591$		0			0
$592$		0			0
$593$		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
$597$	−1.80956e15	+	3.13425e15i	−0.976595	+	1.69151i
$598$		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
$601$		−	2.35266e15i		−	1.22391i	−0.790892	−	0.611956i	$-0.790383\pi$
$601$		−	2.35266e15i		−	1.22391i	0.790892	−	0.611956i	$-0.209617\pi$
$602$		0			0
$603$	−1.05425e15			−0.538517
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.82355e15	+	1.63018e15i	1.39078	+	0.802966i	0.993401	−	0.114689i	$-0.0365873\pi$
$607$	2.82355e15	+	1.63018e15i	1.39078	+	0.802966i	0.397377	+	0.917656i	$0.369921\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.68506e15	−	2.91861e15i	−0.786289	−	1.36189i	−0.928226	−	0.372017i	$-0.878666\pi$
$613$	−1.68506e15	−	2.91861e15i	−0.786289	−	1.36189i	0.141936	−	0.989876i	$-0.454667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−3.01100e15	+	1.73840e15i	−1.33172	+	0.768868i	−0.985563	−	0.169309i	$-0.945846\pi$
$619$	−3.01100e15	+	1.73840e15i	−1.33172	+	0.768868i	−0.346156	+	0.938177i	$0.612513\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.19209e15	+	2.06477e15i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	3.70257e15			1.47347			0.736737	−	0.676180i	$-0.236366\pi$
$631$	3.70257e15			1.47347			0.736737	+	0.676180i	$0.236366\pi$
$632$		0			0
$633$	−5.85561e14	+	3.38074e14i	−0.229009	+	0.132218i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.49744e15	−	1.42190e15i	0.943470	−	0.537157i
$638$		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
$643$		−	4.95474e15i		−	1.77771i	−0.458189	−	0.888855i	$-0.651502\pi$
$643$		−	4.95474e15i		−	1.77771i	0.458189	−	0.888855i	$-0.348498\pi$
$644$		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
$651$	5.12024e15	−	2.93563e15i	1.71631	−	0.984026i
$652$		0			0
$653$		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
$657$		−	5.63992e15i		−	1.79748i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	4.03369e15	−	2.32885e15i	1.24335	−	0.717850i	0.273578	−	0.961850i	$-0.411793\pi$
$661$	4.03369e15	−	2.32885e15i	1.24335	−	0.717850i	0.969775	+	0.244000i	$0.0784596\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.40727e15	−	4.16952e15i	−0.694515	−	1.20293i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.49517e15			−0.696655			−0.348327	−	0.937373i	$-0.613250\pi$
$673$	−2.49517e15			−0.696655			−0.348327	+	0.937373i	$0.613250\pi$
$674$		0			0
$675$	3.15284e15	−	1.82029e15i	0.866025	−	0.500000i
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	2.82037e15	−	4.85121e15i	0.749934	−	1.28993i
$680$		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	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−4.05714e14			−0.101148
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6.56677e15	+	3.79133e15i	1.58571	+	0.915508i	0.994003	+	0.109351i	$0.0348773\pi$
$691$	6.56677e15	+	3.79133e15i	1.58571	+	0.915508i	0.591703	+	0.806156i	$0.298456\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−6.72745e14	+	3.88410e14i	−0.147773	+	0.0853171i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4.76855e15	+	8.25937e15i	−0.999613	+	1.73138i	−0.475722	+	0.879595i	$0.657813\pi$
$709$	−4.76855e15	+	8.25937e15i	−0.999613	+	1.73138i	−0.523891	+	0.851785i	$0.675520\pi$
$710$		0			0
$711$	−4.80921e15	−	8.32979e15i	−0.992637	−	1.71930i
$712$		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
$721$	1.07399e15	+	1.87322e15i	0.205284	+	0.358051i
$722$		0			0
$723$	2.23005e15	−	3.86256e15i	0.419811	−	0.727134i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			5.71865e15i			1.04437i	0.852832	+	0.522185i	$0.174883\pi$
$727$			5.71865e15i			1.04437i	−0.852832	+	0.522185i	$0.825117\pi$
$728$		0			0
$729$	−5.55906e15			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.78768e15	+	4.49622e15i	1.35937	+	0.784830i	0.989538	−	0.144270i	$-0.0460835\pi$
$733$	7.78768e15	+	4.49622e15i	1.35937	+	0.784830i	0.369827	+	0.929100i	$0.379417\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−5.56551e15	−	9.63974e15i	−0.928882	−	1.60887i	−0.785196	−	0.619247i	$-0.787438\pi$
$739$	−5.56551e15	−	9.63974e15i	−0.928882	−	1.60887i	−0.143686	−	0.989623i	$-0.545895\pi$
$740$		0			0
$741$			3.98071e15i			0.654576i
$742$		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
$751$	−5.20385e15	+	9.01333e15i	−0.794886	+	1.37678i	0.128025	+	0.991771i	$0.459136\pi$
$751$	−5.20385e15	+	9.01333e15i	−0.794886	+	1.37678i	−0.922911	+	0.385013i	$0.874197\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.04544e16			−1.52852			−0.764261	−	0.644907i	$-0.776896\pi$
$757$	−1.04544e16			−1.52852			−0.764261	+	0.644907i	$0.776896\pi$
$758$		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
$763$	−3.45654e11	+	1.14703e14i	−4.83901e−5	+	0.0160579i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			1.88680e15i			0.253006i	0.991966	+	0.126503i	$0.0403753\pi$
$769$			1.88680e15i			0.253006i	−0.991966	+	0.126503i	$0.959625\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	1.33352e16	+	7.69911e15i	1.71333	+	0.989193i
$776$		0			0
$777$	1.12292e15	−	1.93148e15i	0.142243	−	0.244667i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−9.21945e15	+	5.32285e15i	−1.08854	+	0.628468i	−0.933187	−	0.359391i	$-0.882985\pi$
$787$	−9.21945e15	+	5.32285e15i	−1.08854	+	0.628468i	−0.155352	+	0.987859i	$0.549651\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.03047e15	+	1.56412e16i	−1.02260	+	1.77120i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		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
$804$		0			0
$805$		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
$811$			1.80139e16i			1.80299i	0.432792	+	0.901494i	$0.357528\pi$
$811$			1.80139e16i			1.80299i	−0.432792	+	0.901494i	$0.642472\pi$
$812$		0			0
$813$	3.81428e15			0.376630
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.23377e15	−	7.12316e14i	−0.118580	−	0.0684623i
$818$		0			0
$819$	−5.69447e15	−	9.93212e15i	−0.539998	−	0.941847i
$820$		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
$823$	9.45387e14	+	1.63746e15i	0.0872792	+	0.151172i	0.906360	−	0.422506i	$-0.138849\pi$
$823$	9.45387e14	+	1.63746e15i	0.0872792	+	0.151172i	−0.819081	+	0.573678i	$0.805516\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1.86921e16	−	1.07919e16i	1.65809	−	0.957301i	0.684501	−	0.729012i	$-0.260020\pi$
$829$	1.86921e16	−	1.07919e16i	1.65809	−	0.957301i	0.973593	−	0.228290i	$-0.0733134\pi$
$830$		0			0
$831$	1.97102e16	+	1.13797e16i	1.72538	+	0.996148i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.17563e16	−	2.03625e16i	−0.989193	−	1.71333i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.22005e16			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.10063e16	+	6.31036e15i	−0.867528	+	0.497388i
$848$		0			0
$849$	−1.25675e16	+	2.17676e16i	−0.977815	+	1.69363i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	6.79606e15i		−	0.515273i	−0.966242	−	0.257637i	$-0.917056\pi$
$853$		−	6.79606e15i		−	0.515273i	0.966242	−	0.257637i	$-0.0829437\pi$
$854$		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
$859$	3.73549e15	+	2.15669e15i	0.272512	+	0.157335i	0.630028	−	0.776572i	$-0.283043\pi$
$859$	3.73549e15	+	2.15669e15i	0.272512	+	0.157335i	−0.357517	+	0.933907i	$0.616377\pi$
$860$		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
$867$		−	1.44246e16i		−	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−7.49078e15	+	4.32480e15i	−0.506323	+	0.292325i
$872$		0			0
$873$	−1.93599e16	−	1.11774e16i	−1.29218	−	0.746043i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.58130e15	+	1.13991e16i	−0.428365	+	0.741949i	−0.996728	−	0.0808282i	$-0.974243\pi$
$877$	−6.58130e15	+	1.13991e16i	−0.428365	+	0.741949i	0.568363	+	0.822778i	$0.307577\pi$
$878$		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
$883$	−1.35313e16			−0.848311			−0.424155	−	0.905589i	$-0.639429\pi$
$883$	−1.35313e16			−0.848311			−0.424155	+	0.905589i	$0.639429\pi$
$884$		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
$889$	1.34269e16	+	7.80604e15i	0.810988	+	0.471489i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	4.09731e15	+	1.23471e13i	0.227100	+	0.000684359i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−7.00345e15	−	1.21303e16i	−0.378854	−	0.656194i	0.612042	−	0.790825i	$-0.290348\pi$
$907$	−7.00345e15	−	1.21303e16i	−0.378854	−	0.656194i	−0.990896	+	0.134631i	$0.957015\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.51341e16	−	2.62130e16i	0.761589	−	1.31911i	−0.180443	−	0.983586i	$-0.557753\pi$
$919$	1.51341e16	−	2.62130e16i	0.761589	−	1.31911i	0.942031	−	0.335525i	$-0.108914\pi$
$920$		0			0
$921$	2.01066e16	+	3.48256e16i	0.999794	+	1.73169i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	5.82884e15			0.283010
$926$		0			0
$927$	7.44955e15	−	4.30100e15i	0.357430	−	0.206363i
$928$		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
$931$	7.75497e13	−	1.28670e16i	0.00363377	−	0.602914i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	3.55712e16i		−	1.60891i	−0.594016	−	0.804453i	$-0.702458\pi$
$937$		−	3.55712e16i		−	1.60891i	0.594016	−	0.804453i	$-0.297542\pi$
$938$		0			0
$939$	2.42947e16			1.08605
$940$		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	$0.666667\pi$
$948$		0			0
$949$	−2.31363e16	−	4.00733e16i	−0.975731	−	1.69002i
$950$		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
$961$	3.70203e16	−	6.41210e16i	1.45701	−	2.52361i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−5.24795e16			−1.99593			−0.997963	−	0.0637934i	$-0.979680\pi$
$967$	−5.24795e16			−1.99593			−0.997963	+	0.0637934i	$0.979680\pi$
$968$		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	$0.166667\pi$
$972$		0			0
$973$	1.75640e16	−	3.02111e16i	0.645658	−	1.11057i
$974$		0			0
$975$	1.49346e16	−	2.58674e16i	0.542834	−	0.940216i
$976$		0			0
$977$		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
$981$	4.56952e14			0.0160580
$982$		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
$988$		0			0
$989$		0			0
$990$		0			0
$991$	2.78356e16	+	4.82127e16i	0.925114	+	1.60235i	0.791377	+	0.611328i	$0.209364\pi$
$991$	2.78356e16	+	4.82127e16i	0.925114	+	1.60235i	0.133737	+	0.991017i	$0.457302\pi$
$992$		0			0
$993$			5.93653e16i			1.95125i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	4.11349e16	−	2.37493e16i	1.32247	−	0.763531i	0.338352	−	0.941020i	$-0.390131\pi$
$997$	4.11349e16	−	2.37493e16i	1.32247	−	0.763531i	0.984123	+	0.177489i	$0.0567974\pi$
$998$		0			0
$999$	−7.70803e15	−	4.45023e15i	−0.245094	−	0.141505i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.k.a.5.1	✓	2
3.2	odd	2	CM	84.12.k.a.5.1	✓	2
7.3	odd	6	inner	84.12.k.a.17.1	yes	2
21.17	even	6	inner	84.12.k.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.k.a.5.1	✓	2	1.1	even	1	trivial
84.12.k.a.5.1	✓	2	3.2	odd	2	CM
84.12.k.a.17.1	yes	2	7.3	odd	6	inner
84.12.k.a.17.1	yes	2	21.17	even	6	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(364.500 - 210.444i) q^{3} +(38442.5 + 22349.5i) q^{7} +(88573.5 - 153414. i) q^{9} +O(q^{10})\)	\(q+(364.500 - 210.444i) q^{3} +(38442.5 + 22349.5i) q^{7} +(88573.5 - 153414. i) q^{9} -1.45340e6i q^{13} +(-5.63558e6 - 3.25371e6i) q^{19} +(1.87156e7 + 56399.0i) q^{21} +(2.44141e7 + 4.22864e7i) q^{25} -7.45591e7i q^{27} +(2.73106e8 - 1.57678e8i) q^{31} +(5.96873e7 - 1.03381e8i) q^{37} +(-3.05860e8 - 5.29764e8i) q^{39} +2.18925e8 q^{43} +(9.78325e8 + 1.71834e9i) q^{49} -2.73889e9 q^{57} +(-1.07618e10 - 6.21334e9i) q^{61} +(6.83371e9 - 3.91804e9i) q^{63} +(-2.97565e9 - 5.15397e9i) q^{67} +(2.75721e10 - 1.59188e10i) q^{73} +(1.77979e10 + 1.02756e10i) q^{75} +(2.71481e10 - 4.70219e10i) q^{79} +(-1.56905e10 - 2.71768e10i) q^{81} +(3.24828e10 - 5.58723e10i) q^{91} +(6.63647e10 - 1.14947e11i) q^{93} -1.26194e11i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9}+O(q^{10}) \) 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9	\( 2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9} - 11271165 q^{19} + 37431234 q^{21} + 48828125 q^{25} + 546211707 q^{31} + 119374607 q^{37} - 611719209 q^{39} + 437849438 q^{43} + 1956649739 q^{49} - 5477786190 q^{57} - 21523645452 q^{61} + 13667422491 q^{63} - 5951291615 q^{67} + 55144201461 q^{73} + 35595703125 q^{75} + 54296224537 q^{79} - 31381059609 q^{81} + 64965586941 q^{91} + 132729444801 q^{93}+O(q^{100}) \) 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9 - 11271165 * q^19 + 37431234 * q^21 + 48828125 * q^25 + 546211707 * q^31 + 119374607 * q^37 - 611719209 * q^39 + 437849438 * q^43 + 1956649739 * q^49 - 5477786190 * q^57 - 21523645452 * q^61 + 13667422491 * q^63 - 5951291615 * q^67 + 55144201461 * q^73 + 35595703125 * q^75 + 54296224537 * q^79 - 31381059609 * q^81 + 64965586941 * q^91 + 132729444801 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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