LMFDB - Embedded newform 84.9.p.a.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.p (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:	$34.2198032451$
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			65.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.65
Dual form			84.9.p.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 70.1481i) q^{3} +(-2136.50 + 1095.52i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})$	$q+(-40.5000 - 70.1481i) q^{3} +(-2136.50 + 1095.52i) q^{7} +(-3280.50 + 5681.99i) q^{9} +56447.0 q^{13} +(-78983.5 + 136803. i) q^{19} +(163377. + 105503. i) q^{21} +(-195312. - 338291. i) q^{25} +531441. q^{27} +(-291720. - 505273. i) q^{31} +(1.73424e6 - 3.00379e6i) q^{37} +(-2.28610e6 - 3.95965e6i) q^{39} +3.34488e6 q^{43} +(3.36446e6 - 4.68117e6i) q^{49} +1.27953e7 q^{57} +(1.19133e7 - 2.06344e7i) q^{61} +(784040. - 1.57334e7i) q^{63} +(1.59374e7 + 2.76044e7i) q^{67} +(-1.95336e7 - 3.38332e7i) q^{73} +(-1.58203e7 + 2.74016e7i) q^{75} +(2.80036e7 - 4.85036e7i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +(-1.20599e8 + 6.18389e7i) q^{91} +(-2.36293e7 + 4.09271e7i) q^{93} +1.76908e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 81 q^{3} - 4273 q^{7} - 6561 q^{9}+O(q^{10}) $ 2 * q - 81 * q^3 - 4273 * q^7 - 6561 * q^9	$ 2 q - 81 q^{3} - 4273 q^{7} - 6561 q^{9} + 112894 q^{13} - 157967 q^{19} + 326754 q^{21} - 390625 q^{25} + 1062882 q^{27} - 583439 q^{31} + 3468481 q^{37} - 4572207 q^{39} + 6689758 q^{43} + 6728927 q^{49} + 25590654 q^{57} + 23826526 q^{61} + 1568079 q^{63} + 31874833 q^{67} - 39067199 q^{73} - 31640625 q^{75} + 56007121 q^{79} - 43046721 q^{81} - 241198031 q^{91} - 47258559 q^{93} + 353816068 q^{97}+O(q^{100}) $ 2 * q - 81 * q^3 - 4273 * q^7 - 6561 * q^9 + 112894 * q^13 - 157967 * q^19 + 326754 * q^21 - 390625 * q^25 + 1062882 * q^27 - 583439 * q^31 + 3468481 * q^37 - 4572207 * q^39 + 6689758 * q^43 + 6728927 * q^49 + 25590654 * q^57 + 23826526 * q^61 + 1568079 * q^63 + 31874833 * q^67 - 39067199 * q^73 - 31640625 * q^75 + 56007121 * q^79 - 43046721 * q^81 - 241198031 * q^91 - 47258559 * q^93 + 353816068 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$3$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	−2136.50	+	1095.52i	−0.889838	+	0.456277i
$7$	−2136.50	+	1095.52i	−0.889838	+	0.456277i
$8$		0			0
$9$	−3280.50	+	5681.99i	−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
$13$	56447.0			1.97637			0.988183	−	0.153277i	$-0.0489828\pi$
$13$	56447.0			1.97637			0.988183	+	0.153277i	$0.0489828\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−78983.5	+	136803.i	−0.606069	+	1.04974i	0.385813	+	0.922577i	$0.373921\pi$
$19$	−78983.5	+	136803.i	−0.606069	+	1.04974i	−0.991882	+	0.127165i	$0.959412\pi$
$20$		0			0
$21$	163377.	+	105503.i	0.840067	+	0.542483i
$22$		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
$25$	−195312.	−	338291.i	−0.500000	−	0.866025i
$26$		0			0
$27$	531441.			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−291720.	−	505273.i	−0.315877	−	0.547116i	0.663746	−	0.747958i	$-0.268966\pi$
$31$	−291720.	−	505273.i	−0.315877	−	0.547116i	−0.979624	+	0.200842i	$0.935632\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.73424e6	−	3.00379e6i	0.925342	−	1.60274i	0.134333	−	0.990936i	$-0.457111\pi$
$37$	1.73424e6	−	3.00379e6i	0.925342	−	1.60274i	0.791010	−	0.611804i	$-0.209556\pi$
$38$		0			0
$39$	−2.28610e6	−	3.95965e6i	−0.988183	−	1.71158i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	3.34488e6			0.978378			0.489189	−	0.872178i	$-0.337293\pi$
$43$	3.34488e6			0.978378			0.489189	+	0.872178i	$0.337293\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	3.36446e6	−	4.68117e6i	0.583622	−	0.812026i
$50$		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
$57$	1.27953e7			1.21214
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	1.19133e7	−	2.06344e7i	0.860422	−	1.49029i	−0.0111006	−	0.999938i	$-0.503533\pi$
$61$	1.19133e7	−	2.06344e7i	0.860422	−	1.49029i	0.871522	−	0.490356i	$-0.163133\pi$
$62$		0			0
$63$	784040.	−	1.57334e7i	0.0497709	−	0.998761i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.59374e7	+	2.76044e7i	0.790895	+	1.36987i	0.925414	+	0.378959i	$0.123718\pi$
$67$	1.59374e7	+	2.76044e7i	0.790895	+	1.36987i	−0.134519	+	0.990911i	$0.542949\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.95336e7	−	3.38332e7i	−0.687845	−	1.19138i	−0.972533	−	0.232763i	$-0.925223\pi$
$73$	−1.95336e7	−	3.38332e7i	−0.687845	−	1.19138i	0.284688	−	0.958620i	$-0.408110\pi$
$74$		0			0
$75$	−1.58203e7	+	2.74016e7i	−0.500000	+	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.80036e7	−	4.85036e7i	0.718960	−	1.24528i	−0.242452	−	0.970163i	$-0.577952\pi$
$79$	2.80036e7	−	4.85036e7i	0.718960	−	1.24528i	0.961412	−	0.275112i	$-0.0887150\pi$
$80$		0			0
$81$	−2.15234e7	−	3.72796e7i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.20599e8	+	6.18389e7i	−1.75865	+	0.901771i
$92$		0			0
$93$	−2.36293e7	+	4.09271e7i	−0.315877	+	0.547116i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.76908e8			1.99830			0.999150	−	0.0412262i	$-0.0131264\pi$
$97$	1.76908e8			1.99830			0.999150	+	0.0412262i	$0.0131264\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−8.44405e7	+	1.46255e8i	−0.750243	+	1.29946i	0.197462	+	0.980311i	$0.436730\pi$
$103$	−8.44405e7	+	1.46255e8i	−0.750243	+	1.29946i	−0.947705	+	0.319149i	$0.896603\pi$
$104$		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
$109$	1.35670e8	+	2.34988e8i	0.961122	+	1.66471i	0.719692	+	0.694293i	$0.244283\pi$
$109$	1.35670e8	+	2.34988e8i	0.961122	+	1.66471i	0.241430	+	0.970418i	$0.422384\pi$
$110$		0			0
$111$	−2.80947e8			−1.85068
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.85174e8	+	3.20731e8i	−0.988183	+	1.71158i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.07179e8	+	1.85640e8i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.87837e8			1.87525			0.937626	−	0.347646i	$-0.113019\pi$
$127$	4.87837e8			1.87525			0.937626	+	0.347646i	$0.113019\pi$
$128$		0			0
$129$	−1.35468e8	−	2.34637e8i	−0.489189	−	0.847300i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	1.88771e7	−	3.78809e8i	0.0603292	−	1.21064i
$134$		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
$139$	−1.53241e8			−0.410503			−0.205252	−	0.978709i	$-0.565801\pi$
$139$	−1.53241e8			−0.410503			−0.205252	+	0.978709i	$0.565801\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.64635e8	−	4.64234e7i	−0.995046	−	0.0994185i
$148$		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
$151$	−1.35117e8	−	2.34030e8i	−0.259898	−	0.450157i	0.706316	−	0.707896i	$-0.250356\pi$
$151$	−1.35117e8	−	2.34030e8i	−0.259898	−	0.450157i	−0.966214	+	0.257740i	$0.917022\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.80867e7	−	3.13271e7i	−0.0297688	−	0.0515611i	0.850757	−	0.525559i	$-0.176144\pi$
$157$	−1.80867e7	−	3.13271e7i	−0.0297688	−	0.0515611i	−0.880526	+	0.473998i	$0.842810\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.85694e8	−	1.01445e9i	0.829698	−	1.43708i	−0.0685768	−	0.997646i	$-0.521846\pi$
$163$	5.85694e8	−	1.01445e9i	0.829698	−	1.43708i	0.898275	−	0.439434i	$-0.144821\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	2.37053e9			2.90602
$170$		0			0
$171$	−5.18211e8	−	8.97567e8i	−0.606069	−	1.04974i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	7.87891e8	+	5.08790e8i	0.840067	+	0.542483i
$176$		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
$181$	2.14644e9			1.99988			0.999939	−	0.0110495i	$-0.00351724\pi$
$181$	2.14644e9			1.99988			0.999939	+	0.0110495i	$0.00351724\pi$
$182$		0			0
$183$	−1.92995e9			−1.72084
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.13542e9	+	5.82205e8i	−0.889838	+	0.456277i
$190$		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
$193$	−7.31592e7	−	1.26715e8i	−0.0527278	−	0.0913272i	0.838457	−	0.544968i	$-0.183458\pi$
$193$	−7.31592e7	−	1.26715e8i	−0.0527278	−	0.0913272i	−0.891185	+	0.453641i	$0.850125\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−8.67362e8	−	1.50232e9i	−0.553080	−	0.957963i	−0.998050	−	0.0624175i	$-0.980119\pi$
$199$	−8.67362e8	−	1.50232e9i	−0.553080	−	0.957963i	0.444970	−	0.895546i	$-0.353214\pi$
$200$		0			0
$201$	1.29093e9	−	2.23596e9i	0.790895	−	1.36987i
$202$		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
$211$	−1.83711e8			−0.0926840			−0.0463420	−	0.998926i	$-0.514756\pi$
$211$	−1.83711e8			−0.0926840			−0.0463420	+	0.998926i	$0.514756\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.17680e9	+	7.59931e8i	0.530716	+	0.342716i
$218$		0			0
$219$	−1.58222e9	+	2.74049e9i	−0.687845	+	1.19138i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.72732e9			−1.91159			−0.955797	−	0.294027i	$-0.905004\pi$
$223$	−4.72732e9			−1.91159			−0.955797	+	0.294027i	$0.905004\pi$
$224$		0			0
$225$	2.56289e9			1.00000
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	1.87615e9	−	3.24958e9i	0.682220	−	1.18164i	−0.292081	−	0.956393i	$-0.594348\pi$
$229$	1.87615e9	−	3.24958e9i	0.682220	−	1.18164i	0.974302	−	0.225247i	$-0.0723189\pi$
$230$		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
$237$	−4.53658e9			−1.43792
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	2.78289e9	+	4.82011e9i	0.824951	+	1.42886i	0.901957	+	0.431826i	$0.142131\pi$
$241$	2.78289e9	+	4.82011e9i	0.824951	+	1.42886i	−0.0770059	+	0.997031i	$0.524536\pi$
$242$		0			0
$243$	−1.74339e9	+	3.01964e9i	−0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.45838e9	+	7.72214e9i	−1.19781	+	2.07467i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−4.14483e8	+	8.31750e9i	−0.0921103	+	1.84839i
$260$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	1.49355e9	−	2.58690e9i	0.276912	−	0.479625i	−0.693704	−	0.720260i	$-0.744022\pi$
$271$	1.49355e9	−	2.58690e9i	0.276912	−	0.479625i	0.970616	+	0.240635i	$0.0773557\pi$
$272$		0			0
$273$	9.22214e9	+	5.95531e9i	1.66028	+	1.07215i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.09899e9	−	3.63556e9i	−0.356526	−	0.617522i	0.630852	−	0.775904i	$-0.282706\pi$
$277$	−2.09899e9	−	3.63556e9i	−0.356526	−	0.617522i	−0.987378	+	0.158382i	$0.949372\pi$
$278$		0			0
$279$	3.82794e9			0.631755
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−5.86273e9	−	1.01545e10i	−0.914016	−	1.58312i	−0.808335	−	0.588723i	$-0.799631\pi$
$283$	−5.86273e9	−	1.01545e10i	−0.914016	−	1.58312i	−0.105681	−	0.994400i	$-0.533702\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.48788e9	+	6.04118e9i	−0.500000	+	0.866025i
$290$		0			0
$291$	−7.16478e9	−	1.24098e10i	−0.999150	−	1.73058i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−7.14633e9	+	3.66439e9i	−0.870597	+	0.446412i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.26824e9			−0.255349			−0.127675	−	0.991816i	$-0.540751\pi$
$307$	−2.26824e9			−0.255349			−0.127675	+	0.991816i	$0.540751\pi$
$308$		0			0
$309$	1.36794e10			1.50049
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	9.51458e9	−	1.64797e10i	0.991316	−	1.71701i	0.381776	−	0.924255i	$-0.375313\pi$
$313$	9.51458e9	−	1.64797e10i	0.991316	−	1.71701i	0.609541	−	0.792755i	$-0.291354\pi$
$314$		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
$325$	−1.10248e10	−	1.90955e10i	−0.988183	−	1.71158i
$326$		0			0
$327$	1.09893e10	−	1.90340e10i	0.961122	−	1.66471i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.58984e9	−	6.21779e9i	0.299064	−	0.517993i	−0.676858	−	0.736113i	$-0.736659\pi$
$331$	3.58984e9	−	6.21779e9i	0.299064	−	0.517993i	0.975922	+	0.218120i	$0.0699924\pi$
$332$		0			0
$333$	1.13784e10	+	1.97079e10i	0.925342	+	1.60274i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.39043e10			1.07803			0.539014	−	0.842297i	$-0.318797\pi$
$337$	1.39043e10			1.07803			0.539014	+	0.842297i	$0.318797\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.05986e9	+	1.36872e10i	−0.148820	+	0.988864i
$344$		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
$349$	−2.96004e10			−1.99524			−0.997621	−	0.0689403i	$-0.978038\pi$
$349$	−2.96004e10			−1.99524			−0.997621	+	0.0689403i	$0.978038\pi$
$350$		0			0
$351$	2.99983e10			1.97637
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		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
$361$	−3.98501e9	−	6.90223e9i	−0.234639	−	0.406407i
$362$		0			0
$363$	1.73631e10			1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−5.58797e9	−	9.67864e9i	−0.308028	−	0.533519i	0.669903	−	0.742448i	$-0.266336\pi$
$367$	−5.58797e9	−	9.67864e9i	−0.308028	−	0.533519i	−0.977931	+	0.208929i	$0.933002\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.53901e10	+	2.66564e10i	−0.795070	+	1.37710i	0.127724	+	0.991810i	$0.459233\pi$
$373$	−1.53901e10	+	2.66564e10i	−0.795070	+	1.37710i	−0.922794	+	0.385293i	$0.874100\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.73548e10			1.81046			0.905230	−	0.424921i	$-0.139698\pi$
$379$	3.73548e10			1.81046			0.905230	+	0.424921i	$0.139698\pi$
$380$		0			0
$381$	−1.97574e10	−	3.42208e10i	−0.937626	−	1.62402i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.09729e10	+	1.90056e10i	−0.489189	+	0.847300i
$388$		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
$397$	−2.43416e10	+	4.21609e10i	−0.979912	+	1.69726i	−0.317245	+	0.948343i	$0.602758\pi$
$397$	−2.43416e10	+	4.21609e10i	−0.979912	+	1.69726i	−0.662667	+	0.748914i	$0.730575\pi$
$398$		0			0
$399$	−2.73372e10	+	1.40176e10i	−1.07861	+	0.553071i
$400$		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
$403$	−1.64667e10	−	2.85211e10i	−0.624290	−	1.08130i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−2.13506e10	−	3.69802e10i	−0.762985	−	1.32153i	−0.941305	−	0.337556i	$-0.890400\pi$
$409$	−2.13506e10	−	3.69802e10i	−0.762985	−	1.32153i	0.178320	−	0.983972i	$-0.442934\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.20627e9	+	1.07496e10i	0.205252	+	0.355506i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	5.92789e10			1.88700			0.943499	−	0.331375i	$-0.107512\pi$
$421$	5.92789e10			1.88700			0.943499	+	0.331375i	$0.107512\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.84727e9	+	5.71366e10i	−0.0856480	+	1.71871i
$428$		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
$433$	−5.54278e10			−1.57680			−0.788399	−	0.615164i	$-0.789090\pi$
$433$	−5.54278e10			−1.57680			−0.788399	+	0.615164i	$0.789090\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−3.37747e10	+	5.84994e10i	−0.909354	+	1.57505i	−0.0943897	+	0.995535i	$0.530090\pi$
$439$	−3.37747e10	+	5.84994e10i	−0.909354	+	1.57505i	−0.814964	+	0.579512i	$0.803243\pi$
$440$		0			0
$441$	1.55612e10	+	3.44734e10i	0.411424	+	0.911444i
$442$		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
$453$	−1.09445e10	+	1.89564e10i	−0.259898	+	0.450157i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.26976e10	+	7.39545e10i	−0.978902	+	1.69551i	−0.312495	+	0.949919i	$0.601165\pi$
$457$	−4.26976e10	+	7.39545e10i	−0.978902	+	1.69551i	−0.666407	+	0.745589i	$0.732168\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	8.71825e10			1.89717			0.948583	−	0.316529i	$-0.102517\pi$
$463$	8.71825e10			1.89717			0.948583	+	0.316529i	$0.102517\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−6.42915e10	−	4.15170e10i	−1.32881	−	0.858094i
$470$		0			0
$471$	−1.46503e9	+	2.53750e9i	−0.0297688	+	0.0515611i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.17059e10			1.21214
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	9.78927e10	−	1.69555e11i	1.82882	−	3.16760i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	5.13912e10	+	8.90122e10i	0.913636	+	1.58246i	0.808887	+	0.587965i	$0.200071\pi$
$487$	5.13912e10	+	8.90122e10i	0.913636	+	1.58246i	0.104749	+	0.994499i	$0.466596\pi$
$488$		0			0
$489$	−9.48824e10			−1.65940
$490$		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
$499$	2.52466e10	−	4.37283e10i	0.407193	−	0.705279i	−0.587381	−	0.809311i	$-0.699841\pi$
$499$	2.52466e10	−	4.37283e10i	0.407193	−	0.705279i	0.994574	+	0.104032i	$0.0331743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.60066e10	−	1.66288e11i	−1.45301	−	2.51669i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	7.87985e10	+	5.08851e10i	1.15567	+	0.746289i
$512$		0			0
$513$	−4.19751e10	+	7.27030e10i	−0.606069	+	1.04974i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	7.00405e10	−	1.21314e11i	0.936143	−	1.62145i	0.163561	−	0.986533i	$-0.447702\pi$
$523$	7.00405e10	−	1.21314e11i	0.936143	−	1.62145i	0.772582	−	0.634915i	$-0.218965\pi$
$524$		0			0
$525$	3.78105e9	−	7.58750e10i	0.0497709	−	0.998761i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.91555e10	−	6.78193e10i	−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
$541$	8.47166e10	−	1.46733e11i	0.988962	−	1.71293i	0.366161	−	0.930551i	$-0.380672\pi$
$541$	8.47166e10	−	1.46733e11i	0.988962	−	1.71293i	0.622801	−	0.782381i	$-0.285995\pi$
$542$		0			0
$543$	−8.69306e10	−	1.50568e11i	−0.999939	−	1.73195i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.18266e10			−0.690599			−0.345300	−	0.938492i	$-0.612223\pi$
$547$	−6.18266e10			−0.690599			−0.345300	+	0.938492i	$0.612223\pi$
$548$		0			0
$549$	7.81629e10	+	1.35382e11i	0.860422	+	1.49029i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−6.69285e9	+	1.34306e11i	−0.0715666	+	1.43614i
$554$		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
$559$	1.88808e11			1.93363
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	8.68252e10	+	5.60684e10i	0.840067	+	0.542483i
$568$		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
$571$	−1.19932e10	−	2.07727e10i	−0.112821	−	0.195411i	0.804086	−	0.594513i	$-0.202655\pi$
$571$	−1.19932e10	−	2.07727e10i	−0.112821	−	0.195411i	−0.916907	+	0.399102i	$0.869322\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.03486e10	+	5.25653e10i	0.273801	+	0.474237i	0.969832	−	0.243775i	$-0.0783857\pi$
$577$	3.03486e10	+	5.25653e10i	0.273801	+	0.474237i	−0.696031	+	0.718012i	$0.745052\pi$
$578$		0			0
$579$	−5.92590e9	+	1.02640e10i	−0.0527278	+	0.0913272i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	9.21641e10			0.765774
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.02563e10	+	1.21688e11i	−0.553080	+	0.957963i
$598$		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
$601$	−1.88317e11			−1.44341			−0.721707	−	0.692199i	$-0.756642\pi$
$601$	−1.88317e11			−1.44341			−0.721707	+	0.692199i	$0.756642\pi$
$602$		0			0
$603$	−2.09131e11			−1.58179
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.29092e10	+	7.43210e10i	−0.316079	+	0.547465i	−0.979666	−	0.200634i	$-0.935700\pi$
$607$	−4.29092e10	+	7.43210e10i	−0.316079	+	0.547465i	0.663587	+	0.748099i	$0.269033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7.89988e9	+	1.36830e10i	0.0559472	+	0.0969034i	0.892643	−	0.450765i	$-0.148849\pi$
$613$	7.89988e9	+	1.36830e10i	0.0559472	+	0.0969034i	−0.836695	+	0.547669i	$0.815515\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	1.37797e11	+	2.38671e11i	0.938592	+	1.62569i	0.768100	+	0.640330i	$0.221203\pi$
$619$	1.37797e11	+	2.38671e11i	0.938592	+	1.62569i	0.170493	+	0.985359i	$0.445464\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.62939e10	+	1.32145e11i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−2.00069e11			−1.26201			−0.631004	−	0.775780i	$-0.717357\pi$
$631$	−2.00069e11			−1.26201			−0.631004	+	0.775780i	$0.717357\pi$
$632$		0			0
$633$	7.44028e9	+	1.28869e10i	0.0463420	+	0.0802667i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.89914e11	−	2.64238e11i	1.15345	−	1.60486i
$638$		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
$643$	−3.41254e11			−1.99633			−0.998167	−	0.0605142i	$-0.980726\pi$
$643$	−3.41254e11			−1.99633			−0.998167	+	0.0605142i	$0.980726\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	5.64740e9	−	1.13327e11i	0.0314430	−	0.630972i
$652$		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
$657$	2.56320e11			1.37569
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	2.92703e10	+	5.06976e10i	0.153328	+	0.265572i	0.932449	−	0.361302i	$-0.117668\pi$
$661$	2.92703e10	+	5.06976e10i	0.153328	+	0.265572i	−0.779121	+	0.626874i	$0.784334\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.91457e11	+	3.31612e11i	0.955797	+	1.65549i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.80397e11			−0.879364			−0.439682	−	0.898153i	$-0.644909\pi$
$673$	−1.80397e11			−0.879364			−0.439682	+	0.898153i	$0.644909\pi$
$674$		0			0
$675$	−1.03797e11	−	1.79782e11i	−0.500000	−	0.866025i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−3.77964e11	+	1.93807e11i	−1.77816	+	0.911779i
$680$		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
$687$	−3.03936e11			−1.36444
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.46610e11	+	2.53936e11i	−0.643059	+	1.11381i	0.341687	+	0.939814i	$0.389002\pi$
$691$	−1.46610e11	+	2.53936e11i	−0.643059	+	1.11381i	−0.984746	+	0.173997i	$0.944332\pi$
$692$		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
$703$	2.73953e11	+	4.74500e11i	1.12164	+	1.94274i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.43843e11	−	4.22348e11i	0.964995	−	1.67142i	0.255367	−	0.966844i	$-0.417804\pi$
$709$	2.43843e11	−	4.22348e11i	0.964995	−	1.67142i	0.709628	−	0.704577i	$-0.248863\pi$
$710$		0			0
$711$	1.83731e11	+	3.18232e11i	0.718960	+	1.24528i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	2.01813e10	−	4.04981e11i	0.0746806	−	1.49863i
$722$		0			0
$723$	2.25414e11	−	3.90429e11i	0.824951	−	1.42886i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.11506e11			1.47312			0.736560	−	0.676372i	$-0.236449\pi$
$727$	4.11506e11			1.47312			0.736560	+	0.676372i	$0.236449\pi$
$728$		0			0
$729$	2.82430e11			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.41810e11	+	2.45621e11i	−0.491235	+	0.850844i	−0.999949	−	0.0100913i	$-0.996788\pi$
$733$	−1.41810e11	+	2.45621e11i	−0.491235	+	0.850844i	0.508714	+	0.860936i	$0.330121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.85342e11	−	4.94226e11i	−0.956726	−	1.65710i	−0.730368	−	0.683054i	$-0.760651\pi$
$739$	−2.85342e11	−	4.94226e11i	−0.956726	−	1.65710i	−0.226358	−	0.974044i	$-0.572682\pi$
$740$		0			0
$741$	7.22258e11			2.39563
$742$		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
$751$	1.45528e10	−	2.52061e10i	0.0457494	−	0.0792403i	−0.842244	−	0.539097i	$-0.818766\pi$
$751$	1.45528e10	−	2.52061e10i	0.0457494	−	0.0792403i	0.887993	+	0.459856i	$0.152099\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4.59764e11			−1.40008			−0.700038	−	0.714106i	$-0.746833\pi$
$757$	−4.59764e11			−1.40008			−0.700038	+	0.714106i	$0.746833\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−5.47293e11	−	3.53421e11i	−1.61481	−	1.04278i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.62887e11			−0.465781			−0.232890	−	0.972503i	$-0.574818\pi$
$769$	−1.62887e11			−0.465781			−0.232890	+	0.972503i	$0.574818\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−1.13953e11	+	1.97372e11i	−0.315877	+	0.547116i
$776$		0			0
$777$	6.00243e11	−	3.07784e11i	1.64681	−	0.844426i
$778$		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
$787$	−3.37710e11	−	5.84931e11i	−0.880329	−	1.52477i	−0.850976	−	0.525205i	$-0.823989\pi$
$787$	−3.37710e11	−	5.84931e11i	−0.880329	−	1.52477i	−0.0293527	−	0.999569i	$-0.509345\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.72468e11	−	1.16475e12i	1.70051	−	2.94537i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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
$811$	8.88545e10			0.205398			0.102699	−	0.994712i	$-0.467252\pi$
$811$	8.88545e10			0.205398			0.102699	+	0.994712i	$0.467252\pi$
$812$		0			0
$813$	−2.41954e11			−0.553824
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.64190e11	+	4.57591e11i	−0.592964	+	1.02704i
$818$		0			0
$819$	4.42567e10	−	8.88105e11i	0.0983656	−	1.97392i
$820$		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
$823$	4.31093e11	+	7.46675e11i	0.939662	+	1.62754i	0.766102	+	0.642719i	$0.222194\pi$
$823$	4.31093e11	+	7.46675e11i	0.939662	+	1.62754i	0.173559	+	0.984823i	$0.444473\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	4.71066e11	+	8.15910e11i	0.997387	+	1.72753i	0.561258	+	0.827641i	$0.310318\pi$
$829$	4.71066e11	+	8.15910e11i	0.997387	+	1.72753i	0.436129	+	0.899884i	$0.356349\pi$
$830$		0			0
$831$	−1.70018e11	+	2.94480e11i	−0.356526	+	0.617522i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.55032e11	−	2.68523e11i	−0.315877	−	0.547116i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	5.00246e11			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.56159e10	−	5.14038e11i	0.0497709	−	0.998761i
$848$		0			0
$849$	−4.74881e11	+	8.22518e11i	−0.914016	+	1.58312i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.05395e12			−1.99077			−0.995387	−	0.0959421i	$-0.969414\pi$
$853$	−1.05395e12			−1.99077			−0.995387	+	0.0959421i	$0.969414\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	4.01376e11	−	6.95204e11i	0.737189	−	1.27685i	−0.216567	−	0.976268i	$-0.569486\pi$
$859$	4.01376e11	−	6.95204e11i	0.737189	−	1.27685i	0.953756	−	0.300581i	$-0.0971807\pi$
$860$		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
$867$	5.65036e11			1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$	8.99619e11	+	1.55819e12i	1.56310	+	2.70736i
$872$		0			0
$873$	−5.80347e11	+	1.00519e12i	−0.999150	+	1.73058i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.05413e11	+	1.82580e11i	−0.178194	+	0.308642i	−0.941262	−	0.337677i	$-0.890359\pi$
$877$	−1.05413e11	+	1.82580e11i	−0.178194	+	0.308642i	0.763068	+	0.646318i	$0.223692\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1.21517e12			−1.99891			−0.999457	−	0.0329612i	$-0.989506\pi$
$883$	−1.21517e12			−1.99891			−0.999457	+	0.0329612i	$0.989506\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−1.04226e12	+	5.34436e11i	−1.66867	+	0.855635i
$890$		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
$903$	5.46476e11	+	3.52894e11i	0.821903	+	0.530754i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.68215e11	+	9.84177e11i	0.839621	+	1.45427i	0.890212	+	0.455547i	$0.150556\pi$
$907$	5.68215e11	+	9.84177e11i	0.839621	+	1.45427i	−0.0505904	+	0.998719i	$0.516110\pi$
$908$		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
$919$	4.34816e11	−	7.53123e11i	0.609597	−	1.05585i	−0.381709	−	0.924282i	$-0.624664\pi$
$919$	4.34816e11	−	7.53123e11i	0.609597	−	1.05585i	0.991307	−	0.131571i	$-0.0420022\pi$
$920$		0			0
$921$	9.18636e10	+	1.59112e11i	0.127675	+	0.221139i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.35488e12			−1.85068
$926$		0			0
$927$	−5.54014e11	−	9.59581e11i	−0.750243	−	1.29946i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	3.74662e11	+	8.30005e11i	0.498702	+	1.10480i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.19893e12			1.55537			0.777685	−	0.628654i	$-0.216394\pi$
$937$	1.19893e12			1.55537			0.777685	+	0.628654i	$0.216394\pi$
$938$		0			0
$939$	−1.54136e12			−1.98263
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		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
$949$	−1.10261e12	−	1.90978e12i	−1.35943	−	2.35461i
$950$		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
$961$	2.56245e11	−	4.43829e11i	0.300443	−	0.520382i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.51346e12			1.73088			0.865438	−	0.501016i	$-0.167040\pi$
$967$	1.51346e12			1.73088			0.865438	+	0.501016i	$0.167040\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	3.27400e11	−	1.67879e11i	0.365281	−	0.187303i
$974$		0			0
$975$	−8.93009e11	+	1.54674e12i	−0.988183	+	1.71158i
$976$		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
$981$	−1.78026e12			−1.92224
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	4.02727e11	+	6.97543e11i	0.417557	+	0.723230i	0.995693	−	0.0927105i	$-0.0295531\pi$
$991$	4.02727e11	+	6.97543e11i	0.417557	+	0.723230i	−0.578136	+	0.815940i	$0.696220\pi$
$992$		0			0
$993$	−5.81555e11			−0.598127
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.63156e11	+	1.66823e12i	0.974801	+	1.68840i	0.680591	+	0.732664i	$0.261723\pi$
$997$	9.63156e11	+	1.66823e12i	0.974801	+	1.68840i	0.294210	+	0.955741i	$0.404944\pi$
$998$		0			0
$999$	9.21647e11	−	1.59634e12i	0.925342	−	1.60274i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.a.65.1	yes	2
3.2	odd	2	CM	84.9.p.a.65.1	yes	2
7.4	even	3	inner	84.9.p.a.53.1	✓	2
21.11	odd	6	inner	84.9.p.a.53.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.a.53.1	✓	2	7.4	even	3	inner
84.9.p.a.53.1	✓	2	21.11	odd	6	inner
84.9.p.a.65.1	yes	2	1.1	even	1	trivial
84.9.p.a.65.1	yes	2	3.2	odd	2	CM

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-2136.50 + 1095.52i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-2136.50 + 1095.52i) q^{7} +(-3280.50 + 5681.99i) q^{9} +56447.0 q^{13} +(-78983.5 + 136803. i) q^{19} +(163377. + 105503. i) q^{21} +(-195312. - 338291. i) q^{25} +531441. q^{27} +(-291720. - 505273. i) q^{31} +(1.73424e6 - 3.00379e6i) q^{37} +(-2.28610e6 - 3.95965e6i) q^{39} +3.34488e6 q^{43} +(3.36446e6 - 4.68117e6i) q^{49} +1.27953e7 q^{57} +(1.19133e7 - 2.06344e7i) q^{61} +(784040. - 1.57334e7i) q^{63} +(1.59374e7 + 2.76044e7i) q^{67} +(-1.95336e7 - 3.38332e7i) q^{73} +(-1.58203e7 + 2.74016e7i) q^{75} +(2.80036e7 - 4.85036e7i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +(-1.20599e8 + 6.18389e7i) q^{91} +(-2.36293e7 + 4.09271e7i) q^{93} +1.76908e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 81 q^{3} - 4273 q^{7} - 6561 q^{9}+O(q^{10}) \) 2 * q - 81 * q^3 - 4273 * q^7 - 6561 * q^9	\( 2 q - 81 q^{3} - 4273 q^{7} - 6561 q^{9} + 112894 q^{13} - 157967 q^{19} + 326754 q^{21} - 390625 q^{25} + 1062882 q^{27} - 583439 q^{31} + 3468481 q^{37} - 4572207 q^{39} + 6689758 q^{43} + 6728927 q^{49} + 25590654 q^{57} + 23826526 q^{61} + 1568079 q^{63} + 31874833 q^{67} - 39067199 q^{73} - 31640625 q^{75} + 56007121 q^{79} - 43046721 q^{81} - 241198031 q^{91} - 47258559 q^{93} + 353816068 q^{97}+O(q^{100}) \) 2 * q - 81 * q^3 - 4273 * q^7 - 6561 * q^9 + 112894 * q^13 - 157967 * q^19 + 326754 * q^21 - 390625 * q^25 + 1062882 * q^27 - 583439 * q^31 + 3468481 * q^37 - 4572207 * q^39 + 6689758 * q^43 + 6728927 * q^49 + 25590654 * q^57 + 23826526 * q^61 + 1568079 * q^63 + 31874833 * q^67 - 39067199 * q^73 - 31640625 * q^75 + 56007121 * q^79 - 43046721 * q^81 - 241198031 * q^91 - 47258559 * q^93 + 353816068 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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