LMFDB - Embedded newform 81.10.c.c.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,Mod(28,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not 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:	$41.7179027293$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.10.c.c.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(256.000 + 443.405i) q^{4} +(6290.00 - 10894.6i) q^{7} +(-59185.0 - 102511. i) q^{13} +(-131072. + 227023. i) q^{16} -976696. q^{19} +(976562. - 1.69146e6i) q^{25} +6.44096e6 q^{28} +(-845614. - 1.46465e6i) q^{31} -1.53845e7 q^{37} +(8.28854e6 - 1.43562e7i) q^{43} +(-5.89514e7 - 1.02107e8i) q^{49} +(3.03027e7 - 5.24859e7i) q^{52} +(5.89515e7 - 1.02107e8i) q^{61} -1.34218e8 q^{64} +(-5.62712e7 - 9.74645e7i) q^{67} +2.96368e8 q^{73} +(-2.50034e8 - 4.33072e8i) q^{76} +(3.08366e8 - 5.34106e8i) q^{79} -1.48909e9 q^{91} +(-6.44464e8 + 1.11624e9i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 512 q^{4} + 12580 q^{7} - 118370 q^{13} - 262144 q^{16} - 1953392 q^{19} + 1953125 q^{25} + 12881920 q^{28} - 1691228 q^{31} - 30768980 q^{37} + 16577080 q^{43} - 117902793 q^{49} + 60605440 q^{52}+ \cdots - 1288928270 q^{97}+O(q^{100}) $ 2 * q + 512 * q^4 + 12580 * q^7 - 118370 * q^13 - 262144 * q^16 - 1953392 * q^19 + 1953125 * q^25 + 12881920 * q^28 - 1691228 * q^31 - 30768980 * q^37 + 16577080 * q^43 - 117902793 * q^49 + 60605440 * q^52 + 117903058 * q^61 - 268435456 * q^64 - 112542320 * q^67 + 592736620 * q^73 - 500068352 * q^76 + 616732324 * q^79 - 2978189200 * q^91 - 1288928270 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	256.000	+	443.405i	0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	6290.00	−	10894.6i	0.990169	−	1.71502i	0.373950	−	0.927449i	$-0.378003\pi$
$7$	6290.00	−	10894.6i	0.990169	−	1.71502i	0.616219	−	0.787575i	$-0.288664\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−59185.0	−	102511.i	−0.574734	−	0.995468i	−0.996071	−	0.0885636i	$-0.971772\pi$
$13$	−59185.0	−	102511.i	−0.574734	−	0.995468i	0.421337	−	0.906904i	$-0.361561\pi$
$14$		0			0
$15$		0			0
$16$	−131072.	+	227023.i	−0.500000	+	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−976696.			−1.71937			−0.859683	−	0.510828i	$-0.829339\pi$
$19$	−976696.			−1.71937			−0.859683	+	0.510828i	$0.829339\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	976562.	−	1.69146e6i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	6.44096e6			1.98034
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−845614.	−	1.46465e6i	−0.164454	−	0.284843i	0.772007	−	0.635614i	$-0.219253\pi$
$31$	−845614.	−	1.46465e6i	−0.164454	−	0.284843i	−0.936461	+	0.350771i	$0.885920\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.53845e7			−1.34951			−0.674754	−	0.738043i	$-0.735750\pi$
$37$	−1.53845e7			−1.34951			−0.674754	+	0.738043i	$0.735750\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	8.28854e6	−	1.43562e7i	0.369717	−	0.640369i	−0.619804	−	0.784757i	$-0.712788\pi$
$43$	8.28854e6	−	1.43562e7i	0.369717	−	0.640369i	0.989521	+	0.144387i	$0.0461211\pi$
$44$		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
$49$	−5.89514e7	−	1.02107e8i	−1.46087	−	2.53030i
$50$		0			0
$51$		0			0
$52$	3.03027e7	−	5.24859e7i	0.574734	−	0.995468i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		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
$61$	5.89515e7	−	1.02107e8i	0.545143	−	0.944216i	−0.453454	−	0.891279i	$-0.649809\pi$
$61$	5.89515e7	−	1.02107e8i	0.545143	−	0.944216i	0.998598	−	0.0529367i	$-0.0168582\pi$
$62$		0			0
$63$		0			0
$64$	−1.34218e8			−1.00000
$65$		0			0
$66$		0			0
$67$	−5.62712e7	−	9.74645e7i	−0.341153	−	0.590894i	0.643494	−	0.765451i	$-0.277484\pi$
$67$	−5.62712e7	−	9.74645e7i	−0.341153	−	0.590894i	−0.984647	+	0.174557i	$0.944151\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.96368e8			1.22146			0.610729	−	0.791839i	$-0.290876\pi$
$73$	2.96368e8			1.22146			0.610729	+	0.791839i	$0.290876\pi$
$74$		0			0
$75$		0			0
$76$	−2.50034e8	−	4.33072e8i	−0.859683	−	1.48901i
$77$		0			0
$78$		0			0
$79$	3.08366e8	−	5.34106e8i	0.890727	−	1.54279i	0.0517223	−	0.998662i	$-0.483529\pi$
$79$	3.08366e8	−	5.34106e8i	0.890727	−	1.54279i	0.839005	−	0.544124i	$-0.183138\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−1.48909e9			−2.27633
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.44464e8	+	1.11624e9i	−0.739139	+	1.28023i	0.213745	+	0.976890i	$0.431434\pi$
$97$	−6.44464e8	+	1.11624e9i	−0.739139	+	1.28023i	−0.952884	+	0.303336i	$0.901899\pi$
$98$		0			0
$99$		0			0
$100$	1.00000e9			1.00000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−3.11393e7	−	5.39348e7i	−0.0272610	−	0.0472174i	0.852073	−	0.523423i	$-0.175345\pi$
$103$	−3.11393e7	−	5.39348e7i	−0.0272610	−	0.0472174i	−0.879334	+	0.476205i	$0.842012\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	2.24018e9			1.52007			0.760035	−	0.649882i	$-0.225182\pi$
$109$	2.24018e9			1.52007			0.760035	+	0.649882i	$0.225182\pi$
$110$		0			0
$111$		0			0
$112$	1.64889e9	+	2.85595e9i	0.990169	+	1.71502i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.17897e9	+	2.04204e9i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	4.32954e8	−	7.49899e8i	0.164454	−	0.284843i
$125$		0			0
$126$		0			0
$127$	−2.28028e9			−0.777808			−0.388904	−	0.921278i	$-0.627146\pi$
$127$	−2.28028e9			−0.777808			−0.388904	+	0.921278i	$0.627146\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−6.14342e9	+	1.06407e10i	−1.70246	+	2.94875i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	1.83035e9	+	3.17026e9i	0.415880	+	0.720325i	0.995520	−	0.0945470i	$-0.0301403\pi$
$139$	1.83035e9	+	3.17026e9i	0.415880	+	0.720325i	−0.579640	+	0.814872i	$0.696807\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−3.93843e9	−	6.82156e9i	−0.674754	−	1.16871i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	6.35691e9	−	1.10105e10i	0.995061	−	1.72350i	0.411562	−	0.911382i	$-0.364983\pi$
$151$	6.35691e9	−	1.10105e10i	0.995061	−	1.72350i	0.583499	−	0.812114i	$-0.301683\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.18468e9	+	1.07122e10i	0.812398	+	1.40711i	0.911181	+	0.412006i	$0.135172\pi$
$157$	6.18468e9	+	1.07122e10i	0.812398	+	1.40711i	−0.0987834	+	0.995109i	$0.531495\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.86962e9			0.540320			0.270160	−	0.962815i	$-0.412923\pi$
$163$	4.86962e9			0.540320			0.270160	+	0.962815i	$0.412923\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−1.70348e9	+	2.95051e9i	−0.160637	+	0.278232i
$170$		0			0
$171$		0			0
$172$	8.48746e9			0.739435
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−1.22852e10	−	2.12785e10i	−0.990169	−	1.71502i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−2.01652e10			−1.39652			−0.698262	−	0.715842i	$-0.746043\pi$
$181$	−2.01652e10			−1.39652			−0.698262	+	0.715842i	$0.746043\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−1.16407e10	−	2.01623e10i	−0.603908	−	1.04600i	−0.992223	−	0.124472i	$-0.960276\pi$
$193$	−1.16407e10	−	2.01623e10i	−0.603908	−	1.04600i	0.388315	−	0.921527i	$-0.373057\pi$
$194$		0			0
$195$		0			0
$196$	3.01831e10	−	5.22787e10i	1.46087	−	2.53030i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−1.97601e10			−0.893205			−0.446602	−	0.894733i	$-0.647366\pi$
$199$	−1.97601e10			−0.893205			−0.446602	+	0.894733i	$0.647366\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	3.10300e10			1.14947
$209$		0			0
$210$		0			0
$211$	−2.47409e10	−	4.28526e10i	−0.859301	−	1.48835i	−0.872597	−	0.488441i	$-0.837566\pi$
$211$	−2.47409e10	−	4.28526e10i	−0.859301	−	1.48835i	0.0132960	−	0.999912i	$-0.495768\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.12756e10			−0.651349
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.68695e10	+	6.38598e10i	−0.998378	+	1.72924i	−0.449890	+	0.893084i	$0.648537\pi$
$223$	−3.68695e10	+	6.38598e10i	−0.998378	+	1.72924i	−0.548488	+	0.836158i	$0.684796\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	3.59107e10	+	6.21991e10i	0.862907	+	1.49460i	0.869110	+	0.494618i	$0.164692\pi$
$229$	3.59107e10	+	6.21991e10i	0.862907	+	1.49460i	−0.00620342	+	0.999981i	$0.501975\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	4.58997e10	−	7.95007e10i	0.876463	−	1.51808i	0.0212660	−	0.999774i	$-0.493230\pi$
$241$	4.58997e10	−	7.95007e10i	0.876463	−	1.51808i	0.855197	−	0.518304i	$-0.173436\pi$
$242$		0			0
$243$		0			0
$244$	6.03664e10			1.09029
$245$		0			0
$246$		0			0
$247$	5.78058e10	+	1.00123e11i	0.988177	+	1.71157i
$248$		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
$256$	−3.43597e10	−	5.95128e10i	−0.500000	−	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−9.67684e10	+	1.67608e11i	−1.33624	+	2.31444i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.88108e10	−	4.99018e10i	0.341153	−	0.590894i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−4.79471e10			−0.540009			−0.270004	−	0.962859i	$-0.587025\pi$
$271$	−4.79471e10			−0.540009			−0.270004	+	0.962859i	$0.587025\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.57546e10	+	1.65852e11i	−0.977239	+	1.69263i	−0.304899	+	0.952385i	$0.598623\pi$
$277$	−9.57546e10	+	1.65852e11i	−0.977239	+	1.69263i	−0.672340	+	0.740243i	$0.734711\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	1.02580e11	+	1.77673e11i	0.950653	+	1.64658i	0.744017	+	0.668161i	$0.232918\pi$
$283$	1.02580e11	+	1.77673e11i	0.950653	+	1.64658i	0.206636	+	0.978418i	$0.433748\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.18588e11			−1.00000
$290$		0			0
$291$		0			0
$292$	7.58703e10	+	1.31411e11i	0.610729	+	1.05781i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.04270e11	−	1.80601e11i	−0.732166	−	1.26815i
$302$		0			0
$303$		0			0
$304$	1.28017e11	−	2.21733e11i	0.859683	−	1.48901i
$305$		0			0
$306$		0			0
$307$	2.80963e11			1.80520			0.902601	−	0.430477i	$-0.141655\pi$
$307$	2.80963e11			1.80520			0.902601	+	0.430477i	$0.141655\pi$
$308$		0			0
$309$		0			0
$310$		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
$313$	−4.10771e10	+	7.11476e10i	−0.241908	+	0.418997i	−0.961258	−	0.275651i	$-0.911107\pi$
$313$	−4.10771e10	+	7.11476e10i	−0.241908	+	0.418997i	0.719350	+	0.694648i	$0.244440\pi$
$314$		0			0
$315$		0			0
$316$	3.15767e11			1.78145
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.31191e11			−1.14947
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5.34738e10	−	9.26193e10i	0.244858	−	0.424107i	−0.717233	−	0.696833i	$-0.754592\pi$
$331$	5.34738e10	−	9.26193e10i	0.244858	−	0.424107i	0.962092	+	0.272726i	$0.0879252\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.23115e11	−	3.86447e11i	−0.942313	−	1.63213i	−0.761044	−	0.648700i	$-0.775313\pi$
$337$	−2.23115e11	−	3.86447e11i	−0.942313	−	1.63213i	−0.181268	−	0.983434i	$-0.558020\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−9.75569e11			−3.80570
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	8.58865e10	−	1.48760e11i	0.309892	−	0.536749i	−0.668446	−	0.743760i	$-0.733040\pi$
$349$	8.58865e10	−	1.48760e11i	0.309892	−	0.536749i	0.978339	+	0.207011i	$0.0663737\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6.31247e11			1.95622
$362$		0			0
$363$		0			0
$364$	−3.81208e11	−	6.60272e11i	−1.13817	−	1.97136i
$365$		0			0
$366$		0			0
$367$	2.80922e11	−	4.86571e11i	0.808329	−	1.40007i	−0.105692	−	0.994399i	$-0.533706\pi$
$367$	2.80922e11	−	4.86571e11i	0.808329	−	1.40007i	0.914021	−	0.405667i	$-0.132961\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.95306e10	+	3.38280e10i	0.0522427	+	0.0904871i	0.890964	−	0.454074i	$-0.150030\pi$
$373$	1.95306e10	+	3.38280e10i	0.0522427	+	0.0904871i	−0.838721	+	0.544561i	$0.816696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.69425e11			1.91553			0.957767	−	0.287546i	$-0.0928396\pi$
$379$	7.69425e11			1.91553			0.957767	+	0.287546i	$0.0928396\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−6.59931e11			−1.47828
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.21693e11			−0.447913			−0.223957	−	0.974599i	$-0.571897\pi$
$397$	−2.21693e11			−0.447913			−0.223957	+	0.974599i	$0.571897\pi$
$398$		0			0
$399$		0			0
$400$	2.56000e11	+	4.43405e11i	0.500000	+	0.866025i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−1.00095e11	+	1.73370e11i	−0.189035	+	0.327417i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	5.65912e11	+	9.80188e11i	0.999986	+	1.73203i	0.504617	+	0.863343i	$0.331634\pi$
$409$	5.65912e11	+	9.80188e11i	0.999986	+	1.73203i	0.495369	+	0.868683i	$0.335033\pi$
$410$		0			0
$411$		0			0
$412$	1.59433e10	−	2.76146e10i	0.0272610	−	0.0472174i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−5.98593e11	+	1.03679e12i	−0.928672	+	1.60851i	−0.143125	+	0.989705i	$0.545715\pi$
$421$	−5.98593e11	+	1.03679e12i	−0.928672	+	1.60851i	−0.785547	+	0.618802i	$0.787618\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.41610e11	−	1.28451e12i	−1.07957	−	1.86987i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	6.13416e11			0.838609			0.419304	−	0.907846i	$-0.362274\pi$
$433$	6.13416e11			0.838609			0.419304	+	0.907846i	$0.362274\pi$
$434$		0			0
$435$		0			0
$436$	5.73486e11	+	9.93307e11i	0.760035	+	1.31642i
$437$		0			0
$438$		0			0
$439$	2.31977e11	−	4.01796e11i	0.298095	−	0.516316i	−0.677605	−	0.735426i	$-0.736982\pi$
$439$	2.31977e11	−	4.01796e11i	0.298095	−	0.516316i	0.975700	+	0.219110i	$0.0703155\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−8.44230e11	+	1.46225e12i	−0.990169	+	1.71502i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.93027e11	−	1.37356e12i	0.850482	−	1.47308i	−0.0302922	−	0.999541i	$-0.509644\pi$
$457$	7.93027e11	−	1.37356e12i	0.850482	−	1.47308i	0.880774	−	0.473537i	$-0.157023\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	9.24615e11	+	1.60148e12i	0.935075	+	1.61960i	0.774501	+	0.632573i	$0.218001\pi$
$463$	9.24615e11	+	1.60148e12i	0.935075	+	1.61960i	0.160574	+	0.987024i	$0.448665\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−1.41578e12			−1.35120
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.53805e11	+	1.65204e12i	−0.859683	+	1.48901i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	9.10531e11	+	1.57709e12i	0.775607	+	1.34339i
$482$		0			0
$483$		0			0
$484$	−6.03635e11	+	1.04553e12i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	1.88740e12			1.52049			0.760244	−	0.649638i	$-0.225079\pi$
$487$	1.88740e12			1.52049			0.760244	+	0.649638i	$0.225079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	4.43345e11			0.328908
$497$		0			0
$498$		0			0
$499$	−1.08719e12	−	1.88307e12i	−0.784969	−	1.35961i	−0.929018	−	0.370035i	$-0.879346\pi$
$499$	−1.08719e12	−	1.88307e12i	−0.784969	−	1.35961i	0.144049	−	0.989571i	$-0.453988\pi$
$500$		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
$507$		0			0
$508$	−5.83753e11	−	1.01109e12i	−0.388904	−	0.673601i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	1.86416e12	−	3.22881e12i	1.20945	−	2.09483i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	3.42183e12			1.99987			0.999934	−	0.0115124i	$-0.00366460\pi$
$523$	3.42183e12			1.99987			0.999934	+	0.0115124i	$0.00366460\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.00576e11	−	1.55984e12i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−6.29086e12			−3.40493
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.90575e11			−0.446974			−0.223487	−	0.974707i	$-0.571744\pi$
$541$	−8.90575e11			−0.446974			−0.223487	+	0.974707i	$0.571744\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.00416e11	−	6.93541e11i	0.191236	−	0.331230i	−0.754424	−	0.656387i	$-0.772084\pi$
$547$	4.00416e11	−	6.93541e11i	0.191236	−	0.331230i	0.945660	+	0.325157i	$0.105417\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.87925e12	−	6.71905e12i	−1.76394	−	3.05524i
$554$		0			0
$555$		0			0
$556$	−9.37140e11	+	1.62317e12i	−0.415880	+	0.720325i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	−1.96223e12			−0.849956
$560$		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
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	1.75526e11	+	3.04021e11i	0.0691003	+	0.119685i	0.898506	−	0.438962i	$-0.144654\pi$
$571$	1.75526e11	+	3.04021e11i	0.0691003	+	0.119685i	−0.829405	+	0.558647i	$0.811320\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4.65164e12			1.74709			0.873544	−	0.486745i	$-0.161816\pi$
$577$	4.65164e12			1.74709			0.873544	+	0.486745i	$0.161816\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$	8.25908e11	+	1.43051e12i	0.282757	+	0.489749i
$590$		0			0
$591$		0			0
$592$	2.01648e12	−	3.49264e12i	0.674754	−	1.16871i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	3.03806e12	−	5.26207e12i	0.949863	−	1.64521i	0.204154	−	0.978939i	$-0.434556\pi$
$601$	3.03806e12	−	5.26207e12i	0.949863	−	1.64521i	0.745709	−	0.666272i	$-0.232111\pi$
$602$		0			0
$603$		0			0
$604$	6.50947e12			1.99012
$605$		0			0
$606$		0			0
$607$	1.94323e12	+	3.36577e12i	0.580998	+	1.00632i	0.995361	+	0.0962067i	$0.0306710\pi$
$607$	1.94323e12	+	3.36577e12i	0.580998	+	1.00632i	−0.414363	+	0.910112i	$0.635996\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.75823e12			−1.93313			−0.966564	−	0.256426i	$-0.917455\pi$
$613$	−6.75823e12			−1.93313			−0.966564	+	0.256426i	$0.917455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$	−1.31633e10	+	2.27995e10i	−0.00360376	+	0.00624190i	−0.867822	−	0.496876i	$-0.834480\pi$
$619$	−1.31633e10	+	2.27995e10i	−0.00360376	+	0.00624190i	0.864218	+	0.503118i	$0.167814\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.90735e12	−	3.30362e12i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−3.16656e12	+	5.48464e12i	−0.812398	+	1.40711i
$629$		0			0
$630$		0			0
$631$	−1.41926e12			−0.356394			−0.178197	−	0.983995i	$-0.557026\pi$
$631$	−1.41926e12			−0.356394			−0.178197	+	0.983995i	$0.557026\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6.97808e12	+	1.20864e13i	−1.67922	+	2.90850i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	4.10827e12	+	7.11574e12i	0.947785	+	1.64161i	0.750077	+	0.661351i	$0.230017\pi$
$643$	4.10827e12	+	7.11574e12i	0.947785	+	1.64161i	0.197709	+	0.980261i	$0.436650\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.24662e12	+	2.15921e12i	0.270160	+	0.467931i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	−4.57672e12	−	7.92712e12i	−0.932498	−	1.61513i	−0.779035	−	0.626980i	$-0.784291\pi$
$661$	−4.57672e12	−	7.92712e12i	−0.932498	−	1.61513i	−0.153463	−	0.988154i	$-0.549043\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	4.04114e12	−	6.99945e12i	0.759339	−	1.31521i	−0.183849	−	0.982954i	$-0.558856\pi$
$673$	4.04114e12	−	6.99945e12i	0.759339	−	1.31521i	0.943188	−	0.332259i	$-0.107811\pi$
$674$		0			0
$675$		0			0
$676$	−1.74436e12			−0.321275
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	8.10736e12	+	1.40424e13i	1.46374	+	2.53528i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	2.17279e12	+	3.76338e12i	0.369717	+	0.640369i
$689$		0			0
$690$		0			0
$691$	−5.87841e12	+	1.01817e13i	−0.980863	+	1.69890i	−0.321817	+	0.946802i	$0.604294\pi$
$691$	−5.87841e12	+	1.01817e13i	−0.980863	+	1.69890i	−0.659046	+	0.752103i	$0.729040\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	6.29000e12	−	1.08946e13i	0.990169	−	1.71502i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	1.50260e13			2.32030
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4.28928e12	+	7.42925e12i	−0.637494	+	1.10417i	0.348486	+	0.937314i	$0.386696\pi$
$709$	−4.28928e12	+	7.42925e12i	−0.637494	+	1.10417i	−0.985981	+	0.166859i	$0.946638\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−7.83465e11			−0.107972
$722$		0			0
$723$		0			0
$724$	−5.16229e12	−	8.94134e12i	−0.698262	−	1.20943i
$725$		0			0
$726$		0			0
$727$	−5.58681e12	+	9.67663e12i	−0.741752	+	1.28475i	0.209945	+	0.977713i	$0.432672\pi$
$727$	−5.58681e12	+	9.67663e12i	−0.741752	+	1.28475i	−0.951697	+	0.307039i	$0.900662\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.18634e12	+	1.24471e13i	0.919476	+	1.59258i	0.800213	+	0.599716i	$0.204720\pi$
$733$	7.18634e12	+	1.24471e13i	0.919476	+	1.59258i	0.119262	+	0.992863i	$0.461947\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.10029e12			−0.875742			−0.437871	−	0.899038i	$-0.644267\pi$
$739$	−7.10029e12			−0.875742			−0.437871	+	0.899038i	$0.644267\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8.45090e12	−	1.46374e13i	−0.969446	−	1.67913i	−0.697164	−	0.716911i	$-0.745555\pi$
$751$	−8.45090e12	−	1.46374e13i	−0.969446	−	1.67913i	−0.272281	−	0.962218i	$-0.587778\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.73497e13			−1.92026			−0.960129	−	0.279557i	$-0.909812\pi$
$757$	−1.73497e13			−1.92026			−0.960129	+	0.279557i	$0.909812\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	1.40907e13	−	2.44059e13i	1.50513	−	2.60696i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−3.09305e12	−	5.35733e12i	−0.318947	−	0.552433i	0.661321	−	0.750103i	$-0.269996\pi$
$769$	−3.09305e12	−	5.35733e12i	−0.318947	−	0.552433i	−0.980269	+	0.197670i	$0.936663\pi$
$770$		0			0
$771$		0			0
$772$	5.96003e12	−	1.03231e13i	0.603908	−	1.04600i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−3.30318e12			−0.328908
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	3.09075e13			2.92174
$785$		0			0
$786$		0			0
$787$	−9.14069e12	−	1.58321e13i	−0.849361	−	1.47114i	−0.881779	−	0.471663i	$-0.843654\pi$
$787$	−9.14069e12	−	1.58321e13i	−0.849361	−	1.47114i	0.0324177	−	0.999474i	$-0.489679\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.39562e13			−1.25325
$794$		0			0
$795$		0			0
$796$	−5.05859e12	−	8.76174e12i	−0.446602	−	0.773538i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		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			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−1.98398e12			−0.161044			−0.0805218	−	0.996753i	$-0.525659\pi$
$811$	−1.98398e12			−0.161044			−0.0805218	+	0.996753i	$0.525659\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.09538e12	+	1.40216e13i	−0.635679	+	1.10103i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	−9.30551e12	−	1.61176e13i	−0.707035	−	1.22462i	−0.965952	−	0.258721i	$-0.916699\pi$
$823$	−9.30551e12	−	1.61176e13i	−0.707035	−	1.22462i	0.258917	−	0.965900i	$-0.416634\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−2.41982e13			−1.77946			−0.889730	−	0.456488i	$-0.849107\pi$
$829$	−2.41982e13			−1.77946			−0.889730	+	0.456488i	$0.849107\pi$
$830$		0			0
$831$		0			0
$832$	7.94368e12	+	1.37589e13i	0.574734	+	0.995468i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	7.25357e12	+	1.25636e13i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	1.26674e13	−	2.19405e13i	0.859301	−	1.48835i
$845$		0			0
$846$		0			0
$847$	2.96630e13			1.98034
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.42382e12	−	1.28584e13i	0.480128	−	0.831606i	−0.519612	−	0.854402i	$-0.673924\pi$
$853$	7.42382e12	−	1.28584e13i	0.480128	−	0.831606i	0.999740	+	0.0227966i	$0.00725700\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	1.44246e13	+	2.49841e13i	0.903927	+	1.56565i	0.822352	+	0.568979i	$0.192662\pi$
$859$	1.44246e13	+	2.49841e13i	0.903927	+	1.56565i	0.0815748	+	0.996667i	$0.474005\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−5.44657e12	−	9.43373e12i	−0.325675	−	0.564085i
$869$		0			0
$870$		0			0
$871$	−6.66082e12	+	1.15369e13i	−0.392144	+	0.679214i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.23230e13	−	2.13440e13i	−0.703424	−	1.21837i	−0.967257	−	0.253797i	$-0.918320\pi$
$877$	−1.23230e13	−	2.13440e13i	−0.703424	−	1.21837i	0.263834	−	0.964568i	$-0.415013\pi$
$878$		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
$883$	3.38683e13			1.87487			0.937434	−	0.348163i	$-0.113194\pi$
$883$	3.38683e13			1.87487			0.937434	+	0.348163i	$0.113194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−1.43430e13	+	2.48428e13i	−0.770161	+	1.33396i
$890$		0			0
$891$		0			0
$892$	−3.77544e13			−1.99676
$893$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.10733e12	−	8.84615e12i	0.250588	−	0.434032i	−0.713100	−	0.701063i	$-0.752709\pi$
$907$	5.10733e12	−	8.84615e12i	0.250588	−	0.434032i	0.963688	+	0.267031i	$0.0860426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.83863e13	+	3.18460e13i	−0.862907	+	1.49460i
$917$		0			0
$918$		0			0
$919$	−6.39629e12			−0.295807			−0.147903	−	0.989002i	$-0.547252\pi$
$919$	−6.39629e12			−0.295807			−0.147903	+	0.989002i	$0.547252\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.50239e13	+	2.60222e13i	−0.674754	+	1.16871i
$926$		0			0
$927$		0			0
$928$		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
$931$	5.75776e13	+	9.97273e13i	2.51177	+	4.35051i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.40492e13			−1.44304			−0.721521	−	0.692392i	$-0.756557\pi$
$937$	−3.40492e13			−1.44304			−0.721521	+	0.692392i	$0.756557\pi$
$938$		0			0
$939$		0			0
$940$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	−1.75406e13	−	3.03811e13i	−0.702013	−	1.21592i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.17897e13	−	2.04203e13i	0.445910	−	0.772338i
$962$		0			0
$963$		0			0
$964$	4.70013e13			1.75293
$965$		0			0
$966$		0			0
$967$	−5.26943e12	−	9.12691e12i	−0.193796	−	0.335664i	0.752709	−	0.658353i	$-0.228747\pi$
$967$	−5.26943e12	−	9.12691e12i	−0.193796	−	0.335664i	−0.946505	+	0.322689i	$0.895413\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	4.60517e13			1.64717
$974$		0			0
$975$		0			0
$976$	1.54538e13	+	2.67667e13i	0.545143	+	0.944216i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.95965e13	+	5.12627e13i	−0.988177	+	1.71157i
$989$		0			0
$990$		0			0
$991$	3.82277e13			1.25906			0.629531	−	0.776976i	$-0.283247\pi$
$991$	3.82277e13			1.25906			0.629531	+	0.776976i	$0.283247\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.08946e13	−	5.35110e13i	0.990271	−	1.71520i	0.374624	−	0.927177i	$-0.377772\pi$
$997$	3.08946e13	−	5.35110e13i	0.990271	−	1.71520i	0.615647	−	0.788022i	$-0.288895\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.c.c.55.1		2
3.2	odd	2	CM	81.10.c.c.55.1		2
9.2	odd	6		9.10.a.b.1.1	✓	1
9.4	even	3	inner	81.10.c.c.28.1		2
9.5	odd	6	inner	81.10.c.c.28.1		2
9.7	even	3		9.10.a.b.1.1	✓	1
36.7	odd	6		144.10.a.h.1.1		1
36.11	even	6		144.10.a.h.1.1		1
45.2	even	12		225.10.b.f.199.1		2
45.7	odd	12		225.10.b.f.199.1		2
45.29	odd	6		225.10.a.d.1.1		1
45.34	even	6		225.10.a.d.1.1		1
45.38	even	12		225.10.b.f.199.2		2
45.43	odd	12		225.10.b.f.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.a.b.1.1	✓	1	9.2	odd	6
9.10.a.b.1.1	✓	1	9.7	even	3
81.10.c.c.28.1		2	9.4	even	3	inner
81.10.c.c.28.1		2	9.5	odd	6	inner
81.10.c.c.55.1		2	1.1	even	1	trivial
81.10.c.c.55.1		2	3.2	odd	2	CM
144.10.a.h.1.1		1	36.7	odd	6
144.10.a.h.1.1		1	36.11	even	6
225.10.a.d.1.1		1	45.29	odd	6
225.10.a.d.1.1		1	45.34	even	6
225.10.b.f.199.1		2	45.2	even	12
225.10.b.f.199.1		2	45.7	odd	12
225.10.b.f.199.2		2	45.38	even	12
225.10.b.f.199.2		2	45.43	odd	12

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(256.000 + 443.405i) q^{4} +(6290.00 - 10894.6i) q^{7} +(-59185.0 - 102511. i) q^{13} +(-131072. + 227023. i) q^{16} -976696. q^{19} +(976562. - 1.69146e6i) q^{25} +6.44096e6 q^{28} +(-845614. - 1.46465e6i) q^{31} -1.53845e7 q^{37} +(8.28854e6 - 1.43562e7i) q^{43} +(-5.89514e7 - 1.02107e8i) q^{49} +(3.03027e7 - 5.24859e7i) q^{52} +(5.89515e7 - 1.02107e8i) q^{61} -1.34218e8 q^{64} +(-5.62712e7 - 9.74645e7i) q^{67} +2.96368e8 q^{73} +(-2.50034e8 - 4.33072e8i) q^{76} +(3.08366e8 - 5.34106e8i) q^{79} -1.48909e9 q^{91} +(-6.44464e8 + 1.11624e9i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 512 q^{4} + 12580 q^{7} - 118370 q^{13} - 262144 q^{16} - 1953392 q^{19} + 1953125 q^{25} + 12881920 q^{28} - 1691228 q^{31} - 30768980 q^{37} + 16577080 q^{43} - 117902793 q^{49} + 60605440 q^{52}+ \cdots - 1288928270 q^{97}+O(q^{100}) \) 2 * q + 512 * q^4 + 12580 * q^7 - 118370 * q^13 - 262144 * q^16 - 1953392 * q^19 + 1953125 * q^25 + 12881920 * q^28 - 1691228 * q^31 - 30768980 * q^37 + 16577080 * q^43 - 117902793 * q^49 + 60605440 * q^52 + 117903058 * q^61 - 268435456 * q^64 - 112542320 * q^67 + 592736620 * q^73 - 500068352 * q^76 + 616732324 * q^79 - 2978189200 * q^91 - 1288928270 * q^97

Introduction

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