LMFDB - Embedded newform 81.8.c.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,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, 8, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
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:	$25.3031870642$
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:	no (minimal twist has level 27)
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			81.8.c.b.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(64.0000 + 110.851i) q^{4} +(-881.500 + 1526.80i) q^{7} +(-6302.50 - 10916.3i) q^{13} +(-8192.00 + 14189.0i) q^{16} +14357.0 q^{19} +(39062.5 - 67658.2i) q^{25} -225664. q^{28} +(-89458.0 - 154946. i) q^{31} -615373. q^{37} +(-517612. + 896530. i) q^{43} +(-1.14231e6 - 1.97854e6i) q^{49} +(806720. - 1.39728e6i) q^{52} +(-768600. + 1.33125e6i) q^{61} -2.09715e6 q^{64} +(2.02923e6 + 3.51473e6i) q^{67} +1.23681e6 q^{73} +(918848. + 1.59149e6i) q^{76} +(2.12271e6 - 3.67665e6i) q^{79} +2.22226e7 q^{91} +(-2.63818e6 + 4.56946e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 128 q^{4} - 1763 q^{7} - 12605 q^{13} - 16384 q^{16} + 28714 q^{19} + 78125 q^{25} - 451328 q^{28} - 178916 q^{31} - 1230746 q^{37} - 1035224 q^{43} - 2284626 q^{49} + 1613440 q^{52} - 1537199 q^{61}+ \cdots - 5276357 q^{97}+O(q^{100}) $ 2 * q + 128 * q^4 - 1763 * q^7 - 12605 * q^13 - 16384 * q^16 + 28714 * q^19 + 78125 * q^25 - 451328 * q^28 - 178916 * q^31 - 1230746 * q^37 - 1035224 * q^43 - 2284626 * q^49 + 1613440 * q^52 - 1537199 * q^61 - 4194304 * q^64 + 4058455 * q^67 + 2473618 * q^73 + 1837696 * q^76 + 4245427 * q^79 + 44445230 * q^91 - 5276357 * 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
$4$	64.0000	+	110.851i	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
$7$	−881.500	+	1526.80i	−0.971358	+	1.68244i	−0.279892	+	0.960031i	$0.590299\pi$
$7$	−881.500	+	1526.80i	−0.971358	+	1.68244i	−0.691466	+	0.722409i	$0.743035\pi$
$8$		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
$13$	−6302.50	−	10916.3i	−0.795630	−	1.37807i	−0.922438	−	0.386144i	$-0.873807\pi$
$13$	−6302.50	−	10916.3i	−0.795630	−	1.37807i	0.126808	−	0.991927i	$-0.459527\pi$
$14$		0			0
$15$		0			0
$16$	−8192.00	+	14189.0i	−0.500000	+	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	14357.0			0.480204			0.240102	−	0.970748i	$-0.422819\pi$
$19$	14357.0			0.480204			0.240102	+	0.970748i	$0.422819\pi$
$20$		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
$25$	39062.5	−	67658.2i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−225664.			−1.94272
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−89458.0	−	154946.i	−0.539328	−	0.934144i	−0.998940	−	0.0460243i	$-0.985345\pi$
$31$	−89458.0	−	154946.i	−0.539328	−	0.934144i	0.459612	−	0.888120i	$-0.347988\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−615373.			−1.99725			−0.998625	−	0.0524236i	$-0.983305\pi$
$37$	−615373.			−1.99725			−0.998625	+	0.0524236i	$0.983305\pi$
$38$		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
$43$	−517612.	+	896530.i	−0.992807	+	1.71959i	−0.392716	+	0.919660i	$0.628465\pi$
$43$	−517612.	+	896530.i	−0.992807	+	1.71959i	−0.600090	+	0.799932i	$0.704869\pi$
$44$		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
$49$	−1.14231e6	−	1.97854e6i	−1.38707	−	2.40248i
$50$		0			0
$51$		0			0
$52$	806720.	−	1.39728e6i	0.795630	−	1.37807i
$53$		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
$61$	−768600.	+	1.33125e6i	−0.433556	+	0.750942i	−0.997177	−	0.0750923i	$-0.976075\pi$
$61$	−768600.	+	1.33125e6i	−0.433556	+	0.750942i	0.563620	+	0.826034i	$0.309408\pi$
$62$		0			0
$63$		0			0
$64$	−2.09715e6			−1.00000
$65$		0			0
$66$		0			0
$67$	2.02923e6	+	3.51473e6i	0.824269	+	1.42768i	0.902477	+	0.430739i	$0.141747\pi$
$67$	2.02923e6	+	3.51473e6i	0.824269	+	1.42768i	−0.0782078	+	0.996937i	$0.524920\pi$
$68$		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
$73$	1.23681e6			0.372111			0.186056	−	0.982539i	$-0.440430\pi$
$73$	1.23681e6			0.372111			0.186056	+	0.982539i	$0.440430\pi$
$74$		0			0
$75$		0			0
$76$	918848.	+	1.59149e6i	0.240102	+	0.415869i
$77$		0			0
$78$		0			0
$79$	2.12271e6	−	3.67665e6i	0.484392	−	0.838991i	−0.515448	−	0.856921i	$-0.672374\pi$
$79$	2.12271e6	−	3.67665e6i	0.484392	−	0.838991i	0.999839	+	0.0179303i	$0.00570769\pi$
$80$		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
$91$	2.22226e7			3.09137
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.63818e6	+	4.56946e6i	−0.293497	+	0.508351i	−0.974634	−	0.223805i	$-0.928152\pi$
$97$	−2.63818e6	+	4.56946e6i	−0.293497	+	0.508351i	0.681137	+	0.732156i	$0.261486\pi$
$98$		0			0
$99$		0			0
$100$	1.00000e7			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
$103$	1.09599e7	+	1.89830e7i	0.988268	+	1.71173i	0.626400	+	0.779502i	$0.284528\pi$
$103$	1.09599e7	+	1.89830e7i	0.988268	+	1.71173i	0.361868	+	0.932229i	$0.382139\pi$
$104$		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
$109$	−1.68240e7			−1.24433			−0.622167	−	0.782884i	$-0.713748\pi$
$109$	−1.68240e7			−1.24433			−0.622167	+	0.782884i	$0.713748\pi$
$110$		0			0
$111$		0			0
$112$	−1.44425e7	−	2.50151e7i	−0.971358	−	1.68244i
$113$		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
$121$	9.74359e6	+	1.68764e7i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	1.14506e7	−	1.98331e7i	0.539328	−	0.934144i
$125$		0			0
$126$		0			0
$127$	−4.51256e7			−1.95484			−0.977418	−	0.211317i	$-0.932225\pi$
$127$	−4.51256e7			−1.95484			−0.977418	+	0.211317i	$0.932225\pi$
$128$		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
$133$	−1.26557e7	+	2.19203e7i	−0.466450	+	0.807915i
$134$		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
$139$	−489774.	−	848313.i	−0.0154683	−	0.0267919i	0.858188	−	0.513336i	$-0.171591\pi$
$139$	−489774.	−	848313.i	−0.0154683	−	0.0267919i	−0.873656	+	0.486544i	$0.838257\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−3.93839e7	−	6.82149e7i	−0.998625	−	1.72967i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	4.10572e6	−	7.11131e6i	0.0970443	−	0.168086i	−0.813416	−	0.581683i	$-0.802394\pi$
$151$	4.10572e6	−	7.11131e6i	0.0970443	−	0.168086i	0.910460	+	0.413597i	$0.135728\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.10057e7	−	7.10240e7i	−0.845659	−	1.46472i	−0.885047	−	0.465501i	$-0.845874\pi$
$157$	−4.10057e7	−	7.10240e7i	−0.845659	−	1.46472i	0.0393881	−	0.999224i	$-0.487459\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.50255e7			−0.271752			−0.135876	−	0.990726i	$-0.543385\pi$
$163$	−1.50255e7			−0.271752			−0.135876	+	0.990726i	$0.543385\pi$
$164$		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
$169$	−4.80688e7	+	8.32575e7i	−0.766054	+	1.32684i
$170$		0			0
$171$		0			0
$172$	−1.32509e8			−1.98561
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	6.88672e7	+	1.19281e8i	0.971358	+	1.68244i
$176$		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
$181$	1.53710e8			1.92676			0.963378	−	0.268146i	$-0.0864109\pi$
$181$	1.53710e8			1.92676			0.963378	+	0.268146i	$0.0864109\pi$
$182$		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
$193$	5.15960e7	+	8.93669e7i	0.516613	+	0.894800i	0.999814	+	0.0192904i	$0.00614071\pi$
$193$	5.15960e7	+	8.93669e7i	0.516613	+	0.894800i	−0.483201	+	0.875509i	$0.660526\pi$
$194$		0			0
$195$		0			0
$196$	1.46216e8	−	2.53254e8i	1.38707	−	2.40248i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−7.37380e7			−0.663293			−0.331646	−	0.943404i	$-0.607604\pi$
$199$	−7.37380e7			−0.663293			−0.331646	+	0.943404i	$0.607604\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2.06520e8			1.59126
$209$		0			0
$210$		0			0
$211$	1.44390e7	+	2.50090e7i	0.105815	+	0.183277i	0.914071	−	0.405554i	$-0.132921\pi$
$211$	1.44390e7	+	2.50090e7i	0.105815	+	0.183277i	−0.808256	+	0.588832i	$0.799588\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.15429e8			2.09552
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.31448e8	+	2.27675e8i	−0.793759	+	1.37483i	0.129866	+	0.991532i	$0.458545\pi$
$223$	−1.31448e8	+	2.27675e8i	−0.793759	+	1.37483i	−0.923624	+	0.383299i	$0.874788\pi$
$224$		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
$229$	1.00593e8	+	1.74232e8i	0.553534	+	0.958749i	0.998016	+	0.0629609i	$0.0200543\pi$
$229$	1.00593e8	+	1.74232e8i	0.553534	+	0.958749i	−0.444482	+	0.895788i	$0.646612\pi$
$230$		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
$241$	−1.72131e8	+	2.98140e8i	−0.792136	+	1.37202i	0.132506	+	0.991182i	$0.457698\pi$
$241$	−1.72131e8	+	2.98140e8i	−0.792136	+	1.37202i	−0.924642	+	0.380838i	$0.875636\pi$
$242$		0			0
$243$		0			0
$244$	−1.96761e8			−0.867113
$245$		0			0
$246$		0			0
$247$	−9.04850e7	−	1.56725e8i	−0.382065	−	0.661756i
$248$		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
$256$	−1.34218e8	−	2.32472e8i	−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
$259$	5.42451e8	−	9.39553e8i	1.94004	−	3.36025i
$260$		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
$268$	−2.59741e8	+	4.49885e8i	−0.824269	+	1.42768i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−6.23847e8			−1.90408			−0.952041	−	0.305969i	$-0.901020\pi$
$271$	−6.23847e8			−1.90408			−0.952041	+	0.305969i	$0.901020\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.92548e6	+	5.06708e6i	−0.00827024	+	0.0143245i	−0.870131	−	0.492821i	$-0.835966\pi$
$277$	−2.92548e6	+	5.06708e6i	−0.00827024	+	0.0143245i	0.861861	+	0.507145i	$0.169299\pi$
$278$		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
$283$	2.23802e8	+	3.87637e8i	0.586964	+	1.01665i	0.994627	+	0.103520i	$0.0330105\pi$
$283$	2.23802e8	+	3.87637e8i	0.586964	+	1.01665i	−0.407663	+	0.913132i	$0.633656\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.10339e8			−1.00000
$290$		0			0
$291$		0			0
$292$	7.91558e7	+	1.37102e8i	0.186056	+	0.322258i
$293$		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
$301$	−9.12550e8	−	1.58058e9i	−1.92874	−	3.34068i
$302$		0			0
$303$		0			0
$304$	−1.17613e8	+	2.03711e8i	−0.240102	+	0.415869i
$305$		0			0
$306$		0			0
$307$	−1.79003e8			−0.353082			−0.176541	−	0.984293i	$-0.556491\pi$
$307$	−1.79003e8			−0.353082			−0.176541	+	0.984293i	$0.556491\pi$
$308$		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
$313$	−2.78504e8	+	4.82383e8i	−0.513365	+	0.889175i	0.486515	+	0.873672i	$0.338268\pi$
$313$	−2.78504e8	+	4.82383e8i	−0.513365	+	0.889175i	−0.999880	+	0.0155022i	$0.995065\pi$
$314$		0			0
$315$		0			0
$316$	5.43415e8			0.968783
$317$		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
$325$	−9.84766e8			−1.59126
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.26161e8	+	2.18517e8i	−0.191217	+	0.331197i	−0.945654	−	0.325175i	$-0.894577\pi$
$331$	−1.26161e8	+	2.18517e8i	−0.191217	+	0.331197i	0.754437	+	0.656372i	$0.227910\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	5.74047e8	+	9.94279e8i	0.817039	+	1.41515i	0.907854	+	0.419286i	$0.137719\pi$
$337$	5.74047e8	+	9.94279e8i	0.817039	+	1.41515i	−0.0908150	+	0.995868i	$0.528947\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.57589e9			3.44665
$344$		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
$349$	4.44540e8	−	7.69967e8i	0.559786	−	0.969578i	−0.437728	−	0.899108i	$-0.644217\pi$
$349$	4.44540e8	−	7.69967e8i	0.559786	−	0.969578i	0.997514	−	0.0704706i	$-0.0224501\pi$
$350$		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
$361$	−6.87748e8			−0.769404
$362$		0			0
$363$		0			0
$364$	1.42225e9	+	2.46340e9i	1.54568	+	2.67720i
$365$		0			0
$366$		0			0
$367$	9.26445e8	−	1.60465e9i	0.978336	−	1.69453i	0.309882	−	0.950775i	$-0.399710\pi$
$367$	9.26445e8	−	1.60465e9i	0.978336	−	1.69453i	0.668454	−	0.743754i	$-0.266956\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.74912e8	−	1.16898e9i	−0.673389	−	1.16634i	−0.976937	−	0.213528i	$-0.931505\pi$
$373$	−6.74912e8	−	1.16898e9i	−0.673389	−	1.16634i	0.303548	−	0.952816i	$-0.401829\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8.28135e8			0.781383			0.390692	−	0.920522i	$-0.372236\pi$
$379$	8.28135e8			0.781383			0.390692	+	0.920522i	$0.372236\pi$
$380$		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
$388$	−6.75374e8			−0.586993
$389$		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
$397$	1.90824e9			1.53062			0.765308	−	0.643665i	$-0.222587\pi$
$397$	1.90824e9			1.53062			0.765308	+	0.643665i	$0.222587\pi$
$398$		0			0
$399$		0			0
$400$	6.40000e8	+	1.10851e9i	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
$403$	−1.12762e9	+	1.95309e9i	−0.858212	+	1.48647i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.34610e8	+	4.06356e8i	0.169557	+	0.293681i	0.938264	−	0.345920i	$-0.112433\pi$
$409$	2.34610e8	+	4.06356e8i	0.169557	+	0.293681i	−0.768707	+	0.639601i	$0.779100\pi$
$410$		0			0
$411$		0			0
$412$	−1.40286e9	+	2.42983e9i	−0.988268	+	1.71173i
$413$		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
$421$	3.46148e8	−	5.99547e8i	0.226087	−	0.391594i	−0.730558	−	0.682850i	$-0.760740\pi$
$421$	3.46148e8	−	5.99547e8i	0.226087	−	0.391594i	0.956645	+	0.291257i	$0.0940734\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.35504e9	−	2.34700e9i	−0.842277	−	1.45887i
$428$		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
$433$	−1.11566e9			−0.660424			−0.330212	−	0.943907i	$-0.607120\pi$
$433$	−1.11566e9			−0.660424			−0.330212	+	0.943907i	$0.607120\pi$
$434$		0			0
$435$		0			0
$436$	−1.07674e9	−	1.86496e9i	−0.622167	−	1.07763i
$437$		0			0
$438$		0			0
$439$	1.62577e9	−	2.81592e9i	0.917135	−	1.58853i	0.113391	−	0.993550i	$-0.463829\pi$
$439$	1.62577e9	−	2.81592e9i	0.917135	−	1.58853i	0.803744	−	0.594975i	$-0.202838\pi$
$440$		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
$448$	1.84864e9	−	3.20194e9i	0.971358	−	1.68244i
$449$		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
$457$	−2.03355e9	+	3.52221e9i	−0.996661	+	1.72627i	−0.427619	+	0.903959i	$0.640647\pi$
$457$	−2.03355e9	+	3.52221e9i	−0.996661	+	1.72627i	−0.569042	+	0.822308i	$0.692686\pi$
$458$		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
$463$	−1.54006e9	−	2.66746e9i	−0.721114	−	1.24901i	−0.960554	−	0.278094i	$-0.910297\pi$
$463$	−1.54006e9	−	2.66746e9i	−0.721114	−	1.24901i	0.239440	−	0.970911i	$-0.423036\pi$
$464$		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
$469$	−7.15506e9			−3.20264
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	5.60820e8	−	9.71369e8i	0.240102	−	0.415869i
$476$		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
$481$	3.87839e9	+	6.71757e9i	1.58907	+	2.75235i
$482$		0			0
$483$		0			0
$484$	−1.24718e9	+	2.16018e9i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−4.48049e9			−1.75782			−0.878910	−	0.476988i	$-0.841728\pi$
$487$	−4.48049e9			−1.75782			−0.878910	+	0.476988i	$0.841728\pi$
$488$		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
$496$	2.93136e9			1.07866
$497$		0			0
$498$		0			0
$499$	−1.77574e9	−	3.07567e9i	−0.639775	−	1.10812i	−0.985482	−	0.169780i	$-0.945694\pi$
$499$	−1.77574e9	−	3.07567e9i	−0.639775	−	1.10812i	0.345707	−	0.938343i	$-0.387639\pi$
$500$		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
$508$	−2.88804e9	−	5.00223e9i	−0.977418	−	1.69294i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−1.09025e9	+	1.88836e9i	−0.361453	+	0.626055i
$512$		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
$523$	4.97447e9			1.52052			0.760258	−	0.649621i	$-0.225072\pi$
$523$	4.97447e9			1.52052			0.760258	+	0.649621i	$0.225072\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.70241e9	−	2.94867e9i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−3.23986e9			−0.932900
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.36581e9			2.00000			1.00000		0.000770940i	$-0.000245398\pi$
$541$	7.36581e9			2.00000			1.00000		0.000770940i	$0.000245398\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−5.70743e8	+	9.88556e8i	−0.149103	+	0.258253i	−0.930896	−	0.365284i	$-0.880972\pi$
$547$	−5.70743e8	+	9.88556e8i	−0.149103	+	0.258253i	0.781793	+	0.623537i	$0.214305\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	3.74234e9	+	6.48193e9i	0.941035	+	1.62992i
$554$		0			0
$555$		0			0
$556$	6.26910e7	−	1.08584e8i	0.0154683	−	0.0267919i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	1.30490e10			3.15963
$560$		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
$571$	4.33318e9	+	7.50529e9i	0.974048	+	1.68710i	0.683042	+	0.730379i	$0.260657\pi$
$571$	4.33318e9	+	7.50529e9i	0.974048	+	1.68710i	0.291006	+	0.956721i	$0.406010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.21890e9			−1.99785			−0.998927	−	0.0463088i	$-0.985254\pi$
$577$	−9.21890e9			−1.99785			−0.998927	+	0.0463088i	$0.985254\pi$
$578$		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
$589$	−1.28435e9	−	2.22456e9i	−0.258988	−	0.448580i
$590$		0			0
$591$		0			0
$592$	5.04114e9	−	8.73150e9i	0.998625	−	1.72967i
$593$		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
$601$	−3.70827e9	+	6.42291e9i	−0.696804	+	1.20690i	0.272765	+	0.962081i	$0.412062\pi$
$601$	−3.70827e9	+	6.42291e9i	−0.696804	+	1.20690i	−0.969569	+	0.244819i	$0.921272\pi$
$602$		0			0
$603$		0			0
$604$	1.05106e9			0.194089
$605$		0			0
$606$		0			0
$607$	4.07074e9	+	7.05073e9i	0.738776	+	1.27960i	0.953047	+	0.302824i	$0.0979293\pi$
$607$	4.07074e9	+	7.05073e9i	0.738776	+	1.27960i	−0.214270	+	0.976774i	$0.568737\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7.09176e9			−1.24349			−0.621745	−	0.783220i	$-0.713576\pi$
$613$	−7.09176e9			−1.24349			−0.621745	+	0.783220i	$0.713576\pi$
$614$		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
$619$	3.80567e9	−	6.59162e9i	0.644932	−	1.11706i	−0.339385	−	0.940648i	$-0.610219\pi$
$619$	3.80567e9	−	6.59162e9i	0.644932	−	1.11706i	0.984317	−	0.176408i	$-0.0564478\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.05176e9	−	5.28580e9i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	5.24873e9	−	9.09107e9i	0.845659	−	1.46472i
$629$		0			0
$630$		0			0
$631$	9.94859e9			1.57637			0.788186	−	0.615437i	$-0.211021\pi$
$631$	9.94859e9			1.57637			0.788186	+	0.615437i	$0.211021\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.43989e10	+	2.49395e10i	−2.20719	+	3.82297i
$638$		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
$643$	−5.24727e8	−	9.08854e8i	−0.0778386	−	0.134820i	0.824479	−	0.565893i	$-0.191469\pi$
$643$	−5.24727e8	−	9.08854e8i	−0.0778386	−	0.134820i	−0.902317	+	0.431073i	$0.858135\pi$
$644$		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
$652$	−9.61634e8	−	1.66560e9i	−0.135876	−	0.235344i
$653$		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
$661$	−4.10174e9	−	7.10442e9i	−0.552412	−	0.956806i	−0.998100	−	0.0616172i	$-0.980374\pi$
$661$	−4.10174e9	−	7.10442e9i	−0.552412	−	0.956806i	0.445688	−	0.895188i	$-0.352959\pi$
$662$		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
$673$	5.24739e9	−	9.08874e9i	0.663576	−	1.14935i	−0.316094	−	0.948728i	$-0.602371\pi$
$673$	5.24739e9	−	9.08874e9i	0.663576	−	1.14935i	0.979669	−	0.200619i	$-0.0642953\pi$
$674$		0			0
$675$		0			0
$676$	−1.23056e10			−1.53211
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−4.65111e9	−	8.05596e9i	−0.570180	−	0.987581i
$680$		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
$688$	−8.48056e9	−	1.46888e10i	−0.992807	−	1.71959i
$689$		0			0
$690$		0			0
$691$	7.60521e9	−	1.31726e10i	0.876876	−	1.51879i	0.0221249	−	0.999755i	$-0.492957\pi$
$691$	7.60521e9	−	1.31726e10i	0.876876	−	1.51879i	0.854751	−	0.519038i	$-0.173710\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−8.81500e9	+	1.52680e10i	−0.971358	+	1.68244i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−8.83491e9			−0.959088
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−7.35457e9	+	1.27385e10i	−0.774989	+	1.34232i	0.159811	+	0.987148i	$0.448912\pi$
$709$	−7.35457e9	+	1.27385e10i	−0.774989	+	1.34232i	−0.934800	+	0.355173i	$0.884422\pi$
$710$		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
$721$	−3.86445e10			−3.83985
$722$		0			0
$723$		0			0
$724$	9.83743e9	+	1.70389e10i	0.963378	+	1.66862i
$725$		0			0
$726$		0			0
$727$	4.00798e9	−	6.94202e9i	0.386861	−	0.670063i	−0.605165	−	0.796100i	$-0.706893\pi$
$727$	4.00798e9	−	6.94202e9i	0.386861	−	0.670063i	0.992025	+	0.126038i	$0.0402261\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	9.88662e9	+	1.71241e10i	0.927223	+	1.60600i	0.787947	+	0.615743i	$0.211144\pi$
$733$	9.88662e9	+	1.71241e10i	0.927223	+	1.60600i	0.139275	+	0.990254i	$0.455523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.90404e10			1.73549			0.867743	−	0.497013i	$-0.165570\pi$
$739$	1.90404e10			1.73549			0.867743	+	0.497013i	$0.165570\pi$
$740$		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
$751$	−4.16816e9	−	7.21947e9i	−0.359091	−	0.621964i	0.628718	−	0.777633i	$-0.283580\pi$
$751$	−4.16816e9	−	7.21947e9i	−0.359091	−	0.621964i	−0.987809	+	0.155669i	$0.950247\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.61656e10			1.35443			0.677213	−	0.735787i	$-0.263187\pi$
$757$	1.61656e10			1.35443			0.677213	+	0.735787i	$0.263187\pi$
$758$		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
$763$	1.48304e10	−	2.56870e10i	1.20869	−	2.09352i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.21501e10	−	2.10447e10i	−0.963472	−	1.66878i	−0.713665	−	0.700487i	$-0.752966\pi$
$769$	−1.21501e10	−	2.10447e10i	−0.963472	−	1.66878i	−0.249807	−	0.968296i	$-0.580367\pi$
$770$		0			0
$771$		0			0
$772$	−6.60429e9	+	1.14390e10i	−0.516613	+	0.894800i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−1.39778e10			−1.07866
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	3.74313e10			2.77414
$785$		0			0
$786$		0			0
$787$	1.11711e10	+	1.93488e10i	0.816926	+	1.41496i	0.907937	+	0.419107i	$0.137657\pi$
$787$	1.11711e10	+	1.93488e10i	0.816926	+	1.41496i	−0.0910110	+	0.995850i	$0.529010\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.93764e10			1.37980
$794$		0			0
$795$		0			0
$796$	−4.71923e9	−	8.17395e9i	−0.331646	−	0.574429i
$797$		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
$811$	8.58179e9			0.564944			0.282472	−	0.959276i	$-0.408846\pi$
$811$	8.58179e9			0.564944			0.282472	+	0.959276i	$0.408846\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.43136e9	+	1.28715e10i	−0.476750	+	0.825755i
$818$		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
$823$	9.17153e9	+	1.58856e10i	0.573512	+	0.993352i	0.996202	+	0.0870772i	$0.0277527\pi$
$823$	9.17153e9	+	1.58856e10i	0.573512	+	0.993352i	−0.422690	+	0.906274i	$0.638914\pi$
$824$		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
$829$	2.47130e10			1.50655			0.753277	−	0.657704i	$-0.228472\pi$
$829$	2.47130e10			1.50655			0.753277	+	0.657704i	$0.228472\pi$
$830$		0			0
$831$		0			0
$832$	1.32173e10	+	2.28930e10i	0.795630	+	1.37807i
$833$		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
$841$	8.62494e9	+	1.49388e10i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−1.84819e9	+	3.20116e9i	−0.105815	+	0.183277i
$845$		0			0
$846$		0			0
$847$	−3.43559e10			−1.94272
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.77096e10	+	3.06739e10i	−0.976981	+	1.69218i	−0.303745	+	0.952754i	$0.598237\pi$
$853$	−1.77096e10	+	3.06739e10i	−0.976981	+	1.69218i	−0.673237	+	0.739427i	$0.735096\pi$
$854$		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
$859$	−1.85768e10	−	3.21759e10i	−0.999986	−	1.73203i	−0.504617	−	0.863343i	$-0.668366\pi$
$859$	−1.85768e10	−	3.21759e10i	−0.999986	−	1.73203i	−0.495368	−	0.868683i	$-0.664967\pi$
$860$		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
$868$	2.01875e10	+	3.49657e10i	1.04776	+	1.81478i
$869$		0			0
$870$		0			0
$871$	2.55784e10	−	4.43031e10i	1.31163	−	2.27180i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.63166e10	−	2.82611e10i	−0.816828	−	1.41479i	−0.908008	−	0.418953i	$-0.862397\pi$
$877$	−1.63166e10	−	2.82611e10i	−0.816828	−	1.41479i	0.0911805	−	0.995834i	$-0.470936\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−4.16837e9			−0.203753			−0.101876	−	0.994797i	$-0.532485\pi$
$883$	−4.16837e9			−0.203753			−0.101876	+	0.994797i	$0.532485\pi$
$884$		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
$889$	3.97782e10	−	6.88979e10i	1.89884	−	3.28889i
$890$		0			0
$891$		0			0
$892$	−3.36508e10			−1.58752
$893$		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
$907$	−1.75341e10	+	3.03699e10i	−0.780293	+	1.35151i	0.151479	+	0.988461i	$0.451597\pi$
$907$	−1.75341e10	+	3.03699e10i	−0.780293	+	1.35151i	−0.931771	+	0.363046i	$0.881737\pi$
$908$		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
$916$	−1.28759e10	+	2.23017e10i	−0.553534	+	0.958749i
$917$		0			0
$918$		0			0
$919$	3.77715e10			1.60532			0.802658	−	0.596440i	$-0.203419\pi$
$919$	3.77715e10			1.60532			0.802658	+	0.596440i	$0.203419\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.40380e10	+	4.16351e10i	−0.998625	+	1.72967i
$926$		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
$931$	−1.64002e10	−	2.84060e10i	−0.666078	−	1.15368i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−4.17131e10			−1.65647			−0.828236	−	0.560379i	$-0.810655\pi$
$937$	−4.17131e10			−1.65647			−0.828236	+	0.560379i	$0.810655\pi$
$938$		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
$949$	−7.79499e9	−	1.35013e10i	−0.296063	−	0.512796i
$950$		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
$961$	−2.24916e9	+	3.89566e9i	−0.0817502	+	0.141595i
$962$		0			0
$963$		0			0
$964$	−4.40656e10			−1.58427
$965$		0			0
$966$		0			0
$967$	−2.66355e10	−	4.61341e10i	−0.947260	−	1.64070i	−0.751163	−	0.660117i	$-0.770507\pi$
$967$	−2.66355e10	−	4.61341e10i	−0.947260	−	1.64070i	−0.196097	−	0.980585i	$-0.562827\pi$
$968$		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
$973$	1.72694e9			0.0601011
$974$		0			0
$975$		0			0
$976$	−1.25927e10	−	2.18113e10i	−0.433556	−	0.750942i
$977$		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
$988$	1.15821e10	−	2.00607e10i	0.382065	−	0.661756i
$989$		0			0
$990$		0			0
$991$	1.10040e10			0.359163			0.179581	−	0.983743i	$-0.442526\pi$
$991$	1.10040e10			0.359163			0.179581	+	0.983743i	$0.442526\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.80718e10	+	4.86218e10i	−0.897093	+	1.55381i	−0.0659013	+	0.997826i	$0.520992\pi$
$997$	−2.80718e10	+	4.86218e10i	−0.897093	+	1.55381i	−0.831192	+	0.555985i	$0.812341\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.c.b.55.1		2
3.2	odd	2	CM	81.8.c.b.55.1		2
9.2	odd	6		27.8.a.a.1.1	✓	1
9.4	even	3	inner	81.8.c.b.28.1		2
9.5	odd	6	inner	81.8.c.b.28.1		2
9.7	even	3		27.8.a.a.1.1	✓	1
36.7	odd	6		432.8.a.d.1.1		1
36.11	even	6		432.8.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.a.1.1	✓	1	9.2	odd	6
27.8.a.a.1.1	✓	1	9.7	even	3
81.8.c.b.28.1		2	9.4	even	3	inner
81.8.c.b.28.1		2	9.5	odd	6	inner
81.8.c.b.55.1		2	1.1	even	1	trivial
81.8.c.b.55.1		2	3.2	odd	2	CM
432.8.a.d.1.1		1	36.7	odd	6
432.8.a.d.1.1		1	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(64.0000 + 110.851i) q^{4} +(-881.500 + 1526.80i) q^{7} +(-6302.50 - 10916.3i) q^{13} +(-8192.00 + 14189.0i) q^{16} +14357.0 q^{19} +(39062.5 - 67658.2i) q^{25} -225664. q^{28} +(-89458.0 - 154946. i) q^{31} -615373. q^{37} +(-517612. + 896530. i) q^{43} +(-1.14231e6 - 1.97854e6i) q^{49} +(806720. - 1.39728e6i) q^{52} +(-768600. + 1.33125e6i) q^{61} -2.09715e6 q^{64} +(2.02923e6 + 3.51473e6i) q^{67} +1.23681e6 q^{73} +(918848. + 1.59149e6i) q^{76} +(2.12271e6 - 3.67665e6i) q^{79} +2.22226e7 q^{91} +(-2.63818e6 + 4.56946e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 128 q^{4} - 1763 q^{7} - 12605 q^{13} - 16384 q^{16} + 28714 q^{19} + 78125 q^{25} - 451328 q^{28} - 178916 q^{31} - 1230746 q^{37} - 1035224 q^{43} - 2284626 q^{49} + 1613440 q^{52} - 1537199 q^{61}+ \cdots - 5276357 q^{97}+O(q^{100}) \) 2 * q + 128 * q^4 - 1763 * q^7 - 12605 * q^13 - 16384 * q^16 + 28714 * q^19 + 78125 * q^25 - 451328 * q^28 - 178916 * q^31 - 1230746 * q^37 - 1035224 * q^43 - 2284626 * q^49 + 1613440 * q^52 - 1537199 * q^61 - 4194304 * q^64 + 4058455 * q^67 + 2473618 * q^73 + 1837696 * q^76 + 4245427 * q^79 + 44445230 * q^91 - 5276357 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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