LMFDB - Embedded newform 81.6.c.b.28.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,6,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, 6, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 6 $
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:	$12.9910894049$
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			28.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.28
Dual form			81.6.c.b.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(16.0000 - 27.7128i) q^{4} +(105.500 + 182.731i) q^{7} +(387.500 - 671.170i) q^{13} +(-512.000 - 886.810i) q^{16} +3143.00 q^{19} +(1562.50 + 2706.33i) q^{25} +6752.00 q^{28} +(5162.00 - 8940.85i) q^{31} -9889.00 q^{37} +(1676.00 + 2902.92i) q^{43} +(-13857.0 + 24001.0i) q^{49} +(-12400.0 - 21477.4i) q^{52} +(9150.50 + 15849.1i) q^{61} -32768.0 q^{64} +(-36737.5 + 63631.2i) q^{67} -78127.0 q^{73} +(50288.0 - 87101.4i) q^{76} +(-4853.50 - 8406.51i) q^{79} +163525. q^{91} +(21669.5 + 37532.7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} + 211 q^{7} + 775 q^{13} - 1024 q^{16} + 6286 q^{19} + 3125 q^{25} + 13504 q^{28} + 10324 q^{31} - 19778 q^{37} + 3352 q^{43} - 27714 q^{49} - 24800 q^{52} + 18301 q^{61} - 65536 q^{64} - 73475 q^{67}+ \cdots + 43339 q^{97}+O(q^{100}) $ 2 * q + 32 * q^4 + 211 * q^7 + 775 * q^13 - 1024 * q^16 + 6286 * q^19 + 3125 * q^25 + 13504 * q^28 + 10324 * q^31 - 19778 * q^37 + 3352 * q^43 - 27714 * q^49 - 24800 * q^52 + 18301 * q^61 - 65536 * q^64 - 73475 * q^67 - 156254 * q^73 + 100576 * q^76 - 9707 * q^79 + 327050 * q^91 + 43339 * q^97

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	16.0000	−	27.7128i	0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	105.500	+	182.731i	0.813781	+	1.40951i	0.910200	+	0.414169i	$0.135928\pi$
$7$	105.500	+	182.731i	0.813781	+	1.40951i	−0.0964195	+	0.995341i	$0.530739\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	387.500	−	671.170i	0.635936	−	1.10147i	−0.350380	−	0.936608i	$-0.613948\pi$
$13$	387.500	−	671.170i	0.635936	−	1.10147i	0.986316	−	0.164866i	$-0.0527191\pi$
$14$		0			0
$15$		0			0
$16$	−512.000	−	886.810i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	3143.00			1.99738			0.998689	−	0.0511835i	$-0.0162993\pi$
$19$	3143.00			1.99738			0.998689	+	0.0511835i	$0.0162993\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	1562.50	+	2706.33i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$	6752.00			1.62756
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	5162.00	−	8940.85i	0.964748	−	1.67099i	0.254456	−	0.967084i	$-0.418103\pi$
$31$	5162.00	−	8940.85i	0.964748	−	1.67099i	0.710291	−	0.703908i	$-0.248563\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−9889.00			−1.18754			−0.593770	−	0.804635i	$-0.702361\pi$
$37$	−9889.00			−1.18754			−0.593770	+	0.804635i	$0.702361\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	1676.00	+	2902.92i	0.138230	+	0.239422i	0.926827	−	0.375489i	$-0.122525\pi$
$43$	1676.00	+	2902.92i	0.138230	+	0.239422i	−0.788597	+	0.614911i	$0.789192\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−13857.0	+	24001.0i	−0.824478	+	1.42804i
$50$		0			0
$51$		0			0
$52$	−12400.0	−	21477.4i	−0.635936	−	1.10147i
$53$		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	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	9150.50	+	15849.1i	0.314862	+	0.545357i	0.979408	−	0.201890i	$-0.0647084\pi$
$61$	9150.50	+	15849.1i	0.314862	+	0.545357i	−0.664546	+	0.747247i	$0.731375\pi$
$62$		0			0
$63$		0			0
$64$	−32768.0			−1.00000
$65$		0			0
$66$		0			0
$67$	−36737.5	+	63631.2i	−0.999822	+	1.73174i	−0.483561	+	0.875310i	$0.660657\pi$
$67$	−36737.5	+	63631.2i	−0.999822	+	1.73174i	−0.516260	+	0.856432i	$0.672676\pi$
$68$		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
$73$	−78127.0			−1.71591			−0.857954	−	0.513727i	$-0.828265\pi$
$73$	−78127.0			−1.71591			−0.857954	+	0.513727i	$0.828265\pi$
$74$		0			0
$75$		0			0
$76$	50288.0	−	87101.4i	0.998689	−	1.72978i
$77$		0			0
$78$		0			0
$79$	−4853.50	−	8406.51i	−0.0874958	−	0.151547i	0.818956	−	0.573856i	$-0.194553\pi$
$79$	−4853.50	−	8406.51i	−0.0874958	−	0.151547i	−0.906452	+	0.422309i	$0.861220\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		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
$91$	163525.			2.07005
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	21669.5	+	37532.7i	0.233840	+	0.405023i	0.958935	−	0.283626i	$-0.0915373\pi$
$97$	21669.5	+	37532.7i	0.233840	+	0.405023i	−0.725095	+	0.688649i	$0.758204\pi$
$98$		0			0
$99$		0			0
$100$	100000.			1.00000
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−35288.5	+	61121.5i	−0.327748	+	0.567676i	−0.982065	−	0.188544i	$-0.939623\pi$
$103$	−35288.5	+	61121.5i	−0.327748	+	0.567676i	0.654317	+	0.756221i	$0.272956\pi$
$104$		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
$109$	114482.			0.922935			0.461467	−	0.887157i	$-0.347323\pi$
$109$	114482.			0.922935			0.461467	+	0.887157i	$0.347323\pi$
$110$		0			0
$111$		0			0
$112$	108032.	−	187117.i	0.813781	−	1.40951i
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	80525.5	−	139474.i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−165184.	−	286107.i	−0.964748	−	1.67099i
$125$		0			0
$126$		0			0
$127$	−267100.			−1.46948			−0.734742	−	0.678347i	$-0.762697\pi$
$127$	−267100.			−1.46948			−0.734742	+	0.678347i	$0.762697\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	331586.	+	574325.i	1.62543	+	2.81532i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	−101096.	+	175104.i	−0.443812	+	0.768705i	−0.997969	−	0.0637074i	$-0.979708\pi$
$139$	−101096.	+	175104.i	−0.443812	+	0.768705i	0.554157	+	0.832412i	$0.313041\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−158224.	+	274052.i	−0.593770	+	1.02844i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−268176.	−	464494.i	−0.957143	−	1.65782i	−0.729387	−	0.684102i	$-0.760194\pi$
$151$	−268176.	−	464494.i	−0.957143	−	1.65782i	−0.227756	−	0.973718i	$-0.573139\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	−0.940785	−	0.339004i	$-0.889910\pi$
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	0.763978	+	0.645242i	$0.223243\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−352375.			−1.03881			−0.519405	−	0.854528i	$-0.673846\pi$
$163$	−352375.			−1.03881			−0.519405	+	0.854528i	$0.673846\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−114666.	−	198607.i	−0.308829	−	0.534907i
$170$		0			0
$171$		0			0
$172$	107264.			0.276460
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−329688.	+	571036.i	−0.813781	+	1.40951i
$176$		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
$181$	−853027.			−1.93538			−0.967690	−	0.252142i	$-0.918865\pi$
$181$	−853027.			−1.93538			−0.967690	+	0.252142i	$0.918865\pi$
$182$		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	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	510588.	−	884365.i	0.986683	−	1.70899i	0.352480	−	0.935820i	$-0.385339\pi$
$193$	510588.	−	884365.i	0.986683	−	1.70899i	0.634204	−	0.773166i	$-0.281328\pi$
$194$		0			0
$195$		0			0
$196$	443424.	+	768033.i	0.824478	+	1.42804i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	1.01476e6			1.81648			0.908241	−	0.418448i	$-0.137426\pi$
$199$	1.01476e6			1.81648			0.908241	+	0.418448i	$0.137426\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−793600.			−1.27187
$209$		0			0
$210$		0			0
$211$	473662.	−	820406.i	0.732423	−	1.26859i	−0.223422	−	0.974722i	$-0.571723\pi$
$211$	473662.	−	820406.i	0.732423	−	1.26859i	0.955845	−	0.293872i	$-0.0949439\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.17836e6			3.14037
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−152026.	−	263317.i	−0.204718	−	0.354582i	0.745325	−	0.666701i	$-0.232294\pi$
$223$	−152026.	−	263317.i	−0.204718	−	0.354582i	−0.950043	+	0.312120i	$0.898961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−634909.	+	1.09969e6i	−0.800060	+	1.38575i	0.119515	+	0.992832i	$0.461866\pi$
$229$	−634909.	+	1.09969e6i	−0.800060	+	1.38575i	−0.919576	+	0.392913i	$0.871467\pi$
$230$		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	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−218288.	−	378087.i	−0.242096	−	0.419323i	0.719215	−	0.694788i	$-0.244502\pi$
$241$	−218288.	−	378087.i	−0.242096	−	0.419323i	−0.961311	+	0.275465i	$0.911168\pi$
$242$		0			0
$243$		0			0
$244$	585632.			0.629724
$245$		0			0
$246$		0			0
$247$	1.21791e6	−	2.10949e6i	1.27020	−	2.20006i
$248$		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
$256$	−524288.	+	908093.i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−1.04329e6	−	1.80703e6i	−0.966397	−	1.67385i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.17560e6	+	2.03620e6i	0.999822	+	1.73174i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−1.88700e6			−1.56081			−0.780403	−	0.625277i	$-0.784986\pi$
$271$	−1.88700e6			−1.56081			−0.780403	+	0.625277i	$0.784986\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.24331e6	+	2.15347e6i	0.973596	+	1.68632i	0.684491	+	0.729021i	$0.260024\pi$
$277$	1.24331e6	+	2.15347e6i	0.973596	+	1.68632i	0.289105	+	0.957297i	$0.406642\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	−976.000	+	1690.48i	−0.000724409	+	0.00125471i	−0.866387	−	0.499373i	$-0.833564\pi$
$283$	−976.000	+	1690.48i	−0.000724409	+	0.00125471i	0.865663	+	0.500627i	$0.166897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.41986e6			−1.00000
$290$		0			0
$291$		0			0
$292$	−1.25003e6	+	2.16512e6i	−0.857954	+	1.48602i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−353636.	+	612516.i	−0.224978	+	0.389673i
$302$		0			0
$303$		0			0
$304$	−1.60922e6	−	2.78724e6i	−0.998689	−	1.72978i
$305$		0			0
$306$		0			0
$307$	−2.30114e6			−1.39347			−0.696733	−	0.717331i	$-0.745364\pi$
$307$	−2.30114e6			−1.39347			−0.696733	+	0.717331i	$0.745364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−1.28354e6	−	2.22315e6i	−0.740539	−	1.28265i	−0.952250	−	0.305318i	$-0.901237\pi$
$313$	−1.28354e6	−	2.22315e6i	−0.740539	−	1.28265i	0.211712	−	0.977332i	$-0.432096\pi$
$314$		0			0
$315$		0			0
$316$	−310624.			−0.174992
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.42188e6			1.27187
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	273076.	+	472981.i	0.136998	+	0.237287i	0.926359	−	0.376642i	$-0.122921\pi$
$331$	273076.	+	472981.i	0.136998	+	0.237287i	−0.789361	+	0.613929i	$0.789588\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.31586e6	+	2.27914e6i	−0.631155	+	1.09319i	0.356161	+	0.934424i	$0.384085\pi$
$337$	−1.31586e6	+	2.27914e6i	−0.631155	+	1.09319i	−0.987316	+	0.158767i	$0.949248\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.30138e6			−1.05622
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	1.74383e6	+	3.02040e6i	0.766373	+	1.32740i	0.939518	+	0.342501i	$0.111274\pi$
$349$	1.74383e6	+	3.02040e6i	0.766373	+	1.32740i	−0.173145	+	0.984896i	$0.555393\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	7.40235e6			2.98952
$362$		0			0
$363$		0			0
$364$	2.61640e6	−	4.53174e6i	1.03502	−	1.79272i
$365$		0			0
$366$		0			0
$367$	1.29159e6	+	2.23710e6i	0.500563	+	0.867000i	1.00000		0.000650122i	$0.000206940\pi$
$367$	1.29159e6	+	2.23710e6i	0.500563	+	0.867000i	−0.499437	+	0.866350i	$0.666460\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.369158	−	0.929367i	$-0.379646\pi$
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.620276	−	0.784384i	$-0.287021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−4.87806e6			−1.74441			−0.872206	−	0.489138i	$-0.837311\pi$
$379$	−4.87806e6			−1.74441			−0.872206	+	0.489138i	$0.837311\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.38685e6			0.467681
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.78949e6			1.84359			0.921794	−	0.387681i	$-0.126724\pi$
$397$	5.78949e6			1.84359			0.921794	+	0.387681i	$0.126724\pi$
$398$		0			0
$399$		0			0
$400$	1.60000e6	−	2.77128e6i	0.500000	−	0.866025i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	−4.00055e6	−	6.92916e6i	−1.22704	−	2.12529i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.18244e6	+	5.51216e6i	−0.940703	+	1.62935i	−0.176569	+	0.984288i	$0.556500\pi$
$409$	−3.18244e6	+	5.51216e6i	−0.940703	+	1.62935i	−0.764134	+	0.645057i	$0.776834\pi$
$410$		0			0
$411$		0			0
$412$	1.12923e6	+	1.95589e6i	0.327748	+	0.567676i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	2.43790e6	+	4.22257e6i	0.670364	+	1.16110i	0.977801	+	0.209536i	$0.0671953\pi$
$421$	2.43790e6	+	4.22257e6i	0.670364	+	1.16110i	−0.307437	+	0.951568i	$0.599471\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.93076e6	+	3.34417e6i	−0.512457	+	0.887602i
$428$		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
$433$	1.85760e6			0.476138			0.238069	−	0.971248i	$-0.423486\pi$
$433$	1.85760e6			0.476138			0.238069	+	0.971248i	$0.423486\pi$
$434$		0			0
$435$		0			0
$436$	1.83171e6	−	3.17262e6i	0.461467	−	0.799285i
$437$		0			0
$438$		0			0
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	0.505505	−	0.862824i	$-0.331306\pi$
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	−0.999980	+	0.00636830i	$0.997973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−3.45702e6	−	5.98774e6i	−0.813781	−	1.40951i
$449$		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
$457$	4.12872e6	+	7.15116e6i	0.924752	+	1.60172i	0.791959	+	0.610574i	$0.209061\pi$
$457$	4.12872e6	+	7.15116e6i	0.924752	+	1.60172i	0.132793	+	0.991144i	$0.457605\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	−1.46109e6	+	2.53068e6i	−0.316755	+	0.548636i	−0.979809	−	0.199936i	$-0.935927\pi$
$463$	−1.46109e6	+	2.53068e6i	−0.316755	+	0.548636i	0.663054	+	0.748572i	$0.269260\pi$
$464$		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
$469$	−1.55032e7			−3.25454
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	4.91094e6	+	8.50599e6i	0.998689	+	1.72978i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−3.83199e6	+	6.63720e6i	−0.755199	+	1.30804i
$482$		0			0
$483$		0			0
$484$	−2.57682e6	−	4.46318e6i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.36487e6			−0.260778			−0.130389	−	0.991463i	$-0.541623\pi$
$487$	−1.36487e6			−0.260778			−0.130389	+	0.991463i	$0.541623\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.05718e7			−1.92950
$497$		0			0
$498$		0			0
$499$	4.18078e6	−	7.24133e6i	0.751634	−	1.30187i	−0.195397	−	0.980724i	$-0.562599\pi$
$499$	4.18078e6	−	7.24133e6i	0.751634	−	1.30187i	0.947030	−	0.321144i	$-0.104067\pi$
$500$		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
$508$	−4.27360e6	+	7.40209e6i	−0.734742	+	1.27261i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−8.24240e6	−	1.42763e7i	−1.39637	−	2.41859i
$512$		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
$523$	2.09363e6			0.334692			0.167346	−	0.985898i	$-0.446480\pi$
$523$	2.09363e6			0.334692			0.167346	+	0.985898i	$0.446480\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.21817e6	+	5.57404e6i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	2.12215e7			3.25086
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.02235e6			−0.443968			−0.221984	−	0.975050i	$-0.571253\pi$
$541$	−3.02235e6			−0.443968			−0.221984	+	0.975050i	$0.571253\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	746656.	+	1.29325e6i	0.106697	+	0.184805i	0.914430	−	0.404744i	$-0.132639\pi$
$547$	746656.	+	1.29325e6i	0.106697	+	0.184805i	−0.807733	+	0.589548i	$0.799306\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.02409e6	−	1.77377e6i	0.142405	−	0.246652i
$554$		0			0
$555$		0			0
$556$	3.23509e6	+	5.60334e6i	0.443812	+	0.768705i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	2.59780e6			0.351622
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−2.94390e6	+	5.09898e6i	−0.377862	+	0.654475i	−0.990751	−	0.135693i	$-0.956674\pi$
$571$	−2.94390e6	+	5.09898e6i	−0.377862	+	0.654475i	0.612889	+	0.790169i	$0.290007\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.46322e7			1.82966			0.914830	−	0.403839i	$-0.132324\pi$
$577$	1.46322e7			1.82966			0.914830	+	0.403839i	$0.132324\pi$
$578$		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	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	1.62242e7	−	2.81011e7i	1.92697	−	3.33760i
$590$		0			0
$591$		0			0
$592$	5.06317e6	+	8.76966e6i	0.593770	+	1.02844i
$593$		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	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−3.01081e6	−	5.21488e6i	−0.340015	−	0.588923i	0.644420	−	0.764671i	$-0.277099\pi$
$601$	−3.01081e6	−	5.21488e6i	−0.340015	−	0.588923i	−0.984435	+	0.175749i	$0.943765\pi$
$602$		0			0
$603$		0			0
$604$	−1.71632e7			−1.91429
$605$		0			0
$606$		0			0
$607$	7.84027e6	−	1.35797e7i	0.863693	−	1.49596i	−0.00464665	−	0.999989i	$-0.501479\pi$
$607$	7.84027e6	−	1.35797e7i	0.863693	−	1.49596i	0.868339	−	0.495970i	$-0.165188\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−566773.			−0.0609197			−0.0304599	−	0.999536i	$-0.509697\pi$
$613$	−566773.			−0.0609197			−0.0304599	+	0.999536i	$0.509697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	−9.39140e6	−	1.62664e7i	−0.985153	−	1.70634i	−0.641253	−	0.767330i	$-0.721585\pi$
$619$	−9.39140e6	−	1.62664e7i	−0.985153	−	1.70634i	−0.343901	−	0.939006i	$-0.611748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.88281e6	+	8.45728e6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	1.74742e6	+	3.02663e6i	0.176807	+	0.306239i
$629$		0			0
$630$		0			0
$631$	1.82315e7			1.82284			0.911422	−	0.411472i	$-0.134985\pi$
$631$	1.82315e7			1.82284			0.911422	+	0.411472i	$0.134985\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.07392e7	+	1.86008e7i	1.04863	+	1.81628i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−1.03351e7	+	1.79009e7i	−0.985796	+	1.70745i	−0.347454	+	0.937697i	$0.612954\pi$
$643$	−1.03351e7	+	1.79009e7i	−0.985796	+	1.70745i	−0.638342	+	0.769753i	$0.720380\pi$
$644$		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
$652$	−5.63800e6	+	9.76530e6i	−0.519405	+	0.899636i
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	345474.	−	598379.i	0.0307548	−	0.0532688i	−0.850238	−	0.526398i	$-0.823542\pi$
$661$	345474.	−	598379.i	0.0307548	−	0.0532688i	0.880993	+	0.473129i	$0.156876\pi$
$662$		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
$673$	−8.65576e6	−	1.49922e7i	−0.736661	−	1.27593i	−0.953991	−	0.299836i	$-0.903068\pi$
$673$	−8.65576e6	−	1.49922e7i	−0.736661	−	1.27593i	0.217330	−	0.976098i	$-0.430265\pi$
$674$		0			0
$675$		0			0
$676$	−7.33862e6			−0.617658
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−4.57226e6	+	7.91939e6i	−0.380590	+	0.659201i
$680$		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
$688$	1.71622e6	−	2.97259e6i	0.138230	−	0.239422i
$689$		0			0
$690$		0			0
$691$	−8.68152e6	−	1.50368e7i	−0.691673	−	1.19801i	−0.971289	−	0.237901i	$-0.923541\pi$
$691$	−8.68152e6	−	1.50368e7i	−0.691673	−	1.19801i	0.279616	−	0.960112i	$-0.409793\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.05500e7	+	1.82731e7i	0.813781	+	1.40951i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−3.10811e7			−2.37197
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.18185e7	−	2.04702e7i	−0.882971	−	1.52935i	−0.848021	−	0.529962i	$-0.822206\pi$
$709$	−1.18185e7	−	2.04702e7i	−0.882971	−	1.52935i	−0.0349502	−	0.999389i	$-0.511127\pi$
$710$		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
$721$	−1.48917e7			−1.06686
$722$		0			0
$723$		0			0
$724$	−1.36484e7	+	2.36398e7i	−0.967690	+	1.67609i
$725$		0			0
$726$		0			0
$727$	1.42256e7	+	2.46395e7i	0.998240	+	1.72900i	0.550474	+	0.834853i	$0.314447\pi$
$727$	1.42256e7	+	2.46395e7i	0.998240	+	1.72900i	0.447767	+	0.894150i	$0.352219\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.65138e6	−	9.78847e6i	0.388503	−	0.672907i	−0.603746	−	0.797177i	$-0.706326\pi$
$733$	5.65138e6	−	9.78847e6i	0.388503	−	0.672907i	0.992248	+	0.124270i	$0.0396590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.96035e7			−1.99403			−0.997017	−	0.0771842i	$-0.975407\pi$
$739$	−2.96035e7			−1.99403			−0.997017	+	0.0771842i	$0.975407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.36610e7	−	2.36615e7i	0.883857	−	1.53088i	0.0368381	−	0.999321i	$-0.488271\pi$
$751$	1.36610e7	−	2.36615e7i	0.883857	−	1.53088i	0.847019	−	0.531563i	$-0.178395\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.12909e7			0.716124			0.358062	−	0.933698i	$-0.383438\pi$
$757$	1.12909e7			0.716124			0.358062	+	0.933698i	$0.383438\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	1.20779e7	+	2.09195e7i	0.751066	+	1.30089i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.25009e7	+	2.16521e7i	−0.762296	+	1.32034i	0.179368	+	0.983782i	$0.442595\pi$
$769$	−1.25009e7	+	2.16521e7i	−0.762296	+	1.32034i	−0.941664	+	0.336554i	$0.890739\pi$
$770$		0			0
$771$		0			0
$772$	−1.63388e7	−	2.82997e7i	−0.986683	−	1.70899i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	3.22625e7			1.92950
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.83791e7			1.64896
$785$		0			0
$786$		0			0
$787$	3.70188e6	−	6.41185e6i	0.213052	−	0.369017i	−0.739616	−	0.673029i	$-0.764993\pi$
$787$	3.70188e6	−	6.41185e6i	0.213052	−	0.369017i	0.952668	+	0.304012i	$0.0983263\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.41833e7			0.800928
$794$		0			0
$795$		0			0
$796$	1.62362e7	−	2.81219e7i	0.908241	−	1.57312i
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		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
$811$	2.27708e7			1.21570			0.607849	−	0.794053i	$-0.292033\pi$
$811$	2.27708e7			1.21570			0.607849	+	0.794053i	$0.292033\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	5.26767e6	+	9.12387e6i	0.276098	+	0.478216i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	−8.21099e6	+	1.42218e7i	−0.422567	+	0.731908i	−0.996190	−	0.0872118i	$-0.972204\pi$
$823$	−8.21099e6	+	1.42218e7i	−0.422567	+	0.731908i	0.573623	+	0.819120i	$0.305538\pi$
$824$		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
$829$	−2.26485e7			−1.14460			−0.572299	−	0.820045i	$-0.693948\pi$
$829$	−2.26485e7			−1.14460			−0.572299	+	0.820045i	$0.693948\pi$
$830$		0			0
$831$		0			0
$832$	−1.26976e7	+	2.19929e7i	−0.635936	+	1.10147i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	1.02556e7	−	1.77632e7i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	−1.51572e7	−	2.62530e7i	−0.732423	−	1.26859i
$845$		0			0
$846$		0			0
$847$	3.39818e7			1.62756
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.50421e6	+	2.60537e6i	0.0707842	+	0.122602i	0.899245	−	0.437445i	$-0.144116\pi$
$853$	1.50421e6	+	2.60537e6i	0.0707842	+	0.122602i	−0.828461	+	0.560047i	$0.810783\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	1.94487e7	−	3.36862e7i	0.899307	−	1.55765i	0.0709259	−	0.997482i	$-0.477405\pi$
$859$	1.94487e7	−	3.36862e7i	0.899307	−	1.55765i	0.828381	−	0.560164i	$-0.189262\pi$
$860$		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
$868$	3.48538e7	−	6.03686e7i	1.57019	−	2.71964i
$869$		0			0
$870$		0			0
$871$	2.84716e7	+	4.93142e7i	1.27165	+	2.20255i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.06152e7	+	3.57066e7i	−0.905084	+	1.56765i	−0.0842800	+	0.996442i	$0.526859\pi$
$877$	−2.06152e7	+	3.57066e7i	−0.905084	+	1.56765i	−0.820804	+	0.571210i	$0.806474\pi$
$878$		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
$883$	−4.58066e7			−1.97709			−0.988545	−	0.150925i	$-0.951775\pi$
$883$	−4.58066e7			−1.97709			−0.988545	+	0.150925i	$0.951775\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−2.81791e7	−	4.88075e7i	−1.19584	−	2.07125i
$890$		0			0
$891$		0			0
$892$	−9.72966e6			−0.409436
$893$		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
$907$	2.47611e7	+	4.28875e7i	0.999428	+	1.73106i	0.528995	+	0.848625i	$0.322569\pi$
$907$	2.47611e7	+	4.28875e7i	0.999428	+	1.73106i	0.470433	+	0.882436i	$0.344098\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	2.03171e7	+	3.51902e7i	0.800060	+	1.38575i
$917$		0			0
$918$		0			0
$919$	4.63039e7			1.80854			0.904271	−	0.426959i	$-0.140415\pi$
$919$	4.63039e7			1.80854			0.904271	+	0.426959i	$0.140415\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.54516e7	−	2.67629e7i	−0.593770	−	1.02844i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−4.35526e7	+	7.54352e7i	−1.64679	+	2.85233i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−4.89709e7			−1.82217			−0.911085	−	0.412219i	$-0.864754\pi$
$937$	−4.89709e7			−1.82217			−0.911085	+	0.412219i	$0.864754\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	−3.02742e7	+	5.24365e7i	−1.09121	+	1.89003i
$950$		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
$961$	−3.89779e7	−	6.75117e7i	−1.36148	−	2.35815i
$962$		0			0
$963$		0			0
$964$	−1.39705e7			−0.484193
$965$		0			0
$966$		0			0
$967$	1.28418e7	−	2.22426e7i	0.441631	−	0.764927i	−0.556180	−	0.831062i	$-0.687733\pi$
$967$	1.28418e7	−	2.22426e7i	0.441631	−	0.764927i	0.997811	+	0.0661347i	$0.0210667\pi$
$968$		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
$973$	−4.26627e7			−1.44466
$974$		0			0
$975$		0			0
$976$	9.37011e6	−	1.62295e7i	0.314862	−	0.545357i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−3.89732e7	−	6.75036e7i	−1.27020	−	2.20006i
$989$		0			0
$990$		0			0
$991$	−1.94402e7			−0.628804			−0.314402	−	0.949290i	$-0.601804\pi$
$991$	−1.94402e7			−0.628804			−0.314402	+	0.949290i	$0.601804\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.24077e7	+	3.88113e7i	0.713937	+	1.23658i	0.963368	+	0.268183i	$0.0864230\pi$
$997$	2.24077e7	+	3.88113e7i	0.713937	+	1.23658i	−0.249431	+	0.968393i	$0.580244\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.6.c.b.28.1		2
3.2	odd	2	CM	81.6.c.b.28.1		2
9.2	odd	6	inner	81.6.c.b.55.1		2
9.4	even	3		27.6.a.a.1.1	✓	1
9.5	odd	6		27.6.a.a.1.1	✓	1
9.7	even	3	inner	81.6.c.b.55.1		2
36.23	even	6		432.6.a.f.1.1		1
36.31	odd	6		432.6.a.f.1.1		1
45.4	even	6		675.6.a.c.1.1		1
45.14	odd	6		675.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.a.a.1.1	✓	1	9.4	even	3
27.6.a.a.1.1	✓	1	9.5	odd	6
81.6.c.b.28.1		2	1.1	even	1	trivial
81.6.c.b.28.1		2	3.2	odd	2	CM
81.6.c.b.55.1		2	9.2	odd	6	inner
81.6.c.b.55.1		2	9.7	even	3	inner
432.6.a.f.1.1		1	36.23	even	6
432.6.a.f.1.1		1	36.31	odd	6
675.6.a.c.1.1		1	45.4	even	6
675.6.a.c.1.1		1	45.14	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(16.0000 - 27.7128i) q^{4} +(105.500 + 182.731i) q^{7} +(387.500 - 671.170i) q^{13} +(-512.000 - 886.810i) q^{16} +3143.00 q^{19} +(1562.50 + 2706.33i) q^{25} +6752.00 q^{28} +(5162.00 - 8940.85i) q^{31} -9889.00 q^{37} +(1676.00 + 2902.92i) q^{43} +(-13857.0 + 24001.0i) q^{49} +(-12400.0 - 21477.4i) q^{52} +(9150.50 + 15849.1i) q^{61} -32768.0 q^{64} +(-36737.5 + 63631.2i) q^{67} -78127.0 q^{73} +(50288.0 - 87101.4i) q^{76} +(-4853.50 - 8406.51i) q^{79} +163525. q^{91} +(21669.5 + 37532.7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} + 211 q^{7} + 775 q^{13} - 1024 q^{16} + 6286 q^{19} + 3125 q^{25} + 13504 q^{28} + 10324 q^{31} - 19778 q^{37} + 3352 q^{43} - 27714 q^{49} - 24800 q^{52} + 18301 q^{61} - 65536 q^{64} - 73475 q^{67}+ \cdots + 43339 q^{97}+O(q^{100}) \) 2 * q + 32 * q^4 + 211 * q^7 + 775 * q^13 - 1024 * q^16 + 6286 * q^19 + 3125 * q^25 + 13504 * q^28 + 10324 * q^31 - 19778 * q^37 + 3352 * q^43 - 27714 * q^49 - 24800 * q^52 + 18301 * q^61 - 65536 * q^64 - 73475 * q^67 - 156254 * q^73 + 100576 * q^76 - 9707 * q^79 + 327050 * q^91 + 43339 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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