LMFDB - Embedded newform 147.4.g.a.68.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,4,Mod(68,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("147.68");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	147.g (of order $6$, 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:	$8.67328077084$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			68.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	147.68
Dual form			147.4.g.a.80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +(36.0000 + 20.7846i) q^{12} +62.3538i q^{13} +(-32.0000 + 55.4256i) q^{16} +(135.000 + 77.9423i) q^{19} +(62.5000 + 108.253i) q^{25} +140.296i q^{27} +(-135.000 + 77.9423i) q^{31} -216.000 q^{36} +(55.0000 - 95.2628i) q^{37} +(-162.000 - 280.592i) q^{39} +520.000 q^{43} -332.554i q^{48} +(432.000 - 249.415i) q^{52} -810.000 q^{57} +(-810.000 - 467.654i) q^{61} +512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-324.000 + 187.061i) q^{73} +(-562.500 - 324.760i) q^{75} -1247.08i q^{76} +(-442.000 + 765.566i) q^{79} +(-364.500 - 631.333i) q^{81} +(405.000 - 701.481i) q^{93} +1371.78i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64}+ \cdots + 810 q^{93}+O(q^{100}) $ 2 * q - 9 * q^3 - 8 * q^4 + 27 * q^9 + 72 * q^12 - 64 * q^16 + 270 * q^19 + 125 * q^25 - 270 * q^31 - 432 * q^36 + 110 * q^37 - 324 * q^39 + 1040 * q^43 + 864 * q^52 - 1620 * q^57 - 1620 * q^61 + 1024 * q^64 + 880 * q^67 - 648 * q^73 - 1125 * q^75 - 884 * q^79 - 729 * q^81 + 810 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−4.50000	+	2.59808i	−0.866025	+	0.500000i
$3$	−4.50000	+	2.59808i	−0.866025	+	0.500000i
$4$	−4.00000	−	6.92820i	−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
$7$		0			0
$7$		0			0
$8$		0			0
$9$	13.5000	−	23.3827i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	36.0000	+	20.7846i	0.866025	+	0.500000i
$13$			62.3538i			1.33030i	0.746712	+	0.665148i	$0.231631\pi$
$13$			62.3538i			1.33030i	−0.746712	+	0.665148i	$0.768369\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	135.000	+	77.9423i	1.63006	+	0.941115i	0.984073	+	0.177766i	$0.0568871\pi$
$19$	135.000	+	77.9423i	1.63006	+	0.941115i	0.645986	+	0.763349i	$0.276446\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	62.5000	+	108.253i	0.500000	+	0.866025i
$26$		0			0
$27$			140.296i			1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−135.000	+	77.9423i	−0.782152	+	0.451576i	−0.837192	−	0.546908i	$-0.815805\pi$
$31$	−135.000	+	77.9423i	−0.782152	+	0.451576i	0.0550403	+	0.998484i	$0.482471\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−216.000			−1.00000
$37$	55.0000	−	95.2628i	0.244377	−	0.423273i	−0.717579	−	0.696477i	$-0.754750\pi$
$37$	55.0000	−	95.2628i	0.244377	−	0.423273i	0.961956	+	0.273204i	$0.0880833\pi$
$38$		0			0
$39$	−162.000	−	280.592i	−0.665148	−	1.15207i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	520.000			1.84417			0.922084	−	0.386989i	$-0.126485\pi$
$43$	520.000			1.84417			0.922084	+	0.386989i	$0.126485\pi$
$44$		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	$0.166667\pi$
$48$		−	332.554i		−	1.00000i
$49$		0			0
$50$		0			0
$51$		0			0
$52$	432.000	−	249.415i	1.15207	−	0.665148i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−810.000			−1.88223
$58$		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
$61$	−810.000	−	467.654i	−1.70016	−	0.981589i	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−810.000	−	467.654i	−1.70016	−	0.981589i	−0.754578	−	0.656210i	$-0.772158\pi$
$62$		0			0
$63$		0			0
$64$	512.000			1.00000
$65$		0			0
$66$		0			0
$67$	440.000	+	762.102i	0.802307	+	1.38964i	0.918094	+	0.396362i	$0.129728\pi$
$67$	440.000	+	762.102i	0.802307	+	1.38964i	−0.115787	+	0.993274i	$0.536939\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−324.000	+	187.061i	−0.519470	+	0.299916i	−0.736718	−	0.676200i	$-0.763625\pi$
$73$	−324.000	+	187.061i	−0.519470	+	0.299916i	0.217248	+	0.976117i	$0.430292\pi$
$74$		0			0
$75$	−562.500	−	324.760i	−0.866025	−	0.500000i
$76$		−	1247.08i		−	1.88223i
$77$		0			0
$78$		0			0
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	0.358177	+	0.933654i	$0.383399\pi$
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	−0.987656	+	0.156637i	$0.949935\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	405.000	−	701.481i	0.451576	−	0.782152i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1371.78i			1.43591i	0.696088	+	0.717957i	$0.254922\pi$
$97$			1371.78i			1.43591i	−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$	500.000	−	866.025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	891.000	+	514.419i	0.852358	+	0.492109i	0.861446	−	0.507850i	$-0.169560\pi$
$103$	891.000	+	514.419i	0.852358	+	0.492109i	−0.00908799	+	0.999959i	$0.502893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	972.000	−	561.184i	0.866025	−	0.500000i
$109$	−323.000	−	559.452i	−0.283833	−	0.491613i	0.688493	−	0.725243i	$-0.258273\pi$
$109$	−323.000	−	559.452i	−0.283833	−	0.491613i	−0.972325	+	0.233630i	$0.924939\pi$
$110$		0			0
$111$			571.577i			0.488754i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1458.00	+	841.777i	1.15207	+	0.665148i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−665.500	+	1152.68i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	1080.00	+	623.538i	0.782152	+	0.451576i
$125$		0			0
$126$		0			0
$127$	380.000			0.265508			0.132754	−	0.991149i	$-0.457618\pi$
$127$	380.000			0.265508			0.132754	+	0.991149i	$0.457618\pi$
$128$		0			0
$129$	−2340.00	+	1351.00i	−1.59710	+	0.922084i
$130$		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
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			2026.50i			1.23659i	0.785948	+	0.618293i	$0.212175\pi$
$139$			2026.50i			1.23659i	−0.785948	+	0.618293i	$0.787825\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	864.000	+	1496.49i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−880.000			−0.488754
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	874.000	+	1513.81i	0.471027	+	0.815843i	0.999451	−	0.0331378i	$-0.0105500\pi$
$151$	874.000	+	1513.81i	0.471027	+	0.815843i	−0.528424	+	0.848981i	$0.677217\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1296.00	+	2244.74i	−0.665148	+	1.15207i
$157$	−702.000	+	405.300i	−0.356852	+	0.206028i	−0.667699	−	0.744432i	$-0.732721\pi$
$157$	−702.000	+	405.300i	−0.356852	+	0.206028i	0.310847	+	0.950460i	$0.399387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1700.00	−	2944.49i	0.816897	−	1.41491i	−0.0910600	−	0.995845i	$-0.529026\pi$
$163$	1700.00	−	2944.49i	0.816897	−	1.41491i	0.907957	−	0.419062i	$-0.137641\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1691.00			−0.769686
$170$		0			0
$171$	3645.00	−	2104.44i	1.63006	−	0.941115i
$172$	−2080.00	−	3602.67i	−0.922084	−	1.59710i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	3429.46i		−	1.40834i	−0.710031	−	0.704171i	$-0.751319\pi$
$181$		−	3429.46i		−	1.40834i	0.710031	−	0.704171i	$-0.248681\pi$
$182$		0			0
$183$	4860.00			1.96318
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−2304.00	+	1330.22i	−0.866025	+	0.500000i
$193$	−575.000	−	995.929i	−0.214453	−	0.371443i	0.738650	−	0.674089i	$-0.235464\pi$
$193$	−575.000	−	995.929i	−0.214453	−	0.371443i	−0.953103	+	0.302646i	$0.902130\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1755.00	−	1013.25i	0.625169	−	0.360942i	−0.153710	−	0.988116i	$-0.549122\pi$
$199$	1755.00	−	1013.25i	0.625169	−	0.360942i	0.778879	+	0.627175i	$0.215789\pi$
$200$		0			0
$201$	−3960.00	−	2286.31i	−1.38964	−	0.802307i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−3456.00	−	1995.32i	−1.15207	−	0.665148i
$209$		0			0
$210$		0			0
$211$	−6032.00			−1.96806			−0.984028	−	0.178011i	$-0.943034\pi$
$211$	−6032.00			−1.96806			−0.984028	+	0.178011i	$0.943034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	972.000	−	1683.55i	0.299916	−	0.519470i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	5830.08i		−	1.75072i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$		−	5830.08i		−	1.75072i	0.483469	−	0.875362i	$-0.339377\pi$
$224$		0			0
$225$	3375.00			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	3240.00	+	5611.84i	0.941115	+	1.63006i
$229$	−4590.00	−	2650.04i	−1.32452	−	0.764714i	−0.340076	−	0.940398i	$-0.610453\pi$
$229$	−4590.00	−	2650.04i	−1.32452	−	0.764714i	−0.984447	+	0.175684i	$0.943786\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	4593.40i		−	1.25896i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1080.00	+	623.538i	−0.288668	+	0.166662i	−0.637341	−	0.770582i	$-0.719966\pi$
$241$	−1080.00	+	623.538i	−0.288668	+	0.166662i	0.348673	+	0.937244i	$0.386632\pi$
$242$		0			0
$243$	3280.50	+	1894.00i	0.866025	+	0.500000i
$244$			7482.46i			1.96318i
$245$		0			0
$246$		0			0
$247$	−4860.00	+	8417.77i	−1.25196	+	2.16846i
$248$		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
$256$	−2048.00	−	3547.24i	−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
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	3520.00	−	6096.82i	0.802307	−	1.38964i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	7695.00	+	4442.71i	1.72486	+	0.995850i	0.907935	+	0.419111i	$0.137658\pi$
$271$	7695.00	+	4442.71i	1.72486	+	0.995850i	0.816928	+	0.576739i	$0.195675\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2015.00	+	3490.08i	0.437074	+	0.757035i	0.997462	−	0.0711951i	$-0.0226813\pi$
$277$	2015.00	+	3490.08i	0.437074	+	0.757035i	−0.560388	+	0.828230i	$0.689348\pi$
$278$		0			0
$279$			4208.88i			0.903151i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	6669.00	−	3850.35i	1.40082	−	0.808761i	0.406340	−	0.913722i	$-0.366805\pi$
$283$	6669.00	−	3850.35i	1.40082	−	0.808761i	0.994476	+	0.104961i	$0.0334717\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2456.50	−	4254.78i	0.500000	−	0.866025i
$290$		0			0
$291$	−3564.00	−	6173.03i	−0.717957	−	1.24354i
$292$	2592.00	+	1496.49i	0.519470	+	0.299916i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$			5196.15i			1.00000i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−8640.00	+	4988.31i	−1.63006	+	0.941115i
$305$		0			0
$306$		0			0
$307$			1590.02i			0.295594i	0.989018	+	0.147797i	$0.0472182\pi$
$307$			1590.02i			0.295594i	−0.989018	+	0.147797i	$0.952782\pi$
$308$		0			0
$309$	−5346.00			−0.984218
$310$		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
$313$	4104.00	+	2369.45i	0.741124	+	0.427888i	0.822478	−	0.568797i	$-0.192591\pi$
$313$	4104.00	+	2369.45i	0.741124	+	0.427888i	−0.0813539	+	0.996685i	$0.525924\pi$
$314$		0			0
$315$		0			0
$316$	7072.00			1.25896
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2916.00	+	5050.66i	−0.500000	+	0.866025i
$325$	−6750.00	+	3897.11i	−1.15207	+	0.665148i
$326$		0			0
$327$	2907.00	+	1678.36i	0.491613	+	0.283833i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	496.000	−	859.097i	0.0823644	−	0.142659i	−0.821901	−	0.569631i	$-0.807086\pi$
$331$	496.000	−	859.097i	0.0823644	−	0.142659i	0.904265	+	0.426971i	$0.140420\pi$
$332$		0			0
$333$	−1485.00	−	2572.10i	−0.244377	−	0.423273i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4930.00			0.796897			0.398448	−	0.917191i	$-0.369549\pi$
$337$	4930.00			0.796897			0.398448	+	0.917191i	$0.369549\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$			5300.08i			0.812913i	0.913670	+	0.406456i	$0.133236\pi$
$349$			5300.08i			0.812913i	−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$	−8748.00			−1.33030
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	8720.50	+	15104.3i	1.27140	+	2.20212i
$362$		0			0
$363$		−	6916.08i		−	1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11583.0	−	6687.45i	1.64749	−	0.951177i	0.669420	−	0.742884i	$-0.266543\pi$
$367$	11583.0	−	6687.45i	1.64749	−	0.951177i	0.978066	−	0.208293i	$-0.0667908\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6480.00			−0.903151
$373$	−6175.00	+	10695.4i	−0.857183	+	1.48469i	0.0174213	+	0.999848i	$0.494454\pi$
$373$	−6175.00	+	10695.4i	−0.857183	+	1.48469i	−0.874605	+	0.484837i	$0.838879\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8584.00			1.16340			0.581702	−	0.813402i	$-0.302387\pi$
$379$	8584.00			1.16340			0.581702	+	0.813402i	$0.302387\pi$
$380$		0			0
$381$	−1710.00	+	987.269i	−0.229937	+	0.132754i
$382$		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
$387$	7020.00	−	12159.0i	0.922084	−	1.59710i
$388$	9504.00	−	5487.14i	1.24354	−	0.717957i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13662.0	−	7887.76i	−1.72714	−	0.997167i	−0.901182	−	0.433441i	$-0.857299\pi$
$397$	−13662.0	−	7887.76i	−1.72714	−	0.997167i	−0.825962	−	0.563726i	$-0.809368\pi$
$398$		0			0
$399$		0			0
$400$	−8000.00			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−4860.00	−	8417.77i	−0.600729	−	1.04049i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12420.0	+	7170.69i	−1.50154	+	0.866914i	−0.501541	+	0.865134i	$0.667233\pi$
$409$	−12420.0	+	7170.69i	−1.50154	+	0.866914i	−0.999998	+	0.00177990i	$0.999433\pi$
$410$		0			0
$411$		0			0
$412$		−	8230.71i		−	0.984218i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5265.00	−	9119.25i	−0.618293	−	1.07091i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	17138.0			1.98398			0.991989	−	0.126322i	$-0.0403172\pi$
$421$	17138.0			1.98398			0.991989	+	0.126322i	$0.0403172\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−7776.00	−	4489.48i	−0.866025	−	0.500000i
$433$		−	17833.2i		−	1.97923i	−0.143727	−	0.989617i	$-0.545909\pi$
$433$		−	17833.2i		−	1.97923i	0.143727	−	0.989617i	$-0.454091\pi$
$434$		0			0
$435$		0			0
$436$	−2584.00	+	4475.62i	−0.283833	+	0.491613i
$437$		0			0
$438$		0			0
$439$	−9315.00	−	5378.02i	−1.01271	−	0.584690i	−0.100728	−	0.994914i	$-0.532117\pi$
$439$	−9315.00	−	5378.02i	−1.01271	−	0.584690i	−0.911985	+	0.410224i	$0.865450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	3960.00	−	2286.31i	0.423273	−	0.244377i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−7866.00	−	4541.44i	−0.815843	−	0.471027i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6355.00	−	11007.2i	0.650491	−	1.12668i	−0.332513	−	0.943099i	$-0.607897\pi$
$457$	6355.00	−	11007.2i	0.650491	−	1.12668i	0.983004	−	0.183585i	$-0.0587702\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−19780.0			−1.98543			−0.992716	−	0.120482i	$-0.961556\pi$
$463$	−19780.0			−1.98543			−0.992716	+	0.120482i	$0.961556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		−	13468.4i		−	1.33030i
$469$		0			0
$470$		0			0
$471$	2106.00	−	3647.70i	0.206028	−	0.356852i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			19485.6i			1.88223i
$476$		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
$481$	5940.00	+	3429.46i	0.563078	+	0.325093i
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	10450.0	+	18099.9i	0.972351	+	1.68416i	0.688415	+	0.725317i	$0.258307\pi$
$487$	10450.0	+	18099.9i	0.972351	+	1.68416i	0.283936	+	0.958843i	$0.408360\pi$
$488$		0			0
$489$			17666.9i			1.63379i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	9976.61i		−	0.903151i
$497$		0			0
$498$		0			0
$499$	−7568.00	+	13108.2i	−0.678938	+	1.17596i	0.296363	+	0.955075i	$0.404226\pi$
$499$	−7568.00	+	13108.2i	−0.678938	+	1.17596i	−0.975301	+	0.220880i	$0.929107\pi$
$500$		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
$507$	7609.50	−	4393.35i	0.666568	−	0.384843i
$508$	−1520.00	−	2632.72i	−0.132754	−	0.229937i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−10935.0	+	18940.0i	−0.941115	+	1.63006i
$514$		0			0
$515$		0			0
$516$	18720.0	+	10808.0i	1.59710	+	0.922084i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	17901.0	+	10335.1i	1.49667	+	0.864100i	0.999993	−	0.00383755i	$-0.00122153\pi$
$523$	17901.0	+	10335.1i	1.49667	+	0.864100i	0.496673	+	0.867938i	$0.334555\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6083.50	−	10536.9i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	11339.0	−	19639.7i	0.901112	−	1.56077i	0.0750596	−	0.997179i	$-0.476085\pi$
$541$	11339.0	−	19639.7i	0.901112	−	1.56077i	0.826053	−	0.563593i	$-0.190581\pi$
$542$		0			0
$543$	8910.00	+	15432.6i	0.704171	+	1.21966i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$	−21870.0	+	12626.7i	−1.70016	+	0.981589i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	14040.0	−	8106.00i	1.07091	−	0.618293i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			32424.0i			2.45329i
$560$		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		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	−0.877320	−	0.479905i	$-0.840671\pi$
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	0.0230498	−	0.999734i	$-0.492662\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	6912.00	−	11971.9i	0.500000	−	0.866025i
$577$	−18468.0	+	10662.5i	−1.33247	+	0.769300i	−0.985677	−	0.168644i	$-0.946061\pi$
$577$	−18468.0	+	10662.5i	−1.33247	+	0.769300i	−0.346789	+	0.937943i	$0.612728\pi$
$578$		0			0
$579$	5175.00	+	2987.79i	0.371443	+	0.214453i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−24300.0			−1.69994
$590$		0			0
$591$		0			0
$592$	3520.00	+	6096.82i	0.244377	+	0.423273i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5265.00	+	9119.25i	−0.360942	+	0.625169i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$			3117.69i			0.211603i	0.994387	+	0.105801i	$0.0337408\pi$
$601$			3117.69i			0.211603i	−0.994387	+	0.105801i	$0.966259\pi$
$602$		0			0
$603$	23760.0			1.60461
$604$	6992.00	−	12110.5i	0.471027	−	0.815843i
$605$		0			0
$606$		0			0
$607$	8073.00	+	4660.95i	0.539824	+	0.311667i	0.745007	−	0.667056i	$-0.232446\pi$
$607$	8073.00	+	4660.95i	0.539824	+	0.311667i	−0.205184	+	0.978723i	$0.565779\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	−0.996266	−	0.0863334i	$-0.972485\pi$
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	0.423366	−	0.905959i	$-0.360848\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−13365.0	+	7716.29i	−0.867827	+	0.501040i	−0.866625	−	0.498959i	$-0.833716\pi$
$619$	−13365.0	+	7716.29i	−0.867827	+	0.501040i	−0.00120126	+	0.999999i	$0.500382\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	20736.0			1.33030
$625$	−7812.50	+	13531.6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	5616.00	+	3242.40i	0.356852	+	0.206028i
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$	27144.0	−	15671.6i	1.70439	−	0.984028i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		−	29836.3i		−	1.82991i	−0.403561	−	0.914953i	$-0.632228\pi$
$643$		−	29836.3i		−	1.82991i	0.403561	−	0.914953i	$-0.367772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−27200.0			−1.63379
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			10101.3i			0.599833i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	23490.0	−	13562.0i	1.38223	−	0.798032i	0.389808	−	0.920896i	$-0.372541\pi$
$661$	23490.0	−	13562.0i	1.38223	−	0.798032i	0.992423	+	0.122864i	$0.0392080\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	15147.0	+	26235.4i	0.875362	+	1.51617i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24050.0			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$673$	−24050.0			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$674$		0			0
$675$	−15187.5	+	8768.51i	−0.866025	+	0.500000i
$676$	6764.00	+	11715.6i	0.384843	+	0.666568i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−29160.0	−	16835.5i	−1.63006	−	0.941115i
$685$		0			0
$686$		0			0
$687$	27540.0			1.52943
$688$	−16640.0	+	28821.3i	−0.922084	+	1.59710i
$689$		0			0
$690$		0			0
$691$	−28215.0	−	16289.9i	−1.55333	−	0.896814i	−0.997868	−	0.0652705i	$-0.979209\pi$
$691$	−28215.0	−	16289.9i	−1.55333	−	0.896814i	−0.555460	−	0.831543i	$-0.687458\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	14850.0	−	8573.65i	0.796698	−	0.459974i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	18073.0	−	31303.4i	0.957328	−	1.65814i	0.228381	−	0.973572i	$-0.426657\pi$
$709$	18073.0	−	31303.4i	0.957328	−	1.65814i	0.728948	−	0.684569i	$-0.240010\pi$
$710$		0			0
$711$	11934.0	+	20670.3i	0.629480	+	1.09029i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	3240.00	−	5611.84i	0.166662	−	0.288668i
$724$	−23760.0	+	13717.8i	−1.21966	+	0.704171i
$725$		0			0
$726$		0			0
$727$		−	37692.9i		−	1.92290i	−0.274971	−	0.961452i	$-0.588668\pi$
$727$		−	37692.9i		−	1.92290i	0.274971	−	0.961452i	$-0.411332\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$		0			0
$732$	−19440.0	−	33671.1i	−0.981589	−	1.70016i
$733$	−31806.0	−	18363.2i	−1.60270	−	0.925321i	−0.990944	−	0.134277i	$-0.957129\pi$
$733$	−31806.0	−	18363.2i	−1.60270	−	0.925321i	−0.611759	−	0.791044i	$-0.709538\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−15688.0	−	27172.4i	−0.780910	−	1.35258i	−0.931412	−	0.363966i	$-0.881422\pi$
$739$	−15688.0	−	27172.4i	−0.780910	−	1.35258i	0.150502	−	0.988610i	$-0.451911\pi$
$740$		0			0
$741$		−	50506.6i		−	2.50392i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11726.0	+	20310.0i	−0.569757	+	0.986849i	0.426832	+	0.904331i	$0.359630\pi$
$751$	−11726.0	+	20310.0i	−0.569757	+	0.986849i	−0.996590	+	0.0825179i	$0.973704\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	41470.0			1.99109			0.995543	−	0.0943039i	$-0.0300625\pi$
$757$	41470.0			1.99109			0.995543	+	0.0943039i	$0.0300625\pi$
$758$		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
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	18432.0	+	10641.7i	0.866025	+	0.500000i
$769$			42400.6i			1.98830i	0.107995	+	0.994151i	$0.465557\pi$
$769$			42400.6i			1.98830i	−0.107995	+	0.994151i	$0.534443\pi$
$770$		0			0
$771$		0			0
$772$	−4600.00	+	7967.43i	−0.214453	+	0.371443i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−16875.0	−	9742.79i	−0.782152	−	0.451576i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7047.00	−	4068.59i	0.319185	−	0.184281i	−0.331844	−	0.943334i	$-0.607671\pi$
$787$	7047.00	−	4068.59i	0.319185	−	0.184281i	0.651029	+	0.759053i	$0.274338\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	29160.0	−	50506.6i	1.30580	−	2.26172i
$794$		0			0
$795$		0			0
$796$	−14040.0	−	8106.00i	−0.625169	−	0.360942i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			36580.9i			1.60461i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	24162.1i		−	1.04617i	−0.852280	−	0.523087i	$-0.824780\pi$
$811$		−	24162.1i		−	1.04617i	0.852280	−	0.523087i	$-0.175220\pi$
$812$		0			0
$813$	−46170.0			−1.99170
$814$		0			0
$815$		0			0
$816$		0			0
$817$	70200.0	+	40530.0i	3.00610	+	1.73558i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	6110.00	+	10582.8i	0.258786	+	0.448231i	0.965917	−	0.258852i	$-0.0833441\pi$
$823$	6110.00	+	10582.8i	0.258786	+	0.448231i	−0.707131	+	0.707083i	$0.750011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	38610.0	−	22291.5i	1.61759	−	0.933915i	0.630047	−	0.776557i	$-0.283036\pi$
$829$	38610.0	−	22291.5i	1.61759	−	0.933915i	0.987542	−	0.157358i	$-0.0502976\pi$
$830$		0			0
$831$	−18135.0	−	10470.2i	−0.757035	−	0.437074i
$832$			31925.2i			1.33030i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−10935.0	−	18940.0i	−0.451576	−	0.782152i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	24128.0	+	41790.9i	0.984028	+	1.70439i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−20007.0	+	34653.1i	−0.808761	+	1.40082i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	17396.7i		−	0.698303i	−0.937066	−	0.349151i	$-0.886470\pi$
$853$		−	17396.7i		−	0.698303i	0.937066	−	0.349151i	$-0.113530\pi$
$854$		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
$859$	34155.0	+	19719.4i	1.35664	+	0.783256i	0.989169	−	0.146779i	$-0.0468906\pi$
$859$	34155.0	+	19719.4i	1.35664	+	0.783256i	0.367470	+	0.930035i	$0.380224\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			25528.7i			1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−47520.0	+	27435.7i	−1.84863	+	1.06731i
$872$		0			0
$873$	32076.0	+	18519.1i	1.24354	+	0.717957i
$874$		0			0
$875$		0			0
$876$	−15552.0			−0.599833
$877$	−25075.0	+	43431.2i	−0.965476	+	1.67225i	−0.257146	+	0.966373i	$0.582782\pi$
$877$	−25075.0	+	43431.2i	−0.965476	+	1.67225i	−0.708330	+	0.705881i	$0.750551\pi$
$878$		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
$883$	20680.0			0.788151			0.394076	−	0.919078i	$-0.371065\pi$
$883$	20680.0			0.788151			0.394076	+	0.919078i	$0.371065\pi$
$884$		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
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−40392.0	+	23320.3i	−1.51617	+	0.875362i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−13500.0	−	23382.7i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	22420.0	+	38832.6i	0.820776	+	1.42163i	0.905105	+	0.425188i	$0.139792\pi$
$907$	22420.0	+	38832.6i	0.820776	+	1.42163i	−0.0843291	+	0.996438i	$0.526875\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	25920.0	−	44894.8i	0.941115	−	1.63006i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			42400.6i			1.52943i
$917$		0			0
$918$		0			0
$919$	1378.00	−	2386.77i	0.0494625	−	0.0856715i	−0.840234	−	0.542224i	$-0.817583\pi$
$919$	1378.00	−	2386.77i	0.0494625	−	0.0856715i	0.889697	+	0.456552i	$0.150916\pi$
$920$		0			0
$921$	−4131.00	−	7155.10i	−0.147797	−	0.255992i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	13750.0			0.488754
$926$		0			0
$927$	24057.0	−	13889.3i	0.852358	−	0.492109i
$928$		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
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			14466.1i			0.504361i	0.967680	+	0.252181i	$0.0811477\pi$
$937$			14466.1i			0.504361i	−0.967680	+	0.252181i	$0.918852\pi$
$938$		0			0
$939$	−24624.0			−0.855776
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−31824.0	+	18373.6i	−1.09029	+	0.629480i
$949$	−11664.0	−	20202.6i	−0.398977	−	0.691049i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2745.50	+	4755.35i	−0.0921587	+	0.159624i
$962$		0			0
$963$		0			0
$964$	8640.00	+	4988.31i	0.288668	+	0.166662i
$965$		0			0
$966$		0			0
$967$	−50020.0			−1.66343			−0.831714	−	0.555204i	$-0.812640\pi$
$967$	−50020.0			−1.66343			−0.831714	+	0.555204i	$0.812640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		−	30304.0i		−	1.00000i
$973$		0			0
$974$		0			0
$975$	20250.0	−	35074.0i	0.665148	−	1.15207i
$976$	51840.0	−	29929.8i	1.70016	−	0.981589i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−17442.0			−0.567666
$982$		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
$988$	77760.0			2.50392
$989$		0			0
$990$		0			0
$991$	−22814.0	−	39515.0i	−0.731292	−	1.26664i	−0.956331	−	0.292286i	$-0.905584\pi$
$991$	−22814.0	−	39515.0i	−0.731292	−	1.26664i	0.225039	−	0.974350i	$-0.427749\pi$
$992$		0			0
$993$			5154.58i			0.164729i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	48438.0	−	27965.7i	1.53866	−	0.888347i	0.539745	−	0.841828i	$-0.318520\pi$
$997$	48438.0	−	27965.7i	1.53866	−	0.888347i	0.998917	−	0.0465191i	$-0.0148128\pi$
$998$		0			0
$999$	13365.0	+	7716.29i	0.423273	+	0.244377i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.g.a.68.1		2
3.2	odd	2	CM	147.4.g.a.68.1		2
7.2	even	3		21.4.c.a.20.1	✓	2
7.3	odd	6	inner	147.4.g.a.80.1		2
7.4	even	3		147.4.g.b.80.1		2
7.5	odd	6		21.4.c.a.20.2	yes	2
7.6	odd	2		147.4.g.b.68.1		2
21.2	odd	6		21.4.c.a.20.1	✓	2
21.5	even	6		21.4.c.a.20.2	yes	2
21.11	odd	6		147.4.g.b.80.1		2
21.17	even	6	inner	147.4.g.a.80.1		2
21.20	even	2		147.4.g.b.68.1		2
28.19	even	6		336.4.k.a.209.1		2
28.23	odd	6		336.4.k.a.209.2		2
84.23	even	6		336.4.k.a.209.2		2
84.47	odd	6		336.4.k.a.209.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.c.a.20.1	✓	2	7.2	even	3
21.4.c.a.20.1	✓	2	21.2	odd	6
21.4.c.a.20.2	yes	2	7.5	odd	6
21.4.c.a.20.2	yes	2	21.5	even	6
147.4.g.a.68.1		2	1.1	even	1	trivial
147.4.g.a.68.1		2	3.2	odd	2	CM
147.4.g.a.80.1		2	7.3	odd	6	inner
147.4.g.a.80.1		2	21.17	even	6	inner
147.4.g.b.68.1		2	7.6	odd	2
147.4.g.b.68.1		2	21.20	even	2
147.4.g.b.80.1		2	7.4	even	3
147.4.g.b.80.1		2	21.11	odd	6
336.4.k.a.209.1		2	28.19	even	6
336.4.k.a.209.1		2	84.47	odd	6
336.4.k.a.209.2		2	28.23	odd	6
336.4.k.a.209.2		2	84.23	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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +(36.0000 + 20.7846i) q^{12} +62.3538i q^{13} +(-32.0000 + 55.4256i) q^{16} +(135.000 + 77.9423i) q^{19} +(62.5000 + 108.253i) q^{25} +140.296i q^{27} +(-135.000 + 77.9423i) q^{31} -216.000 q^{36} +(55.0000 - 95.2628i) q^{37} +(-162.000 - 280.592i) q^{39} +520.000 q^{43} -332.554i q^{48} +(432.000 - 249.415i) q^{52} -810.000 q^{57} +(-810.000 - 467.654i) q^{61} +512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-324.000 + 187.061i) q^{73} +(-562.500 - 324.760i) q^{75} -1247.08i q^{76} +(-442.000 + 765.566i) q^{79} +(-364.500 - 631.333i) q^{81} +(405.000 - 701.481i) q^{93} +1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64}+ \cdots + 810 q^{93}+O(q^{100}) \) 2 * q - 9 * q^3 - 8 * q^4 + 27 * q^9 + 72 * q^12 - 64 * q^16 + 270 * q^19 + 125 * q^25 - 270 * q^31 - 432 * q^36 + 110 * q^37 - 324 * q^39 + 1040 * q^43 + 864 * q^52 - 1620 * q^57 - 1620 * q^61 + 1024 * q^64 + 880 * q^67 - 648 * q^73 - 1125 * q^75 - 884 * q^79 - 729 * q^81 + 810 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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