LMFDB - Embedded newform 81.3.d.a.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,3,Mod(26,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.26"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.20709014132$
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:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			53.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.53
Dual form			81.3.d.a.26.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{4} +(6.50000 + 11.2583i) q^{7} +(0.500000 - 0.866025i) q^{13} +(-8.00000 - 13.8564i) q^{16} +11.0000 q^{19} +(-12.5000 - 21.6506i) q^{25} -52.0000 q^{28} +(23.0000 - 39.8372i) q^{31} +47.0000 q^{37} +(11.0000 + 19.0526i) q^{43} +(-60.0000 + 103.923i) q^{49} +(2.00000 + 3.46410i) q^{52} +(60.5000 + 104.789i) q^{61} +64.0000 q^{64} +(54.5000 - 94.3968i) q^{67} -97.0000 q^{73} +(-22.0000 + 38.1051i) q^{76} +(-65.5000 - 113.449i) q^{79} +13.0000 q^{91} +(-83.5000 - 144.626i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 13 q^{7} + q^{13} - 16 q^{16} + 22 q^{19} - 25 q^{25} - 104 q^{28} + 46 q^{31} + 94 q^{37} + 22 q^{43} - 120 q^{49} + 4 q^{52} + 121 q^{61} + 128 q^{64} + 109 q^{67} - 194 q^{73} - 44 q^{76}+ \cdots - 167 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 13 * q^7 + q^13 - 16 * q^16 + 22 * q^19 - 25 * q^25 - 104 * q^28 + 46 * q^31 + 94 * q^37 + 22 * q^43 - 120 * q^49 + 4 * q^52 + 121 * q^61 + 128 * q^64 + 109 * q^67 - 194 * q^73 - 44 * q^76 - 131 * q^79 + 26 * q^91 - 167 * q^97

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

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	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	6.50000	+	11.2583i	0.928571	+	1.60833i	0.785714	+	0.618590i	$0.212296\pi$
$7$	6.50000	+	11.2583i	0.928571	+	1.60833i	0.142857	+	0.989743i	$0.454371\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.0384615	−	0.0666173i	−0.846154	−	0.532939i	$-0.821088\pi$
$13$	0.500000	−	0.866025i	0.0384615	−	0.0666173i	0.884615	+	0.466321i	$0.154421\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	11.0000			0.578947			0.289474	−	0.957186i	$-0.406520\pi$
$19$	11.0000			0.578947			0.289474	+	0.957186i	$0.406520\pi$
$20$		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
$25$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−52.0000			−1.85714
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	23.0000	−	39.8372i	0.741935	−	1.28507i	−0.209677	−	0.977771i	$-0.567241\pi$
$31$	23.0000	−	39.8372i	0.741935	−	1.28507i	0.951613	−	0.307299i	$-0.0994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	47.0000			1.27027			0.635135	−	0.772401i	$-0.280944\pi$
$37$	47.0000			1.27027			0.635135	+	0.772401i	$0.280944\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	11.0000	+	19.0526i	0.255814	+	0.443083i	0.965116	−	0.261822i	$-0.0843232\pi$
$43$	11.0000	+	19.0526i	0.255814	+	0.443083i	−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−60.0000	+	103.923i	−1.22449	+	2.12088i
$50$		0			0
$51$		0			0
$52$	2.00000	+	3.46410i	0.0384615	+	0.0666173i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	60.5000	+	104.789i	0.991803	+	1.71785i	0.606557	+	0.795040i	$0.292550\pi$
$61$	60.5000	+	104.789i	0.991803	+	1.71785i	0.385246	+	0.922814i	$0.374117\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$		0			0
$66$		0			0
$67$	54.5000	−	94.3968i	0.813433	−	1.40891i	−0.0970149	−	0.995283i	$-0.530929\pi$
$67$	54.5000	−	94.3968i	0.813433	−	1.40891i	0.910448	−	0.413624i	$-0.135737\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−97.0000			−1.32877			−0.664384	−	0.747392i	$-0.731306\pi$
$73$	−97.0000			−1.32877			−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$		0			0
$76$	−22.0000	+	38.1051i	−0.289474	+	0.501383i
$77$		0			0
$78$		0			0
$79$	−65.5000	−	113.449i	−0.829114	−	1.43607i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−65.5000	−	113.449i	−0.829114	−	1.43607i	0.0696203	−	0.997574i	$-0.477821\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	13.0000			0.142857
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−83.5000	−	144.626i	−0.860825	−	1.49099i	−0.871134	−	0.491045i	$-0.836615\pi$
$97$	−83.5000	−	144.626i	−0.860825	−	1.49099i	0.0103093	−	0.999947i	$-0.496718\pi$
$98$		0			0
$99$		0			0
$100$	100.000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	18.5000	−	32.0429i	0.179612	−	0.311097i	−0.762136	−	0.647417i	$-0.775849\pi$
$103$	18.5000	−	32.0429i	0.179612	−	0.311097i	0.941748	+	0.336321i	$0.109183\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−214.000			−1.96330			−0.981651	−	0.190684i	$-0.938929\pi$
$109$	−214.000			−1.96330			−0.981651	+	0.190684i	$0.938929\pi$
$110$		0			0
$111$		0			0
$112$	104.000	−	180.133i	0.928571	−	1.60833i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	+	104.789i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	92.0000	+	159.349i	0.741935	+	1.28507i
$125$		0			0
$126$		0			0
$127$	146.000			1.14961			0.574803	−	0.818292i	$-0.305079\pi$
$127$	146.000			1.14961			0.574803	+	0.818292i	$0.305079\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	71.5000	+	123.842i	0.537594	+	0.931140i
$134$		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
$139$	−125.500	+	217.372i	−0.902878	+	1.56383i	−0.0791367	+	0.996864i	$0.525216\pi$
$139$	−125.500	+	217.372i	−0.902878	+	1.56383i	−0.823741	+	0.566966i	$0.808117\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−94.0000	+	162.813i	−0.635135	+	1.10009i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−29.5000	−	51.0955i	−0.195364	−	0.338381i	0.751656	−	0.659556i	$-0.229256\pi$
$151$	−29.5000	−	51.0955i	−0.195364	−	0.338381i	−0.947020	+	0.321175i	$0.895922\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	299.000			1.83436			0.917178	−	0.398478i	$-0.130461\pi$
$163$	299.000			1.83436			0.917178	+	0.398478i	$0.130461\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	84.0000	+	145.492i	0.497041	+	0.860901i
$170$		0			0
$171$		0			0
$172$	−88.0000			−0.511628
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	162.500	−	281.458i	0.928571	−	1.60833i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−313.000			−1.72928			−0.864641	−	0.502390i	$-0.832454\pi$
$181$	−313.000			−1.72928			−0.864641	+	0.502390i	$0.832454\pi$
$182$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−71.5000	+	123.842i	−0.370466	+	0.641666i	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−71.5000	+	123.842i	−0.370466	+	0.641666i	0.619171	+	0.785256i	$0.287469\pi$
$194$		0			0
$195$		0			0
$196$	−240.000	−	415.692i	−1.22449	−	2.12088i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−277.000			−1.39196			−0.695980	−	0.718061i	$-0.745030\pi$
$199$	−277.000			−1.39196			−0.695980	+	0.718061i	$0.745030\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−16.0000			−0.0769231
$209$		0			0
$210$		0			0
$211$	126.500	−	219.104i	0.599526	−	1.03841i	−0.393365	−	0.919382i	$-0.628689\pi$
$211$	126.500	−	219.104i	0.599526	−	1.03841i	0.992891	−	0.119027i	$-0.0379776\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	598.000			2.75576
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−169.000	−	292.717i	−0.757848	−	1.31263i	−0.943946	−	0.330099i	$-0.892918\pi$
$223$	−169.000	−	292.717i	−0.757848	−	1.31263i	0.186099	−	0.982531i	$-0.440416\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−13.0000	+	22.5167i	−0.0567686	+	0.0983260i	−0.893013	−	0.450031i	$-0.851413\pi$
$229$	−13.0000	+	22.5167i	−0.0567686	+	0.0983260i	0.836245	+	0.548357i	$0.184746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	96.5000	+	167.143i	0.400415	+	0.693539i	0.993776	−	0.111397i	$-0.0355327\pi$
$241$	96.5000	+	167.143i	0.400415	+	0.693539i	−0.593361	+	0.804936i	$0.702199\pi$
$242$		0			0
$243$		0			0
$244$	−484.000			−1.98361
$245$		0			0
$246$		0			0
$247$	5.50000	−	9.52628i	0.0222672	−	0.0385679i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	305.500	+	529.142i	1.17954	+	2.04302i
$260$		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
$268$	218.000	+	377.587i	0.813433	+	1.40891i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	299.000			1.10332			0.551661	−	0.834069i	$-0.313994\pi$
$271$	299.000			1.10332			0.551661	+	0.834069i	$0.313994\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−61.0000	−	105.655i	−0.220217	−	0.381426i	0.734657	−	0.678439i	$-0.237343\pi$
$277$	−61.0000	−	105.655i	−0.220217	−	0.381426i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	−229.000	+	396.640i	−0.809187	+	1.40155i	0.104240	+	0.994552i	$0.466759\pi$
$283$	−229.000	+	396.640i	−0.809187	+	1.40155i	−0.913428	+	0.407001i	$0.866574\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$		0			0
$292$	194.000	−	336.018i	0.664384	−	1.15075i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−143.000	+	247.683i	−0.475083	+	0.822868i
$302$		0			0
$303$		0			0
$304$	−88.0000	−	152.420i	−0.289474	−	0.501383i
$305$		0			0
$306$		0			0
$307$	−358.000			−1.16612			−0.583062	−	0.812428i	$-0.698145\pi$
$307$	−358.000			−1.16612			−0.583062	+	0.812428i	$0.698145\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−299.500	−	518.749i	−0.956869	−	1.65735i	−0.730032	−	0.683413i	$-0.760495\pi$
$313$	−299.500	−	518.749i	−0.956869	−	1.65735i	−0.226837	−	0.973933i	$-0.572838\pi$
$314$		0			0
$315$		0			0
$316$	524.000			1.65823
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−25.0000			−0.0769231
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	330.500	+	572.443i	0.998489	+	1.72943i	0.546828	+	0.837245i	$0.315835\pi$
$331$	330.500	+	572.443i	0.998489	+	1.72943i	0.451662	+	0.892189i	$0.350831\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	324.500	−	562.050i	0.962908	−	1.66781i	0.247774	−	0.968818i	$-0.420301\pi$
$337$	324.500	−	562.050i	0.962908	−	1.66781i	0.715134	−	0.698988i	$-0.246366\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−923.000			−2.69096
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−335.500	−	581.103i	−0.961318	−	1.66505i	−0.719198	−	0.694805i	$-0.755490\pi$
$349$	−335.500	−	581.103i	−0.961318	−	1.66505i	−0.242120	−	0.970246i	$-0.577843\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−240.000			−0.664820
$362$		0			0
$363$		0			0
$364$	−26.0000	+	45.0333i	−0.0714286	+	0.123718i
$365$		0			0
$366$		0			0
$367$	−245.500	−	425.218i	−0.668937	−	1.15863i	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−245.500	−	425.218i	−0.668937	−	1.15863i	0.309264	−	0.950976i	$-0.399917\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	288.500	−	499.697i	0.773458	−	1.33967i	−0.162198	−	0.986758i	$-0.551858\pi$
$373$	288.500	−	499.697i	0.773458	−	1.33967i	0.935657	−	0.352911i	$-0.114808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	83.0000			0.218997			0.109499	−	0.993987i	$-0.465075\pi$
$379$	83.0000			0.218997			0.109499	+	0.993987i	$0.465075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	668.000			1.72165
$389$		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
$397$	362.000			0.911839			0.455919	−	0.890021i	$-0.349311\pi$
$397$	362.000			0.911839			0.455919	+	0.890021i	$0.349311\pi$
$398$		0			0
$399$		0			0
$400$	−200.000	+	346.410i	−0.500000	+	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−23.0000	−	39.8372i	−0.0570720	−	0.0988515i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−71.5000	+	123.842i	−0.174817	+	0.302791i	−0.940098	−	0.340905i	$-0.889267\pi$
$409$	−71.5000	+	123.842i	−0.174817	+	0.302791i	0.765281	+	0.643696i	$0.222600\pi$
$410$		0			0
$411$		0			0
$412$	74.0000	+	128.172i	0.179612	+	0.311097i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	240.500	+	416.558i	0.571259	+	0.989449i	0.996437	+	0.0843398i	$0.0268781\pi$
$421$	240.500	+	416.558i	0.571259	+	0.989449i	−0.425178	+	0.905110i	$0.639789\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−786.500	+	1362.26i	−1.84192	+	3.19030i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−862.000			−1.99076			−0.995381	−	0.0960028i	$-0.969394\pi$
$433$	−862.000			−1.99076			−0.995381	+	0.0960028i	$0.969394\pi$
$434$		0			0
$435$		0			0
$436$	428.000	−	741.318i	0.981651	−	1.70027i
$437$		0			0
$438$		0			0
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	0.914579	−	0.404408i	$-0.132522\pi$
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	−0.807517	+	0.589844i	$0.799189\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	416.000	+	720.533i	0.928571	+	1.60833i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	407.000	+	704.945i	0.890591	+	1.54255i	0.839168	+	0.543872i	$0.183042\pi$
$457$	407.000	+	704.945i	0.890591	+	1.54255i	0.0514223	+	0.998677i	$0.483625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	198.500	−	343.812i	0.428726	−	0.742575i	−0.568035	−	0.823005i	$-0.692296\pi$
$463$	198.500	−	343.812i	0.428726	−	0.742575i	0.996760	+	0.0804300i	$0.0256293\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	1417.00			3.02132
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−137.500	−	238.157i	−0.289474	−	0.501383i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	23.5000	−	40.7032i	0.0488565	−	0.0846220i
$482$		0			0
$483$		0			0
$484$	−242.000	−	419.156i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−349.000			−0.716632			−0.358316	−	0.933600i	$-0.616649\pi$
$487$	−349.000			−0.716632			−0.358316	+	0.933600i	$0.616649\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−736.000			−1.48387
$497$		0			0
$498$		0			0
$499$	−13.0000	+	22.5167i	−0.0260521	+	0.0451236i	−0.878758	−	0.477269i	$-0.841627\pi$
$499$	−13.0000	+	22.5167i	−0.0260521	+	0.0451236i	0.852705	+	0.522392i	$0.174960\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−292.000	+	505.759i	−0.574803	+	0.995588i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−630.500	−	1092.06i	−1.23386	−	2.13710i
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	803.000			1.53537			0.767686	−	0.640826i	$-0.221408\pi$
$523$	803.000			1.53537			0.767686	+	0.640826i	$0.221408\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−572.000			−1.07519
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−241.000			−0.445471			−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000			−0.445471			−0.222736	+	0.974879i	$0.571499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	546.500	+	946.566i	0.999086	+	1.73047i	0.536563	+	0.843860i	$0.319722\pi$
$547$	546.500	+	946.566i	0.999086	+	1.73047i	0.462523	+	0.886607i	$0.346944\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	851.500	−	1474.84i	1.53978	−	2.66698i
$554$		0			0
$555$		0			0
$556$	−502.000	−	869.490i	−0.902878	−	1.56383i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	22.0000			0.0393560
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	90.5000	−	156.751i	0.158494	−	0.274519i	−0.775832	−	0.630940i	$-0.782670\pi$
$571$	90.5000	−	156.751i	0.158494	−	0.274519i	0.934326	+	0.356420i	$0.116003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1033.00			−1.79029			−0.895147	−	0.445770i	$-0.852930\pi$
$577$	−1033.00			−1.79029			−0.895147	+	0.445770i	$0.852930\pi$
$578$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	253.000	−	438.209i	0.429542	−	0.743988i
$590$		0			0
$591$		0			0
$592$	−376.000	−	651.251i	−0.635135	−	1.10009i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	263.000	+	455.529i	0.437604	+	0.757952i	0.997504	−	0.0706077i	$-0.0224939\pi$
$601$	263.000	+	455.529i	0.437604	+	0.757952i	−0.559900	+	0.828560i	$0.689161\pi$
$602$		0			0
$603$		0			0
$604$	236.000			0.390728
$605$		0			0
$606$		0			0
$607$	−593.500	+	1027.97i	−0.977759	+	1.69353i	−0.307249	+	0.951629i	$0.599408\pi$
$607$	−593.500	+	1027.97i	−0.977759	+	1.69353i	−0.670511	+	0.741900i	$0.733925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	983.000			1.60359			0.801794	−	0.597600i	$-0.203879\pi$
$613$	983.000			1.60359			0.801794	+	0.597600i	$0.203879\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	474.500	+	821.858i	0.766559	+	1.32772i	0.939418	+	0.342773i	$0.111366\pi$
$619$	474.500	+	821.858i	0.766559	+	1.32772i	−0.172859	+	0.984947i	$0.555301\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	236.000	+	408.764i	0.375796	+	0.650898i
$629$		0			0
$630$		0			0
$631$	587.000			0.930269			0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000			0.930269			0.465135	+	0.885240i	$0.346006\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	60.0000	+	103.923i	0.0941915	+	0.163145i
$638$		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
$643$	−157.000	+	271.932i	−0.244168	+	0.422911i	−0.961897	−	0.273411i	$-0.911848\pi$
$643$	−157.000	+	271.932i	−0.244168	+	0.422911i	0.717729	+	0.696322i	$0.245181\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−598.000	+	1035.77i	−0.917178	+	1.58860i
$653$		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
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−539.500	+	934.441i	−0.816188	+	1.41368i	0.0922844	+	0.995733i	$0.470583\pi$
$661$	−539.500	+	934.441i	−0.816188	+	1.41368i	−0.908472	+	0.417946i	$0.862750\pi$
$662$		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
$673$	−11.5000	−	19.9186i	−0.0170877	−	0.0295967i	0.857355	−	0.514725i	$-0.172106\pi$
$673$	−11.5000	−	19.9186i	−0.0170877	−	0.0295967i	−0.874443	+	0.485129i	$0.838773\pi$
$674$		0			0
$675$		0			0
$676$	−672.000			−0.994083
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	1085.50	−	1880.14i	1.59867	−	2.76899i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	176.000	−	304.841i	0.255814	−	0.443083i
$689$		0			0
$690$		0			0
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.737337	+	0.675525i	$0.236083\pi$
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.216353	+	0.976315i	$0.430584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	650.000	+	1125.83i	0.928571	+	1.60833i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	517.000			0.735420
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−695.500	−	1204.64i	−0.980959	−	1.69907i	−0.658674	−	0.752428i	$-0.728882\pi$
$709$	−695.500	−	1204.64i	−0.980959	−	1.69907i	−0.322285	−	0.946643i	$-0.604451\pi$
$710$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	481.000			0.667129
$722$		0			0
$723$		0			0
$724$	626.000	−	1084.26i	0.864641	−	1.49760i
$725$		0			0
$726$		0			0
$727$	−241.000	−	417.424i	−0.331499	−	0.574174i	0.651307	−	0.758815i	$-0.274221\pi$
$727$	−241.000	−	417.424i	−0.331499	−	0.574174i	−0.982806	+	0.184641i	$0.940888\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−517.000	+	895.470i	−0.705321	+	1.22165i	0.261255	+	0.965270i	$0.415864\pi$
$733$	−517.000	+	895.470i	−0.705321	+	1.22165i	−0.966576	+	0.256381i	$0.917470\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1222.00			−1.65359			−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−1222.00			−1.65359			−0.826793	+	0.562506i	$0.809837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−89.5000	+	155.019i	−0.119174	+	0.206416i	−0.919441	−	0.393229i	$-0.871358\pi$
$751$	−89.5000	+	155.019i	−0.119174	+	0.206416i	0.800266	+	0.599645i	$0.204691\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−673.000			−0.889036			−0.444518	−	0.895770i	$-0.646625\pi$
$757$	−673.000			−0.889036			−0.444518	+	0.895770i	$0.646625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−1391.00	−	2409.28i	−1.82307	−	3.15764i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−431.500	+	747.380i	−0.561118	+	0.971885i	0.436281	+	0.899811i	$0.356295\pi$
$769$	−431.500	+	747.380i	−0.561118	+	0.971885i	−0.997399	+	0.0720749i	$0.977038\pi$
$770$		0			0
$771$		0			0
$772$	−286.000	−	495.367i	−0.370466	−	0.641666i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−1150.00			−1.48387
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1920.00			2.44898
$785$		0			0
$786$		0			0
$787$	306.500	−	530.874i	0.389454	−	0.674553i	−0.602922	−	0.797800i	$-0.705997\pi$
$787$	306.500	−	530.874i	0.389454	−	0.674553i	0.992376	+	0.123246i	$0.0393305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	121.000			0.152585
$794$		0			0
$795$		0			0
$796$	554.000	−	959.556i	0.695980	−	1.20547i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		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		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1514.00			1.86683			0.933416	−	0.358797i	$-0.116813\pi$
$811$	1514.00			1.86683			0.933416	+	0.358797i	$0.116813\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	121.000	+	209.578i	0.148103	+	0.256522i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	810.500	−	1403.83i	0.984812	−	1.70574i	0.342041	−	0.939685i	$-0.388882\pi$
$823$	810.500	−	1403.83i	0.984812	−	1.70574i	0.642770	−	0.766059i	$-0.277785\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−1609.00			−1.94089			−0.970446	−	0.241317i	$-0.922421\pi$
$829$	−1609.00			−1.94089			−0.970446	+	0.241317i	$0.922421\pi$
$830$		0			0
$831$		0			0
$832$	32.0000	−	55.4256i	0.0384615	−	0.0666173i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	506.000	+	876.418i	0.599526	+	1.03841i
$845$		0			0
$846$		0			0
$847$	−1573.00			−1.85714
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	240.500	+	416.558i	0.281946	+	0.488345i	0.971864	−	0.235543i	$-0.0756867\pi$
$853$	240.500	+	416.558i	0.281946	+	0.488345i	−0.689918	+	0.723888i	$0.742353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	774.500	−	1341.47i	0.901630	−	1.56167i	0.0762515	−	0.997089i	$-0.475705\pi$
$859$	774.500	−	1341.47i	0.901630	−	1.56167i	0.825378	−	0.564580i	$-0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−1196.00	+	2071.53i	−1.37788	+	2.38656i
$869$		0			0
$870$		0			0
$871$	−54.5000	−	94.3968i	−0.0625718	−	0.108377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−863.500	+	1495.63i	−0.984607	+	1.70539i	−0.340935	+	0.940087i	$0.610744\pi$
$877$	−863.500	+	1495.63i	−0.984607	+	1.70539i	−0.643672	+	0.765302i	$0.722590\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	443.000			0.501699			0.250849	−	0.968026i	$-0.419290\pi$
$883$	443.000			0.501699			0.250849	+	0.968026i	$0.419290\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	949.000	+	1643.72i	1.06749	+	1.84895i
$890$		0			0
$891$		0			0
$892$	1352.00			1.51570
$893$		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
$907$	726.500	+	1258.33i	0.800992	+	1.38736i	0.918964	+	0.394342i	$0.129028\pi$
$907$	726.500	+	1258.33i	0.800992	+	1.38736i	−0.117971	+	0.993017i	$0.537639\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−52.0000	−	90.0666i	−0.0567686	−	0.0983260i
$917$		0			0
$918$		0			0
$919$	866.000			0.942329			0.471164	−	0.882045i	$-0.343834\pi$
$919$	866.000			0.942329			0.471164	+	0.882045i	$0.343834\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−587.500	−	1017.58i	−0.635135	−	1.10009i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−660.000	+	1143.15i	−0.708915	+	1.22788i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1847.00			1.97118			0.985592	−	0.169138i	$-0.0540985\pi$
$937$	1847.00			1.97118			0.985592	+	0.169138i	$0.0540985\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−48.5000	+	84.0045i	−0.0511064	+	0.0885189i
$950$		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
$961$	−577.500	−	1000.26i	−0.600937	−	1.04085i
$962$		0			0
$963$		0			0
$964$	−772.000			−0.800830
$965$		0			0
$966$		0			0
$967$	126.500	−	219.104i	0.130817	−	0.226582i	−0.793175	−	0.608994i	$-0.791573\pi$
$967$	126.500	−	219.104i	0.130817	−	0.226582i	0.923992	+	0.382412i	$0.124907\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−3263.00			−3.35355
$974$		0			0
$975$		0			0
$976$	968.000	−	1676.63i	0.991803	−	1.71785i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	22.0000	+	38.1051i	0.0222672	+	0.0385679i
$989$		0			0
$990$		0			0
$991$	1739.00			1.75479			0.877397	−	0.479766i	$-0.159278\pi$
$991$	1739.00			1.75479			0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	947.000	+	1640.25i	0.949850	+	1.64519i	0.745737	+	0.666240i	$0.232097\pi$
$997$	947.000	+	1640.25i	0.949850	+	1.64519i	0.204112	+	0.978947i	$0.434569\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.d.a.53.1		2
3.2	odd	2	CM	81.3.d.a.53.1		2
4.3	odd	2		1296.3.q.a.1025.1		2
9.2	odd	6	inner	81.3.d.a.26.1		2
9.4	even	3		27.3.b.a.26.1	✓	1
9.5	odd	6		27.3.b.a.26.1	✓	1
9.7	even	3	inner	81.3.d.a.26.1		2
12.11	even	2		1296.3.q.a.1025.1		2
36.7	odd	6		1296.3.q.a.593.1		2
36.11	even	6		1296.3.q.a.593.1		2
36.23	even	6		432.3.e.b.161.1		1
36.31	odd	6		432.3.e.b.161.1		1
45.4	even	6		675.3.c.c.26.1		1
45.13	odd	12		675.3.d.c.674.2		2
45.14	odd	6		675.3.c.c.26.1		1
45.22	odd	12		675.3.d.c.674.1		2
45.23	even	12		675.3.d.c.674.2		2
45.32	even	12		675.3.d.c.674.1		2
72.5	odd	6		1728.3.e.a.1025.1		1
72.13	even	6		1728.3.e.a.1025.1		1
72.59	even	6		1728.3.e.d.1025.1		1
72.67	odd	6		1728.3.e.d.1025.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.a.26.1	✓	1	9.4	even	3
27.3.b.a.26.1	✓	1	9.5	odd	6
81.3.d.a.26.1		2	9.2	odd	6	inner
81.3.d.a.26.1		2	9.7	even	3	inner
81.3.d.a.53.1		2	1.1	even	1	trivial
81.3.d.a.53.1		2	3.2	odd	2	CM
432.3.e.b.161.1		1	36.23	even	6
432.3.e.b.161.1		1	36.31	odd	6
675.3.c.c.26.1		1	45.4	even	6
675.3.c.c.26.1		1	45.14	odd	6
675.3.d.c.674.1		2	45.22	odd	12
675.3.d.c.674.1		2	45.32	even	12
675.3.d.c.674.2		2	45.13	odd	12
675.3.d.c.674.2		2	45.23	even	12
1296.3.q.a.593.1		2	36.7	odd	6
1296.3.q.a.593.1		2	36.11	even	6
1296.3.q.a.1025.1		2	4.3	odd	2
1296.3.q.a.1025.1		2	12.11	even	2
1728.3.e.a.1025.1		1	72.5	odd	6
1728.3.e.a.1025.1		1	72.13	even	6
1728.3.e.d.1025.1		1	72.59	even	6
1728.3.e.d.1025.1		1	72.67	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{4} +(6.50000 + 11.2583i) q^{7} +(0.500000 - 0.866025i) q^{13} +(-8.00000 - 13.8564i) q^{16} +11.0000 q^{19} +(-12.5000 - 21.6506i) q^{25} -52.0000 q^{28} +(23.0000 - 39.8372i) q^{31} +47.0000 q^{37} +(11.0000 + 19.0526i) q^{43} +(-60.0000 + 103.923i) q^{49} +(2.00000 + 3.46410i) q^{52} +(60.5000 + 104.789i) q^{61} +64.0000 q^{64} +(54.5000 - 94.3968i) q^{67} -97.0000 q^{73} +(-22.0000 + 38.1051i) q^{76} +(-65.5000 - 113.449i) q^{79} +13.0000 q^{91} +(-83.5000 - 144.626i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 13 q^{7} + q^{13} - 16 q^{16} + 22 q^{19} - 25 q^{25} - 104 q^{28} + 46 q^{31} + 94 q^{37} + 22 q^{43} - 120 q^{49} + 4 q^{52} + 121 q^{61} + 128 q^{64} + 109 q^{67} - 194 q^{73} - 44 q^{76}+ \cdots - 167 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 13 * q^7 + q^13 - 16 * q^16 + 22 * q^19 - 25 * q^25 - 104 * q^28 + 46 * q^31 + 94 * q^37 + 22 * q^43 - 120 * q^49 + 4 * q^52 + 121 * q^61 + 128 * q^64 + 109 * q^67 - 194 * q^73 - 44 * q^76 - 131 * q^79 + 26 * q^91 - 167 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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