LMFDB - Embedded newform 441.4.p.b.215.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(80,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("441.80");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.p (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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			215.2
Root			$1.00781 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	441.215
Dual form			441.4.p.b.80.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.44168 - 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} +22.0220i q^{8} +O(q^{10})$	$q+(-1.44168 - 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} +22.0220i q^{8} +(25.4395 - 14.6875i) q^{11} +(-2.58497 + 4.47731i) q^{16} -48.9007 q^{22} +(157.350 + 90.8461i) q^{23} +(62.5000 + 108.253i) q^{25} -304.463i q^{29} +(160.026 - 92.3911i) q^{32} +(5.29150 - 9.16515i) q^{37} -534.442 q^{43} +(-133.017 - 76.7973i) q^{44} +(-151.232 - 261.942i) q^{46} -208.088i q^{50} +(665.895 - 384.454i) q^{53} +(-253.420 + 438.937i) q^{58} -266.248 q^{64} +(-370.000 - 640.859i) q^{67} -1178.70i q^{71} +(-15.2573 + 8.80879i) q^{74} +(692.000 - 1198.58i) q^{79} +(770.492 + 444.844i) q^{86} +(323.448 + 560.228i) q^{88} -950.024i q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{4}+O(q^{10}) $ 8 * q + 32 * q^4	$ 8 q + 32 q^{4} - 444 q^{16} - 1640 q^{22} + 500 q^{25} + 748 q^{46} - 260 q^{58} - 7104 q^{64} - 2960 q^{67} + 5536 q^{79} - 8260 q^{88}+O(q^{100}) $ 8 * q + 32 * q^4 - 444 * q^16 - 1640 * q^22 + 500 * q^25 + 748 * q^46 - 260 * q^58 - 7104 * q^64 - 2960 * q^67 + 5536 * q^79 - 8260 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

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 $

$2$	−1.44168	−	0.832353i	−0.509710	−	0.294281i	0.223005	−	0.974817i	$-0.428414\pi$
$2$	−1.44168	−	0.832353i	−0.509710	−	0.294281i	−0.732714	+	0.680536i	$0.761747\pi$
$3$		0			0
$3$		0			0
$4$	−2.61438	−	4.52824i	−0.326797	−	0.566030i
$5$		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
$8$			22.0220i			0.973243i
$9$		0			0
$10$		0			0
$11$	25.4395	−	14.6875i	0.697299	−	0.402586i	−0.109042	−	0.994037i	$-0.534778\pi$
$11$	25.4395	−	14.6875i	0.697299	−	0.402586i	0.806341	+	0.591451i	$0.201445\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−2.58497	+	4.47731i	−0.0403902	+	0.0699579i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$	−48.9007			−0.473894
$23$	157.350	+	90.8461i	1.42651	+	0.823597i	0.996844	−	0.0793904i	$-0.0252974\pi$
$23$	157.350	+	90.8461i	1.42651	+	0.823597i	0.429668	+	0.902987i	$0.358631\pi$
$24$		0			0
$25$	62.5000	+	108.253i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	304.463i		−	1.94956i	−0.223165	−	0.974781i	$-0.571639\pi$
$29$		−	304.463i		−	1.94956i	0.223165	−	0.974781i	$-0.428361\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	160.026	−	92.3911i	0.884028	−	0.510394i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.29150	−	9.16515i	0.0235113	−	0.0407227i	−0.854030	−	0.520223i	$-0.825849\pi$
$37$	5.29150	−	9.16515i	0.0235113	−	0.0407227i	0.877542	+	0.479500i	$0.159182\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−534.442			−1.89539			−0.947693	−	0.319183i	$-0.896592\pi$
$43$	−534.442			−1.89539			−0.947693	+	0.319183i	$0.896592\pi$
$44$	−133.017	−	76.7973i	−0.455751	−	0.263128i
$45$		0			0
$46$	−151.232	−	261.942i	−0.484738	−	0.839591i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$		−	208.088i		−	0.588562i
$51$		0			0
$52$		0			0
$53$	665.895	−	384.454i	1.72580	−	0.996394i	0.820478	−	0.571678i	$-0.193707\pi$
$53$	665.895	−	384.454i	1.72580	−	0.996394i	0.905327	−	0.424716i	$-0.139626\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−253.420	+	438.937i	−0.573719	+	0.993711i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−266.248			−0.520016
$65$		0			0
$66$		0			0
$67$	−370.000	−	640.859i	−0.674667	−	1.16856i	−0.976566	−	0.215218i	$-0.930954\pi$
$67$	−370.000	−	640.859i	−0.674667	−	1.16856i	0.301899	−	0.953340i	$-0.402379\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1178.70i		−	1.97022i	−0.171931	−	0.985109i	$-0.555000\pi$
$71$		−	1178.70i		−	1.97022i	0.171931	−	0.985109i	$-0.445000\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	−15.2573	+	8.80879i	−0.0239679	+	0.0138379i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	692.000	−	1198.58i	0.985520	−	1.70697i	0.345918	−	0.938265i	$-0.387568\pi$
$79$	692.000	−	1198.58i	0.985520	−	1.70697i	0.639602	−	0.768706i	$-0.279099\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	770.492	+	444.844i	0.966097	+	0.557776i
$87$		0			0
$88$	323.448	+	560.228i	0.391814	+	0.678642i
$89$		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
$92$		−	950.024i		−	1.07660i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	326.797	−	566.030i	0.326797	−	0.566030i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−1280.01			−1.17288
$107$	−18.1199	−	10.4615i	−0.0163712	−	0.00945193i	0.491792	−	0.870713i	$-0.336342\pi$
$107$	−18.1199	−	10.4615i	−0.0163712	−	0.00945193i	−0.508163	+	0.861261i	$0.669675\pi$
$108$		0			0
$109$	−1137.67	−	1970.51i	−0.999718	−	1.73156i	−0.520407	−	0.853919i	$-0.674220\pi$
$109$	−1137.67	−	1970.51i	−0.999718	−	1.73156i	−0.479312	−	0.877645i	$-0.659114\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1157.60i			0.963699i	0.876254	+	0.481849i	$0.160035\pi$
$113$			1157.60i			0.963699i	−0.876254	+	0.481849i	$0.839965\pi$
$114$		0			0
$115$		0			0
$116$	−1378.68	+	795.980i	−1.10351	+	0.637111i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−234.055	+	405.396i	−0.175849	+	0.304580i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2000.00			1.39741			0.698706	−	0.715409i	$-0.253760\pi$
$127$	2000.00			1.39741			0.698706	+	0.715409i	$0.253760\pi$
$128$	−896.365	−	517.517i	−0.618970	−	0.357363i
$129$		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
$133$		0			0
$134$			1231.88i			0.794167i
$135$		0			0
$136$		0			0
$137$	2384.05	−	1376.43i	1.48674	−	0.858370i	0.486854	−	0.873483i	$-0.338144\pi$
$137$	2384.05	−	1376.43i	1.48674	−	0.858370i	0.999886	+	0.0151133i	$0.00481091\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−981.091	+	1699.30i	−0.579798	+	1.00424i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−55.3360			−0.0307337
$149$	−1672.58	−	965.663i	−0.919616	−	0.530941i	−0.0361037	−	0.999348i	$-0.511495\pi$
$149$	−1672.58	−	965.663i	−0.919616	−	0.530941i	−0.883513	+	0.468407i	$0.844828\pi$
$150$		0			0
$151$	1124.44	+	1947.59i	0.606000	+	1.04962i	0.991893	+	0.127079i	$0.0405601\pi$
$151$	1124.44	+	1947.59i	0.606000	+	1.04962i	−0.385893	+	0.922544i	$0.626107\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−1995.28	+	1151.98i	−1.00466	+	0.580040i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	890.000	−	1541.53i	0.427670	−	0.740746i	−0.568996	−	0.822340i	$-0.692668\pi$
$163$	890.000	−	1541.53i	0.427670	−	0.740746i	0.996666	+	0.0815946i	$0.0260013\pi$
$164$		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
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$	1397.23	+	2420.08i	0.619407	+	1.07284i
$173$		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
$176$			151.867i			0.0650421i
$177$		0			0
$178$		0			0
$179$	−1364.72	+	787.920i	−0.569854	+	0.329005i	−0.757091	−	0.653310i	$-0.773380\pi$
$179$	−1364.72	+	787.920i	−0.569854	+	0.329005i	0.187237	+	0.982315i	$0.440047\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−2000.61	+	3465.16i	−0.801560	+	1.38834i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4551.22	+	2627.65i	1.72416	+	0.995444i	0.909755	+	0.415146i	$0.136270\pi$
$191$	4551.22	+	2627.65i	1.72416	+	0.995444i	0.814405	+	0.580298i	$0.197064\pi$
$192$		0			0
$193$	1386.37	+	2401.27i	0.517064	+	0.895581i	0.999804	+	0.0198172i	$0.00630844\pi$
$193$	1386.37	+	2401.27i	0.517064	+	0.895581i	−0.482740	+	0.875764i	$0.660358\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5147.21i			1.86154i	0.365603	+	0.930771i	$0.380863\pi$
$197$			5147.21i			1.86154i	−0.365603	+	0.930771i	$0.619137\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−2383.95	+	1376.37i	−0.842853	+	0.486622i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1772.65			−0.578363			−0.289181	−	0.957274i	$-0.593383\pi$
$211$	−1772.65			−0.578363			−0.289181	+	0.957274i	$0.593383\pi$
$212$	−3481.80	−	2010.22i	−1.12798	−	0.651238i
$213$		0			0
$214$	17.4154	+	30.1644i	0.00556305	+	0.00963548i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			3787.78i			1.17679i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	963.532	−	1668.89i	0.283598	−	0.491207i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	6704.87			1.89740
$233$	356.810	+	206.004i	0.100324	+	0.0579219i	0.549322	−	0.835611i	$-0.314886\pi$
$233$	356.810	+	206.004i	0.100324	+	0.0579219i	−0.448999	+	0.893532i	$0.648219\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5533.66i			1.49767i	0.662757	+	0.748834i	$0.269386\pi$
$239$			5533.66i			1.49767i	−0.662757	+	0.748834i	$0.730614\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	674.865	−	389.633i	0.179264	−	0.103498i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		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
$253$	5337.20			1.32627
$254$	−2883.35	−	1664.71i	−0.712274	−	0.411232i
$255$		0			0
$256$	1926.51	+	3336.81i	0.470338	+	0.814650i
$257$		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
$263$	2139.12	−	1235.02i	0.501535	−	0.289561i	−0.227812	−	0.973705i	$-0.573157\pi$
$263$	2139.12	−	1235.02i	0.501535	−	0.289561i	0.729347	+	0.684144i	$0.239824\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1934.64	+	3350.89i	−0.440959	+	0.763763i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	−4582.71			−1.01041
$275$	3179.93	+	1835.94i	0.697299	+	0.402586i
$276$		0			0
$277$	−3655.00	−	6330.65i	−0.792807	−	1.37318i	−0.924222	−	0.381855i	$-0.875285\pi$
$277$	−3655.00	−	6330.65i	−0.792807	−	1.37318i	0.131415	−	0.991327i	$-0.458048\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8982.08i		−	1.90685i	−0.301625	−	0.953427i	$-0.597529\pi$
$281$		−	8982.08i		−	1.90685i	0.301625	−	0.953427i	$-0.402471\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	−5337.41	+	3081.56i	−1.11520	+	0.643862i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2456.50	−	4254.78i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	201.835	+	116.529i	0.0396331	+	0.0228822i
$297$		0			0
$298$	1607.54	+	2784.35i	0.312492	+	0.541251i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		−	3743.74i		−	0.713337i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$	−7236.60			−1.28826
$317$	−1169.11	−	674.983i	−0.207140	−	0.119593i	0.392841	−	0.919606i	$-0.371492\pi$
$317$	−1169.11	−	674.983i	−0.207140	−	0.119593i	−0.599982	+	0.800014i	$0.704825\pi$
$318$		0			0
$319$	−4471.79	−	7745.37i	−0.784866	−	1.35943i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−2566.19	+	1481.59i	−0.435975	+	0.251710i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2553.15	−	4422.19i	0.423969	−	0.734336i	−0.572354	−	0.820006i	$-0.693970\pi$
$331$	2553.15	−	4422.19i	0.423969	−	0.734336i	0.996324	+	0.0856702i	$0.0273031\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11916.5			−1.92621			−0.963103	−	0.269135i	$-0.913262\pi$
$337$	−11916.5			−1.92621			−0.963103	+	0.269135i	$0.913262\pi$
$338$	−3167.36	−	1828.68i	−0.509710	−	0.294281i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		−	11769.5i		−	1.84467i
$345$		0			0
$346$		0			0
$347$	10018.7	−	5784.28i	1.54994	−	0.894860i	0.551798	−	0.833978i	$-0.313942\pi$
$347$	10018.7	−	5784.28i	1.54994	−	0.894860i	0.998145	−	0.0608819i	$-0.0193913\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	2713.99	−	4700.76i	0.410955	−	0.711794i
$353$		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
$358$	2623.31			0.387280
$359$	−1728.70	−	998.065i	−0.254143	−	0.146729i	0.367517	−	0.930017i	$-0.380208\pi$
$359$	−1728.70	−	998.065i	−0.254143	−	0.146729i	−0.621660	+	0.783287i	$0.713541\pi$
$360$		0			0
$361$	−3429.50	−	5940.07i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	−813.492	+	469.670i	−0.115234	+	0.0665305i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6985.00	+	12098.4i	−0.969624	+	1.67944i	−0.272980	+	0.962020i	$0.588009\pi$
$373$	−6985.00	+	12098.4i	−0.969624	+	1.67944i	−0.696643	+	0.717418i	$0.745324\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8704.52			1.17974			0.589870	−	0.807498i	$-0.299179\pi$
$379$	8704.52			1.17974			0.589870	+	0.807498i	$0.299179\pi$
$380$		0			0
$381$		0			0
$382$	−4374.26	−	7576.43i	−0.585881	−	1.01477i
$383$		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
$386$		−	4615.81i		−	0.608649i
$387$		0			0
$388$		0			0
$389$	13283.0	−	7668.95i	1.73130	−	0.999566i	0.850925	−	0.525287i	$-0.176042\pi$
$389$	13283.0	−	7668.95i	1.73130	−	0.999566i	0.880375	−	0.474279i	$-0.157291\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	4284.30	−	7420.62i	0.547817	−	0.948846i
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−646.243			−0.0807804
$401$	8807.35	+	5084.92i	1.09680	+	0.633239i	0.935379	−	0.353646i	$-0.115058\pi$
$401$	8807.35	+	5084.92i	1.09680	+	0.633239i	0.161423	+	0.986885i	$0.448392\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	310.876i		−	0.0378612i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−15262.0			−1.76680			−0.883402	−	0.468616i	$-0.844753\pi$
$421$	−15262.0			−1.76680			−0.883402	+	0.468616i	$0.844753\pi$
$422$	2555.59	+	1475.47i	0.294797	+	0.170201i
$423$		0			0
$424$	8466.45	+	14664.3i	0.969734	+	1.67963i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			109.402i			0.0123555i
$429$		0			0
$430$		0			0
$431$	4336.40	−	2503.62i	0.484633	−	0.279803i	−0.237712	−	0.971336i	$-0.576398\pi$
$431$	4336.40	−	2503.62i	0.484633	−	0.279803i	0.722345	+	0.691533i	$0.243064\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−5948.62	+	10303.3i	−0.653411	+	1.13174i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10402.5	+	6005.90i	1.11566	+	0.644129i	0.940290	−	0.340373i	$-0.110553\pi$
$443$	10402.5	+	6005.90i	1.11566	+	0.644129i	0.175373	+	0.984502i	$0.443887\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	15219.6i		−	1.59968i	−0.600213	−	0.799841i	$-0.704917\pi$
$449$		−	15219.6i		−	1.59968i	0.600213	−	0.799841i	$-0.295083\pi$
$450$		0			0
$451$		0			0
$452$	5241.89	−	3026.41i	0.545482	−	0.314934i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8910.89	−	15434.1i	0.912109	−	1.57982i	0.101030	−	0.994883i	$-0.467786\pi$
$457$	8910.89	−	15434.1i	0.912109	−	1.57982i	0.811079	−	0.584936i	$-0.198880\pi$
$458$		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
$463$	−8440.00			−0.847171			−0.423585	−	0.905856i	$-0.639229\pi$
$463$	−8440.00			−0.847171			−0.423585	+	0.905856i	$0.639229\pi$
$464$	1363.17	+	787.028i	0.136387	+	0.0787432i
$465$		0			0
$466$	−342.937	−	593.984i	−0.0340906	−	0.0590467i
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13595.9	+	7849.61i	−1.32165	+	0.763056i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	4605.96	−	7977.75i	0.440735	−	0.763376i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2447.64			0.229868
$485$		0			0
$486$		0			0
$487$	1648.30	+	2854.94i	0.153371	+	0.265647i	0.932465	−	0.361261i	$-0.117654\pi$
$487$	1648.30	+	2854.94i	0.153371	+	0.265647i	−0.779094	+	0.626908i	$0.784320\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8998.48i			0.827079i	0.910486	+	0.413540i	$0.135708\pi$
$491$			8998.48i			0.827079i	−0.910486	+	0.413540i	$0.864292\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10543.3	+	18261.6i	−0.945859	+	1.63828i	−0.191838	+	0.981427i	$0.561445\pi$
$499$	−10543.3	+	18261.6i	−0.945859	+	1.63828i	−0.754022	+	0.656850i	$0.771889\pi$
$500$		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
$506$	−7694.52	−	4442.44i	−0.676015	−	0.390297i
$507$		0			0
$508$	−5228.76	−	9056.47i	−0.456670	−	0.790976i
$509$		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
$512$			1866.13i			0.161079i
$513$		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		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	−4111.89			−0.340850
$527$		0			0
$528$		0			0
$529$	10422.5	+	18052.4i	0.856623	+	1.48371i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	14113.0	−	8148.13i	1.13729	−	0.656615i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7939.00	+	13750.8i	−0.630914	+	1.09277i	0.356452	+	0.934314i	$0.383986\pi$
$541$	−7939.00	+	13750.8i	−0.630914	+	1.09277i	−0.987365	+	0.158461i	$0.949347\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12980.0			1.01460			0.507299	−	0.861770i	$-0.330644\pi$
$547$	12980.0			1.01460			0.507299	+	0.861770i	$0.330644\pi$
$548$	−12465.6	−	7197.04i	−0.971725	−	0.561026i
$549$		0			0
$550$	−3056.29	−	5293.65i	−0.236947	−	0.410404i
$551$		0			0
$552$		0			0
$553$		0			0
$554$			12169.0i			0.933233i
$555$		0			0
$556$		0			0
$557$	−2431.79	+	1403.99i	−0.184988	+	0.106803i	−0.589634	−	0.807671i	$-0.700728\pi$
$557$	−2431.79	+	1403.99i	−0.184988	+	0.106803i	0.404646	+	0.914473i	$0.367395\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−7476.25	+	12949.3i	−0.561151	+	0.971942i
$563$		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
$568$	25957.2			1.91750
$569$	−14273.0	−	8240.54i	−1.05159	−	0.607138i	−0.128499	−	0.991710i	$-0.541016\pi$
$569$	−14273.0	−	8240.54i	−1.05159	−	0.607138i	−0.923095	+	0.384572i	$0.874349\pi$
$570$		0			0
$571$	−3394.00	−	5878.58i	−0.248747	−	0.430842i	0.714431	−	0.699705i	$-0.246685\pi$
$571$	−3394.00	−	5878.58i	−0.248747	−	0.430842i	−0.963178	+	0.268863i	$0.913352\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			22711.5i			1.64719i
$576$		0			0
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−7082.96	+	4089.35i	−0.509710	+	0.294281i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	11293.3	−	19560.6i	0.802268	−	1.38957i
$584$		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
$589$		0			0
$590$		0			0
$591$		0			0
$592$	27.3568	+	47.3834i	0.00189925	+	0.00328960i
$593$		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
$596$			10098.4i			0.694040i
$597$		0			0
$598$		0			0
$599$	−24787.7	+	14311.2i	−1.69082	+	0.976194i	−0.736960	+	0.675937i	$0.763739\pi$
$599$	−24787.7	+	14311.2i	−1.69082	+	0.976194i	−0.953858	+	0.300257i	$0.902927\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	5879.45	−	10183.5i	0.396078	−	0.686028i
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7505.00	+	12999.0i	0.494493	+	0.856487i	0.999980	−	0.00634752i	$-0.00202049\pi$
$613$	7505.00	+	12999.0i	0.494493	+	0.856487i	−0.505487	+	0.862834i	$0.668687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19836.0i		−	1.29428i	−0.762372	−	0.647139i	$-0.775965\pi$
$617$		−	19836.0i		−	1.29428i	0.762372	−	0.647139i	$-0.224035\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	+	13531.6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	26192.0			1.65244			0.826218	−	0.563351i	$-0.190488\pi$
$631$	26192.0			1.65244			0.826218	+	0.563351i	$0.190488\pi$
$632$	26395.1	+	15239.2i	1.66130	+	0.959151i
$633$		0			0
$634$	1123.65	+	1946.22i	0.0703877	+	0.121915i
$635$		0			0
$636$		0			0
$637$		0			0
$638$			14888.4i			0.923885i
$639$		0			0
$640$		0			0
$641$	−24554.8	+	14176.7i	−1.51304	+	0.873553i	−0.513154	+	0.858296i	$0.671523\pi$
$641$	−24554.8	+	14176.7i	−1.51304	+	0.873553i	−0.999884	+	0.0152566i	$0.995143\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		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
$652$	−9307.19			−0.559045
$653$	−28534.6	−	16474.5i	−1.71002	−	0.987282i	−0.934500	−	0.355964i	$-0.884153\pi$
$653$	−28534.6	−	16474.5i	−1.71002	−	0.987282i	−0.775524	−	0.631318i	$-0.782514\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			22614.9i			1.33680i	0.743803	+	0.668399i	$0.233020\pi$
$659$			22614.9i			1.33680i	−0.743803	+	0.668399i	$0.766980\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	−7361.64	+	4250.24i	−0.432203	+	0.249532i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	27659.2	−	47907.2i	1.60565	−	2.78107i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9609.37			0.550392			0.275196	−	0.961388i	$-0.411257\pi$
$673$	9609.37			0.550392			0.275196	+	0.961388i	$0.411257\pi$
$674$	17179.7	+	9918.70i	0.981806	+	0.566846i
$675$		0			0
$676$	−5743.79	−	9948.53i	−0.326797	−	0.566030i
$677$		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
$683$	−27406.2	+	15823.0i	−1.53539	+	0.886456i	−0.536286	+	0.844036i	$0.680173\pi$
$683$	−27406.2	+	15823.0i	−1.53539	+	0.886456i	−0.999100	+	0.0424197i	$0.986493\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1381.52	−	2392.86i	0.0765551	−	0.132597i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−19258.2			−1.05336
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			23110.9i			1.24520i	0.782539	+	0.622602i	$0.213924\pi$
$701$			23110.9i			1.24520i	−0.782539	+	0.622602i	$0.786076\pi$
$702$		0			0
$703$		0			0
$704$	−6773.22	+	3910.52i	−0.362607	+	0.209351i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17805.9	+	30840.7i	−0.943180	+	1.63364i	−0.183826	+	0.982959i	$0.558848\pi$
$709$	−17805.9	+	30840.7i	−0.943180	+	1.63364i	−0.759354	+	0.650677i	$0.774485\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	7135.78	+	4119.84i	0.372453	+	0.215036i
$717$		0			0
$718$	1661.48	+	2877.77i	0.0863593	+	0.149579i
$719$		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
$722$			11418.2i			0.588562i
$723$		0			0
$724$		0			0
$725$	32959.0	−	19028.9i	1.68837	−	0.974781i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$	33573.5			1.68143
$737$	−18825.2	−	10868.7i	−0.940889	−	0.543223i
$738$		0			0
$739$	12662.0	+	21931.2i	0.630283	+	1.09168i	0.987494	+	0.157658i	$0.0503945\pi$
$739$	12662.0	+	21931.2i	0.630283	+	1.09168i	−0.357211	+	0.934024i	$0.616272\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	4655.40i		−	0.229865i	−0.993373	−	0.114933i	$-0.963335\pi$
$743$		−	4655.40i		−	0.229865i	0.993373	−	0.114933i	$-0.0366652\pi$
$744$		0			0
$745$		0			0
$746$	20140.2	−	11628.0i	0.988453	−	0.570684i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−20544.3	+	35583.7i	−0.998230	+	1.72898i	−0.447610	+	0.894229i	$0.647725\pi$
$751$	−20544.3	+	35583.7i	−0.998230	+	1.72898i	−0.550620	+	0.834756i	$0.685609\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22848.7			1.09703			0.548514	−	0.836141i	$-0.315194\pi$
$757$	22848.7			1.09703			0.548514	+	0.836141i	$0.315194\pi$
$758$	−12549.1	−	7245.23i	−0.601325	−	0.347175i
$759$		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
$764$		−	27478.6i		−	1.30123i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	7249.01	−	12555.7i	0.337950	−	0.585347i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−25533.1			−1.17661
$779$		0			0
$780$		0			0
$781$	−17312.1	−	29985.4i	−0.793182	−	1.37383i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	23307.8	−	13456.8i	1.05369	−	0.608347i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	20003.3	+	11548.9i	0.884028	+	0.510394i
$801$		0			0
$802$	−8464.90	−	14661.6i	−0.372701	−	0.645537i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6413.48	+	3702.82i	−0.278722	+	0.160920i	−0.632845	−	0.774279i	$-0.718113\pi$
$809$	−6413.48	+	3702.82i	−0.278722	+	0.160920i	0.354123	+	0.935199i	$0.384779\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−258.758	+	448.182i	−0.0111418	+	0.0192982i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37581.5	−	21697.7i	−1.59757	−	0.922357i	−0.991954	−	0.126599i	$-0.959594\pi$
$821$	−37581.5	−	21697.7i	−1.59757	−	0.922357i	−0.605615	−	0.795758i	$-0.707073\pi$
$822$		0			0
$823$	23120.0	+	40045.0i	0.979238	+	1.69609i	0.665176	+	0.746687i	$0.268357\pi$
$823$	23120.0	+	40045.0i	0.979238	+	1.69609i	0.314062	+	0.949403i	$0.398310\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	46002.9i		−	1.93431i	−0.254179	−	0.967157i	$-0.581805\pi$
$827$		−	46002.9i		−	1.93431i	0.254179	−	0.967157i	$-0.418195\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−68308.5			−2.80079
$842$	22002.9	+	12703.4i	0.900557	+	0.519937i
$843$		0			0
$844$	4634.39	+	8026.99i	0.189007	+	0.327370i
$845$		0			0
$846$		0			0
$847$		0			0
$848$			3975.22i			0.160978i
$849$		0			0
$850$		0			0
$851$	1665.24	−	961.425i	0.0670782	−	0.0387276i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	230.384	−	399.037i	0.00919902	−	0.0159332i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$	−8335.58			−0.329363
$863$	40849.5	+	23584.5i	1.61128	+	0.930273i	0.989074	+	0.147419i	$0.0470966\pi$
$863$	40849.5	+	23584.5i	1.61128	+	0.930273i	0.622206	+	0.782854i	$0.286237\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	40655.0i		−	1.58703i
$870$		0			0
$871$		0			0
$872$	43394.5	−	25053.8i	1.68523	−	0.972969i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3275.00	−	5672.47i	0.126099	−	0.218410i	−0.796063	−	0.605214i	$-0.793088\pi$
$877$	3275.00	−	5672.47i	0.126099	−	0.218410i	0.922162	+	0.386804i	$0.126421\pi$
$878$		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
$883$	−43014.6			−1.63936			−0.819681	−	0.572820i	$-0.805850\pi$
$883$	−43014.6			−1.63936			−0.819681	+	0.572820i	$0.805850\pi$
$884$		0			0
$885$		0			0
$886$	−9998.06	−	17317.1i	−0.379110	−	0.656637i
$887$		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
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−12668.1	+	21941.7i	−0.470756	+	0.815373i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−25492.7			−0.937913
$905$		0			0
$906$		0			0
$907$	7125.01	+	12340.9i	0.260840	+	0.451788i	0.966465	−	0.256797i	$-0.0826671\pi$
$907$	7125.01	+	12340.9i	0.260840	+	0.451788i	−0.705625	+	0.708585i	$0.749334\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1065.41i			0.0387473i	0.999812	+	0.0193736i	$0.00616720\pi$
$911$			1065.41i			0.0387473i	−0.999812	+	0.0193736i	$0.993833\pi$
$912$		0			0
$913$		0			0
$914$	−25693.3	+	14834.0i	−0.929822	+	0.536833i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	25650.6	−	44428.1i	0.920711	−	1.59472i	0.122395	−	0.992482i	$-0.460943\pi$
$919$	25650.6	−	44428.1i	0.920711	−	1.59472i	0.798317	−	0.602238i	$-0.205724\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1322.88			0.0470226
$926$	12167.8	+	7025.06i	0.431811	+	0.249306i
$927$		0			0
$928$	−28129.6	−	48722.0i	−0.995044	−	1.72347i
$929$		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
$932$		−	2154.29i		−	0.0757149i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	26134.6			0.898211
$947$	49393.3	+	28517.3i	1.69490	+	0.978549i	0.950455	+	0.310862i	$0.100618\pi$
$947$	49393.3	+	28517.3i	1.69490	+	0.978549i	0.744442	+	0.667687i	$0.232715\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	56795.7i		−	1.93053i	−0.261276	−	0.965264i	$-0.584143\pi$
$953$		−	56795.7i		−	1.93053i	0.261276	−	0.965264i	$-0.415857\pi$
$954$		0			0
$955$		0			0
$956$	25057.7	−	14467.1i	0.847724	−	0.489434i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14895.5	+	25799.8i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52040.0			1.73060			0.865302	−	0.501251i	$-0.167127\pi$
$967$	52040.0			1.73060			0.865302	+	0.501251i	$0.167127\pi$
$968$	−8927.62	−	5154.36i	−0.296430	−	0.171144i
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$		0			0
$974$		−	5487.88i		−	0.180537i
$975$		0			0
$976$		0			0
$977$	52484.1	−	30301.7i	1.71864	−	0.992259i	0.797231	−	0.603674i	$-0.206297\pi$
$977$	52484.1	−	30301.7i	1.71864	−	0.992259i	0.921413	−	0.388585i	$-0.127036\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	7489.91	−	12972.9i	0.243394	−	0.421570i
$983$		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
$988$		0			0
$989$	−84094.5	−	48552.0i	−2.70379	−	1.56103i
$990$		0			0
$991$	12077.9	+	20919.5i	0.387150	+	0.670564i	0.992065	−	0.125727i	$-0.0401262\pi$
$991$	12077.9	+	20919.5i	0.387150	+	0.670564i	−0.604915	+	0.796290i	$0.706793\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	30400.1	−	17551.5i	0.964227	−	0.556697i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.p.b.215.2		8
3.2	odd	2	inner	441.4.p.b.215.3		8
7.2	even	3		63.4.c.a.62.3	yes	4
7.3	odd	6	inner	441.4.p.b.80.3		8
7.4	even	3	inner	441.4.p.b.80.3		8
7.5	odd	6		63.4.c.a.62.3	yes	4
7.6	odd	2	CM	441.4.p.b.215.2		8
21.2	odd	6		63.4.c.a.62.2	✓	4
21.5	even	6		63.4.c.a.62.2	✓	4
21.11	odd	6	inner	441.4.p.b.80.2		8
21.17	even	6	inner	441.4.p.b.80.2		8
21.20	even	2	inner	441.4.p.b.215.3		8
28.19	even	6		1008.4.k.a.881.1		4
28.23	odd	6		1008.4.k.a.881.1		4
84.23	even	6		1008.4.k.a.881.2		4
84.47	odd	6		1008.4.k.a.881.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.c.a.62.2	✓	4	21.2	odd	6
63.4.c.a.62.2	✓	4	21.5	even	6
63.4.c.a.62.3	yes	4	7.2	even	3
63.4.c.a.62.3	yes	4	7.5	odd	6
441.4.p.b.80.2		8	21.11	odd	6	inner
441.4.p.b.80.2		8	21.17	even	6	inner
441.4.p.b.80.3		8	7.3	odd	6	inner
441.4.p.b.80.3		8	7.4	even	3	inner
441.4.p.b.215.2		8	1.1	even	1	trivial
441.4.p.b.215.2		8	7.6	odd	2	CM
441.4.p.b.215.3		8	3.2	odd	2	inner
441.4.p.b.215.3		8	21.20	even	2	inner
1008.4.k.a.881.1		4	28.19	even	6
1008.4.k.a.881.1		4	28.23	odd	6
1008.4.k.a.881.2		4	84.23	even	6
1008.4.k.a.881.2		4	84.47	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.44168 - 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} +22.0220i q^{8} +O(q^{10})\)	\(q+(-1.44168 - 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} +22.0220i q^{8} +(25.4395 - 14.6875i) q^{11} +(-2.58497 + 4.47731i) q^{16} -48.9007 q^{22} +(157.350 + 90.8461i) q^{23} +(62.5000 + 108.253i) q^{25} -304.463i q^{29} +(160.026 - 92.3911i) q^{32} +(5.29150 - 9.16515i) q^{37} -534.442 q^{43} +(-133.017 - 76.7973i) q^{44} +(-151.232 - 261.942i) q^{46} -208.088i q^{50} +(665.895 - 384.454i) q^{53} +(-253.420 + 438.937i) q^{58} -266.248 q^{64} +(-370.000 - 640.859i) q^{67} -1178.70i q^{71} +(-15.2573 + 8.80879i) q^{74} +(692.000 - 1198.58i) q^{79} +(770.492 + 444.844i) q^{86} +(323.448 + 560.228i) q^{88} -950.024i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{4}+O(q^{10}) \) 8 * q + 32 * q^4	\( 8 q + 32 q^{4} - 444 q^{16} - 1640 q^{22} + 500 q^{25} + 748 q^{46} - 260 q^{58} - 7104 q^{64} - 2960 q^{67} + 5536 q^{79} - 8260 q^{88}+O(q^{100}) \) 8 * q + 32 * q^4 - 444 * q^16 - 1640 * q^22 + 500 * q^25 + 748 * q^46 - 260 * q^58 - 7104 * q^64 - 2960 * q^67 + 5536 * q^79 - 8260 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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