LMFDB - Embedded newform 441.4.e.a.226.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(226,441)]

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([0, 2]))

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

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.0198423125$
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 49)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			226.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.226
Dual form			441.4.e.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{2} +(-8.50000 - 14.7224i) q^{4} +45.0000 q^{8} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{2} +(-8.50000 - 14.7224i) q^{4} +45.0000 q^{8} +(-34.0000 - 58.8897i) q^{11} +(-44.5000 + 77.0763i) q^{16} +340.000 q^{22} +(-20.0000 + 34.6410i) q^{23} +(62.5000 + 108.253i) q^{25} +166.000 q^{29} +(-42.5000 - 73.6122i) q^{32} +(-225.000 + 389.711i) q^{37} -180.000 q^{43} +(-578.000 + 1001.13i) q^{44} +(-100.000 - 173.205i) q^{46} -625.000 q^{50} +(295.000 + 510.955i) q^{53} +(-415.000 + 718.801i) q^{58} -287.000 q^{64} +(370.000 + 640.859i) q^{67} -688.000 q^{71} +(-1125.00 - 1948.56i) q^{74} +(692.000 - 1198.58i) q^{79} +(450.000 - 779.423i) q^{86} +(-1530.00 - 2650.04i) q^{88} +680.000 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} - 17 q^{4} + 90 q^{8}+O(q^{10}) $ 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8	$ 2 q - 5 q^{2} - 17 q^{4} + 90 q^{8} - 68 q^{11} - 89 q^{16} + 680 q^{22} - 40 q^{23} + 125 q^{25} + 332 q^{29} - 85 q^{32} - 450 q^{37} - 360 q^{43} - 1156 q^{44} - 200 q^{46} - 1250 q^{50} + 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} - 1376 q^{71} - 2250 q^{74} + 1384 q^{79} + 900 q^{86} - 3060 q^{88} + 1360 q^{92}+O(q^{100}) $ 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8 - 68 * q^11 - 89 * q^16 + 680 * q^22 - 40 * q^23 + 125 * q^25 + 332 * q^29 - 85 * q^32 - 450 * q^37 - 360 * q^43 - 1156 * q^44 - 200 * q^46 - 1250 * q^50 + 590 * q^53 - 830 * q^58 - 574 * q^64 + 740 * q^67 - 1376 * q^71 - 2250 * q^74 + 1384 * q^79 + 900 * q^86 - 3060 * q^88 + 1360 * q^92

Display 100 coefficients 10 coefficients

Character values

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

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−2.50000	+	4.33013i	−0.883883	+	1.53093i	−0.0368954	+	0.999319i	$0.511747\pi$
$2$	−2.50000	+	4.33013i	−0.883883	+	1.53093i	−0.846988	+	0.531612i	$0.821586\pi$
$3$		0			0
$3$		0			0
$4$	−8.50000	−	14.7224i	−1.06250	−	1.84030i
$5$		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
$8$	45.0000			1.98874
$9$		0			0
$10$		0			0
$11$	−34.0000	−	58.8897i	−0.931944	−	1.61417i	−0.779994	−	0.625786i	$-0.784778\pi$
$11$	−34.0000	−	58.8897i	−0.931944	−	1.61417i	−0.151950	−	0.988388i	$-0.548555\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−44.5000	+	77.0763i	−0.695312	+	1.20432i
$17$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$	340.000			3.29492
$23$	−20.0000	+	34.6410i	−0.181317	+	0.314050i	−0.942329	−	0.334687i	$-0.891369\pi$
$23$	−20.0000	+	34.6410i	−0.181317	+	0.314050i	0.761012	+	0.648737i	$0.224703\pi$
$24$		0			0
$25$	62.5000	+	108.253i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	166.000			1.06295			0.531473	−	0.847075i	$-0.321639\pi$
$29$	166.000			1.06295			0.531473	+	0.847075i	$0.321639\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−42.5000	−	73.6122i	−0.234782	−	0.406654i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−225.000	+	389.711i	−0.999724	+	1.73157i	−0.479500	+	0.877542i	$0.659182\pi$
$37$	−225.000	+	389.711i	−0.999724	+	1.73157i	−0.520223	+	0.854030i	$0.674151\pi$
$38$		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
$43$	−180.000			−0.638366			−0.319183	−	0.947693i	$-0.603408\pi$
$43$	−180.000			−0.638366			−0.319183	+	0.947693i	$0.603408\pi$
$44$	−578.000	+	1001.13i	−1.98038	+	3.43012i
$45$		0			0
$46$	−100.000	−	173.205i	−0.320526	−	0.555167i
$47$		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
$50$	−625.000			−1.76777
$51$		0			0
$52$		0			0
$53$	295.000	+	510.955i	0.764554	+	1.32425i	0.940482	+	0.339843i	$0.110374\pi$
$53$	295.000	+	510.955i	0.764554	+	1.32425i	−0.175928	+	0.984403i	$0.556293\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−415.000	+	718.801i	−0.939520	+	1.62730i
$59$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	−287.000			−0.560547
$65$		0			0
$66$		0			0
$67$	370.000	+	640.859i	0.674667	+	1.16856i	0.976566	+	0.215218i	$0.0690461\pi$
$67$	370.000	+	640.859i	0.674667	+	1.16856i	−0.301899	+	0.953340i	$0.597621\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−688.000			−1.15001			−0.575004	−	0.818151i	$-0.695000\pi$
$71$	−688.000			−1.15001			−0.575004	+	0.818151i	$0.695000\pi$
$72$		0			0
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$	−1125.00	−	1948.56i	−1.76728	−	3.06102i
$75$		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
$86$	450.000	−	779.423i	0.564241	−	0.977295i
$87$		0			0
$88$	−1530.00	−	2650.04i	−1.85339	−	3.21017i
$89$		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
$92$	680.000			0.770597
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	1062.50	−	1840.30i	1.06250	−	1.84030i
$101$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	−2950.00			−2.70311
$107$	−790.000	+	1368.32i	−0.713759	+	1.23627i	0.249678	+	0.968329i	$0.419675\pi$
$107$	−790.000	+	1368.32i	−0.713759	+	1.23627i	−0.963436	+	0.267937i	$0.913658\pi$
$108$		0			0
$109$	27.0000	+	46.7654i	0.0237260	+	0.0410946i	0.877645	−	0.479312i	$-0.159114\pi$
$109$	27.0000	+	46.7654i	0.0237260	+	0.0410946i	−0.853919	+	0.520407i	$0.825780\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	670.000			0.557773			0.278886	−	0.960324i	$-0.410035\pi$
$113$	670.000			0.557773			0.278886	+	0.960324i	$0.410035\pi$
$114$		0			0
$115$		0			0
$116$	−1411.00	−	2443.92i	−1.12938	−	1.95614i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1646.50	+	2851.82i	−1.23704	+	2.14262i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2000.00			−1.39741			−0.698706	−	0.715409i	$-0.746240\pi$
$127$	−2000.00			−1.39741			−0.698706	+	0.715409i	$0.746240\pi$
$128$	1057.50	−	1831.64i	0.730240	−	1.26481i
$129$		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
$134$	−3700.00			−2.38531
$135$		0			0
$136$		0			0
$137$	1555.00	+	2693.34i	0.969727	+	1.67962i	0.696339	+	0.717713i	$0.254811\pi$
$137$	1555.00	+	2693.34i	0.969727	+	1.67962i	0.273388	+	0.961904i	$0.411856\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	1720.00	−	2979.13i	1.01647	−	1.76058i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	7650.00			4.24883
$149$	407.000	−	704.945i	0.223777	−	0.387593i	−0.732175	−	0.681117i	$-0.761495\pi$
$149$	407.000	−	704.945i	0.223777	−	0.387593i	0.955952	+	0.293524i	$0.0948280\pi$
$150$		0			0
$151$	1476.00	+	2556.51i	0.795465	+	1.37779i	0.922544	+	0.385893i	$0.126107\pi$
$151$	1476.00	+	2556.51i	0.795465	+	1.37779i	−0.127079	+	0.991893i	$0.540560\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	3460.00	+	5992.90i	1.74217	+	3.01753i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−890.000	+	1541.53i	−0.427670	+	0.740746i	−0.996666	−	0.0815946i	$-0.973999\pi$
$163$	−890.000	+	1541.53i	−0.427670	+	0.740746i	0.568996	+	0.822340i	$0.307332\pi$
$164$		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
$172$	1530.00	+	2650.04i	0.678264	+	1.17479i
$173$		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
$176$	6052.00			2.59197
$177$		0			0
$178$		0			0
$179$	−1042.00	−	1804.80i	−0.435099	−	0.753614i	0.562205	−	0.826998i	$-0.309953\pi$
$179$	−1042.00	−	1804.80i	−0.435099	−	0.753614i	−0.997304	+	0.0733844i	$0.976620\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−900.000	+	1558.85i	−0.360592	+	0.624563i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2036.00	+	3526.46i	−0.771308	+	1.33594i	0.165539	+	0.986203i	$0.447064\pi$
$191$	−2036.00	+	3526.46i	−0.771308	+	1.33594i	−0.936846	+	0.349741i	$0.886270\pi$
$192$		0			0
$193$	2295.00	+	3975.06i	0.855947	+	1.48254i	0.875764	+	0.482740i	$0.160358\pi$
$193$	2295.00	+	3975.06i	0.855947	+	1.48254i	−0.0198172	+	0.999804i	$0.506308\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2210.00			0.799269			0.399634	−	0.916675i	$-0.369137\pi$
$197$	2210.00			0.799269			0.399634	+	0.916675i	$0.369137\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	2812.50	+	4871.39i	0.994369	+	1.72230i
$201$		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
$211$	5868.00			1.91455			0.957274	−	0.289181i	$-0.0933830\pi$
$211$	5868.00			1.91455			0.957274	+	0.289181i	$0.0933830\pi$
$212$	5015.00	−	8686.23i	1.62468	−	2.81402i
$213$		0			0
$214$	−3950.00	−	6841.60i	−1.26176	−	2.18543i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−270.000			−0.0838840
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−1675.00	+	2901.19i	−0.493006	+	0.853911i
$227$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	7470.00			2.11392
$233$	−2365.00	+	4096.30i	−0.664963	+	1.15175i	0.314333	+	0.949313i	$0.398219\pi$
$233$	−2365.00	+	4096.30i	−0.664963	+	1.15175i	−0.979295	+	0.202436i	$0.935114\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7376.00			1.99629			0.998146	−	0.0608655i	$-0.0193861\pi$
$239$	7376.00			1.99629			0.998146	+	0.0608655i	$0.0193861\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	−8232.50	−	14259.1i	−2.18680	−	3.78765i
$243$		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
$253$	2720.00			0.675909
$254$	5000.00	−	8660.25i	1.23515	−	2.13934i
$255$		0			0
$256$	4139.50	+	7169.82i	1.01062	+	1.75045i
$257$		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
$263$	3760.00	+	6512.51i	0.881565	+	1.52691i	0.849601	+	0.527426i	$0.176843\pi$
$263$	3760.00	+	6512.51i	0.881565	+	1.52691i	0.0319637	+	0.999489i	$0.489824\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	6290.00	−	10894.6i	1.43367	−	2.48319i
$269$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	−15550.0			−3.42850
$275$	4250.00	−	7361.22i	0.931944	−	1.61417i
$276$		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
$281$	−4342.00			−0.921786			−0.460893	−	0.887456i	$-0.652471\pi$
$281$	−4342.00			−0.921786			−0.460893	+	0.887456i	$0.652471\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$	5848.00	+	10129.0i	1.22188	+	2.11636i
$285$		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
$296$	−10125.0	+	17537.0i	−1.98819	+	3.44364i
$297$		0			0
$298$	2035.00	+	3524.72i	0.395585	+	0.685174i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−14760.0			−2.81239
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$	−23528.0			−4.18846
$317$	−3485.00	+	6036.20i	−0.617467	+	1.06948i	0.372479	+	0.928041i	$0.378508\pi$
$317$	−3485.00	+	6036.20i	−0.617467	+	1.06948i	−0.989946	+	0.141444i	$0.954825\pi$
$318$		0			0
$319$	−5644.00	−	9775.69i	−0.990606	−	1.71578i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−4450.00	−	7707.63i	−0.756021	−	1.30947i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5454.00	+	9446.61i	−0.905677	+	1.56868i	−0.0856702	+	0.996324i	$0.527303\pi$
$331$	−5454.00	+	9446.61i	−0.905677	+	1.56868i	−0.820006	+	0.572354i	$0.806030\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3330.00			−0.538269			−0.269135	−	0.963103i	$-0.586738\pi$
$337$	−3330.00			−0.538269			−0.269135	+	0.963103i	$0.586738\pi$
$338$	5492.50	−	9513.29i	0.883883	−	1.53093i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−8100.00			−1.26954
$345$		0			0
$346$		0			0
$347$	−2050.00	−	3550.70i	−0.317146	−	0.549314i	0.662745	−	0.748845i	$-0.269391\pi$
$347$	−2050.00	−	3550.70i	−0.317146	−	0.549314i	−0.979891	+	0.199532i	$0.936058\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−2890.00	+	5005.63i	−0.437607	+	0.757957i
$353$		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
$358$	10420.0			1.53831
$359$	−4052.00	+	7018.27i	−0.595700	+	1.03178i	0.397747	+	0.917495i	$0.369792\pi$
$359$	−4052.00	+	7018.27i	−0.595700	+	1.03178i	−0.993448	+	0.114288i	$0.963541\pi$
$360$		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		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−1780.00	−	3083.05i	−0.252144	−	0.436726i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6985.00	−	12098.4i	0.969624	−	1.67944i	0.272980	−	0.962020i	$-0.411991\pi$
$373$	6985.00	−	12098.4i	0.969624	−	1.67944i	0.696643	−	0.717418i	$-0.254676\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11916.0			1.61500			0.807498	−	0.589870i	$-0.200821\pi$
$379$	11916.0			1.61500			0.807498	+	0.589870i	$0.200821\pi$
$380$		0			0
$381$		0			0
$382$	−10180.0	−	17632.3i	−1.36349	−	2.36164i
$383$		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
$386$	−22950.0			−3.02623
$387$		0			0
$388$		0			0
$389$	−5263.00	−	9115.78i	−0.685976	−	1.18815i	−0.973129	−	0.230261i	$-0.926042\pi$
$389$	−5263.00	−	9115.78i	−0.685976	−	1.18815i	0.287153	−	0.957885i	$-0.407291\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−5525.00	+	9569.58i	−0.706461	+	1.22363i
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−11125.0			−1.39062
$401$	799.000	−	1383.91i	0.0995016	−	0.172342i	−0.811977	−	0.583690i	$-0.801608\pi$
$401$	799.000	−	1383.91i	0.0995016	−	0.172342i	0.911478	+	0.411348i	$0.134942\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	30600.0			3.72675
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		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	$-0.155247\pi$
$421$	15262.0			1.76680			0.883402	+	0.468616i	$0.155247\pi$
$422$	−14670.0	+	25409.2i	−1.69224	+	2.93104i
$423$		0			0
$424$	13275.0	+	22993.0i	1.52050	+	2.63358i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	26860.0			3.03347
$429$		0			0
$430$		0			0
$431$	−4304.00	−	7454.75i	−0.481012	−	0.833138i	0.518750	−	0.854926i	$-0.326398\pi$
$431$	−4304.00	−	7454.75i	−0.481012	−	0.833138i	−0.999763	+	0.0217878i	$0.993064\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	459.000	−	795.011i	0.0504177	−	0.0873260i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9290.00	−	16090.8i	0.996346	−	1.72572i	0.424205	−	0.905566i	$-0.360553\pi$
$443$	9290.00	−	16090.8i	0.996346	−	1.72572i	0.572140	−	0.820156i	$-0.306113\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2686.00			0.282317			0.141158	−	0.989987i	$-0.454917\pi$
$449$	2686.00			0.282317			0.141158	+	0.989987i	$0.454917\pi$
$450$		0			0
$451$		0			0
$452$	−5695.00	−	9864.03i	−0.592633	−	1.02647i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4005.00	+	6936.86i	−0.409947	+	0.710050i	−0.994883	−	0.101030i	$-0.967786\pi$
$457$	−4005.00	+	6936.86i	−0.409947	+	0.710050i	0.584936	+	0.811079i	$0.301120\pi$
$458$		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	$0.639229\pi$
$464$	−7387.00	+	12794.7i	−0.739079	+	1.28012i
$465$		0			0
$466$	−11825.0	−	20481.5i	−1.17550	−	2.03602i
$467$		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
$473$	6120.00	+	10600.2i	0.594922	+	1.03043i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−18440.0	+	31939.0i	−1.76449	+	3.05619i
$479$		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
$484$	55981.0			5.25742
$485$		0			0
$486$		0			0
$487$	−10620.0	−	18394.4i	−0.988169	−	1.71156i	−0.626908	−	0.779094i	$-0.715680\pi$
$487$	−10620.0	−	18394.4i	−0.988169	−	1.71156i	−0.361261	−	0.932465i	$-0.617654\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20372.0			−1.87246			−0.936228	−	0.351394i	$-0.885708\pi$
$491$	−20372.0			−1.87246			−0.936228	+	0.351394i	$0.885708\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3618.00	−	6266.56i	0.324577	−	0.562184i	−0.656850	−	0.754022i	$-0.728111\pi$
$499$	3618.00	−	6266.56i	0.324577	−	0.562184i	0.981427	+	0.191838i	$0.0614447\pi$
$500$		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
$506$	−6800.00	+	11777.9i	−0.597425	+	1.03477i
$507$		0			0
$508$	17000.0	+	29444.9i	1.48475	+	2.57166i
$509$		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
$512$	−24475.0			−2.11260
$513$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$	−37600.0			−3.11680
$527$		0			0
$528$		0			0
$529$	5283.50	+	9151.29i	0.434248	+	0.752140i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	16650.0	+	28838.6i	1.34174	+	2.32395i
$537$		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	$0.330644\pi$
$548$	26435.0	−	45786.8i	2.06067	−	3.56919i
$549$		0			0
$550$	21250.0	+	36806.1i	1.64746	+	2.85348i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	36550.0			2.80300
$555$		0			0
$556$		0			0
$557$	10235.0	+	17727.5i	0.778583	+	1.34855i	0.932758	+	0.360502i	$0.117395\pi$
$557$	10235.0	+	17727.5i	0.778583	+	1.34855i	−0.154175	+	0.988044i	$0.549272\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10855.0	−	18801.4i	0.814752	−	1.41119i
$563$		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
$568$	−30960.0			−2.28706
$569$	−13453.0	+	23301.3i	−0.991176	+	1.71677i	−0.380794	+	0.924660i	$0.624349\pi$
$569$	−13453.0	+	23301.3i	−0.991176	+	1.71677i	−0.610382	+	0.792107i	$0.708984\pi$
$570$		0			0
$571$	3394.00	+	5878.58i	0.248747	+	0.430842i	0.963178	−	0.268863i	$-0.0866479\pi$
$571$	3394.00	+	5878.58i	0.248747	+	0.430842i	−0.714431	+	0.699705i	$0.753315\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5000.00			−0.362634
$576$		0			0
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	12282.5	+	21273.9i	0.883883	+	1.53093i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20060.0	−	34744.9i	1.42504	−	2.46825i
$584$		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
$592$	−20025.0	−	34684.3i	−1.39024	−	2.40797i
$593$		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
$596$	−13838.0			−0.951051
$597$		0			0
$598$		0			0
$599$	−12368.0	−	21422.0i	−0.843644	−	1.46123i	−0.886794	−	0.462166i	$-0.847073\pi$
$599$	−12368.0	−	21422.0i	−0.843644	−	1.46123i	0.0431495	−	0.999069i	$-0.486261\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$	25092.0	−	43460.6i	1.69036	−	2.92779i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7505.00	−	12999.0i	−0.494493	−	0.856487i	0.505487	−	0.862834i	$-0.331313\pi$
$613$	−7505.00	−	12999.0i	−0.494493	−	0.856487i	−0.999980	+	0.00634752i	$0.997980\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30550.0			−1.99335			−0.996675	−	0.0814823i	$-0.974035\pi$
$617$	−30550.0			−1.99335			−0.996675	+	0.0814823i	$0.974035\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		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	$-0.809512\pi$
$631$	−26192.0			−1.65244			−0.826218	+	0.563351i	$0.809512\pi$
$632$	31140.0	−	53936.1i	1.95994	−	3.39472i
$633$		0			0
$634$	−17425.0	−	30181.0i	−1.09154	−	1.89060i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	56440.0			3.50232
$639$		0			0
$640$		0			0
$641$	4439.00	+	7688.57i	0.273526	+	0.473760i	0.969762	−	0.244052i	$-0.0784768\pi$
$641$	4439.00	+	7688.57i	0.273526	+	0.473760i	−0.696236	+	0.717813i	$0.745143\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		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
$652$	30260.0			1.81760
$653$	13525.0	−	23426.0i	0.810527	−	1.40387i	−0.101969	−	0.994788i	$-0.532514\pi$
$653$	13525.0	−	23426.0i	0.810527	−	1.40387i	0.912496	−	0.409086i	$-0.134153\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1804.00			0.106637			0.0533186	−	0.998578i	$-0.483020\pi$
$659$	1804.00			0.106637			0.0533186	+	0.998578i	$0.483020\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	−27270.0	−	47233.0i	−1.60103	−	2.77306i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3320.00	+	5750.41i	−0.192730	+	0.333818i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33570.0			−1.92278			−0.961388	−	0.275196i	$-0.911257\pi$
$673$	−33570.0			−1.92278			−0.961388	+	0.275196i	$0.911257\pi$
$674$	8325.00	−	14419.3i	0.475767	−	0.824053i
$675$		0			0
$676$	18674.5	+	32345.2i	1.06250	+	1.84030i
$677$		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
$683$	−17030.0	−	29496.8i	−0.954077	−	1.65251i	−0.736466	−	0.676475i	$-0.763507\pi$
$683$	−17030.0	−	29496.8i	−0.954077	−	1.65251i	−0.217612	−	0.976035i	$-0.569827\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	8010.00	−	13873.7i	0.443864	−	0.768795i
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	20500.0			1.12128
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4198.00			0.226186			0.113093	−	0.993584i	$-0.463924\pi$
$701$	4198.00			0.226186			0.113093	+	0.993584i	$0.463924\pi$
$702$		0			0
$703$		0			0
$704$	9758.00	+	16901.4i	0.522398	+	0.904821i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6273.00	+	10865.2i	−0.332281	+	0.575528i	−0.982959	−	0.183826i	$-0.941152\pi$
$709$	−6273.00	+	10865.2i	−0.332281	+	0.575528i	0.650677	+	0.759354i	$0.274485\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−17714.0	+	30681.5i	−0.924586	+	1.60143i
$717$		0			0
$718$	−20260.0	−	35091.3i	−1.05306	−	1.82395i
$719$		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
$722$	−34295.0			−1.76777
$723$		0			0
$724$		0			0
$725$	10375.0	+	17970.0i	0.531473	+	0.920538i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$	3400.00			0.170279
$737$	25160.0	−	43578.4i	1.25750	−	2.17806i
$738$		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
$743$	−25160.0			−1.24230			−0.621151	−	0.783691i	$-0.713335\pi$
$743$	−25160.0			−1.24230			−0.621151	+	0.783691i	$0.713335\pi$
$744$		0			0
$745$		0			0
$746$	34925.0	+	60491.9i	1.71407	+	2.96885i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1224.00	−	2120.03i	0.0594732	−	0.103011i	−0.834756	−	0.550620i	$-0.814391\pi$
$751$	1224.00	−	2120.03i	0.0594732	−	0.103011i	0.894229	+	0.447610i	$0.147725\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34830.0			−1.67228			−0.836141	−	0.548514i	$-0.815194\pi$
$757$	−34830.0			−1.67228			−0.836141	+	0.548514i	$0.815194\pi$
$758$	−29790.0	+	51597.8i	−1.42747	+	2.47245i
$759$		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
$764$	69224.0			3.27806
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	39015.0	−	67576.0i	1.81889	−	3.15040i
$773$		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
$778$	52630.0			2.42529
$779$		0			0
$780$		0			0
$781$	23392.0	+	40516.1i	1.07174	+	1.85631i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	−18785.0	−	32536.6i	−0.849223	−	1.47090i
$789$		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
$800$	5312.50	−	9201.52i	0.234782	−	0.406654i
$801$		0			0
$802$	3995.00	+	6919.54i	0.175896	+	0.304660i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18677.0	+	32349.5i	0.811679	+	1.40587i	0.911688	+	0.410883i	$0.134779\pi$
$809$	18677.0	+	32349.5i	0.811679	+	1.40587i	−0.100009	+	0.994987i	$0.531887\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−76500.0	+	132502.i	−3.29401	+	5.70539i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21769.0	+	37705.0i	−0.925388	+	1.60282i	−0.134451	+	0.990920i	$0.542927\pi$
$821$	−21769.0	+	37705.0i	−0.925388	+	1.60282i	−0.790936	+	0.611898i	$0.790406\pi$
$822$		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
$827$	23980.0			1.00830			0.504151	−	0.863615i	$-0.331805\pi$
$827$	23980.0			1.00830			0.504151	+	0.863615i	$0.331805\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		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
$841$	3167.00			0.129854
$842$	−38155.0	+	66086.4i	−1.56165	+	2.70485i
$843$		0			0
$844$	−49878.0	−	86391.2i	−2.03421	−	3.52335i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−52510.0			−2.12642
$849$		0			0
$850$		0			0
$851$	−9000.00	−	15588.5i	−0.362534	−	0.627926i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−35550.0	+	61574.4i	−1.41948	+	2.45861i
$857$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	43040.0			1.70064
$863$	−10100.0	+	17493.7i	−0.398387	+	0.690027i	−0.993527	−	0.113595i	$-0.963763\pi$
$863$	−10100.0	+	17493.7i	−0.398387	+	0.690027i	0.595140	+	0.803622i	$0.297097\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−94112.0			−3.67380
$870$		0			0
$871$		0			0
$872$	1215.00	+	2104.44i	0.0471847	+	0.0817264i
$873$		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
$883$	30060.0			1.14564			0.572820	−	0.819681i	$-0.305850\pi$
$883$	30060.0			1.14564			0.572820	+	0.819681i	$0.305850\pi$
$884$		0			0
$885$		0			0
$886$	46450.0	+	80453.8i	1.76131	+	3.05067i
$887$		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
$898$	−6715.00	+	11630.7i	−0.249535	+	0.432207i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	30150.0			1.10926
$905$		0			0
$906$		0			0
$907$	−26370.0	−	45674.2i	−0.965382	−	1.67209i	−0.708585	−	0.705625i	$-0.750666\pi$
$907$	−26370.0	−	45674.2i	−0.965382	−	1.67209i	−0.256797	−	0.966465i	$-0.582667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	39632.0			1.44135			0.720673	−	0.693275i	$-0.243833\pi$
$911$	39632.0			1.44135			0.720673	+	0.693275i	$0.243833\pi$
$912$		0			0
$913$		0			0
$914$	−20025.0	−	34684.3i	−0.724692	−	1.25520i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10872.0	+	18830.9i	−0.390244	+	0.675922i	−0.992482	−	0.122395i	$-0.960943\pi$
$919$	−10872.0	+	18830.9i	−0.390244	+	0.675922i	0.602238	+	0.798317i	$0.294276\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−56250.0			−1.99945
$926$	21100.0	−	36546.3i	0.748800	−	1.29696i
$927$		0			0
$928$	−7055.00	−	12219.6i	−0.249560	−	0.432251i
$929$		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
$932$	80410.0			2.82609
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		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
$946$	−61200.0			−2.10337
$947$	24410.0	−	42279.4i	0.837612	−	1.45079i	−0.0542742	−	0.998526i	$-0.517285\pi$
$947$	24410.0	−	42279.4i	0.837612	−	1.45079i	0.891886	−	0.452260i	$-0.149382\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−29290.0			−0.995589			−0.497794	−	0.867295i	$-0.665857\pi$
$953$	−29290.0			−0.995589			−0.497794	+	0.867295i	$0.665857\pi$
$954$		0			0
$955$		0			0
$956$	−62696.0	−	108593.i	−2.12106	−	3.67378i
$957$		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	$0.167127\pi$
$968$	−74092.5	+	128332.i	−2.46015	+	4.26110i
$969$		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
$974$	106200.			3.49370
$975$		0			0
$976$		0			0
$977$	−18745.0	−	32467.3i	−0.613824	−	1.06317i	−0.990590	−	0.136865i	$-0.956297\pi$
$977$	−18745.0	−	32467.3i	−0.613824	−	1.06317i	0.376766	−	0.926308i	$-0.377036\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	50930.0	−	88213.3i	1.65503	−	2.86660i
$983$		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
$989$	3600.00	−	6235.38i	0.115747	−	0.200479i
$990$		0			0
$991$	−28764.0	−	49820.7i	−0.922017	−	1.59698i	−0.796290	−	0.604915i	$-0.793207\pi$
$991$	−28764.0	−	49820.7i	−0.922017	−	1.59698i	−0.125727	−	0.992065i	$-0.540126\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$	18090.0	+	31332.8i	0.573776	+	0.993810i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.e.a.226.1		2
3.2	odd	2		49.4.c.d.30.1		2
7.2	even	3		441.4.a.m.1.1		1
7.3	odd	6	inner	441.4.e.a.361.1		2
7.4	even	3	inner	441.4.e.a.361.1		2
7.5	odd	6		441.4.a.m.1.1		1
7.6	odd	2	CM	441.4.e.a.226.1		2
21.2	odd	6		49.4.a.a.1.1	✓	1
21.5	even	6		49.4.a.a.1.1	✓	1
21.11	odd	6		49.4.c.d.18.1		2
21.17	even	6		49.4.c.d.18.1		2
21.20	even	2		49.4.c.d.30.1		2
84.23	even	6		784.4.a.k.1.1		1
84.47	odd	6		784.4.a.k.1.1		1
105.44	odd	6		1225.4.a.l.1.1		1
105.89	even	6		1225.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.a.1.1	✓	1	21.2	odd	6
49.4.a.a.1.1	✓	1	21.5	even	6
49.4.c.d.18.1		2	21.11	odd	6
49.4.c.d.18.1		2	21.17	even	6
49.4.c.d.30.1		2	3.2	odd	2
49.4.c.d.30.1		2	21.20	even	2
441.4.a.m.1.1		1	7.2	even	3
441.4.a.m.1.1		1	7.5	odd	6
441.4.e.a.226.1		2	1.1	even	1	trivial
441.4.e.a.226.1		2	7.6	odd	2	CM
441.4.e.a.361.1		2	7.3	odd	6	inner
441.4.e.a.361.1		2	7.4	even	3	inner
784.4.a.k.1.1		1	84.23	even	6
784.4.a.k.1.1		1	84.47	odd	6
1225.4.a.l.1.1		1	105.44	odd	6
1225.4.a.l.1.1		1	105.89	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{2} +(-8.50000 - 14.7224i) q^{4} +45.0000 q^{8} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{2} +(-8.50000 - 14.7224i) q^{4} +45.0000 q^{8} +(-34.0000 - 58.8897i) q^{11} +(-44.5000 + 77.0763i) q^{16} +340.000 q^{22} +(-20.0000 + 34.6410i) q^{23} +(62.5000 + 108.253i) q^{25} +166.000 q^{29} +(-42.5000 - 73.6122i) q^{32} +(-225.000 + 389.711i) q^{37} -180.000 q^{43} +(-578.000 + 1001.13i) q^{44} +(-100.000 - 173.205i) q^{46} -625.000 q^{50} +(295.000 + 510.955i) q^{53} +(-415.000 + 718.801i) q^{58} -287.000 q^{64} +(370.000 + 640.859i) q^{67} -688.000 q^{71} +(-1125.00 - 1948.56i) q^{74} +(692.000 - 1198.58i) q^{79} +(450.000 - 779.423i) q^{86} +(-1530.00 - 2650.04i) q^{88} +680.000 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} - 17 q^{4} + 90 q^{8}+O(q^{10}) \) 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8	\( 2 q - 5 q^{2} - 17 q^{4} + 90 q^{8} - 68 q^{11} - 89 q^{16} + 680 q^{22} - 40 q^{23} + 125 q^{25} + 332 q^{29} - 85 q^{32} - 450 q^{37} - 360 q^{43} - 1156 q^{44} - 200 q^{46} - 1250 q^{50} + 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} - 1376 q^{71} - 2250 q^{74} + 1384 q^{79} + 900 q^{86} - 3060 q^{88} + 1360 q^{92}+O(q^{100}) \) 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8 - 68 * q^11 - 89 * q^16 + 680 * q^22 - 40 * q^23 + 125 * q^25 + 332 * q^29 - 85 * q^32 - 450 * q^37 - 360 * q^43 - 1156 * q^44 - 200 * q^46 - 1250 * q^50 + 590 * q^53 - 830 * q^58 - 574 * q^64 + 740 * q^67 - 1376 * q^71 - 2250 * q^74 + 1384 * q^79 + 900 * q^86 - 3060 * q^88 + 1360 * q^92

Introduction

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