LMFDB - Embedded newform 504.2.cj.b.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(37,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("504.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.cj (of order $6$, degree $2$, 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:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			109.4
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	504.109
Dual form			504.2.cj.b.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(1.37333 - 0.792893i) q^{5} +(1.62132 + 2.09077i) q^{7} -2.82843i q^{8} +O(q^{10})$	$q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(1.37333 - 0.792893i) q^{5} +(1.62132 + 2.09077i) q^{7} -2.82843i q^{8} +(1.12132 - 1.94218i) q^{10} +(0.148586 + 0.0857864i) q^{11} +(3.46410 + 1.41421i) q^{14} +(-2.00000 - 3.46410i) q^{16} -3.17157i q^{20} +0.242641 q^{22} +(-1.24264 + 2.15232i) q^{25} +(5.24264 - 0.717439i) q^{28} -10.4142i q^{29} +(-4.62132 + 8.00436i) q^{31} +(-4.89898 - 2.82843i) q^{32} +(3.88437 + 1.58579i) q^{35} +(-2.24264 - 3.88437i) q^{40} +(0.297173 - 0.171573i) q^{44} +(-1.74264 + 6.77962i) q^{49} +3.51472i q^{50} +(-8.72180 - 5.03553i) q^{53} +0.272078 q^{55} +(5.91359 - 4.58579i) q^{56} +(-7.36396 - 12.7548i) q^{58} +(12.6932 + 7.32843i) q^{59} +13.0711i q^{62} -8.00000 q^{64} +(5.87868 - 0.804479i) q^{70} +(-7.00000 + 12.1244i) q^{73} +(0.0615465 + 0.449747i) q^{77} +(3.86396 + 6.69258i) q^{79} +(-5.49333 - 3.17157i) q^{80} +17.8284i q^{83} +(0.242641 - 0.420266i) q^{88} +15.9706 q^{97} +(2.65962 + 9.53553i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 4 q^{7}+O(q^{10}) $ 8 * q + 8 * q^4 - 4 * q^7	$ 8 q + 8 q^{4} - 4 q^{7} - 8 q^{10} - 16 q^{16} - 32 q^{22} + 24 q^{25} + 8 q^{28} - 20 q^{31} + 16 q^{40} + 20 q^{49} + 104 q^{55} - 8 q^{58} - 64 q^{64} + 64 q^{70} - 56 q^{73} - 20 q^{79} - 32 q^{88} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 4 * q^7 - 8 * q^10 - 16 * q^16 - 32 * q^22 + 24 * q^25 + 8 * q^28 - 20 * q^31 + 16 * q^40 + 20 * q^49 + 104 * q^55 - 8 * q^58 - 64 * q^64 + 64 * q^70 - 56 * q^73 - 20 * q^79 - 32 * q^88 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$	$1$

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 $

$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	1.37333	−	0.792893i	0.614172	−	0.354593i	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	1.37333	−	0.792893i	0.614172	−	0.354593i	0.774597	+	0.632456i	$0.217953\pi$
$6$		0			0
$7$	1.62132	+	2.09077i	0.612801	+	0.790237i
$7$	1.62132	+	2.09077i	0.612801	+	0.790237i
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$	1.12132	−	1.94218i	0.354593	−	0.614172i
$11$	0.148586	+	0.0857864i	0.0448005	+	0.0258656i	0.522233	−	0.852803i	$-0.325099\pi$
$11$	0.148586	+	0.0857864i	0.0448005	+	0.0258656i	−0.477432	+	0.878668i	$0.658432\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.46410	+	1.41421i	0.925820	+	0.377964i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		−	3.17157i		−	0.709185i
$21$		0			0
$22$	0.242641			0.0517312
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−1.24264	+	2.15232i	−0.248528	+	0.430463i
$26$		0			0
$27$		0			0
$28$	5.24264	−	0.717439i	0.990766	−	0.135583i
$29$		−	10.4142i		−	1.93387i	−0.255021	−	0.966935i	$-0.582082\pi$
$29$		−	10.4142i		−	1.93387i	0.255021	−	0.966935i	$-0.417918\pi$
$30$		0			0
$31$	−4.62132	+	8.00436i	−0.830014	+	1.43763i	0.0680129	+	0.997684i	$0.478334\pi$
$31$	−4.62132	+	8.00436i	−0.830014	+	1.43763i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−4.89898	−	2.82843i	−0.866025	−	0.500000i
$33$		0			0
$34$		0			0
$35$	3.88437	+	1.58579i	0.656578	+	0.268047i
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$	−2.24264	−	3.88437i	−0.354593	−	0.614172i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.297173	−	0.171573i	0.0448005	−	0.0258656i
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−1.74264	+	6.77962i	−0.248949	+	0.968517i
$50$			3.51472i			0.497056i
$51$		0			0
$52$		0			0
$53$	−8.72180	−	5.03553i	−1.19803	−	0.691684i	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−8.72180	−	5.03553i	−1.19803	−	0.691684i	−0.960116	+	0.279602i	$0.909797\pi$
$54$		0			0
$55$	0.272078			0.0366870
$56$	5.91359	−	4.58579i	0.790237	−	0.612801i
$57$		0			0
$58$	−7.36396	−	12.7548i	−0.966935	−	1.67478i
$59$	12.6932	+	7.32843i	1.65251	+	0.954080i	0.976034	+	0.217620i	$0.0698294\pi$
$59$	12.6932	+	7.32843i	1.65251	+	0.954080i	0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$			13.0711i			1.66003i
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	5.87868	−	0.804479i	0.702637	−	0.0961536i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−7.00000	+	12.1244i	−0.819288	+	1.41905i	0.0869195	+	0.996215i	$0.472298\pi$
$73$	−7.00000	+	12.1244i	−0.819288	+	1.41905i	−0.906208	+	0.422833i	$0.861036\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0615465	+	0.449747i	0.00701388	+	0.0512535i
$78$		0			0
$79$	3.86396	+	6.69258i	0.434730	+	0.752974i	0.997274	−	0.0737937i	$-0.0235106\pi$
$79$	3.86396	+	6.69258i	0.434730	+	0.752974i	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−5.49333	−	3.17157i	−0.614172	−	0.354593i
$81$		0			0
$82$		0			0
$83$			17.8284i			1.95692i	0.206427	+	0.978462i	$0.433816\pi$
$83$			17.8284i			1.95692i	−0.206427	+	0.978462i	$0.566184\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.242641	−	0.420266i	0.0258656	−	0.0448005i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	15.9706			1.62156			0.810782	−	0.585348i	$-0.199042\pi$
$97$	15.9706			1.62156			0.810782	+	0.585348i	$0.199042\pi$
$98$	2.65962	+	9.53553i	0.268662	+	0.963234i
$99$		0			0
$100$	2.48528	+	4.30463i	0.248528	+	0.430463i
$101$	−17.1464	−	9.89949i	−1.70613	−	0.985037i	−0.939239	−	0.343263i	$-0.888468\pi$
$101$	−17.1464	−	9.89949i	−1.70613	−	0.985037i	−0.766894	−	0.641774i	$-0.778199\pi$
$102$		0			0
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	0.282194	−	0.959357i	$-0.408938\pi$
$104$		0			0
$105$		0			0
$106$	−14.2426			−1.38337
$107$	−17.8894	+	10.3284i	−1.72943	+	0.998487i	−0.837218	+	0.546869i	$0.815820\pi$
$107$	−17.8894	+	10.3284i	−1.72943	+	0.998487i	−0.892211	+	0.451618i	$0.850847\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$	0.333226	−	0.192388i	0.0317719	−	0.0183435i
$111$		0			0
$112$	4.00000	−	9.79796i	0.377964	−	0.925820i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	−18.0379	−	10.4142i	−1.67478	−	0.966935i
$117$		0			0
$118$	20.7279			1.90816
$119$		0			0
$120$		0			0
$121$	−5.48528	−	9.50079i	−0.498662	−	0.863708i
$122$		0			0
$123$		0			0
$124$	9.24264	+	16.0087i	0.830014	+	1.43763i
$125$			11.8701i			1.06169i
$126$		0			0
$127$	6.75736			0.599619			0.299809	−	0.953999i	$-0.403077\pi$
$127$	6.75736			0.599619			0.299809	+	0.953999i	$0.403077\pi$
$128$	−9.79796	+	5.65685i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	7.19988	−	4.15685i	0.629057	−	0.363186i	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	7.19988	−	4.15685i	0.629057	−	0.363186i	0.780387	+	0.625297i	$0.215022\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	6.63103	−	5.14214i	0.560424	−	0.434590i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−8.25736	−	14.3022i	−0.685736	−	1.18773i
$146$			19.7990i			1.63858i
$147$		0			0
$148$		0			0
$149$	2.44949	−	1.41421i	0.200670	−	0.115857i	−0.396298	−	0.918122i	$-0.629705\pi$
$149$	2.44949	−	1.41421i	0.200670	−	0.115857i	0.596968	+	0.802265i	$0.296372\pi$
$150$		0			0
$151$	11.1066	−	19.2372i	0.903842	−	1.56550i	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	11.1066	−	19.2372i	0.903842	−	1.56550i	0.822464	−	0.568818i	$-0.192599\pi$
$152$		0			0
$153$		0			0
$154$	0.393398	+	0.507306i	0.0317009	+	0.0408799i
$155$			14.6569i			1.17727i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	9.46473	+	5.46447i	0.752974	+	0.434730i
$159$		0			0
$160$	−8.97056			−0.709185
$161$		0			0
$162$		0			0
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	12.6066	+	21.8353i	0.978462	+	1.69475i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.1464	−	9.89949i	1.30362	−	0.752645i	0.322596	−	0.946537i	$-0.395445\pi$
$173$	17.1464	−	9.89949i	1.30362	−	0.752645i	0.981023	+	0.193892i	$0.0621112\pi$
$174$		0			0
$175$	−6.51472	+	0.891519i	−0.492466	+	0.0673925i
$176$		−	0.686292i		−	0.0517312i
$177$		0			0
$178$		0			0
$179$	−9.79796	−	5.65685i	−0.732334	−	0.422813i	0.0869415	−	0.996213i	$-0.472291\pi$
$179$	−9.79796	−	5.65685i	−0.732334	−	0.422813i	−0.819275	+	0.573400i	$0.805624\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	10.7426	−	18.6068i	0.773272	−	1.33935i	−0.162488	−	0.986710i	$-0.551952\pi$
$193$	10.7426	−	18.6068i	0.773272	−	1.33935i	0.935760	−	0.352636i	$-0.114715\pi$
$194$	19.5599	−	11.2929i	1.40432	−	0.810782i
$195$		0			0
$196$	10.0000	+	9.79796i	0.714286	+	0.699854i
$197$		−	14.1421i		−	1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$		−	14.1421i		−	1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$		0			0
$199$	−7.00000	+	12.1244i	−0.496217	+	0.859473i	−0.999990	−	0.00436292i	$-0.998611\pi$
$199$	−7.00000	+	12.1244i	−0.496217	+	0.859473i	0.503774	+	0.863836i	$0.331945\pi$
$200$	6.08767	+	3.51472i	0.430463	+	0.248528i
$201$		0			0
$202$	−28.0000			−1.97007
$203$	21.7737	−	16.8848i	1.52822	−	1.18508i
$204$		0			0
$205$		0			0
$206$	−17.1464	−	9.89949i	−1.19465	−	0.689730i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−17.4436	+	10.0711i	−1.19803	+	0.691684i
$213$		0			0
$214$	−14.6066	+	25.2994i	−0.998487	+	1.72943i
$215$		0			0
$216$		0			0
$217$	−24.2279	+	3.31552i	−1.64470	+	0.225072i
$218$		0			0
$219$		0			0
$220$	0.272078	−	0.471253i	0.0183435	−	0.0317719i
$221$		0			0
$222$		0			0
$223$	−0.272078			−0.0182197			−0.00910984	−	0.999959i	$-0.502900\pi$
$223$	−0.272078			−0.0182197			−0.00910984	+	0.999959i	$0.502900\pi$
$224$	−2.02922	−	14.8284i	−0.135583	−	0.990766i
$225$		0			0
$226$		0			0
$227$	4.45322	+	2.57107i	0.295571	+	0.170648i	0.640451	−	0.767999i	$-0.278747\pi$
$227$	4.45322	+	2.57107i	0.295571	+	0.170648i	−0.344881	+	0.938647i	$0.612081\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	−29.4558			−1.93387
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$	25.3864	−	14.6569i	1.65251	−	0.954080i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	10.2279	−	17.7153i	0.658838	−	1.14114i	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	10.2279	−	17.7153i	0.658838	−	1.14114i	0.980917	−	0.194429i	$-0.0622852\pi$
$242$	−13.4361	−	7.75736i	−0.863708	−	0.498662i
$243$		0			0
$244$		0			0
$245$	2.98229	+	10.6924i	0.190531	+	0.683112i
$246$		0			0
$247$		0			0
$248$	22.6398	+	13.0711i	1.43763	+	0.830014i
$249$		0			0
$250$	8.39340	+	14.5378i	0.530845	+	0.919451i
$251$		−	24.1716i		−	1.52570i	−0.646578	−	0.762848i	$-0.723800\pi$
$251$		−	24.1716i		−	1.52570i	0.646578	−	0.762848i	$-0.276200\pi$
$252$		0			0
$253$		0			0
$254$	8.27604	−	4.77817i	0.519285	−	0.299809i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	5.87868	−	10.1822i	0.363186	−	0.629057i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−15.9706			−0.981064
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.2664	−	12.2782i	−1.29664	−	0.748614i	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−21.2664	−	12.2782i	−1.29664	−	0.748614i	−0.979822	+	0.199874i	$0.935947\pi$
$270$		0			0
$271$	−16.1066	−	27.8975i	−0.978406	−	1.69465i	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−16.1066	−	27.8975i	−0.978406	−	1.69465i	−0.310204	−	0.950670i	$-0.600397\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.369279	+	0.213203i	−0.0222684	+	0.0128567i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	4.48528	−	10.9867i	0.268047	−	0.656578i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$	−20.2263	−	11.6777i	−1.18773	−	0.685736i
$291$		0			0
$292$	14.0000	+	24.2487i	0.819288	+	1.41905i
$293$			34.0711i			1.99045i	0.0975919	+	0.995227i	$0.468886\pi$
$293$			34.0711i			1.99045i	−0.0975919	+	0.995227i	$0.531114\pi$
$294$		0			0
$295$	23.2426			1.35324
$296$		0			0
$297$		0			0
$298$	2.00000	−	3.46410i	0.115857	−	0.200670i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		−	31.4142i		−	1.80768i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	0.840532	+	0.343146i	0.0478938	+	0.0195525i
$309$		0			0
$310$	10.3640	+	17.9509i	0.588633	+	1.01954i
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−12.7426	−	22.0709i	−0.720257	−	1.24752i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−12.7426	−	22.0709i	−0.720257	−	1.24752i	0.240640	−	0.970614i	$-0.422643\pi$
$314$		0			0
$315$		0			0
$316$	15.4558			0.869459
$317$	26.4626	−	15.2782i	1.48629	−	0.858108i	0.486408	−	0.873732i	$-0.338307\pi$
$317$	26.4626	−	15.2782i	1.48629	−	0.858108i	0.999878	+	0.0156238i	$0.00497343\pi$
$318$		0			0
$319$	0.893398	−	1.54741i	0.0500207	−	0.0866384i
$320$	−10.9867	+	6.34315i	−0.614172	+	0.354593i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	30.8797	+	17.8284i	1.69475	+	0.978462i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	36.4558			1.98588			0.992938	−	0.118633i	$-0.0378512\pi$
$337$	36.4558			1.98588			0.992938	+	0.118633i	$0.0378512\pi$
$338$	15.9217	−	9.19239i	0.866025	−	0.500000i
$339$		0			0
$340$		0			0
$341$	−1.37333	+	0.792893i	−0.0743701	+	0.0429376i
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$		0			0
$345$		0			0
$346$	14.0000	−	24.2487i	0.752645	−	1.30362i
$347$	−24.4949	−	14.1421i	−1.31495	−	0.759190i	−0.332043	−	0.943264i	$-0.607738\pi$
$347$	−24.4949	−	14.1421i	−1.31495	−	0.759190i	−0.982912	+	0.184075i	$0.941071\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−7.34847	+	5.69848i	−0.392792	+	0.304597i
$351$		0			0
$352$	−0.485281	−	0.840532i	−0.0258656	−	0.0448005i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−16.0000			−0.845626
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−9.50000	+	16.4545i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			22.2010i			1.16205i
$366$		0			0
$367$	11.6213	−	20.1287i	0.606628	−	1.05071i	−0.385164	−	0.922848i	$-0.625855\pi$
$367$	11.6213	−	20.1287i	0.606628	−	1.05071i	0.991792	−	0.127862i	$-0.0408116\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.61269	−	26.3995i	−0.187561	−	1.37059i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	0.441125	+	0.568852i	0.0224818	+	0.0289914i
$386$		−	30.3848i		−	1.54654i
$387$		0			0
$388$	15.9706	−	27.6618i	0.810782	−	1.40432i
$389$	26.9444	+	15.5563i	1.36613	+	0.788738i	0.990432	−	0.138002i	$-0.0440680\pi$
$389$	26.9444	+	15.5563i	1.36613	+	0.788738i	0.375703	+	0.926740i	$0.377401\pi$
$390$		0			0
$391$		0			0
$392$	19.1757	+	4.92893i	0.968517	+	0.248949i
$393$		0			0
$394$	−10.0000	−	17.3205i	−0.503793	−	0.872595i
$395$	10.6130	+	6.12742i	0.533998	+	0.308304i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$			19.7990i			0.992434i
$399$		0			0
$400$	9.94113			0.497056
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$		0			0
$404$	−34.2929	+	19.7990i	−1.70613	+	0.985037i
$405$		0			0
$406$	14.7279	−	36.0759i	0.730934	−	1.79042i
$407$		0			0
$408$		0			0
$409$	−19.4706	+	33.7240i	−0.962757	+	1.66754i	−0.247234	+	0.968956i	$0.579522\pi$
$409$	−19.4706	+	33.7240i	−0.962757	+	1.66754i	−0.715523	+	0.698589i	$0.753812\pi$
$410$		0			0
$411$		0			0
$412$	−28.0000			−1.37946
$413$	5.25770	+	38.4203i	0.258714	+	1.89054i
$414$		0			0
$415$	14.1360	+	24.4843i	0.693911	+	1.20189i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	39.5980i		−	1.93449i	−0.253849	−	0.967244i	$-0.581697\pi$
$419$		−	39.5980i		−	1.93449i	0.253849	−	0.967244i	$-0.418303\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−14.2426	+	24.6690i	−0.691684	+	1.19803i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			41.3137i			1.99697i
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	−27.3286	+	21.1924i	−1.31181	+	1.01727i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	2.10660	+	3.64874i	0.100543	+	0.174145i	0.911908	−	0.410394i	$-0.134609\pi$
$439$	2.10660	+	3.64874i	0.100543	+	0.174145i	−0.811366	+	0.584539i	$0.801275\pi$
$440$		−	0.769553i		−	0.0366870i
$441$		0			0
$442$		0			0
$443$	−35.6301	+	20.5711i	−1.69284	+	0.977361i	−0.740630	+	0.671913i	$0.765473\pi$
$443$	−35.6301	+	20.5711i	−1.69284	+	0.977361i	−0.952209	+	0.305448i	$0.901194\pi$
$444$		0			0
$445$		0			0
$446$	−0.333226	+	0.192388i	−0.0157787	+	0.00910984i
$447$		0			0
$448$	−12.9706	−	16.7262i	−0.612801	−	0.790237i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	7.27208			0.341296
$455$		0			0
$456$		0			0
$457$	17.9853	+	31.1514i	0.841316	+	1.45720i	0.888783	+	0.458329i	$0.151552\pi$
$457$	17.9853	+	31.1514i	0.841316	+	1.45720i	−0.0474665	+	0.998873i	$0.515115\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			19.7990i			0.922131i	0.887366	+	0.461065i	$0.152533\pi$
$461$			19.7990i			0.922131i	−0.887366	+	0.461065i	$0.847467\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$	−36.0759	+	20.8284i	−1.67478	+	0.966935i
$465$		0			0
$466$		0			0
$467$	−34.2929	+	19.7990i	−1.58688	+	0.916188i	−0.593068	+	0.805153i	$0.702083\pi$
$467$	−34.2929	+	19.7990i	−1.58688	+	0.916188i	−0.993816	+	0.111035i	$0.964583\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	20.7279	−	35.9018i	0.954080	−	1.65251i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		−	28.9289i		−	1.31768i
$483$		0			0
$484$	−21.9411			−0.997324
$485$	21.9329	−	12.6630i	0.995921	−	0.574995i
$486$		0			0
$487$	−18.5919	+	32.2021i	−0.842479	+	1.45922i	0.0453143	+	0.998973i	$0.485571\pi$
$487$	−18.5919	+	32.2021i	−0.842479	+	1.45922i	−0.887793	+	0.460243i	$0.847762\pi$
$488$		0			0
$489$		0			0
$490$	11.2132	+	10.9867i	0.506561	+	0.496326i
$491$		−	21.6863i		−	0.978689i	−0.872091	−	0.489344i	$-0.837236\pi$
$491$		−	21.6863i		−	0.978689i	0.872091	−	0.489344i	$-0.162764\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	36.9706			1.66003
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	20.5595	+	11.8701i	0.919451	+	0.530845i
$501$		0			0
$502$	−17.0919	−	29.6040i	−0.762848	−	1.32129i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−31.3970			−1.39715
$506$		0			0
$507$		0			0
$508$	6.75736	−	11.7041i	0.299809	−	0.519285i
$509$	−34.9997	+	20.2071i	−1.55134	+	0.895664i	−0.553303	+	0.832980i	$0.686633\pi$
$509$	−34.9997	+	20.2071i	−1.55134	+	0.895664i	−0.998033	+	0.0626839i	$0.980034\pi$
$510$		0			0
$511$	−36.6985	+	5.02207i	−1.62345	+	0.222163i
$512$			22.6274i			1.00000i
$513$		0			0
$514$		0			0
$515$	−19.2266	−	11.1005i	−0.847227	−	0.489147i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		−	16.6274i		−	0.726372i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$	−19.5599	+	11.2929i	−0.849626	+	0.490532i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−16.3787	+	28.3687i	−0.708112	+	1.22649i
$536$		0			0
$537$		0			0
$538$	−34.7279			−1.49723
$539$	−0.840532	+	0.857864i	−0.0362043	+	0.0369508i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$	−39.4530	−	22.7782i	−1.69465	−	0.978406i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	−0.301515	+	0.522240i	−0.0128567	+	0.0222684i
$551$		0			0
$552$		0			0
$553$	−7.72792	+	18.9295i	−0.328625	+	0.804963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	27.6513	+	15.9645i	1.17162	+	0.676436i	0.954062	−	0.299611i	$-0.0968568\pi$
$557$	27.6513	+	15.9645i	1.17162	+	0.676436i	0.217560	+	0.976047i	$0.430190\pi$
$558$		0			0
$559$		0			0
$560$	−2.27541	−	16.6274i	−0.0961536	−	0.702637i
$561$		0			0
$562$		0			0
$563$	40.8263	+	23.5711i	1.72062	+	0.993402i	0.917653	+	0.397382i	$0.130081\pi$
$563$	40.8263	+	23.5711i	1.72062	+	0.993402i	0.802970	+	0.596020i	$0.203252\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.22792	+	5.59093i	−0.134380	+	0.232753i	−0.925361	−	0.379088i	$-0.876238\pi$
$577$	−3.22792	+	5.59093i	−0.134380	+	0.232753i	0.790980	+	0.611842i	$0.209571\pi$
$578$	20.8207	+	12.0208i	0.866025	+	0.500000i
$579$		0			0
$580$	−33.0294			−1.37147
$581$	−37.2751	+	28.9056i	−1.54643	+	1.19921i
$582$		0			0
$583$	−0.863961	−	1.49642i	−0.0357816	−	0.0619756i
$584$	34.2929	+	19.7990i	1.41905	+	0.819288i
$585$		0			0
$586$	24.0919	+	41.7284i	0.995227	+	1.72378i
$587$		−	37.6274i		−	1.55305i	−0.630087	−	0.776525i	$-0.716981\pi$
$587$		−	37.6274i		−	1.55305i	0.630087	−	0.776525i	$-0.283019\pi$
$588$		0			0
$589$		0			0
$590$	28.4663	−	16.4350i	1.17194	−	0.676619i
$591$		0			0
$592$		0			0
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		−	5.65685i		−	0.231714i
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−43.4264			−1.77140			−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−43.4264			−1.77140			−0.885700	+	0.464258i	$0.846321\pi$
$602$		0			0
$603$		0			0
$604$	−22.2132	−	38.4744i	−0.903842	−	1.56550i
$605$	−15.0662	−	8.69848i	−0.612529	−	0.353644i
$606$		0			0
$607$	13.5919	+	23.5418i	0.551678	+	0.955533i	0.998154	+	0.0607380i	$0.0193454\pi$
$607$	13.5919	+	23.5418i	0.551678	+	0.955533i	−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$	1.27208	−	0.174080i	0.0512535	−	0.00701388i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	25.3864	+	14.6569i	1.01954	+	0.588633i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3.19848	+	5.53994i	0.127939	+	0.221598i
$626$	−31.2130	−	18.0208i	−1.24752	−	0.720257i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−20.7574			−0.826337			−0.413169	−	0.910654i	$-0.635578\pi$
$631$	−20.7574			−0.826337			−0.413169	+	0.910654i	$0.635578\pi$
$632$	18.9295	−	10.9289i	0.752974	−	0.434730i
$633$		0			0
$634$	21.6066	−	37.4237i	0.858108	−	1.48629i
$635$	9.28009	−	5.35786i	0.368269	−	0.212620i
$636$		0			0
$637$		0			0
$638$		−	2.52691i		−	0.100041i
$639$		0			0
$640$	−8.97056	+	15.5375i	−0.354593	+	0.614172i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		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	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	1.25736	+	2.17781i	0.0493557	+	0.0854865i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.83028	−	4.52082i	0.306423	−	0.176913i	−0.338902	−	0.940822i	$-0.610055\pi$
$653$	7.83028	−	4.52082i	0.306423	−	0.176913i	0.645325	+	0.763909i	$0.276722\pi$
$654$		0			0
$655$	6.59188	−	11.4175i	0.257566	−	0.446118i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			45.2548i			1.76288i	0.472298	+	0.881439i	$0.343425\pi$
$659$			45.2548i			1.76288i	−0.472298	+	0.881439i	$0.656575\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	50.4264			1.95692
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	50.9411			1.96364			0.981818	−	0.189824i	$-0.0607919\pi$
$673$	50.9411			1.96364			0.981818	+	0.189824i	$0.0607919\pi$
$674$	44.6491	−	25.7782i	1.71982	−	0.992938i
$675$		0			0
$676$	13.0000	−	22.5167i	0.500000	−	0.866025i
$677$	37.7464	−	21.7929i	1.45071	−	0.837569i	0.452190	−	0.891922i	$-0.350643\pi$
$677$	37.7464	−	21.7929i	1.45071	−	0.837569i	0.998522	+	0.0543526i	$0.0173095\pi$
$678$		0			0
$679$	25.8934	+	33.3908i	0.993697	+	1.28142i
$680$		0			0
$681$		0			0
$682$	−1.12132	+	1.94218i	−0.0429376	+	0.0743701i
$683$	36.5217	+	21.0858i	1.39746	+	0.806825i	0.994126	−	0.108227i	$-0.0345173\pi$
$683$	36.5217	+	21.0858i	1.39746	+	0.806825i	0.403336	+	0.915052i	$0.367851\pi$
$684$		0			0
$685$		0			0
$686$	−15.6245	+	21.0208i	−0.596547	+	0.802578i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		−	39.5980i		−	1.50529i
$693$		0			0
$694$	−40.0000			−1.51838
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−4.97056	+	12.1753i	−0.187870	+	0.460185i
$701$		−	51.3848i		−	1.94078i	−0.241551	−	0.970388i	$-0.577656\pi$
$701$		−	51.3848i		−	1.94078i	0.241551	−	0.970388i	$-0.422344\pi$
$702$		0			0
$703$		0			0
$704$	−1.18869	−	0.686292i	−0.0448005	−	0.0258656i
$705$		0			0
$706$		0			0
$707$	−7.10228	−	51.8995i	−0.267109	−	1.95188i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−19.5959	+	11.3137i	−0.732334	+	0.422813i
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	14.0000	−	34.2929i	0.521387	−	1.27713i
$722$			26.8701i			1.00000i
$723$		0			0
$724$		0			0
$725$	22.4147	+	12.9411i	0.832461	+	0.480621i
$726$		0			0
$727$	45.6690			1.69377			0.846886	−	0.531775i	$-0.178475\pi$
$727$	45.6690			1.69377			0.846886	+	0.531775i	$0.178475\pi$
$728$		0			0
$729$		0			0
$730$	15.6985	+	27.1906i	0.581027	+	1.00637i
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		−	32.8701i		−	1.21326i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$	−23.0919	−	29.7781i	−0.847730	−	1.09319i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	2.24264	−	3.88437i	0.0821640	−	0.142312i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−50.5988	−	20.6569i	−1.84884	−	0.754785i
$750$		0			0
$751$	−25.8345	−	44.7467i	−0.942715	−	1.63283i	−0.760263	−	0.649616i	$-0.774930\pi$
$751$	−25.8345	−	44.7467i	−0.942715	−	1.63283i	−0.182453	−	0.983215i	$-0.558404\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	35.2254i		−	1.28198i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	29.4264			1.06114			0.530572	−	0.847640i	$-0.321977\pi$
$769$	29.4264			1.06114			0.530572	+	0.847640i	$0.321977\pi$
$770$	0.942506	+	0.384776i	0.0339655	+	0.0138664i
$771$		0			0
$772$	−21.4853	−	37.2136i	−0.773272	−	1.33935i
$773$	−17.1464	−	9.89949i	−0.616714	−	0.356060i	0.158874	−	0.987299i	$-0.449213\pi$
$773$	−17.1464	−	9.89949i	−0.616714	−	0.356060i	−0.775589	+	0.631239i	$0.782547\pi$
$774$		0			0
$775$	−11.4853	−	19.8931i	−0.412563	−	0.714581i
$776$		−	45.1716i		−	1.62156i
$777$		0			0
$778$	44.0000			1.57748
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	26.9706	−	7.52255i	0.963234	−	0.268662i
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−24.4949	−	14.1421i	−0.872595	−	0.503793i
$789$		0			0
$790$	17.3310			0.616608
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	14.0000	+	24.2487i	0.496217	+	0.859473i
$797$		−	11.8701i		−	0.420459i	−0.977652	−	0.210230i	$-0.932579\pi$
$797$		−	11.8701i		−	0.420459i	0.977652	−	0.210230i	$-0.0674211\pi$
$798$		0			0
$799$		0			0
$800$	12.1753	−	7.02944i	0.430463	−	0.248528i
$801$		0			0
$802$		0			0
$803$	−2.08021	+	1.20101i	−0.0734090	+	0.0423827i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−28.0000	+	48.4974i	−0.985037	+	1.70613i
$809$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−7.47156	−	54.5980i	−0.262200	−	1.91601i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			55.0711i			1.92551i
$819$		0			0
$820$		0			0
$821$	44.2033	−	25.5208i	1.54271	−	0.890683i	0.544041	−	0.839059i	$-0.316894\pi$
$821$	44.2033	−	25.5208i	1.54271	−	0.890683i	0.998667	−	0.0516239i	$-0.0164397\pi$
$822$		0			0
$823$	23.0000	−	39.8372i	0.801730	−	1.38864i	−0.116747	−	0.993162i	$-0.537247\pi$
$823$	23.0000	−	39.8372i	0.801730	−	1.38864i	0.918477	−	0.395475i	$-0.129420\pi$
$824$	−34.2929	+	19.7990i	−1.19465	+	0.689730i
$825$		0			0
$826$	33.6066	+	43.3373i	1.16932	+	1.50790i
$827$			19.2843i			0.670580i	0.942115	+	0.335290i	$0.108834\pi$
$827$			19.2843i			0.670580i	−0.942115	+	0.335290i	$0.891166\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$	34.6261	+	19.9914i	1.20189	+	0.693911i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−28.0000	−	48.4974i	−0.967244	−	1.67532i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−79.4558			−2.73986
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.8533	−	10.3076i	0.614172	−	0.354593i
$846$		0			0
$847$	10.9706	−	26.8723i	0.376953	−	0.923342i
$848$			40.2843i			1.38337i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	29.2132	+	50.5988i	0.998487	+	1.72943i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$	15.6985	−	27.1906i	0.533764	−	0.924507i
$866$	17.1464	−	9.89949i	0.582659	−	0.336399i
$867$		0			0
$868$	−18.4853	+	45.2795i	−0.627431	+	1.53689i
$869$			1.32590i			0.0449781i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.8176	+	19.2452i	−0.838987	+	0.650605i
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	5.16010	+	2.97918i	0.174145	+	0.100543i
$879$		0			0
$880$	−0.544156	−	0.942506i	−0.0183435	−	0.0317719i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−29.0919	+	50.3886i	−0.977361	+	1.69284i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	10.9558	+	14.1281i	0.367447	+	0.473841i
$890$		0			0
$891$		0			0
$892$	−0.272078	+	0.471253i	−0.00910984	+	0.0157787i
$893$		0			0
$894$		0			0
$895$	−17.9411			−0.599706
$896$	−27.7128	−	11.3137i	−0.925820	−	0.377964i
$897$		0			0
$898$		0			0
$899$	83.3591	+	48.1274i	2.78018	+	1.60514i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	8.90644	−	5.14214i	0.295571	−	0.170648i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−1.52944	+	2.64906i	−0.0506170	+	0.0876712i
$914$	44.0548	+	25.4350i	1.45720	+	0.841316i
$915$		0			0
$916$		0			0
$917$	20.3643	+	8.31371i	0.672490	+	0.274543i
$918$		0			0
$919$	−25.0000	−	43.3013i	−0.824674	−	1.42838i	−0.902168	−	0.431384i	$-0.858025\pi$
$919$	−25.0000	−	43.3013i	−0.824674	−	1.42838i	0.0774944	−	0.996993i	$-0.475308\pi$
$920$		0			0
$921$		0			0
$922$	14.0000	+	24.2487i	0.461065	+	0.798589i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	31.8434	−	18.3848i	1.04644	−	0.604161i
$927$		0			0
$928$	−29.4558	+	51.0190i	−0.966935	+	1.67478i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−28.0000	+	48.4974i	−0.916188	+	1.58688i
$935$		0			0
$936$		0			0
$937$	12.0294			0.392985			0.196492	−	0.980505i	$-0.437045\pi$
$937$	12.0294			0.392985			0.196492	+	0.980505i	$0.437045\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−49.3995	−	28.5208i	−1.61038	−	0.929752i	−0.989282	−	0.146017i	$-0.953354\pi$
$941$	−49.3995	−	28.5208i	−1.61038	−	0.929752i	−0.621096	−	0.783735i	$-0.713312\pi$
$942$		0			0
$943$		0			0
$944$		−	58.6274i		−	1.90816i
$945$		0			0
$946$		0			0
$947$	−48.9898	+	28.2843i	−1.59195	+	0.919115i	−0.598983	+	0.800762i	$0.704428\pi$
$947$	−48.9898	+	28.2843i	−1.59195	+	0.919115i	−0.992972	+	0.118354i	$0.962238\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.2132	−	47.1347i	−0.877845	−	1.52047i
$962$		0			0
$963$		0			0
$964$	−20.4558	−	35.4306i	−0.658838	−	1.14114i
$965$		−	34.0711i		−	1.09679i
$966$		0			0
$967$	−35.2426			−1.13333			−0.566663	−	0.823949i	$-0.691766\pi$
$967$	−35.2426			−1.13333			−0.566663	+	0.823949i	$0.691766\pi$
$968$	−26.8723	+	15.5147i	−0.863708	+	0.498662i
$969$		0			0
$970$	17.9081	−	31.0178i	0.574995	−	0.995921i
$971$	−24.3463	+	14.0563i	−0.781310	+	0.451090i	−0.836894	−	0.547364i	$-0.815631\pi$
$971$	−24.3463	+	14.0563i	−0.781310	+	0.451090i	0.0555842	+	0.998454i	$0.482298\pi$
$972$		0			0
$973$		0			0
$974$			52.5858i			1.68496i
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$	21.5020	+	5.52691i	0.686858	+	0.176551i
$981$		0			0
$982$	−15.3345	−	26.5602i	−0.489344	−	0.847569i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−11.2132	−	19.4218i	−0.357282	−	0.618831i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−25.1066	+	43.4859i	−0.797537	+	1.38138i	0.123678	+	0.992322i	$0.460531\pi$
$991$	−25.1066	+	43.4859i	−0.797537	+	1.38138i	−0.921215	+	0.389053i	$0.872802\pi$
$992$	45.2795	−	26.1421i	1.43763	−	0.830014i
$993$		0			0
$994$		0			0
$995$			22.2010i			0.703819i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cj.b.109.4	yes	8
3.2	odd	2	inner	504.2.cj.b.109.1	yes	8
4.3	odd	2		2016.2.cr.b.1873.3		8
7.2	even	3	inner	504.2.cj.b.37.1	✓	8
8.3	odd	2		2016.2.cr.b.1873.2		8
8.5	even	2	inner	504.2.cj.b.109.1	yes	8
12.11	even	2		2016.2.cr.b.1873.2		8
21.2	odd	6	inner	504.2.cj.b.37.4	yes	8
24.5	odd	2	CM	504.2.cj.b.109.4	yes	8
24.11	even	2		2016.2.cr.b.1873.3		8
28.23	odd	6		2016.2.cr.b.1297.2		8
56.37	even	6	inner	504.2.cj.b.37.4	yes	8
56.51	odd	6		2016.2.cr.b.1297.3		8
84.23	even	6		2016.2.cr.b.1297.3		8
168.107	even	6		2016.2.cr.b.1297.2		8
168.149	odd	6	inner	504.2.cj.b.37.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.cj.b.37.1	✓	8	7.2	even	3	inner
504.2.cj.b.37.1	✓	8	168.149	odd	6	inner
504.2.cj.b.37.4	yes	8	21.2	odd	6	inner
504.2.cj.b.37.4	yes	8	56.37	even	6	inner
504.2.cj.b.109.1	yes	8	3.2	odd	2	inner
504.2.cj.b.109.1	yes	8	8.5	even	2	inner
504.2.cj.b.109.4	yes	8	1.1	even	1	trivial
504.2.cj.b.109.4	yes	8	24.5	odd	2	CM
2016.2.cr.b.1297.2		8	28.23	odd	6
2016.2.cr.b.1297.2		8	168.107	even	6
2016.2.cr.b.1297.3		8	56.51	odd	6
2016.2.cr.b.1297.3		8	84.23	even	6
2016.2.cr.b.1873.2		8	8.3	odd	2
2016.2.cr.b.1873.2		8	12.11	even	2
2016.2.cr.b.1873.3		8	4.3	odd	2
2016.2.cr.b.1873.3		8	24.11	even	2

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(1.37333 - 0.792893i) q^{5} +(1.62132 + 2.09077i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(1.37333 - 0.792893i) q^{5} +(1.62132 + 2.09077i) q^{7} -2.82843i q^{8} +(1.12132 - 1.94218i) q^{10} +(0.148586 + 0.0857864i) q^{11} +(3.46410 + 1.41421i) q^{14} +(-2.00000 - 3.46410i) q^{16} -3.17157i q^{20} +0.242641 q^{22} +(-1.24264 + 2.15232i) q^{25} +(5.24264 - 0.717439i) q^{28} -10.4142i q^{29} +(-4.62132 + 8.00436i) q^{31} +(-4.89898 - 2.82843i) q^{32} +(3.88437 + 1.58579i) q^{35} +(-2.24264 - 3.88437i) q^{40} +(0.297173 - 0.171573i) q^{44} +(-1.74264 + 6.77962i) q^{49} +3.51472i q^{50} +(-8.72180 - 5.03553i) q^{53} +0.272078 q^{55} +(5.91359 - 4.58579i) q^{56} +(-7.36396 - 12.7548i) q^{58} +(12.6932 + 7.32843i) q^{59} +13.0711i q^{62} -8.00000 q^{64} +(5.87868 - 0.804479i) q^{70} +(-7.00000 + 12.1244i) q^{73} +(0.0615465 + 0.449747i) q^{77} +(3.86396 + 6.69258i) q^{79} +(-5.49333 - 3.17157i) q^{80} +17.8284i q^{83} +(0.242641 - 0.420266i) q^{88} +15.9706 q^{97} +(2.65962 + 9.53553i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) 8 * q + 8 * q^4 - 4 * q^7	\( 8 q + 8 q^{4} - 4 q^{7} - 8 q^{10} - 16 q^{16} - 32 q^{22} + 24 q^{25} + 8 q^{28} - 20 q^{31} + 16 q^{40} + 20 q^{49} + 104 q^{55} - 8 q^{58} - 64 q^{64} + 64 q^{70} - 56 q^{73} - 20 q^{79} - 32 q^{88} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 4 * q^7 - 8 * q^10 - 16 * q^16 - 32 * q^22 + 24 * q^25 + 8 * q^28 - 20 * q^31 + 16 * q^40 + 20 * q^49 + 104 * q^55 - 8 * q^58 - 64 * q^64 + 64 * q^70 - 56 * q^73 - 20 * q^79 - 32 * q^88 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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