LMFDB - Embedded newform 504.2.cj.b.37.3

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			37.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	504.37
Dual form			504.2.cj.b.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.82282 - 2.20711i) q^{5} +(-2.62132 + 0.358719i) q^{7} +2.82843i q^{8} +O(q^{10})$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.82282 - 2.20711i) q^{5} +(-2.62132 + 0.358719i) q^{7} +2.82843i q^{8} +(-3.12132 - 5.40629i) q^{10} +(-5.04757 + 2.91421i) q^{11} +(-3.46410 - 1.41421i) q^{14} +(-2.00000 + 3.46410i) q^{16} -8.82843i q^{20} -8.24264 q^{22} +(7.24264 + 12.5446i) q^{25} +(-3.24264 - 4.18154i) q^{28} -7.58579i q^{29} +(-0.378680 - 0.655892i) q^{31} +(-4.89898 + 2.82843i) q^{32} +(10.8126 + 4.41421i) q^{35} +(6.24264 - 10.8126i) q^{40} +(-10.0951 - 5.82843i) q^{44} +(6.74264 - 1.88064i) q^{49} +20.4853i q^{50} +(-3.52565 + 2.03553i) q^{53} +25.7279 q^{55} +(-1.01461 - 7.41421i) q^{56} +(5.36396 - 9.29065i) q^{58} +(-2.89525 + 1.67157i) q^{59} -1.07107i q^{62} -8.00000 q^{64} +(10.1213 + 13.0519i) q^{70} +(-7.00000 - 12.1244i) q^{73} +(12.1859 - 9.44975i) q^{77} +(-8.86396 + 15.3528i) q^{79} +(15.2913 - 8.82843i) q^{80} +12.1716i q^{83} +(-8.24264 - 14.2767i) q^{88} -17.9706 q^{97} +(9.58783 + 2.46447i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

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

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
$5$	−3.82282	−	2.20711i	−1.70962	−	0.987048i	−0.935021	−	0.354593i	$-0.884620\pi$
$5$	−3.82282	−	2.20711i	−1.70962	−	0.987048i	−0.774597	−	0.632456i	$-0.782047\pi$
$6$		0			0
$7$	−2.62132	+	0.358719i	−0.990766	+	0.135583i
$7$	−2.62132	+	0.358719i	−0.990766	+	0.135583i
$8$			2.82843i			1.00000i
$9$		0			0
$10$	−3.12132	−	5.40629i	−0.987048	−	1.70962i
$11$	−5.04757	+	2.91421i	−1.52190	+	0.878668i	−0.522233	+	0.852803i	$0.674901\pi$
$11$	−5.04757	+	2.91421i	−1.52190	+	0.878668i	−0.999665	+	0.0258656i	$0.991766\pi$
$12$		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	$-0.166667\pi$
$17$		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	$0.666667\pi$
$20$		−	8.82843i		−	1.97410i
$21$		0			0
$22$	−8.24264			−1.75734
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	7.24264	+	12.5446i	1.44853	+	2.50892i
$26$		0			0
$27$		0			0
$28$	−3.24264	−	4.18154i	−0.612801	−	0.790237i
$29$		−	7.58579i		−	1.40865i	−0.709880	−	0.704323i	$-0.751251\pi$
$29$		−	7.58579i		−	1.40865i	0.709880	−	0.704323i	$-0.248749\pi$
$30$		0			0
$31$	−0.378680	−	0.655892i	−0.0680129	−	0.117802i	0.830014	−	0.557743i	$-0.188333\pi$
$31$	−0.378680	−	0.655892i	−0.0680129	−	0.117802i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−4.89898	+	2.82843i	−0.866025	+	0.500000i
$33$		0			0
$34$		0			0
$35$	10.8126	+	4.41421i	1.82766	+	0.746138i
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$	6.24264	−	10.8126i	0.987048	−	1.70962i
$41$		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			$\pi$
$44$	−10.0951	−	5.82843i	−1.52190	−	0.878668i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	6.74264	−	1.88064i	0.963234	−	0.268662i
$50$			20.4853i			2.89706i
$51$		0			0
$52$		0			0
$53$	−3.52565	+	2.03553i	−0.484285	+	0.279602i	−0.722200	−	0.691684i	$-0.756869\pi$
$53$	−3.52565	+	2.03553i	−0.484285	+	0.279602i	0.237915	+	0.971286i	$0.423536\pi$
$54$		0			0
$55$	25.7279			3.46915
$56$	−1.01461	−	7.41421i	−0.135583	−	0.990766i
$57$		0			0
$58$	5.36396	−	9.29065i	0.704323	−	1.21992i
$59$	−2.89525	+	1.67157i	−0.376929	+	0.217620i	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−2.89525	+	1.67157i	−0.376929	+	0.217620i	0.299552	+	0.954080i	$0.403163\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		−	1.07107i		−	0.136026i
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	10.1213	+	13.0519i	1.20973	+	1.56000i
$71$		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	−0.906208	−	0.422833i	$-0.861036\pi$
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	0.0869195	−	0.996215i	$-0.472298\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.1859	−	9.44975i	1.38871	−	1.07690i
$78$		0			0
$79$	−8.86396	+	15.3528i	−0.997274	+	1.72733i	−0.434730	+	0.900561i	$0.643156\pi$
$79$	−8.86396	+	15.3528i	−0.997274	+	1.72733i	−0.562544	+	0.826767i	$0.690177\pi$
$80$	15.2913	−	8.82843i	1.70962	−	0.987048i
$81$		0			0
$82$		0			0
$83$			12.1716i			1.33600i	0.744160	+	0.668002i	$0.232850\pi$
$83$			12.1716i			1.33600i	−0.744160	+	0.668002i	$0.767150\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−8.24264	−	14.2767i	−0.878668	−	1.52190i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−17.9706			−1.82463			−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−17.9706			−1.82463			−0.912317	+	0.409484i	$0.865709\pi$
$98$	9.58783	+	2.46447i	0.968517	+	0.248949i
$99$		0			0
$100$	−14.4853	+	25.0892i	−1.44853	+	2.50892i
$101$	−17.1464	+	9.89949i	−1.70613	+	0.985037i	−0.766894	+	0.641774i	$0.778199\pi$
$101$	−17.1464	+	9.89949i	−1.70613	+	0.985037i	−0.939239	+	0.343263i	$0.888468\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$		0			0
$105$		0			0
$106$	−5.75736			−0.559204
$107$	8.09140	+	4.67157i	0.782225	+	0.451618i	0.837218	−	0.546869i	$-0.184180\pi$
$107$	8.09140	+	4.67157i	0.782225	+	0.451618i	−0.0549930	+	0.998487i	$0.517514\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	31.5101	+	18.1924i	3.00437	+	1.73458i
$111$		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
$116$	13.1390	−	7.58579i	1.21992	−	0.704323i
$117$		0			0
$118$	−4.72792			−0.435241
$119$		0			0
$120$		0			0
$121$	11.4853	−	19.8931i	1.04412	−	1.80846i
$122$		0			0
$123$		0			0
$124$	0.757359	−	1.31178i	0.0680129	−	0.117802i
$125$		−	41.8701i		−	3.74497i
$126$		0			0
$127$	15.2426			1.35257			0.676283	−	0.736642i	$-0.263590\pi$
$127$	15.2426			1.35257			0.676283	+	0.736642i	$0.263590\pi$
$128$	−9.79796	−	5.65685i	−0.866025	−	0.500000i
$129$		0			0
$130$		0			0
$131$	12.3960	+	7.15685i	1.08305	+	0.625297i	0.931717	−	0.363186i	$-0.118311\pi$
$131$	12.3960	+	7.15685i	1.08305	+	0.625297i	0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	3.16693	+	23.1421i	0.267654	+	1.95587i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−16.7426	+	28.9991i	−1.39040	+	2.40824i
$146$		−	19.7990i		−	1.63858i
$147$		0			0
$148$		0			0
$149$	2.44949	+	1.41421i	0.200670	+	0.115857i	0.596968	−	0.802265i	$-0.296372\pi$
$149$	2.44949	+	1.41421i	0.200670	+	0.115857i	−0.396298	+	0.918122i	$0.629705\pi$
$150$		0			0
$151$	−10.1066	−	17.5051i	−0.822464	−	1.42455i	−0.903842	−	0.427865i	$-0.859266\pi$
$151$	−10.1066	−	17.5051i	−0.822464	−	1.42455i	0.0813788	−	0.996683i	$-0.474068\pi$
$152$		0			0
$153$		0			0
$154$	21.6066	−	2.95680i	1.74111	−	0.238265i
$155$			3.34315i			0.268528i
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−21.7122	+	12.5355i	−1.72733	+	0.997274i
$159$		0			0
$160$	24.9706			1.97410
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$	−8.60660	+	14.9071i	−0.668002	+	1.15701i
$167$		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	0.981023	−	0.193892i	$-0.0621112\pi$
$173$	17.1464	+	9.89949i	1.30362	+	0.752645i	0.322596	+	0.946537i	$0.395445\pi$
$174$		0			0
$175$	−23.4853	−	30.2854i	−1.77532	−	2.28936i
$176$		−	23.3137i		−	1.75734i
$177$		0			0
$178$		0			0
$179$	−9.79796	+	5.65685i	−0.732334	+	0.422813i	−0.819275	−	0.573400i	$-0.805624\pi$
$179$	−9.79796	+	5.65685i	−0.732334	+	0.422813i	0.0869415	+	0.996213i	$0.472291\pi$
$180$		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	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	2.25736	+	3.90986i	0.162488	+	0.281438i	0.935760	−	0.352636i	$-0.114715\pi$
$193$	2.25736	+	3.90986i	0.162488	+	0.281438i	−0.773272	+	0.634074i	$0.781381\pi$
$194$	−22.0094	−	12.7071i	−1.58018	−	0.912317i
$195$		0			0
$196$	10.0000	+	9.79796i	0.714286	+	0.699854i
$197$			14.1421i			1.00759i	0.863825	+	0.503793i	$0.168062\pi$
$197$			14.1421i			1.00759i	−0.863825	+	0.503793i	$0.831938\pi$
$198$		0			0
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	0.503774	−	0.863836i	$-0.331945\pi$
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	−0.999990	+	0.00436292i	$0.998611\pi$
$200$	−35.4815	+	20.4853i	−2.50892	+	1.44853i
$201$		0			0
$202$	−28.0000			−1.97007
$203$	2.72117	+	19.8848i	0.190989	+	1.39564i
$204$		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			$\pi$
$212$	−7.05130	−	4.07107i	−0.484285	−	0.279602i
$213$		0			0
$214$	6.60660	+	11.4430i	0.451618	+	0.782225i
$215$		0			0
$216$		0			0
$217$	1.22792	+	1.58346i	0.0833568	+	0.107493i
$218$		0			0
$219$		0			0
$220$	25.7279	+	44.5621i	1.73458	+	3.00437i
$221$		0			0
$222$		0			0
$223$	−25.7279			−1.72287			−0.861435	−	0.507869i	$-0.830434\pi$
$223$	−25.7279			−1.72287			−0.861435	+	0.507869i	$0.830434\pi$
$224$	11.8272	−	9.17157i	0.790237	−	0.612801i
$225$		0			0
$226$		0			0
$227$	20.0417	−	11.5711i	1.33021	−	0.767999i	0.344881	−	0.938647i	$-0.387919\pi$
$227$	20.0417	−	11.5711i	1.33021	−	0.767999i	0.985332	+	0.170648i	$0.0545860\pi$
$228$		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
$232$	21.4558			1.40865
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$	−5.79050	−	3.34315i	−0.376929	−	0.217620i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−15.2279	−	26.3755i	−0.980917	−	1.69900i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	−15.2279	−	26.3755i	−0.980917	−	1.69900i	−0.322078	−	0.946713i	$-0.604381\pi$
$242$	28.1331	−	16.2426i	1.80846	−	1.04412i
$243$		0			0
$244$		0			0
$245$	−29.9267	−	7.69239i	−1.91195	−	0.491449i
$246$		0			0
$247$		0			0
$248$	1.85514	−	1.07107i	0.117802	−	0.0680129i
$249$		0			0
$250$	29.6066	−	51.2801i	1.87249	−	3.24324i
$251$		−	29.8284i		−	1.88275i	−0.337357	−	0.941377i	$-0.609533\pi$
$251$		−	29.8284i		−	1.88275i	0.337357	−	0.941377i	$-0.390467\pi$
$252$		0			0
$253$		0			0
$254$	18.6683	+	10.7782i	1.17136	+	0.676283i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	10.1213	+	17.5306i	0.625297	+	1.08305i
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	17.9706			1.10392
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.67796	+	3.27817i	−0.346192	+	0.199874i	−0.663007	−	0.748614i	$-0.730720\pi$
$269$	−5.67796	+	3.27817i	−0.346192	+	0.199874i	0.316815	+	0.948487i	$0.397387\pi$
$270$		0			0
$271$	5.10660	−	8.84489i	0.310204	−	0.537289i	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	5.10660	−	8.84489i	0.310204	−	0.537289i	0.978406	+	0.206691i	$0.0662693\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−73.1154	−	42.2132i	−4.40903	−	2.54555i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$	−12.4853	+	30.5826i	−0.746138	+	1.82766i
$281$		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	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$	−41.0109	+	23.6777i	−2.40824	+	1.39040i
$291$		0			0
$292$	14.0000	−	24.2487i	0.819288	−	1.41905i
$293$			19.9289i			1.16426i	0.813095	+	0.582130i	$0.197781\pi$
$293$			19.9289i			1.16426i	−0.813095	+	0.582130i	$0.802219\pi$
$294$		0			0
$295$	14.7574			0.859207
$296$		0			0
$297$		0			0
$298$	2.00000	+	3.46410i	0.115857	+	0.200670i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		−	28.5858i		−	1.64493i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	28.5533	+	11.6569i	1.62698	+	0.664211i
$309$		0			0
$310$	−2.36396	+	4.09450i	−0.134264	+	0.232552i
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−4.25736	+	7.37396i	−0.240640	+	0.416801i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−4.25736	+	7.37396i	−0.240640	+	0.416801i	0.720257	+	0.693708i	$0.244024\pi$
$314$		0			0
$315$		0			0
$316$	−35.4558			−1.99455
$317$	0.481813	+	0.278175i	0.0270613	+	0.0156238i	0.513470	−	0.858108i	$-0.328360\pi$
$317$	0.481813	+	0.278175i	0.0270613	+	0.0156238i	−0.486408	+	0.873732i	$0.661693\pi$
$318$		0			0
$319$	22.1066	+	38.2898i	1.23773	+	2.14381i
$320$	30.5826	+	17.6569i	1.70962	+	0.987048i
$321$		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	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−21.0818	+	12.1716i	−1.15701	+	0.668002i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.4558			−0.787460			−0.393730	−	0.919226i	$-0.628816\pi$
$337$	−14.4558			−0.787460			−0.393730	+	0.919226i	$0.628816\pi$
$338$	15.9217	+	9.19239i	0.866025	+	0.500000i
$339$		0			0
$340$		0			0
$341$	3.82282	+	2.20711i	0.207017	+	0.119522i
$342$		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	−0.982912	−	0.184075i	$-0.941071\pi$
$347$	−24.4949	+	14.1421i	−1.31495	+	0.759190i	−0.332043	+	0.943264i	$0.607738\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−7.34847	−	53.6985i	−0.392792	−	2.87030i
$351$		0			0
$352$	16.4853	−	28.5533i	0.878668	−	1.52190i
$353$		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
$358$	−16.0000			−0.845626
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−9.50000	−	16.4545i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			61.7990i			3.23471i
$366$		0			0
$367$	7.37868	+	12.7802i	0.385164	+	0.667124i	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	7.37868	+	12.7802i	0.385164	+	0.667124i	−0.606628	+	0.794986i	$0.707478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.51167	−	6.60051i	0.441904	−	0.342681i
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		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	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	−67.4411	+	9.22911i	−3.43712	+	0.470359i
$386$			6.38478i			0.324977i
$387$		0			0
$388$	−17.9706	−	31.1259i	−0.912317	−	1.58018i
$389$	26.9444	−	15.5563i	1.36613	−	0.788738i	0.375703	−	0.926740i	$-0.377401\pi$
$389$	26.9444	−	15.5563i	1.36613	−	0.788738i	0.990432	+	0.138002i	$0.0440680\pi$
$390$		0			0
$391$		0			0
$392$	5.31925	+	19.0711i	0.268662	+	0.963234i
$393$		0			0
$394$	−10.0000	+	17.3205i	−0.503793	+	0.872595i
$395$	67.7707	−	39.1274i	3.40991	−	1.96871i
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		−	19.7990i		−	0.992434i
$399$		0			0
$400$	−57.9411			−2.89706
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$	−34.2929	−	19.7990i	−1.70613	−	0.985037i
$405$		0			0
$406$	−10.7279	+	26.2779i	−0.532418	+	1.30415i
$407$		0			0
$408$		0			0
$409$	14.4706	+	25.0637i	0.715523	+	1.23932i	0.962757	+	0.270367i	$0.0871450\pi$
$409$	14.4706	+	25.0637i	0.715523	+	1.23932i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$	−28.0000			−1.37946
$413$	6.98975	−	5.42031i	0.343943	−	0.266716i
$414$		0			0
$415$	26.8640	−	46.5297i	1.31870	−	2.28406i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			39.5980i			1.93449i	0.253849	+	0.967244i	$0.418303\pi$
$419$			39.5980i			1.93449i	−0.253849	+	0.967244i	$0.581697\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−5.75736	−	9.97204i	−0.279602	−	0.484285i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			18.6863i			0.903236i
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	0.384213	+	2.80761i	0.0184428	+	0.134770i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−19.1066	+	33.0936i	−0.911908	+	1.57947i	−0.100543	+	0.994933i	$0.532058\pi$
$439$	−19.1066	+	33.0936i	−0.911908	+	1.57947i	−0.811366	+	0.584539i	$0.801275\pi$
$440$			72.7696i			3.46915i
$441$		0			0
$442$		0			0
$443$	11.1352	+	6.42893i	0.529051	+	0.305448i	0.740630	−	0.671913i	$-0.234527\pi$
$443$	11.1352	+	6.42893i	0.529051	+	0.305448i	−0.211579	+	0.977361i	$0.567861\pi$
$444$		0			0
$445$		0			0
$446$	−31.5101	−	18.1924i	−1.49205	−	0.861435i
$447$		0			0
$448$	20.9706	−	2.86976i	0.990766	−	0.135583i
$449$		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
$454$	32.7279			1.53600
$455$		0			0
$456$		0			0
$457$	1.01472	−	1.75754i	0.0474665	−	0.0822145i	−0.841316	−	0.540544i	$-0.818219\pi$
$457$	1.01472	−	1.75754i	0.0474665	−	0.0822145i	0.888783	+	0.458329i	$0.151552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	19.7990i		−	0.922131i	−0.887366	−	0.461065i	$-0.847467\pi$
$461$		−	19.7990i		−	0.922131i	0.887366	−	0.461065i	$-0.152533\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$	26.2779	+	15.1716i	1.21992	+	0.704323i
$465$		0			0
$466$		0			0
$467$	−34.2929	−	19.7990i	−1.58688	−	0.916188i	−0.993816	−	0.111035i	$-0.964583\pi$
$467$	−34.2929	−	19.7990i	−1.58688	−	0.916188i	−0.593068	−	0.805153i	$-0.702083\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−4.72792	−	8.18900i	−0.217620	−	0.376929i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		−	43.0711i		−	1.96183i
$483$		0			0
$484$	45.9411			2.08823
$485$	68.6982	+	39.6630i	3.11943	+	1.80100i
$486$		0			0
$487$	19.5919	+	33.9341i	0.887793	+	1.53770i	0.842479	+	0.538730i	$0.181096\pi$
$487$	19.5919	+	33.9341i	0.887793	+	1.53770i	0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$	−31.2132	−	30.5826i	−1.41007	−	1.38158i
$491$		−	44.3137i		−	1.99985i	−0.0122607	−	0.999925i	$-0.503903\pi$
$491$		−	44.3137i		−	1.99985i	0.0122607	−	0.999925i	$-0.496097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	3.02944			0.136026
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	72.5211	−	41.8701i	3.24324	−	1.87249i
$501$		0			0
$502$	21.0919	−	36.5322i	0.941377	−	1.63051i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	87.3970			3.88911
$506$		0			0
$507$		0			0
$508$	15.2426	+	26.4010i	0.676283	+	1.17136i
$509$	32.5502	+	18.7929i	1.44276	+	0.832980i	0.998033	−	0.0626839i	$-0.0199660\pi$
$509$	32.5502	+	18.7929i	1.44276	+	0.832980i	0.444731	+	0.895664i	$0.353299\pi$
$510$		0			0
$511$	22.6985	+	29.2708i	1.00412	+	1.29486i
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$		0			0
$515$	53.5195	−	30.8995i	2.35835	−	1.36159i
$516$		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	$0.666667\pi$
$524$			28.6274i			1.25059i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$	22.0094	+	12.7071i	0.956025	+	0.551961i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−20.6213	−	35.7172i	−0.891537	−	1.54419i
$536$		0			0
$537$		0			0
$538$	−9.27208			−0.399748
$539$	−28.5533	+	29.1421i	−1.22988	+	1.25524i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	12.5086	−	7.22183i	0.537289	−	0.310204i
$543$		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
$550$	−59.6985	−	103.401i	−2.54555	−	4.40903i
$551$		0			0
$552$		0			0
$553$	17.7279	−	43.4244i	0.753868	−	1.84659i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−39.8987	+	23.0355i	−1.69056	+	0.976047i	−0.736501	+	0.676436i	$0.763523\pi$
$557$	−39.8987	+	23.0355i	−1.69056	+	0.976047i	−0.954062	+	0.299611i	$0.903143\pi$
$558$		0			0
$559$		0			0
$560$	−36.9164	+	28.6274i	−1.56000	+	1.20973i
$561$		0			0
$562$		0			0
$563$	−16.3314	+	9.42893i	−0.688286	+	0.397382i	−0.802970	−	0.596020i	$-0.796748\pi$
$563$	−16.3314	+	9.42893i	−0.688286	+	0.397382i	0.114684	+	0.993402i	$0.463415\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	22.2279	+	38.4999i	0.925361	+	1.60277i	0.790980	+	0.611842i	$0.209571\pi$
$577$	22.2279	+	38.4999i	0.925361	+	1.60277i	0.134380	+	0.990930i	$0.457096\pi$
$578$	20.8207	−	12.0208i	0.866025	−	0.500000i
$579$		0			0
$580$	−66.9706			−2.78080
$581$	−4.36618	−	31.9056i	−0.181140	−	1.32367i
$582$		0			0
$583$	11.8640	−	20.5490i	0.491355	−	0.851052i
$584$	34.2929	−	19.7990i	1.41905	−	0.819288i
$585$		0			0
$586$	−14.0919	+	24.4079i	−0.582130	+	1.00828i
$587$			7.62742i			0.314817i	0.987534	+	0.157409i	$0.0503140\pi$
$587$			7.62742i			0.314817i	−0.987534	+	0.157409i	$0.949686\pi$
$588$		0			0
$589$		0			0
$590$	18.0740	+	10.4350i	0.744095	+	0.429603i
$591$		0			0
$592$		0			0
$593$		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
$596$			5.65685i			0.231714i
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	41.4264			1.68982			0.844909	−	0.534910i	$-0.179654\pi$
$601$	41.4264			1.68982			0.844909	+	0.534910i	$0.179654\pi$
$602$		0			0
$603$		0			0
$604$	20.2132	−	35.0103i	0.822464	−	1.42455i
$605$	−87.8124	+	50.6985i	−3.57008	+	2.06119i
$606$		0			0
$607$	−24.5919	+	42.5944i	−0.998154	+	1.72885i	−0.446476	+	0.894795i	$0.647321\pi$
$607$	−24.5919	+	42.5944i	−0.998154	+	1.72885i	−0.551678	+	0.834058i	$0.686012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$	26.7279	+	34.4669i	1.07690	+	1.38871i
$617$		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	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	−5.79050	+	3.34315i	−0.232552	+	0.134264i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−56.1985	+	97.3386i	−2.24794	+	3.89355i
$626$	−10.4284	+	6.02082i	−0.416801	+	0.240640i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29.2426			−1.16413			−0.582066	−	0.813142i	$-0.697755\pi$
$631$	−29.2426			−1.16413			−0.582066	+	0.813142i	$0.697755\pi$
$632$	−43.4244	−	25.0711i	−1.72733	−	0.997274i
$633$		0			0
$634$	0.393398	+	0.681386i	0.0156238	+	0.0270613i
$635$	−58.2699	−	33.6421i	−2.31237	−	1.33505i
$636$		0			0
$637$		0			0
$638$			62.5269i			2.47546i
$639$		0			0
$640$	24.9706	+	43.2503i	0.987048	+	1.70962i
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		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
$649$	9.74264	−	16.8747i	0.382432	−	0.662392i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	33.8110	+	19.5208i	1.32313	+	0.763909i	0.984226	−	0.176913i	$-0.0566112\pi$
$653$	33.8110	+	19.5208i	1.32313	+	0.763909i	0.338902	+	0.940822i	$0.389945\pi$
$654$		0			0
$655$	−31.5919	−	54.7187i	−1.23440	−	2.13804i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	45.2548i		−	1.76288i	−0.472298	−	0.881439i	$-0.656575\pi$
$659$		−	45.2548i		−	1.76288i	0.472298	−	0.881439i	$-0.343425\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	−34.4264			−1.33600
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−16.9411			−0.653032			−0.326516	−	0.945192i	$-0.605875\pi$
$673$	−16.9411			−0.653032			−0.326516	+	0.945192i	$0.605875\pi$
$674$	−17.7047	−	10.2218i	−0.681960	−	0.393730i
$675$		0			0
$676$	13.0000	+	22.5167i	0.500000	+	0.866025i
$677$	−40.1959	−	23.2071i	−1.54485	−	0.891922i	−0.998522	−	0.0543526i	$-0.982690\pi$
$677$	−40.1959	−	23.2071i	−1.54485	−	0.891922i	−0.546332	−	0.837569i	$-0.683976\pi$
$678$		0			0
$679$	47.1066	−	6.44639i	1.80779	−	0.247390i
$680$		0			0
$681$		0			0
$682$	3.12132	+	5.40629i	0.119522	+	0.207017i
$683$	−41.4206	+	23.9142i	−1.58492	+	0.915052i	−0.590790	+	0.806825i	$0.701184\pi$
$683$	−41.4206	+	23.9142i	−1.58492	+	0.915052i	−0.994126	+	0.108227i	$0.965483\pi$
$684$		0			0
$685$		0			0
$686$	−26.0168	−	3.02082i	−0.993327	−	0.115335i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$			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
$700$	28.9706	−	70.9631i	1.09498	−	2.68215i
$701$		−	14.6152i		−	0.552009i	−0.961156	−	0.276005i	$-0.910989\pi$
$701$		−	14.6152i		−	0.552009i	0.961156	−	0.276005i	$-0.0890105\pi$
$702$		0			0
$703$		0			0
$704$	40.3805	−	23.3137i	1.52190	−	0.878668i
$705$		0			0
$706$		0			0
$707$	41.3951	−	32.1005i	1.55682	−	1.20726i
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		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	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	14.0000	−	34.2929i	0.521387	−	1.27713i
$722$		−	26.8701i		−	1.00000i
$723$		0			0
$724$		0			0
$725$	95.1608	−	54.9411i	3.53418	−	2.04046i
$726$		0			0
$727$	−47.6690			−1.76795			−0.883974	−	0.467537i	$-0.845142\pi$
$727$	−47.6690			−1.76795			−0.883974	+	0.467537i	$0.845142\pi$
$728$		0			0
$729$		0			0
$730$	−43.6985	+	75.6880i	−1.61735	+	2.80134i
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$			20.8701i			0.770328i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$	15.0919	−	2.06528i	0.554040	−	0.0758187i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−6.24264	−	10.8126i	−0.228713	−	0.396142i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.8859	−	9.34315i	−0.836234	−	0.341391i
$750$		0			0
$751$	20.8345	−	36.0865i	0.760263	−	1.31681i	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	20.8345	−	36.0865i	0.760263	−	1.31681i	0.942715	−	0.333599i	$-0.108263\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			89.2254i			3.24724i
$756$		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	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−55.4264			−1.99873			−0.999364	−	0.0356685i	$-0.988644\pi$
$769$	−55.4264			−1.99873			−0.999364	+	0.0356685i	$0.988644\pi$
$770$	−89.1241	−	36.3848i	−3.21181	−	1.31122i
$771$		0			0
$772$	−4.51472	+	7.81972i	−0.162488	+	0.281438i
$773$	−17.1464	+	9.89949i	−0.616714	+	0.356060i	−0.775589	−	0.631239i	$-0.782547\pi$
$773$	−17.1464	+	9.89949i	−0.616714	+	0.356060i	0.158874	+	0.987299i	$0.449213\pi$
$774$		0			0
$775$	5.48528	−	9.50079i	0.197037	−	0.341278i
$776$		−	50.8284i		−	1.82463i
$777$		0			0
$778$	44.0000			1.57748
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6.97056	+	27.1185i	−0.248949	+	0.968517i
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−24.4949	+	14.1421i	−0.872595	+	0.503793i
$789$		0			0
$790$	110.669			3.93743
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	14.0000	−	24.2487i	0.496217	−	0.859473i
$797$			41.8701i			1.48311i	0.670890	+	0.741557i	$0.265912\pi$
$797$			41.8701i			1.48311i	−0.670890	+	0.741557i	$0.734088\pi$
$798$		0			0
$799$		0			0
$800$	−70.9631	−	40.9706i	−2.50892	−	1.44853i
$801$		0			0
$802$		0			0
$803$	70.6659	+	40.7990i	2.49375	+	1.43977i
$804$		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	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−31.7203	+	24.5980i	−1.11316	+	0.863220i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			40.9289i			1.43105i
$819$		0			0
$820$		0			0
$821$	−2.56202	−	1.47918i	−0.0894152	−	0.0516239i	0.454626	−	0.890683i	$-0.349773\pi$
$821$	−2.56202	−	1.47918i	−0.0894152	−	0.0516239i	−0.544041	+	0.839059i	$0.683106\pi$
$822$		0			0
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	0.918477	+	0.395475i	$0.129420\pi$
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	−0.116747	+	0.993162i	$0.537247\pi$
$824$	−34.2929	−	19.7990i	−1.19465	−	0.689730i
$825$		0			0
$826$	12.3934	−	1.69600i	0.431221	−	0.0590113i
$827$		−	37.2843i		−	1.29650i	−0.761427	−	0.648251i	$-0.775501\pi$
$827$		−	37.2843i		−	1.29650i	0.761427	−	0.648251i	$-0.224499\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$	65.8030	−	37.9914i	2.28406	−	1.31870i
$831$		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
$841$	−28.5442			−0.984281
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−49.6967	−	28.6924i	−1.70962	−	0.987048i
$846$		0			0
$847$	−22.9706	+	56.2662i	−0.789278	+	1.93333i
$848$		−	16.2843i		−	0.559204i
$849$		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
$856$	−13.2132	+	22.8859i	−0.451618	+	0.782225i
$857$		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
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$	−43.6985	−	75.6880i	−1.48579	−	2.57347i
$866$	17.1464	+	9.89949i	0.582659	+	0.336399i
$867$		0			0
$868$	−1.51472	+	3.71029i	−0.0514129	+	0.125935i
$869$		−	103.326i		−	3.50509i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.0196	+	109.755i	0.507755	+	3.71039i
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	−46.8014	+	27.0208i	−1.57947	+	0.911908i
$879$		0			0
$880$	−51.4558	+	89.1241i	−1.73458	+	3.00437i
$881$		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
$886$	9.09188	+	15.7476i	0.305448	+	0.529051i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−39.9558	+	5.46783i	−1.34008	+	0.183385i
$890$		0			0
$891$		0			0
$892$	−25.7279	−	44.5621i	−0.861435	−	1.49205i
$893$		0			0
$894$		0			0
$895$	49.9411			1.66935
$896$	27.7128	+	11.3137i	0.925820	+	0.377964i
$897$		0			0
$898$		0			0
$899$	−4.97546	+	2.87258i	−0.165941	+	0.0958060i
$900$		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	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	40.0834	+	23.1421i	1.33021	+	0.767999i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−35.4706	−	61.4368i	−1.17390	−	2.03326i
$914$	2.48554	−	1.43503i	0.0822145	−	0.0474665i
$915$		0			0
$916$		0			0
$917$	−35.0613	−	14.3137i	−1.15783	−	0.472680i
$918$		0			0
$919$	−25.0000	+	43.3013i	−0.824674	+	1.42838i	0.0774944	+	0.996993i	$0.475308\pi$
$919$	−25.0000	+	43.3013i	−0.824674	+	1.42838i	−0.902168	+	0.431384i	$0.858025\pi$
$920$		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
$928$	21.4558	+	37.1626i	0.704323	+	1.21992i
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−28.0000	−	48.4974i	−0.916188	−	1.58688i
$935$		0			0
$936$		0			0
$937$	45.9706			1.50179			0.750896	−	0.660420i	$-0.229622\pi$
$937$	45.9706			1.50179			0.750896	+	0.660420i	$0.229622\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.75818	−	4.47918i	0.252909	−	0.146017i	−0.368186	−	0.929752i	$-0.620021\pi$
$941$	7.75818	−	4.47918i	0.252909	−	0.146017i	0.621096	+	0.783735i	$0.286688\pi$
$942$		0			0
$943$		0			0
$944$		−	13.3726i		−	0.435241i
$945$		0			0
$946$		0			0
$947$	−48.9898	−	28.2843i	−1.59195	−	0.919115i	−0.992972	−	0.118354i	$-0.962238\pi$
$947$	−48.9898	−	28.2843i	−1.59195	−	0.919115i	−0.598983	−	0.800762i	$-0.704428\pi$
$948$		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
$961$	15.2132	−	26.3500i	0.490748	−	0.850001i
$962$		0			0
$963$		0			0
$964$	30.4558	−	52.7511i	0.980917	−	1.69900i
$965$		−	19.9289i		−	0.641535i
$966$		0			0
$967$	−26.7574			−0.860459			−0.430229	−	0.902720i	$-0.641567\pi$
$967$	−26.7574			−0.860459			−0.430229	+	0.902720i	$0.641567\pi$
$968$	56.2662	+	32.4853i	1.80846	+	1.04412i
$969$		0			0
$970$	56.0919	+	97.1540i	1.80100	+	3.11943i
$971$	−29.5425	−	17.0563i	−0.948063	−	0.547364i	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−29.5425	−	17.0563i	−0.948063	−	0.547364i	−0.892479	+	0.451090i	$0.851035\pi$
$972$		0			0
$973$		0			0
$974$			55.4142i			1.77559i
$975$		0			0
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$	−16.6031	−	59.5269i	−0.530366	−	1.90152i
$981$		0			0
$982$	31.3345	−	54.2730i	0.999925	−	1.73192i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	31.2132	−	54.0629i	0.994535	−	1.72259i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−3.89340	−	6.74356i	−0.123678	−	0.214216i	0.797537	−	0.603269i	$-0.206136\pi$
$991$	−3.89340	−	6.74356i	−0.123678	−	0.214216i	−0.921215	+	0.389053i	$0.872802\pi$
$992$	3.71029	+	2.14214i	0.117802	+	0.0680129i
$993$		0			0
$994$		0			0
$995$			61.7990i			1.95916i
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		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.37.3	yes	8
3.2	odd	2	inner	504.2.cj.b.37.2	✓	8
4.3	odd	2		2016.2.cr.b.1297.1		8
7.4	even	3	inner	504.2.cj.b.109.2	yes	8
8.3	odd	2		2016.2.cr.b.1297.4		8
8.5	even	2	inner	504.2.cj.b.37.2	✓	8
12.11	even	2		2016.2.cr.b.1297.4		8
21.11	odd	6	inner	504.2.cj.b.109.3	yes	8
24.5	odd	2	CM	504.2.cj.b.37.3	yes	8
24.11	even	2		2016.2.cr.b.1297.1		8
28.11	odd	6		2016.2.cr.b.1873.4		8
56.11	odd	6		2016.2.cr.b.1873.1		8
56.53	even	6	inner	504.2.cj.b.109.3	yes	8
84.11	even	6		2016.2.cr.b.1873.1		8
168.11	even	6		2016.2.cr.b.1873.4		8
168.53	odd	6	inner	504.2.cj.b.109.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.cj.b.37.2	✓	8	3.2	odd	2	inner
504.2.cj.b.37.2	✓	8	8.5	even	2	inner
504.2.cj.b.37.3	yes	8	1.1	even	1	trivial
504.2.cj.b.37.3	yes	8	24.5	odd	2	CM
504.2.cj.b.109.2	yes	8	7.4	even	3	inner
504.2.cj.b.109.2	yes	8	168.53	odd	6	inner
504.2.cj.b.109.3	yes	8	21.11	odd	6	inner
504.2.cj.b.109.3	yes	8	56.53	even	6	inner
2016.2.cr.b.1297.1		8	4.3	odd	2
2016.2.cr.b.1297.1		8	24.11	even	2
2016.2.cr.b.1297.4		8	8.3	odd	2
2016.2.cr.b.1297.4		8	12.11	even	2
2016.2.cr.b.1873.1		8	56.11	odd	6
2016.2.cr.b.1873.1		8	84.11	even	6
2016.2.cr.b.1873.4		8	28.11	odd	6
2016.2.cr.b.1873.4		8	168.11	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.82282 - 2.20711i) q^{5} +(-2.62132 + 0.358719i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.82282 - 2.20711i) q^{5} +(-2.62132 + 0.358719i) q^{7} +2.82843i q^{8} +(-3.12132 - 5.40629i) q^{10} +(-5.04757 + 2.91421i) q^{11} +(-3.46410 - 1.41421i) q^{14} +(-2.00000 + 3.46410i) q^{16} -8.82843i q^{20} -8.24264 q^{22} +(7.24264 + 12.5446i) q^{25} +(-3.24264 - 4.18154i) q^{28} -7.58579i q^{29} +(-0.378680 - 0.655892i) q^{31} +(-4.89898 + 2.82843i) q^{32} +(10.8126 + 4.41421i) q^{35} +(6.24264 - 10.8126i) q^{40} +(-10.0951 - 5.82843i) q^{44} +(6.74264 - 1.88064i) q^{49} +20.4853i q^{50} +(-3.52565 + 2.03553i) q^{53} +25.7279 q^{55} +(-1.01461 - 7.41421i) q^{56} +(5.36396 - 9.29065i) q^{58} +(-2.89525 + 1.67157i) q^{59} -1.07107i q^{62} -8.00000 q^{64} +(10.1213 + 13.0519i) q^{70} +(-7.00000 - 12.1244i) q^{73} +(12.1859 - 9.44975i) q^{77} +(-8.86396 + 15.3528i) q^{79} +(15.2913 - 8.82843i) q^{80} +12.1716i q^{83} +(-8.24264 - 14.2767i) q^{88} -17.9706 q^{97} +(9.58783 + 2.46447i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) 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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