LMFDB - Embedded newform 72.3.p.a.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,3,Mod(43,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("72.43");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	72.p (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:	$1.96185790339$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			67.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	72.67
Dual form			72.3.p.a.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-1.94949 + 2.28024i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-5.89898 - 1.09638i) q^{6} -8.00000 q^{8} +(-1.39898 - 8.89060i) q^{9} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-1.94949 + 2.28024i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-5.89898 - 1.09638i) q^{6} -8.00000 q^{8} +(-1.39898 - 8.89060i) q^{9} +(10.8485 + 18.7901i) q^{11} +(-4.00000 - 11.3137i) q^{12} +(-8.00000 - 13.8564i) q^{16} +28.3939 q^{17} +(14.0000 - 11.3137i) q^{18} +2.30306 q^{19} +(-21.6969 + 37.5802i) q^{22} +(15.5959 - 18.2419i) q^{24} +(-12.5000 - 21.6506i) q^{25} +(23.0000 + 14.1421i) q^{27} +(16.0000 - 27.7128i) q^{32} +(-63.9949 - 11.8940i) q^{33} +(28.3939 + 49.1796i) q^{34} +(33.5959 + 12.9350i) q^{36} +(2.30306 + 3.98902i) q^{38} +(17.8939 - 30.9931i) q^{41} +(-33.2423 - 57.5774i) q^{43} -86.7878 q^{44} +(47.1918 + 8.77101i) q^{48} +(-24.5000 + 42.4352i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-55.3536 + 64.7448i) q^{51} +(-1.49490 + 53.9793i) q^{54} +(-4.48979 + 5.25153i) q^{57} +(-57.2423 + 99.1467i) q^{59} +64.0000 q^{64} +(-43.3939 - 122.736i) q^{66} +(66.9393 - 115.942i) q^{67} +(-56.7878 + 98.3593i) q^{68} +(11.1918 + 71.1248i) q^{72} +100.394 q^{73} +(73.7372 + 13.7047i) q^{75} +(-4.60612 + 7.97804i) q^{76} +(-77.0857 + 24.8755i) q^{81} +71.5755 q^{82} +(-79.0000 - 136.832i) q^{83} +(66.4847 - 115.155i) q^{86} +(-86.7878 - 150.321i) q^{88} +146.000 q^{89} +(32.0000 + 90.5097i) q^{96} +(49.9847 + 86.5760i) q^{97} -98.0000 q^{98} +(151.879 - 122.736i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 2 * q^3 - 8 * q^4 - 4 * q^6 - 32 * q^8 + 14 * q^9	$ 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9} + 14 q^{11} - 16 q^{12} - 32 q^{16} - 4 q^{17} + 56 q^{18} + 68 q^{19} - 28 q^{22} - 16 q^{24} - 50 q^{25} + 92 q^{27} + 64 q^{32} - 158 q^{33} - 4 q^{34} + 56 q^{36} + 68 q^{38} - 46 q^{41} + 14 q^{43} - 112 q^{44} + 32 q^{48} - 98 q^{49} + 100 q^{50} - 290 q^{51} + 92 q^{54} + 178 q^{57} - 82 q^{59} + 256 q^{64} - 56 q^{66} + 62 q^{67} + 8 q^{68} - 112 q^{72} + 284 q^{73} + 50 q^{75} - 136 q^{76} - 34 q^{81} - 184 q^{82} - 316 q^{83} - 28 q^{86} - 112 q^{88} + 584 q^{89} + 128 q^{96} - 94 q^{97} - 392 q^{98} + 196 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 2 * q^3 - 8 * q^4 - 4 * q^6 - 32 * q^8 + 14 * q^9 + 14 * q^11 - 16 * q^12 - 32 * q^16 - 4 * q^17 + 56 * q^18 + 68 * q^19 - 28 * q^22 - 16 * q^24 - 50 * q^25 + 92 * q^27 + 64 * q^32 - 158 * q^33 - 4 * q^34 + 56 * q^36 + 68 * q^38 - 46 * q^41 + 14 * q^43 - 112 * q^44 + 32 * q^48 - 98 * q^49 + 100 * q^50 - 290 * q^51 + 92 * q^54 + 178 * q^57 - 82 * q^59 + 256 * q^64 - 56 * q^66 + 62 * q^67 + 8 * q^68 - 112 * q^72 + 284 * q^73 + 50 * q^75 - 136 * q^76 - 34 * q^81 - 184 * q^82 - 316 * q^83 - 28 * q^86 - 112 * q^88 + 584 * q^89 + 128 * q^96 - 94 * q^97 - 392 * q^98 + 196 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	+	1.73205i	0.500000	+	0.866025i
$2$	1.00000	+	1.73205i	0.500000	+	0.866025i
$3$	−1.94949	+	2.28024i	−0.649830	+	0.760080i
$3$	−1.94949	+	2.28024i	−0.649830	+	0.760080i
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−5.89898	−	1.09638i	−0.983163	−	0.182729i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−8.00000			−1.00000
$9$	−1.39898	−	8.89060i	−0.155442	−	0.987845i
$10$		0			0
$11$	10.8485	+	18.7901i	0.986224	+	1.70819i	0.636364	+	0.771389i	$0.280438\pi$
$11$	10.8485	+	18.7901i	0.986224	+	1.70819i	0.349861	+	0.936802i	$0.386229\pi$
$12$	−4.00000	−	11.3137i	−0.333333	−	0.942809i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	28.3939			1.67023			0.835114	−	0.550077i	$-0.185402\pi$
$17$	28.3939			1.67023			0.835114	+	0.550077i	$0.185402\pi$
$18$	14.0000	−	11.3137i	0.777778	−	0.628539i
$19$	2.30306			0.121214			0.0606069	−	0.998162i	$-0.480696\pi$
$19$	2.30306			0.121214			0.0606069	+	0.998162i	$0.480696\pi$
$20$		0			0
$21$		0			0
$22$	−21.6969	+	37.5802i	−0.986224	+	1.70819i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	15.5959	−	18.2419i	0.649830	−	0.760080i
$25$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$26$		0			0
$27$	23.0000	+	14.1421i	0.851852	+	0.523783i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	16.0000	−	27.7128i	0.500000	−	0.866025i
$33$	−63.9949	−	11.8940i	−1.93924	−	0.360424i
$34$	28.3939	+	49.1796i	0.835114	+	1.44646i
$35$		0			0
$36$	33.5959	+	12.9350i	0.933220	+	0.359306i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	2.30306	+	3.98902i	0.0606069	+	0.104974i
$39$		0			0
$40$		0			0
$41$	17.8939	−	30.9931i	0.436436	−	0.755929i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	17.8939	−	30.9931i	0.436436	−	0.755929i	0.997412	+	0.0719030i	$0.0229072\pi$
$42$		0			0
$43$	−33.2423	−	57.5774i	−0.773078	−	1.33901i	−0.935869	−	0.352349i	$-0.885383\pi$
$43$	−33.2423	−	57.5774i	−0.773078	−	1.33901i	0.162791	−	0.986661i	$-0.447950\pi$
$44$	−86.7878			−1.97245
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	47.1918	+	8.77101i	0.983163	+	0.182729i
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$	25.0000	−	43.3013i	0.500000	−	0.866025i
$51$	−55.3536	+	64.7448i	−1.08536	+	1.26951i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−1.49490	+	53.9793i	−0.0276833	+	0.999617i
$55$		0			0
$56$		0			0
$57$	−4.48979	+	5.25153i	−0.0787683	+	0.0921321i
$58$		0			0
$59$	−57.2423	+	99.1467i	−0.970209	+	1.68045i	−0.275294	+	0.961360i	$0.588775\pi$
$59$	−57.2423	+	99.1467i	−0.970209	+	1.68045i	−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$		0			0
$66$	−43.3939	−	122.736i	−0.657483	−	1.85964i
$67$	66.9393	−	115.942i	0.999094	−	1.73048i	0.462687	−	0.886522i	$-0.346886\pi$
$67$	66.9393	−	115.942i	0.999094	−	1.73048i	0.536407	−	0.843959i	$-0.319781\pi$
$68$	−56.7878	+	98.3593i	−0.835114	+	1.44646i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	11.1918	+	71.1248i	0.155442	+	0.987845i
$73$	100.394			1.37526			0.687629	−	0.726062i	$-0.258651\pi$
$73$	100.394			1.37526			0.687629	+	0.726062i	$0.258651\pi$
$74$		0			0
$75$	73.7372	+	13.7047i	0.983163	+	0.182729i
$76$	−4.60612	+	7.97804i	−0.0606069	+	0.104974i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−77.0857	+	24.8755i	−0.951675	+	0.307106i
$82$	71.5755			0.872872
$83$	−79.0000	−	136.832i	−0.951807	−	1.64858i	−0.741511	−	0.670941i	$-0.765890\pi$
$83$	−79.0000	−	136.832i	−0.951807	−	1.64858i	−0.210296	−	0.977638i	$-0.567443\pi$
$84$		0			0
$85$		0			0
$86$	66.4847	−	115.155i	0.773078	−	1.33901i
$87$		0			0
$88$	−86.7878	−	150.321i	−0.986224	−	1.70819i
$89$	146.000			1.64045			0.820225	−	0.572041i	$-0.193848\pi$
$89$	146.000			1.64045			0.820225	+	0.572041i	$0.193848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	32.0000	+	90.5097i	0.333333	+	0.942809i
$97$	49.9847	+	86.5760i	0.515306	+	0.892536i	0.999842	+	0.0177651i	$0.00565510\pi$
$97$	49.9847	+	86.5760i	0.515306	+	0.892536i	−0.484536	+	0.874771i	$0.661012\pi$
$98$	−98.0000			−1.00000
$99$	151.879	−	122.736i	1.53413	−	1.23976i
$100$	100.000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	−167.495	−	31.1304i	−1.64211	−	0.305200i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.8786			−0.129706			−0.0648531	−	0.997895i	$-0.520658\pi$
$107$	−13.8786			−0.129706			−0.0648531	+	0.997895i	$0.520658\pi$
$108$	−94.9898	+	51.3901i	−0.879535	+	0.475834i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−49.0000	+	84.8705i	−0.433628	+	0.751066i	−0.997183	−	0.0750128i	$-0.976100\pi$
$113$	−49.0000	+	84.8705i	−0.433628	+	0.751066i	0.563554	+	0.826079i	$0.309434\pi$
$114$	−13.5857	−	2.52502i	−0.119173	−	0.0221493i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−228.969			−1.94042
$119$		0			0
$120$		0			0
$121$	−174.879	+	302.899i	−1.44528	+	2.50329i
$122$		0			0
$123$	35.7878	+	101.223i	0.290957	+	0.822952i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	64.0000	+	110.851i	0.500000	+	0.866025i
$129$	196.096	+	36.4461i	1.52012	+	0.282528i
$130$		0			0
$131$	−31.0000	+	53.6936i	−0.236641	+	0.409875i	−0.959748	−	0.280861i	$-0.909380\pi$
$131$	−31.0000	+	53.6936i	−0.236641	+	0.409875i	0.723107	+	0.690736i	$0.242713\pi$
$132$	169.192	−	197.897i	1.28176	−	1.49922i
$133$		0			0
$134$	267.757			1.99819
$135$		0			0
$136$	−227.151			−1.67023
$137$	−118.288	−	204.880i	−0.863414	−	1.49548i	−0.868613	−	0.495491i	$-0.834988\pi$
$137$	−118.288	−	204.880i	−0.863414	−	1.49548i	0.00519888	−	0.999986i	$-0.498345\pi$
$138$		0			0
$139$	−29.3332	+	50.8065i	−0.211030	+	0.365515i	−0.952037	−	0.305983i	$-0.901015\pi$
$139$	−29.3332	+	50.8065i	−0.211030	+	0.365515i	0.741007	+	0.671497i	$0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−112.000	+	90.5097i	−0.777778	+	0.628539i
$145$		0			0
$146$	100.394	+	173.887i	0.687629	+	1.19101i
$147$	−49.0000	−	138.593i	−0.333333	−	0.942809i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	50.0000	+	141.421i	0.333333	+	0.942809i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−18.4245			−0.121214
$153$	−39.7225	−	252.439i	−0.259624	−	1.64993i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−120.171	−	108.641i	−0.741799	−	0.670622i
$163$	−322.000			−1.97546			−0.987730	−	0.156171i	$-0.950085\pi$
$163$	−322.000			−1.97546			−0.987730	+	0.156171i	$0.950085\pi$
$164$	71.5755	+	123.972i	0.436436	+	0.755929i
$165$		0			0
$166$	158.000	−	273.664i	0.951807	−	1.64858i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−84.5000	−	146.358i	−0.500000	−	0.866025i
$170$		0			0
$171$	−3.22194	−	20.4756i	−0.0188417	−	0.119740i
$172$	265.939			1.54616
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	173.576	−	300.642i	0.986224	−	1.70819i
$177$	−114.485	−	323.812i	−0.646806	−	1.82944i
$178$	146.000	+	252.879i	0.820225	+	1.42067i
$179$	−34.0000			−0.189944			−0.0949721	−	0.995480i	$-0.530276\pi$
$179$	−34.0000			−0.189944			−0.0949721	+	0.995480i	$0.530276\pi$
$180$		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
$187$	308.030	+	533.524i	1.64722	+	2.85307i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−124.767	+	145.935i	−0.649830	+	0.760080i
$193$	186.166	−	322.450i	0.964592	−	1.67072i	0.253886	−	0.967234i	$-0.418291\pi$
$193$	186.166	−	322.450i	0.964592	−	1.67072i	0.710706	−	0.703489i	$-0.248375\pi$
$194$	−99.9694	+	173.152i	−0.515306	+	0.892536i
$195$		0			0
$196$	−98.0000	−	169.741i	−0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	364.464	+	140.325i	1.84073	+	0.708712i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	100.000	+	173.205i	0.500000	+	0.866025i
$201$	133.879	+	378.666i	0.666063	+	1.88391i
$202$		0			0
$203$		0			0
$204$	−113.576	−	321.240i	−0.556743	−	1.57471i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.9847	+	43.2748i	0.119544	+	0.207056i
$210$		0			0
$211$	113.000	−	195.722i	0.535545	−	0.927591i	−0.463592	−	0.886049i	$-0.653440\pi$
$211$	113.000	−	195.722i	0.535545	−	0.927591i	0.999137	−	0.0415423i	$-0.0132271\pi$
$212$		0			0
$213$		0			0
$214$	−13.8786	−	24.0384i	−0.0648531	−	0.112329i
$215$		0			0
$216$	−184.000	−	113.137i	−0.851852	−	0.523783i
$217$		0			0
$218$		0			0
$219$	−195.717	+	228.922i	−0.893684	+	1.04531i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$	−175.000	+	141.421i	−0.777778	+	0.628539i
$226$	−196.000			−0.867257
$227$	74.7577	+	129.484i	0.329329	+	0.570414i	0.982379	−	0.186900i	$-0.0598442\pi$
$227$	74.7577	+	129.484i	0.329329	+	0.570414i	−0.653050	+	0.757315i	$0.726511\pi$
$228$	−9.21225	−	26.0562i	−0.0404046	−	0.114281i
$229$		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
$232$		0			0
$233$	−363.969			−1.56210			−0.781050	−	0.624468i	$-0.785316\pi$
$233$	−363.969			−1.56210			−0.781050	+	0.624468i	$0.785316\pi$
$234$		0			0
$235$		0			0
$236$	−228.969	−	396.587i	−0.970209	−	1.68045i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−142.560	−	246.922i	−0.591536	−	1.02457i	−0.994026	−	0.109146i	$-0.965188\pi$
$241$	−142.560	−	246.922i	−0.591536	−	1.02457i	0.402490	−	0.915425i	$-0.368145\pi$
$242$	−699.514			−2.89055
$243$	93.5556	−	224.268i	0.385003	−	0.922916i
$244$		0			0
$245$		0			0
$246$	−139.536	+	163.209i	−0.567218	+	0.663452i
$247$		0			0
$248$		0			0
$249$	466.019	+	86.6137i	1.87156	+	0.347846i
$250$		0			0
$251$	394.666			1.57238			0.786188	−	0.617988i	$-0.212052\pi$
$251$	394.666			1.57238			0.786188	+	0.617988i	$0.212052\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$	−50.4694	+	87.4155i	−0.196379	+	0.340138i	−0.947352	−	0.320195i	$-0.896252\pi$
$257$	−50.4694	+	87.4155i	−0.196379	+	0.340138i	0.750973	+	0.660333i	$0.229585\pi$
$258$	132.969	+	376.094i	0.515385	+	1.45773i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−124.000			−0.473282
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	511.959	+	95.1520i	1.93924	+	0.360424i
$265$		0			0
$266$		0			0
$267$	−284.626	+	332.915i	−1.06601	+	1.24687i
$268$	267.757	+	463.769i	0.999094	+	1.73048i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−227.151	−	393.437i	−0.835114	−	1.44646i
$273$		0			0
$274$	236.576	−	409.761i	0.863414	−	1.49548i
$275$	271.212	−	469.752i	0.986224	−	1.70819i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$	−117.333			−0.422060
$279$		0			0
$280$		0			0
$281$	119.000	+	206.114i	0.423488	+	0.733502i	0.996278	−	0.0862000i	$-0.0274724\pi$
$281$	119.000	+	206.114i	0.423488	+	0.733502i	−0.572790	+	0.819702i	$0.694139\pi$
$282$		0			0
$283$	41.0000	−	71.0141i	0.144876	−	0.250933i	−0.784450	−	0.620191i	$-0.787055\pi$
$283$	41.0000	−	71.0141i	0.144876	−	0.250933i	0.929327	+	0.369258i	$0.120388\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−268.767	−	103.480i	−0.933220	−	0.359306i
$289$	517.212			1.78966
$290$		0			0
$291$	−294.859	−	54.8020i	−1.01326	−	0.188323i
$292$	−200.788	+	347.775i	−0.687629	+	1.19101i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$	191.050	−	223.463i	0.649830	−	0.760080i
$295$		0			0
$296$		0			0
$297$	−16.2173	+	585.593i	−0.0546039	+	1.97169i
$298$		0			0
$299$		0			0
$300$	−194.949	+	228.024i	−0.649830	+	0.760080i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−18.4245	−	31.9122i	−0.0606069	−	0.104974i
$305$		0			0
$306$	397.514	−	321.240i	1.29907	−	1.04980i
$307$	−21.1520			−0.0688992			−0.0344496	−	0.999406i	$-0.510968\pi$
$307$	−21.1520			−0.0688992			−0.0344496	+	0.999406i	$0.510968\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−278.469	−	482.323i	−0.889679	−	1.54097i	−0.840256	−	0.542191i	$-0.817595\pi$
$313$	−278.469	−	482.323i	−0.889679	−	1.54097i	−0.0494230	−	0.998778i	$-0.515738\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	27.0561	−	31.6465i	0.0842870	−	0.0985871i
$322$		0			0
$323$	65.3928			0.202455
$324$	68.0000	−	316.784i	0.209877	−	0.977728i
$325$		0			0
$326$	−322.000	−	557.720i	−0.987730	−	1.71080i
$327$		0			0
$328$	−143.151	+	247.945i	−0.436436	+	0.755929i
$329$		0			0
$330$		0			0
$331$	−7.00000	−	12.1244i	−0.0211480	−	0.0366295i	0.855258	−	0.518203i	$-0.173399\pi$
$331$	−7.00000	−	12.1244i	−0.0211480	−	0.0366295i	−0.876406	+	0.481573i	$0.840065\pi$
$332$	632.000			1.90361
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	86.2571	−	149.402i	0.255956	−	0.443329i	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	86.2571	−	149.402i	0.255956	−	0.443329i	0.965155	+	0.261680i	$0.0842765\pi$
$338$	169.000	−	292.717i	0.500000	−	0.866025i
$339$	−98.0000	−	277.186i	−0.289086	−	0.817657i
$340$		0			0
$341$		0			0
$342$	32.2429	−	26.0562i	0.0942774	−	0.0761876i
$343$		0			0
$344$	265.939	+	460.619i	0.773078	+	1.33901i
$345$		0			0
$346$		0			0
$347$	−68.9699	+	119.459i	−0.198761	+	0.344263i	−0.948127	−	0.317892i	$-0.897025\pi$
$347$	−68.9699	+	119.459i	−0.198761	+	0.344263i	0.749366	+	0.662156i	$0.230358\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	694.302			1.97245
$353$	342.439	+	593.121i	0.970081	+	1.68023i	0.695294	+	0.718725i	$0.255274\pi$
$353$	342.439	+	593.121i	0.970081	+	1.68023i	0.274788	+	0.961505i	$0.411392\pi$
$354$	446.373	−	522.105i	1.26094	−	1.47487i
$355$		0			0
$356$	−292.000	+	505.759i	−0.820225	+	1.42067i
$357$		0			0
$358$	−34.0000	−	58.8897i	−0.0949721	−	0.164496i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−355.696			−0.985307
$362$		0			0
$363$	−349.757	−	989.263i	−0.963518	−	2.72524i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	−300.581	−	115.729i	−0.814582	−	0.313628i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$	−616.060	+	1067.05i	−1.64722	+	2.85307i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−622.242			−1.64180			−0.820900	−	0.571073i	$-0.806527\pi$
$379$	−622.242			−1.64180			−0.820900	+	0.571073i	$0.806527\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	−377.535	−	70.1681i	−0.983163	−	0.182729i
$385$		0			0
$386$	744.665			1.92918
$387$	−465.393	+	376.094i	−1.20257	+	0.971820i
$388$	−399.878			−1.03061
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	196.000	−	339.482i	0.500000	−	0.866025i
$393$	−62.0000	−	175.362i	−0.157761	−	0.446215i
$394$		0			0
$395$		0			0
$396$	121.414	+	771.596i	0.306602	+	1.94847i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−200.000	+	346.410i	−0.500000	+	0.866025i
$401$	−294.379	+	509.879i	−0.734111	+	1.27152i	0.221001	+	0.975274i	$0.429068\pi$
$401$	−294.379	+	509.879i	−0.734111	+	1.27152i	−0.955112	+	0.296244i	$0.904266\pi$
$402$	−521.990	+	610.550i	−1.29848	+	1.51878i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	442.829	−	517.959i	1.08536	−	1.26951i
$409$	−406.833	+	704.655i	−0.994701	+	1.72287i	−0.408313	+	0.912842i	$0.633883\pi$
$409$	−406.833	+	704.655i	−0.994701	+	1.72287i	−0.586388	+	0.810030i	$0.699451\pi$
$410$		0			0
$411$	697.777	+	129.688i	1.69775	+	0.315542i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−58.6663	−	165.933i	−0.140687	−	0.397922i
$418$	−49.9694	+	86.5495i	−0.119544	+	0.207056i
$419$	257.000	−	445.137i	0.613365	−	1.06238i	−0.377304	−	0.926090i	$-0.623149\pi$
$419$	257.000	−	445.137i	0.613365	−	1.06238i	0.990669	−	0.136290i	$-0.0435179\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	452.000			1.07109
$423$		0			0
$424$		0			0
$425$	−354.923	−	614.745i	−0.835114	−	1.44646i
$426$		0			0
$427$		0			0
$428$	27.7571	−	48.0768i	0.0648531	−	0.112329i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	11.9592	−	431.834i	0.0276833	−	0.999617i
$433$	269.484			0.622364			0.311182	−	0.950350i	$-0.399275\pi$
$433$	269.484			0.622364			0.311182	+	0.950350i	$0.399275\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−592.221	−	110.069i	−1.35210	−	0.251300i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	411.550	+	158.454i	0.933220	+	0.359306i
$442$		0			0
$443$	270.939	+	469.281i	0.611601	+	1.05932i	0.990971	+	0.134079i	$0.0428076\pi$
$443$	270.939	+	469.281i	0.611601	+	1.05932i	−0.379370	+	0.925245i	$0.623859\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−227.243			−0.506109			−0.253054	−	0.967452i	$-0.581435\pi$
$449$	−227.243			−0.506109			−0.253054	+	0.967452i	$0.581435\pi$
$450$	−419.949	−	161.688i	−0.933220	−	0.359306i
$451$	776.485			1.72170
$452$	−196.000	−	339.482i	−0.433628	−	0.751066i
$453$		0			0
$454$	−149.515	+	258.968i	−0.329329	+	0.570414i
$455$		0			0
$456$	35.9184	−	42.0122i	0.0787683	−	0.0921321i
$457$	322.620	+	558.795i	0.705953	+	1.22275i	0.966347	+	0.257244i	$0.0828143\pi$
$457$	322.620	+	558.795i	0.705953	+	1.22275i	−0.260394	+	0.965502i	$0.583852\pi$
$458$		0			0
$459$	653.059	+	401.550i	1.42279	+	0.874837i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$	−363.969	−	630.413i	−0.781050	−	1.35282i
$467$	−791.332			−1.69450			−0.847250	−	0.531194i	$-0.821743\pi$
$467$	−791.332			−1.69450			−0.847250	+	0.531194i	$0.821743\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	457.939	−	793.173i	0.970209	−	1.68045i
$473$	721.257	−	1249.25i	1.52486	−	2.64113i
$474$		0			0
$475$	−28.7883	−	49.8627i	−0.0606069	−	0.104974i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$	285.120	−	493.843i	0.591536	−	1.02457i
$483$		0			0
$484$	−699.514	−	1211.59i	−1.44528	−	2.50329i
$485$		0			0
$486$	482.000	−	62.2254i	0.991770	−	0.128036i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	627.736	−	734.237i	1.28371	−	1.50151i
$490$		0			0
$491$	−61.6964	+	106.861i	−0.125655	+	0.217640i	−0.921989	−	0.387217i	$-0.873436\pi$
$491$	−61.6964	+	106.861i	−0.125655	+	0.217640i	0.796334	+	0.604857i	$0.206770\pi$
$492$	−422.222	−	78.4737i	−0.858176	−	0.159499i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	316.000	+	893.783i	0.634538	+	1.79474i
$499$	−457.696	+	792.753i	−0.917227	+	1.58868i	−0.113620	+	0.993524i	$0.536245\pi$
$499$	−457.696	+	792.753i	−0.917227	+	1.58868i	−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$	394.666	+	683.582i	0.786188	+	1.36172i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	498.464	+	92.6438i	0.983163	+	0.182729i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$	52.9704	+	32.5702i	0.103256	+	0.0634897i
$514$	−201.878			−0.392758
$515$		0			0
$516$	−518.445	+	606.404i	−1.00474	+	1.17520i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	267.849			0.514106			0.257053	−	0.966397i	$-0.417249\pi$
$521$	267.849			0.514106			0.257053	+	0.966397i	$0.417249\pi$
$522$		0			0
$523$	398.000			0.760994			0.380497	−	0.924782i	$-0.375753\pi$
$523$	398.000			0.760994			0.380497	+	0.924782i	$0.375753\pi$
$524$	−124.000	−	214.774i	−0.236641	−	0.409875i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	347.151	+	981.891i	0.657483	+	1.85964i
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$	961.555	+	370.215i	1.81084	+	0.697203i
$532$		0			0
$533$		0			0
$534$	−861.251	−	160.071i	−1.61283	−	0.299758i
$535$		0			0
$536$	−535.514	+	927.538i	−0.999094	+	1.73048i
$537$	66.2827	−	77.5281i	0.123431	−	0.144373i
$538$		0			0
$539$	−1063.15			−1.97245
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	454.302	−	786.874i	0.835114	−	1.44646i
$545$		0			0
$546$		0			0
$547$	86.4852	+	149.797i	0.158108	+	0.273852i	0.934186	−	0.356785i	$-0.116127\pi$
$547$	86.4852	+	149.797i	0.158108	+	0.273852i	−0.776078	+	0.630637i	$0.782794\pi$
$548$	946.302			1.72683
$549$		0			0
$550$	1084.85			1.97245
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−117.333	−	203.226i	−0.211030	−	0.365515i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1817.06	−	337.717i	−3.23897	−	0.601991i
$562$	−238.000	+	412.228i	−0.423488	+	0.733502i
$563$	−534.150	+	925.176i	−0.948758	+	1.64330i	−0.200710	+	0.979651i	$0.564325\pi$
$563$	−534.150	+	925.176i	−0.948758	+	1.64330i	−0.748047	+	0.663646i	$0.769008\pi$
$564$		0			0
$565$		0			0
$566$	164.000			0.289753
$567$		0			0
$568$		0			0
$569$	−255.014	−	441.698i	−0.448180	−	0.776270i	0.550088	−	0.835107i	$-0.314594\pi$
$569$	−255.014	−	441.698i	−0.448180	−	0.776270i	−0.998268	+	0.0588367i	$0.981261\pi$
$570$		0			0
$571$	239.666	−	415.113i	0.419730	−	0.726994i	−0.576182	−	0.817321i	$-0.695458\pi$
$571$	239.666	−	415.113i	0.419730	−	0.726994i	0.995912	+	0.0903277i	$0.0287914\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−89.5347	−	568.999i	−0.155442	−	0.987845i
$577$	998.392			1.73032			0.865158	−	0.501500i	$-0.167218\pi$
$577$	998.392			1.73032			0.865158	+	0.501500i	$0.167218\pi$
$578$	517.212	+	895.838i	0.894831	+	1.54989i
$579$	372.333	+	1053.12i	0.643062	+	1.81885i
$580$		0			0
$581$		0			0
$582$	−199.939	−	565.512i	−0.343537	−	0.971671i
$583$		0			0
$584$	−803.151			−1.37526
$585$		0			0
$586$		0			0
$587$	−409.424	−	709.143i	−0.697485	−	1.20808i	−0.969336	−	0.245741i	$-0.920969\pi$
$587$	−409.424	−	709.143i	−0.697485	−	1.20808i	0.271850	−	0.962340i	$-0.412365\pi$
$588$	578.100	+	107.445i	0.983163	+	0.182729i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−862.000			−1.45363			−0.726813	−	0.686836i	$-0.758999\pi$
$593$	−862.000			−1.45363			−0.726813	+	0.686836i	$0.758999\pi$
$594$	−1030.49	+	557.504i	−1.73484	+	0.938558i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−589.898	−	109.638i	−0.983163	−	0.182729i
$601$	566.530	+	981.258i	0.942645	+	1.63271i	0.760399	+	0.649456i	$0.225003\pi$
$601$	566.530	+	981.258i	0.942645	+	1.63271i	0.182246	+	0.983253i	$0.441663\pi$
$602$		0			0
$603$	−1124.44	−	432.930i	−1.86475	−	0.717960i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	36.8490	−	63.8243i	0.0606069	−	0.104974i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	953.918	+	367.275i	1.55869	+	0.600122i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−21.1520	−	36.6364i	−0.0344496	−	0.0596684i
$615$		0			0
$616$		0			0
$617$	430.893	−	746.328i	0.698368	−	1.20961i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$	430.893	−	746.328i	0.698368	−	1.20961i	0.969032	−	0.246935i	$-0.0794233\pi$
$618$		0			0
$619$	−618.150	−	1070.67i	−0.998628	−	1.72967i	−0.544670	−	0.838651i	$-0.683345\pi$
$619$	−618.150	−	1070.67i	−0.998628	−	1.72967i	−0.453958	−	0.891023i	$-0.649988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$	556.939	−	964.646i	0.889679	−	1.54097i
$627$	−147.384	−	27.3926i	−0.235062	−	0.0436884i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	226.000	+	639.225i	0.357030	+	1.00983i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	634.893	+	1099.67i	0.990472	+	1.71555i	0.614497	+	0.788919i	$0.289359\pi$
$641$	634.893	+	1099.67i	0.990472	+	1.71555i	0.375975	+	0.926630i	$0.377308\pi$
$642$	81.8694	+	15.2161i	0.127522	+	0.0237011i
$643$	487.212	−	843.875i	0.757717	−	1.31240i	−0.186296	−	0.982494i	$-0.559648\pi$
$643$	487.212	−	843.875i	0.757717	−	1.31240i	0.944012	−	0.329910i	$-0.107018\pi$
$644$		0			0
$645$		0			0
$646$	65.3928	+	113.264i	0.101227	+	0.175331i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	616.686	−	199.004i	0.951675	−	0.307106i
$649$	−2483.97			−3.82738
$650$		0			0
$651$		0			0
$652$	644.000	−	1115.44i	0.987730	−	1.71080i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−572.604			−0.872872
$657$	−140.449	−	892.562i	−0.213773	−	1.35854i
$658$		0			0
$659$	497.000	+	860.829i	0.754173	+	1.30627i	0.945784	+	0.324795i	$0.105295\pi$
$659$	497.000	+	860.829i	0.754173	+	1.30627i	−0.191611	+	0.981471i	$0.561371\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	14.0000	−	24.2487i	0.0211480	−	0.0366295i
$663$		0			0
$664$	632.000	+	1094.66i	0.951807	+	1.64858i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	623.000	+	1079.07i	0.925706	+	1.60337i	0.790422	+	0.612563i	$0.209861\pi$
$673$	623.000	+	1079.07i	0.925706	+	1.60337i	0.135284	+	0.990807i	$0.456805\pi$
$674$	345.029			0.511912
$675$	18.6862	−	674.741i	0.0276833	−	0.999617i
$676$	676.000			1.00000
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	382.100	−	446.927i	0.563569	−	0.659184i
$679$		0			0
$680$		0			0
$681$	−440.994	−	81.9625i	−0.647568	−	0.120356i
$682$		0			0
$683$	932.664			1.36554			0.682770	−	0.730633i	$-0.260775\pi$
$683$	932.664			1.36554			0.682770	+	0.730633i	$0.260775\pi$
$684$	77.3735	+	29.7901i	0.113119	+	0.0435528i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−531.878	+	921.239i	−0.773078	+	1.33901i
$689$		0			0
$690$		0			0
$691$	−367.000	−	635.663i	−0.531114	−	0.919917i	−0.999341	−	0.0363084i	$-0.988440\pi$
$691$	−367.000	−	635.663i	−0.531114	−	0.919917i	0.468226	−	0.883609i	$-0.344893\pi$
$692$		0			0
$693$		0			0
$694$	−275.880			−0.397521
$695$		0			0
$696$		0			0
$697$	508.077	−	880.014i	0.728948	−	1.26257i
$698$		0			0
$699$	709.555	−	829.937i	1.01510	−	1.18732i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	694.302	+	1202.57i	0.986224	+	1.70819i
$705$		0			0
$706$	−684.878	+	1186.24i	−0.970081	+	1.68023i
$707$		0			0
$708$	1350.69	+	251.037i	1.90775	+	0.354571i
$709$		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
$712$	−1168.00			−1.64045
$713$		0			0
$714$		0			0
$715$		0			0
$716$	68.0000	−	117.779i	0.0949721	−	0.164496i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−355.696	−	616.083i	−0.492654	−	0.853301i
$723$	840.960	+	156.300i	1.16315	+	0.216182i
$724$		0			0
$725$		0			0
$726$	1363.70	−	1595.06i	1.87837	−	2.19705i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$	−943.879	−	1634.85i	−1.29122	−	2.23645i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2904.76			3.94132
$738$	−100.133	−	636.350i	−0.135681	−	0.862262i
$739$	−1088.24			−1.47258			−0.736292	−	0.676664i	$-0.763425\pi$
$739$	−1088.24			−1.47258			−0.736292	+	0.676664i	$0.763425\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1106.00	+	893.783i	−1.48059	+	1.19650i
$748$	−2464.24			−3.29444
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	−769.398	+	899.934i	−1.02178	+	1.19513i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−622.242	−	1077.75i	−0.820900	−	1.42184i
$759$		0			0
$760$		0			0
$761$	−697.000	+	1207.24i	−0.915900	+	1.58639i	−0.110322	+	0.993896i	$0.535188\pi$
$761$	−697.000	+	1207.24i	−0.915900	+	1.58639i	−0.805578	+	0.592490i	$0.798145\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−256.000	−	724.077i	−0.333333	−	0.942809i
$769$	527.000	−	912.791i	0.685306	−	1.18698i	−0.288035	−	0.957620i	$-0.593002\pi$
$769$	527.000	−	912.791i	0.685306	−	1.18698i	0.973341	−	0.229364i	$-0.0736647\pi$
$770$		0			0
$771$	−100.939	−	285.498i	−0.130919	−	0.370296i
$772$	744.665	+	1289.80i	0.964592	+	1.67072i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−1116.81	−	429.990i	−1.44290	−	0.555542i
$775$		0			0
$776$	−399.878	−	692.608i	−0.515306	−	0.892536i
$777$		0			0
$778$		0			0
$779$	41.2107	−	71.3790i	0.0529021	−	0.0916290i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$	241.737	−	282.750i	0.307553	−	0.359732i
$787$	−463.000	+	801.940i	−0.588310	+	1.01898i	0.406144	+	0.913809i	$0.366873\pi$
$787$	−463.000	+	801.940i	−0.588310	+	1.01898i	−0.994454	+	0.105174i	$0.966460\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1215.03	+	981.891i	−1.53413	+	1.23976i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	−800.000			−1.00000
$801$	−204.251	−	1298.03i	−0.254995	−	1.62051i
$802$	−1177.51			−1.46822
$803$	1089.12	+	1886.41i	1.35631	+	2.34920i
$804$	−1579.49	−	293.563i	−1.96454	−	0.365128i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1084.94			1.34109			0.670543	−	0.741871i	$-0.266061\pi$
$809$	1084.94			1.34109			0.670543	+	0.741871i	$0.266061\pi$
$810$		0			0
$811$	1485.21			1.83133			0.915665	−	0.401942i	$-0.131665\pi$
$811$	1485.21			1.83133			0.915665	+	0.401942i	$0.131665\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1339.96	+	249.043i	1.64211	+	0.305200i
$817$	−76.5592	−	132.604i	−0.0937077	−	0.162306i
$818$	−1627.33			−1.98940
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	473.151	+	1338.27i	0.575610	+	1.62807i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	542.423	+	1534.21i	0.657483	+	1.85964i
$826$		0			0
$827$	1262.00			1.52600			0.762999	−	0.646400i	$-0.223726\pi$
$827$	1262.00			1.52600			0.762999	+	0.646400i	$0.223726\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−695.650	+	1204.90i	−0.835114	+	1.44646i
$834$	228.739	−	267.546i	0.274267	−	0.320799i
$835$		0			0
$836$	−199.878			−0.239088
$837$		0			0
$838$	1028.00			1.22673
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$	−701.979	−	130.469i	−0.832715	−	0.154767i
$844$	452.000	+	782.887i	0.535545	+	0.927591i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	82.0000	+	231.931i	0.0965842	+	0.273181i
$850$	709.847	−	1229.49i	0.835114	−	1.44646i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	111.029			0.129706
$857$	−601.000	−	1040.96i	−0.701284	−	1.21466i	−0.968016	−	0.250888i	$-0.919277\pi$
$857$	−601.000	−	1040.96i	−0.701284	−	1.21466i	0.266733	−	0.963771i	$-0.414056\pi$
$858$		0			0
$859$	198.394	−	343.629i	0.230960	−	0.400034i	−0.727131	−	0.686499i	$-0.759147\pi$
$859$	198.394	−	343.629i	0.230960	−	0.400034i	0.958091	+	0.286465i	$0.0924801\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	759.918	−	411.121i	0.879535	−	0.475834i
$865$		0			0
$866$	269.484	+	466.759i	0.311182	+	0.538983i
$867$	−1008.30	+	1179.37i	−1.16298	+	1.36029i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	699.786	−	565.512i	0.801587	−	0.647780i
$874$		0			0
$875$		0			0
$876$	−401.576	−	1135.83i	−0.458420	−	1.29661i
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1438.00			−1.63224			−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−1438.00			−1.63224			−0.816118	+	0.577885i	$0.803878\pi$
$882$	137.100	+	871.279i	0.155442	+	0.987845i
$883$	778.121			0.881225			0.440612	−	0.897697i	$-0.354761\pi$
$883$	778.121			0.881225			0.440612	+	0.897697i	$0.354761\pi$
$884$		0			0
$885$		0			0
$886$	−541.879	+	938.561i	−0.611601	+	1.05932i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1303.68	−	1178.59i	−1.46316	−	1.32277i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−227.243	−	393.596i	−0.253054	−	0.438303i
$899$		0			0
$900$	−139.898	−	889.060i	−0.155442	−	0.987845i
$901$		0			0
$902$	776.485	+	1344.91i	0.860848	+	1.49103i
$903$		0			0
$904$	392.000	−	678.964i	0.433628	−	0.751066i
$905$		0			0
$906$		0			0
$907$	−685.696	−	1187.66i	−0.756005	−	1.30944i	−0.944873	−	0.327436i	$-0.893815\pi$
$907$	−685.696	−	1187.66i	−0.756005	−	1.30944i	0.188868	−	0.982002i	$-0.439518\pi$
$908$	−598.061			−0.658658
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$	108.686	+	20.2002i	0.119173	+	0.0221493i
$913$	1714.06	−	2968.84i	1.87739	−	3.25174i
$914$	−645.241	+	1117.59i	−0.705953	+	1.22275i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−42.4459	+	1532.68i	−0.0462374	+	1.66959i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	41.2357	−	48.2317i	0.0447727	−	0.0523689i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−529.000	−	916.255i	−0.569429	−	0.986281i	−0.996622	−	0.0821203i	$-0.973831\pi$
$929$	−529.000	−	916.255i	−0.569429	−	0.986281i	0.427193	−	0.904160i	$-0.359503\pi$
$930$		0			0
$931$	−56.4250	+	97.7310i	−0.0606069	+	0.104974i
$932$	727.939	−	1260.83i	0.781050	−	1.35282i
$933$		0			0
$934$	−791.332	−	1370.63i	−0.847250	−	1.46748i
$935$		0			0
$936$		0			0
$937$	−718.000			−0.766275			−0.383138	−	0.923691i	$-0.625157\pi$
$937$	−718.000			−0.766275			−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$	1642.69	+	305.307i	1.74940	+	0.325141i
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	1831.76			1.94042
$945$		0			0
$946$	2885.03			3.04971
$947$	−946.605	−	1639.57i	−0.999582	−	1.73133i	−0.524815	−	0.851216i	$-0.675866\pi$
$947$	−946.605	−	1639.57i	−0.999582	−	1.73133i	−0.474767	−	0.880111i	$-0.657468\pi$
$948$		0			0
$949$		0			0
$950$	57.5765	−	99.7255i	0.0606069	−	0.104974i
$951$		0			0
$952$		0			0
$953$	−1575.06			−1.65274			−0.826368	−	0.563131i	$-0.809597\pi$
$953$	−1575.06			−1.65274			−0.826368	+	0.563131i	$0.809597\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$	19.4158	+	123.389i	0.0201618	+	0.128130i
$964$	1140.48			1.18307
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	1399.03	−	2423.19i	1.44528	−	2.50329i
$969$	−127.483	+	149.111i	−0.131561	+	0.153882i
$970$		0			0
$971$	974.000			1.00309			0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000			1.00309			0.501545	+	0.865132i	$0.332765\pi$
$972$	589.778	+	772.623i	0.606767	+	0.794880i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−317.834	+	550.504i	−0.325316	+	0.563464i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−317.834	+	550.504i	−0.325316	+	0.563464i	0.656260	+	0.754535i	$0.272137\pi$
$978$	1899.47	+	353.033i	1.94220	+	0.360975i
$979$	1583.88	+	2743.35i	1.61785	+	2.80220i
$980$		0			0
$981$		0			0
$982$	−246.786			−0.251309
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$	−286.302	−	809.784i	−0.290957	−	0.822952i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	41.2929	+	7.67463i	0.0415839	+	0.00772874i
$994$		0			0
$995$		0			0
$996$	−1232.08	+	1441.11i	−1.23703	+	1.44690i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	−1830.79			−1.83445
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.a.67.1	yes	4
3.2	odd	2		216.3.p.a.91.1		4
4.3	odd	2		288.3.t.a.175.2		4
8.3	odd	2	CM	72.3.p.a.67.1	yes	4
8.5	even	2		288.3.t.a.175.2		4
9.2	odd	6		216.3.p.a.19.1		4
9.4	even	3		648.3.b.a.163.1		2
9.5	odd	6		648.3.b.b.163.2		2
9.7	even	3	inner	72.3.p.a.43.1	✓	4
12.11	even	2		864.3.t.a.847.2		4
24.5	odd	2		864.3.t.a.847.2		4
24.11	even	2		216.3.p.a.91.1		4
36.7	odd	6		288.3.t.a.79.2		4
36.11	even	6		864.3.t.a.559.2		4
36.23	even	6		2592.3.b.a.1135.1		2
36.31	odd	6		2592.3.b.b.1135.2		2
72.5	odd	6		2592.3.b.a.1135.1		2
72.11	even	6		216.3.p.a.19.1		4
72.13	even	6		2592.3.b.b.1135.2		2
72.29	odd	6		864.3.t.a.559.2		4
72.43	odd	6	inner	72.3.p.a.43.1	✓	4
72.59	even	6		648.3.b.b.163.2		2
72.61	even	6		288.3.t.a.79.2		4
72.67	odd	6		648.3.b.a.163.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.a.43.1	✓	4	9.7	even	3	inner
72.3.p.a.43.1	✓	4	72.43	odd	6	inner
72.3.p.a.67.1	yes	4	1.1	even	1	trivial
72.3.p.a.67.1	yes	4	8.3	odd	2	CM
216.3.p.a.19.1		4	9.2	odd	6
216.3.p.a.19.1		4	72.11	even	6
216.3.p.a.91.1		4	3.2	odd	2
216.3.p.a.91.1		4	24.11	even	2
288.3.t.a.79.2		4	36.7	odd	6
288.3.t.a.79.2		4	72.61	even	6
288.3.t.a.175.2		4	4.3	odd	2
288.3.t.a.175.2		4	8.5	even	2
648.3.b.a.163.1		2	9.4	even	3
648.3.b.a.163.1		2	72.67	odd	6
648.3.b.b.163.2		2	9.5	odd	6
648.3.b.b.163.2		2	72.59	even	6
864.3.t.a.559.2		4	36.11	even	6
864.3.t.a.559.2		4	72.29	odd	6
864.3.t.a.847.2		4	12.11	even	2
864.3.t.a.847.2		4	24.5	odd	2
2592.3.b.a.1135.1		2	36.23	even	6
2592.3.b.a.1135.1		2	72.5	odd	6
2592.3.b.b.1135.2		2	36.31	odd	6
2592.3.b.b.1135.2		2	72.13	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-1.94949 + 2.28024i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-5.89898 - 1.09638i) q^{6} -8.00000 q^{8} +(-1.39898 - 8.89060i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-1.94949 + 2.28024i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-5.89898 - 1.09638i) q^{6} -8.00000 q^{8} +(-1.39898 - 8.89060i) q^{9} +(10.8485 + 18.7901i) q^{11} +(-4.00000 - 11.3137i) q^{12} +(-8.00000 - 13.8564i) q^{16} +28.3939 q^{17} +(14.0000 - 11.3137i) q^{18} +2.30306 q^{19} +(-21.6969 + 37.5802i) q^{22} +(15.5959 - 18.2419i) q^{24} +(-12.5000 - 21.6506i) q^{25} +(23.0000 + 14.1421i) q^{27} +(16.0000 - 27.7128i) q^{32} +(-63.9949 - 11.8940i) q^{33} +(28.3939 + 49.1796i) q^{34} +(33.5959 + 12.9350i) q^{36} +(2.30306 + 3.98902i) q^{38} +(17.8939 - 30.9931i) q^{41} +(-33.2423 - 57.5774i) q^{43} -86.7878 q^{44} +(47.1918 + 8.77101i) q^{48} +(-24.5000 + 42.4352i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-55.3536 + 64.7448i) q^{51} +(-1.49490 + 53.9793i) q^{54} +(-4.48979 + 5.25153i) q^{57} +(-57.2423 + 99.1467i) q^{59} +64.0000 q^{64} +(-43.3939 - 122.736i) q^{66} +(66.9393 - 115.942i) q^{67} +(-56.7878 + 98.3593i) q^{68} +(11.1918 + 71.1248i) q^{72} +100.394 q^{73} +(73.7372 + 13.7047i) q^{75} +(-4.60612 + 7.97804i) q^{76} +(-77.0857 + 24.8755i) q^{81} +71.5755 q^{82} +(-79.0000 - 136.832i) q^{83} +(66.4847 - 115.155i) q^{86} +(-86.7878 - 150.321i) q^{88} +146.000 q^{89} +(32.0000 + 90.5097i) q^{96} +(49.9847 + 86.5760i) q^{97} -98.0000 q^{98} +(151.879 - 122.736i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 2 * q^3 - 8 * q^4 - 4 * q^6 - 32 * q^8 + 14 * q^9	\( 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9} + 14 q^{11} - 16 q^{12} - 32 q^{16} - 4 q^{17} + 56 q^{18} + 68 q^{19} - 28 q^{22} - 16 q^{24} - 50 q^{25} + 92 q^{27} + 64 q^{32} - 158 q^{33} - 4 q^{34} + 56 q^{36} + 68 q^{38} - 46 q^{41} + 14 q^{43} - 112 q^{44} + 32 q^{48} - 98 q^{49} + 100 q^{50} - 290 q^{51} + 92 q^{54} + 178 q^{57} - 82 q^{59} + 256 q^{64} - 56 q^{66} + 62 q^{67} + 8 q^{68} - 112 q^{72} + 284 q^{73} + 50 q^{75} - 136 q^{76} - 34 q^{81} - 184 q^{82} - 316 q^{83} - 28 q^{86} - 112 q^{88} + 584 q^{89} + 128 q^{96} - 94 q^{97} - 392 q^{98} + 196 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 2 * q^3 - 8 * q^4 - 4 * q^6 - 32 * q^8 + 14 * q^9 + 14 * q^11 - 16 * q^12 - 32 * q^16 - 4 * q^17 + 56 * q^18 + 68 * q^19 - 28 * q^22 - 16 * q^24 - 50 * q^25 + 92 * q^27 + 64 * q^32 - 158 * q^33 - 4 * q^34 + 56 * q^36 + 68 * q^38 - 46 * q^41 + 14 * q^43 - 112 * q^44 + 32 * q^48 - 98 * q^49 + 100 * q^50 - 290 * q^51 + 92 * q^54 + 178 * q^57 - 82 * q^59 + 256 * q^64 - 56 * q^66 + 62 * q^67 + 8 * q^68 - 112 * q^72 + 284 * q^73 + 50 * q^75 - 136 * q^76 - 34 * q^81 - 184 * q^82 - 316 * q^83 - 28 * q^86 - 112 * q^88 + 584 * q^89 + 128 * q^96 - 94 * q^97 - 392 * q^98 + 196 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more