LMFDB - Embedded newform 72.3.p.a.43.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [72,3,Mod(43,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			43.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	72.43
Dual form			72.3.p.a.67.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(2.94949 + 0.548188i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(3.89898 - 4.56048i) q^{6} -8.00000 q^{8} +(8.39898 + 3.23375i) q^{9} +(-3.84847 + 6.66574i) q^{11} +(-4.00000 - 11.3137i) q^{12} +(-8.00000 + 13.8564i) q^{16} -30.3939 q^{17} +(14.0000 - 11.3137i) q^{18} +31.6969 q^{19} +(7.69694 + 13.3315i) q^{22} +(-23.5959 - 4.38551i) q^{24} +(-12.5000 + 21.6506i) q^{25} +(23.0000 + 14.1421i) q^{27} +(16.0000 + 27.7128i) q^{32} +(-15.0051 + 17.5509i) q^{33} +(-30.3939 + 52.6437i) q^{34} +(-5.59592 - 35.5624i) q^{36} +(31.6969 - 54.9007i) q^{38} +(-40.8939 - 70.8303i) q^{41} +(40.2423 - 69.7018i) q^{43} +30.7878 q^{44} +(-31.1918 + 36.4838i) q^{48} +(-24.5000 - 42.4352i) q^{49} +(25.0000 + 43.3013i) q^{50} +(-89.6464 - 16.6616i) q^{51} +(47.4949 - 25.6950i) q^{54} +(93.4898 + 17.3759i) q^{57} +(16.2423 + 28.1326i) q^{59} +64.0000 q^{64} +(15.3939 + 43.5405i) q^{66} +(-35.9393 - 62.2487i) q^{67} +(60.7878 + 105.287i) q^{68} +(-67.1918 - 25.8700i) q^{72} +41.6061 q^{73} +(-48.7372 + 57.0060i) q^{75} +(-63.3939 - 109.801i) q^{76} +(60.0857 + 54.3204i) q^{81} -163.576 q^{82} +(-79.0000 + 136.832i) q^{83} +(-80.4847 - 139.404i) q^{86} +(30.7878 - 53.3260i) q^{88} +146.000 q^{89} +(32.0000 + 90.5097i) q^{96} +(-96.9847 + 167.982i) q^{97} -98.0000 q^{98} +(-53.8786 + 43.5405i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9} + 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}+ \cdots + 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

Character values

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

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{3}\right)$

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
$3$	2.94949	+	0.548188i	0.983163	+	0.182729i
$3$	2.94949	+	0.548188i	0.983163	+	0.182729i
$4$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	3.89898	−	4.56048i	0.649830	−	0.760080i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−8.00000			−1.00000
$9$	8.39898	+	3.23375i	0.933220	+	0.359306i
$10$		0			0
$11$	−3.84847	+	6.66574i	−0.349861	+	0.605977i	−0.986224	−	0.165412i	$-0.947104\pi$
$11$	−3.84847	+	6.66574i	−0.349861	+	0.605977i	0.636364	+	0.771389i	$0.280438\pi$
$12$	−4.00000	−	11.3137i	−0.333333	−	0.942809i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$17$	−30.3939			−1.78788			−0.893938	−	0.448192i	$-0.852068\pi$
$17$	−30.3939			−1.78788			−0.893938	+	0.448192i	$0.852068\pi$
$18$	14.0000	−	11.3137i	0.777778	−	0.628539i
$19$	31.6969			1.66826			0.834130	−	0.551568i	$-0.185970\pi$
$19$	31.6969			1.66826			0.834130	+	0.551568i	$0.185970\pi$
$20$		0			0
$21$		0			0
$22$	7.69694	+	13.3315i	0.349861	+	0.605977i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−23.5959	−	4.38551i	−0.983163	−	0.182729i
$25$	−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	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	16.0000	+	27.7128i	0.500000	+	0.866025i
$33$	−15.0051	+	17.5509i	−0.454700	+	0.531844i
$34$	−30.3939	+	52.6437i	−0.893938	+	1.54835i
$35$		0			0
$36$	−5.59592	−	35.5624i	−0.155442	−	0.987845i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	31.6969	−	54.9007i	0.834130	−	1.44476i
$39$		0			0
$40$		0			0
$41$	−40.8939	−	70.8303i	−0.997412	−	1.72757i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−40.8939	−	70.8303i	−0.997412	−	1.72757i	−0.436436	−	0.899735i	$-0.643759\pi$
$42$		0			0
$43$	40.2423	−	69.7018i	0.935869	−	1.62097i	0.162791	−	0.986661i	$-0.447950\pi$
$43$	40.2423	−	69.7018i	0.935869	−	1.62097i	0.773078	−	0.634311i	$-0.218716\pi$
$44$	30.7878			0.699722
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	−31.1918	+	36.4838i	−0.649830	+	0.760080i
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$	25.0000	+	43.3013i	0.500000	+	0.866025i
$51$	−89.6464	−	16.6616i	−1.75777	−	0.326697i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	47.4949	−	25.6950i	0.879535	−	0.475834i
$55$		0			0
$56$		0			0
$57$	93.4898	+	17.3759i	1.64017	+	0.304840i
$58$		0			0
$59$	16.2423	+	28.1326i	0.275294	+	0.476823i	0.970209	−	0.242268i	$-0.0778915\pi$
$59$	16.2423	+	28.1326i	0.275294	+	0.476823i	−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$		0			0
$66$	15.3939	+	43.5405i	0.233241	+	0.659704i
$67$	−35.9393	−	62.2487i	−0.536407	−	0.929085i	−0.999094	−	0.0425626i	$-0.986448\pi$
$67$	−35.9393	−	62.2487i	−0.536407	−	0.929085i	0.462687	−	0.886522i	$-0.346886\pi$
$68$	60.7878	+	105.287i	0.893938	+	1.54835i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−67.1918	−	25.8700i	−0.933220	−	0.359306i
$73$	41.6061			0.569947			0.284973	−	0.958535i	$-0.408015\pi$
$73$	41.6061			0.569947			0.284973	+	0.958535i	$0.408015\pi$
$74$		0			0
$75$	−48.7372	+	57.0060i	−0.649830	+	0.760080i
$76$	−63.3939	−	109.801i	−0.834130	−	1.44476i
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	60.0857	+	54.3204i	0.741799	+	0.670622i
$82$	−163.576			−1.99482
$83$	−79.0000	+	136.832i	−0.951807	+	1.64858i	−0.210296	+	0.977638i	$0.567443\pi$
$83$	−79.0000	+	136.832i	−0.951807	+	1.64858i	−0.741511	+	0.670941i	$0.765890\pi$
$84$		0			0
$85$		0			0
$86$	−80.4847	−	139.404i	−0.935869	−	1.62097i
$87$		0			0
$88$	30.7878	−	53.3260i	0.349861	−	0.605977i
$89$	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
$97$	−96.9847	+	167.982i	−0.999842	+	1.73178i	−0.484536	+	0.874771i	$0.661012\pi$
$97$	−96.9847	+	167.982i	−0.999842	+	1.73178i	−0.515306	+	0.857006i	$0.672322\pi$
$98$	−98.0000			−1.00000
$99$	−53.8786	+	43.5405i	−0.544228	+	0.439803i
$100$	100.000			1.00000
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	−118.505	+	138.611i	−1.16181	+	1.35893i
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	191.879			1.79326			0.896629	−	0.442783i	$-0.146009\pi$
$107$	191.879			1.79326			0.896629	+	0.442783i	$0.146009\pi$
$108$	2.98979	−	107.959i	0.0276833	−	0.999617i
$109$		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	0.563554	−	0.826079i	$-0.309434\pi$
$113$	−49.0000	−	84.8705i	−0.433628	−	0.751066i	−0.997183	+	0.0750128i	$0.976100\pi$
$114$	123.586	−	144.553i	1.08409	−	1.26801i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	64.9694			0.550588
$119$		0			0
$120$		0			0
$121$	30.8786	+	53.4833i	0.255195	+	0.442010i
$122$		0			0
$123$	−81.7878	−	231.331i	−0.664941	−	1.88074i
$124$		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
$129$	156.904	−	183.524i	1.21631	−	1.42267i
$130$		0			0
$131$	−31.0000	−	53.6936i	−0.236641	−	0.409875i	0.723107	−	0.690736i	$-0.242713\pi$
$131$	−31.0000	−	53.6936i	−0.236641	−	0.409875i	−0.959748	+	0.280861i	$0.909380\pi$
$132$	90.8082	+	16.8775i	0.687941	+	0.127860i
$133$		0			0
$134$	−143.757			−1.07281
$135$		0			0
$136$	243.151			1.78788
$137$	−0.712246	+	1.23365i	−0.00519888	+	0.00900472i	−0.868613	−	0.495491i	$-0.834988\pi$
$137$	−0.712246	+	1.23365i	−0.00519888	+	0.00900472i	0.863414	+	0.504496i	$0.168322\pi$
$138$		0			0
$139$	132.333	+	229.208i	0.952037	+	1.64898i	0.741007	+	0.671497i	$0.234348\pi$
$139$	132.333	+	229.208i	0.952037	+	1.64898i	0.211030	+	0.977480i	$0.432318\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−112.000	+	90.5097i	−0.777778	+	0.628539i
$145$		0			0
$146$	41.6061	−	72.0639i	0.284973	−	0.493588i
$147$	−49.0000	−	138.593i	−0.333333	−	0.942809i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	50.0000	+	141.421i	0.333333	+	0.942809i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−253.576			−1.66826
$153$	−255.278	−	98.2862i	−1.66848	−	0.642394i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	154.171	−	49.7511i	0.951675	−	0.307106i
$163$	−322.000			−1.97546			−0.987730	−	0.156171i	$-0.950085\pi$
$163$	−322.000			−1.97546			−0.987730	+	0.156171i	$0.950085\pi$
$164$	−163.576	+	283.321i	−0.997412	+	1.72757i
$165$		0			0
$166$	158.000	+	273.664i	0.951807	+	1.64858i
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−84.5000	+	146.358i	−0.500000	+	0.866025i
$170$		0			0
$171$	266.222	+	102.500i	1.55685	+	0.599415i
$172$	−321.939			−1.87174
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	−61.5755	−	106.652i	−0.349861	−	0.605977i
$177$	32.4847	+	91.8806i	0.183529	+	0.519099i
$178$	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
$187$	116.970	−	202.598i	0.625507	−	1.08341i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	188.767	+	35.0840i	0.983163	+	0.182729i
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	−0.964592	−	0.263745i	$-0.915042\pi$
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	0.253886	−	0.967234i	$-0.418291\pi$
$194$	193.969	+	335.965i	0.999842	+	1.73178i
$195$		0			0
$196$	−98.0000	+	169.741i	−0.500000	+	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	21.5357	+	136.861i	0.108766	+	0.691217i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	100.000	−	173.205i	0.500000	−	0.866025i
$201$	−71.8786	−	203.303i	−0.357605	−	1.01146i
$202$		0			0
$203$		0			0
$204$	121.576	+	343.867i	0.595958	+	1.68562i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−121.985	+	211.284i	−0.583659	+	1.01093i
$210$		0			0
$211$	113.000	+	195.722i	0.535545	+	0.927591i	0.999137	+	0.0415423i	$0.0132271\pi$
$211$	113.000	+	195.722i	0.535545	+	0.927591i	−0.463592	+	0.886049i	$0.653440\pi$
$212$		0			0
$213$		0			0
$214$	191.879	−	332.343i	0.896629	−	1.55301i
$215$		0			0
$216$	−184.000	−	113.137i	−0.851852	−	0.523783i
$217$		0			0
$218$		0			0
$219$	122.717	+	22.8080i	0.560351	+	0.104146i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−175.000	+	141.421i	−0.777778	+	0.628539i
$226$	−196.000			−0.867257
$227$	148.242	−	256.763i	0.653050	−	1.13112i	−0.329329	−	0.944215i	$-0.606822\pi$
$227$	148.242	−	256.763i	0.653050	−	1.13112i	0.982379	−	0.186900i	$-0.0598442\pi$
$228$	−126.788	−	358.610i	−0.556087	−	1.57285i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−70.0306			−0.300561			−0.150280	−	0.988643i	$-0.548018\pi$
$233$	−70.0306			−0.300561			−0.150280	+	0.988643i	$0.548018\pi$
$234$		0			0
$235$		0			0
$236$	64.9694	−	112.530i	0.275294	−	0.476823i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.402490	−	0.915425i	$-0.368145\pi$
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.591536	−	0.806279i	$-0.298522\pi$
$242$	123.514			0.510390
$243$	147.444	+	193.156i	0.606767	+	0.794880i
$244$		0			0
$245$		0			0
$246$	−482.464	−	89.6702i	−1.96124	−	0.364513i
$247$		0			0
$248$		0			0
$249$	−308.019	+	360.278i	−1.23703	+	1.44690i
$250$		0			0
$251$	71.3337			0.284198			0.142099	−	0.989852i	$-0.454615\pi$
$251$	71.3337			0.284198			0.142099	+	0.989852i	$0.454615\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	−	221.703i	−0.500000	−	0.866025i
$257$	243.469	+	421.701i	0.947352	+	1.64086i	0.750973	+	0.660333i	$0.229585\pi$
$257$	243.469	+	421.701i	0.947352	+	1.64086i	0.196379	+	0.980528i	$0.437082\pi$
$258$	−160.969	−	455.290i	−0.623912	−	1.76469i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−124.000			−0.473282
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	120.041	−	140.407i	0.454700	−	0.531844i
$265$		0			0
$266$		0			0
$267$	430.626	+	80.0355i	1.61283	+	0.299758i
$268$	−143.757	+	248.995i	−0.536407	+	0.929085i
$269$		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			$\pi$
$272$	243.151	−	421.150i	0.893938	−	1.54835i
$273$		0			0
$274$	1.42449	+	2.46729i	0.00519888	+	0.00900472i
$275$	−96.2117	−	166.644i	−0.349861	−	0.605977i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	529.333			1.90407
$279$		0			0
$280$		0			0
$281$	119.000	−	206.114i	0.423488	−	0.733502i	−0.572790	−	0.819702i	$-0.694139\pi$
$281$	119.000	−	206.114i	0.423488	−	0.733502i	0.996278	+	0.0862000i	$0.0274724\pi$
$282$		0			0
$283$	41.0000	+	71.0141i	0.144876	+	0.250933i	0.929327	−	0.369258i	$-0.120388\pi$
$283$	41.0000	+	71.0141i	0.144876	+	0.250933i	−0.784450	+	0.620191i	$0.787055\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	44.7673	+	284.499i	0.155442	+	0.987845i
$289$	634.788			2.19650
$290$		0			0
$291$	−378.141	+	442.297i	−1.29945	+	1.51992i
$292$	−83.2122	−	144.128i	−0.284973	−	0.493588i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$	−289.050	−	53.7224i	−0.983163	−	0.182729i
$295$		0			0
$296$		0			0
$297$	−182.783	+	98.8865i	−0.615430	+	0.332951i
$298$		0			0
$299$		0			0
$300$	294.949	+	54.8188i	0.983163	+	0.182729i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−253.576	+	439.206i	−0.834130	+	1.44476i
$305$		0			0
$306$	−425.514	+	343.867i	−1.39057	+	1.12375i
$307$	−520.848			−1.69657			−0.848287	−	0.529537i	$-0.822366\pi$
$307$	−520.848			−1.69657			−0.848287	+	0.529537i	$0.822366\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	15.4694	−	26.7938i	0.0494230	−	0.0856031i	−0.840256	−	0.542191i	$-0.817595\pi$
$313$	15.4694	−	26.7938i	0.0494230	−	0.0856031i	0.889679	+	0.456587i	$0.150928\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	565.944	+	105.186i	1.76307	+	0.327681i
$322$		0			0
$323$	−963.393			−2.98264
$324$	68.0000	−	316.784i	0.209877	−	0.977728i
$325$		0			0
$326$	−322.000	+	557.720i	−0.987730	+	1.71080i
$327$		0			0
$328$	327.151	+	566.642i	0.997412	+	1.72757i
$329$		0			0
$330$		0			0
$331$	−7.00000	+	12.1244i	−0.0211480	+	0.0366295i	−0.876406	−	0.481573i	$-0.840065\pi$
$331$	−7.00000	+	12.1244i	−0.0211480	+	0.0366295i	0.855258	+	0.518203i	$0.173399\pi$
$332$	632.000			1.90361
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−325.257	−	563.362i	−0.965155	−	1.67170i	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	−325.257	−	563.362i	−0.965155	−	1.67170i	−0.255956	−	0.966688i	$-0.582390\pi$
$338$	169.000	+	292.717i	0.500000	+	0.866025i
$339$	−98.0000	−	277.186i	−0.289086	−	0.817657i
$340$		0			0
$341$		0			0
$342$	443.757	−	358.610i	1.29754	−	1.04857i
$343$		0			0
$344$	−321.939	+	557.614i	−0.935869	+	1.62097i
$345$		0			0
$346$		0			0
$347$	−260.030	−	450.385i	−0.749366	−	1.29794i	−0.948127	−	0.317892i	$-0.897025\pi$
$347$	−260.030	−	450.385i	−0.749366	−	1.29794i	0.198761	−	0.980048i	$-0.436308\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$	−246.302			−0.699722
$353$	−245.439	+	425.112i	−0.695294	+	1.20428i	0.274788	+	0.961505i	$0.411392\pi$
$353$	−245.439	+	425.112i	−0.695294	+	1.20428i	−0.970081	+	0.242779i	$0.921941\pi$
$354$	191.627	+	35.6154i	0.541318	+	0.100609i
$355$		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
$361$	643.696			1.78309
$362$		0			0
$363$	61.7571	+	174.676i	0.170130	+	0.481200i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	−114.419	−	727.143i	−0.310080	−	1.97058i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	−233.940	−	405.196i	−0.625507	−	1.08341i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−63.7582			−0.168227			−0.0841137	−	0.996456i	$-0.526806\pi$
$379$	−63.7582			−0.168227			−0.0841137	+	0.996456i	$0.526806\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	249.535	−	291.871i	0.649830	−	0.760080i
$385$		0			0
$386$	−548.665			−1.42141
$387$	563.393	−	455.290i	1.45580	−	1.17646i
$388$	775.878			1.99968
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		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
$396$	258.586	+	99.5599i	0.652994	+	0.251414i
$397$		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
$401$	−88.6214	−	153.497i	−0.221001	−	0.382785i	0.734111	−	0.679029i	$-0.237599\pi$
$401$	−88.6214	−	153.497i	−0.221001	−	0.382785i	−0.955112	+	0.296244i	$0.904266\pi$
$402$	−424.010	−	78.8060i	−1.05475	−	0.196035i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	717.171	+	133.293i	1.75777	+	0.326697i
$409$	239.833	+	415.402i	0.586388	+	1.01565i	0.994701	+	0.102812i	$0.0327839\pi$
$409$	239.833	+	415.402i	0.586388	+	1.01565i	−0.408313	+	0.912842i	$0.633883\pi$
$410$		0			0
$411$	−2.77703	+	3.24818i	−0.00675677	+	0.00790312i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	264.666	+	748.589i	0.634691	+	1.79518i
$418$	243.969	+	422.567i	0.583659	+	1.01093i
$419$	257.000	+	445.137i	0.613365	+	1.06238i	0.990669	+	0.136290i	$0.0435179\pi$
$419$	257.000	+	445.137i	0.613365	+	1.06238i	−0.377304	+	0.926090i	$0.623149\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	452.000			1.07109
$423$		0			0
$424$		0			0
$425$	379.923	−	658.047i	0.893938	−	1.54835i
$426$		0			0
$427$		0			0
$428$	−383.757	−	664.687i	−0.896629	−	1.55301i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−379.959	+	205.560i	−0.879535	+	0.475834i
$433$	−847.484			−1.95724			−0.978619	−	0.205684i	$-0.934058\pi$
$433$	−847.484			−1.95724			−0.978619	+	0.205684i	$0.934058\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	162.221	−	189.744i	0.370369	−	0.433205i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−68.5500	−	435.640i	−0.155442	−	0.987845i
$442$		0			0
$443$	168.061	−	291.090i	0.379370	−	0.657087i	−0.611601	−	0.791166i	$-0.709474\pi$
$443$	168.061	−	291.090i	0.379370	−	0.657087i	0.990971	+	0.134079i	$0.0428076\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−638.757			−1.42262			−0.711311	−	0.702878i	$-0.751898\pi$
$449$	−638.757			−1.42262			−0.711311	+	0.702878i	$0.751898\pi$
$450$	69.9490	+	444.530i	0.155442	+	0.987845i
$451$	629.515			1.39582
$452$	−196.000	+	339.482i	−0.433628	+	0.751066i
$453$		0			0
$454$	−296.485	−	513.527i	−0.653050	−	1.13112i
$455$		0			0
$456$	−747.918	−	139.007i	−1.64017	−	0.304840i
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.260394	+	0.965502i	$0.583852\pi$
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.705953	+	0.708259i	$0.749481\pi$
$458$		0			0
$459$	−699.059	−	429.834i	−1.52300	−	0.936458i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$	−70.0306	+	121.297i	−0.150280	+	0.260293i
$467$	825.332			1.76731			0.883653	−	0.468143i	$-0.155077\pi$
$467$	825.332			1.76731			0.883653	+	0.468143i	$0.155077\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−129.939	−	225.061i	−0.275294	−	0.476823i
$473$	309.743	+	536.490i	0.654847	+	1.13423i
$474$		0			0
$475$	−396.212	+	686.259i	−0.834130	+	1.44476i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	−479.120	−	829.861i	−0.994026	−	1.72170i
$483$		0			0
$484$	123.514	−	213.933i	0.255195	−	0.442010i
$485$		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
$489$	−949.736	−	176.517i	−1.94220	−	0.360975i
$490$		0			0
$491$	452.696	+	784.093i	0.921989	+	1.59693i	0.796334	+	0.604857i	$0.206770\pi$
$491$	452.696	+	784.093i	0.921989	+	1.59693i	0.125655	+	0.992074i	$0.459897\pi$
$492$	−637.778	+	745.982i	−1.29630	+	1.51622i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	316.000	+	893.783i	0.634538	+	1.79474i
$499$	56.6964	+	98.2011i	0.113620	+	0.196796i	0.917227	−	0.398364i	$-0.130422\pi$
$499$	56.6964	+	98.2011i	0.113620	+	0.196796i	−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$	71.3337	−	123.554i	0.142099	−	0.246123i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−329.464	+	385.360i	−0.649830	+	0.760080i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$	729.030	+	448.262i	1.42111	+	0.873806i
$514$	973.878			1.89470
$515$		0			0
$516$	−949.555	−	176.483i	−1.84022	−	0.342021i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	738.151			1.41680			0.708398	−	0.705813i	$-0.249418\pi$
$521$	738.151			1.41680			0.708398	+	0.705813i	$0.249418\pi$
$522$		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
$528$	−123.151	−	348.324i	−0.233241	−	0.659704i
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		0			0
$531$	45.4454	+	288.809i	0.0855846	+	0.543896i
$532$		0			0
$533$		0			0
$534$	569.251	−	665.830i	1.06601	−	1.24687i
$535$		0			0
$536$	287.514	+	497.989i	0.536407	+	0.929085i
$537$	−100.283	−	18.6384i	−0.186746	−	0.0347084i
$538$		0			0
$539$	377.150			0.699722
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−486.302	−	842.300i	−0.893938	−	1.54835i
$545$		0			0
$546$		0			0
$547$	424.515	−	735.281i	0.776078	−	1.34421i	−0.158108	−	0.987422i	$-0.550539\pi$
$547$	424.515	−	735.281i	0.776078	−	1.34421i	0.934186	−	0.356785i	$-0.116127\pi$
$548$	5.69797			0.0103978
$549$		0			0
$550$	−384.847			−0.699722
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	529.333	−	916.831i	0.952037	−	1.64898i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	456.063	−	533.439i	0.812947	−	0.950871i
$562$	−238.000	−	412.228i	−0.423488	−	0.733502i
$563$	421.150	+	729.454i	0.748047	+	1.29566i	0.948758	+	0.316005i	$0.102342\pi$
$563$	421.150	+	729.454i	0.748047	+	1.29566i	−0.200710	+	0.979651i	$0.564325\pi$
$564$		0			0
$565$		0			0
$566$	164.000			0.289753
$567$		0			0
$568$		0			0
$569$	568.014	−	983.830i	0.998268	−	1.72905i	0.448180	−	0.893943i	$-0.352072\pi$
$569$	568.014	−	983.830i	0.998268	−	1.72905i	0.550088	−	0.835107i	$-0.314594\pi$
$570$		0			0
$571$	−568.666	−	984.958i	−0.995912	−	1.72497i	−0.576182	−	0.817321i	$-0.695458\pi$
$571$	−568.666	−	984.958i	−0.995912	−	1.72497i	−0.419730	−	0.907649i	$-0.637875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	537.535	+	206.960i	0.933220	+	0.359306i
$577$	−1000.39			−1.73378			−0.866891	−	0.498498i	$-0.833885\pi$
$577$	−1000.39			−1.73378			−0.866891	+	0.498498i	$0.833885\pi$
$578$	634.788	−	1099.48i	1.09825	−	1.90222i
$579$	−274.333	−	775.930i	−0.473804	−	1.34012i
$580$		0			0
$581$		0			0
$582$	387.939	+	1097.26i	0.666561	+	1.88532i
$583$		0			0
$584$	−332.849			−0.569947
$585$		0			0
$586$		0			0
$587$	−159.576	+	276.394i	−0.271850	+	0.470858i	−0.969336	−	0.245741i	$-0.920969\pi$
$587$	−159.576	+	276.394i	−0.271850	+	0.470858i	0.697485	+	0.716599i	$0.254302\pi$
$588$	−382.100	+	446.927i	−0.649830	+	0.760080i
$589$		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	$0.758999\pi$
$594$	−11.5061	+	415.475i	−0.0193706	+	0.699454i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	389.898	−	456.048i	0.649830	−	0.760080i
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	−0.942645	−	0.333797i	$-0.891670\pi$
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	0.760399	+	0.649456i	$0.225003\pi$
$602$		0			0
$603$	−100.557	−	639.044i	−0.166761	−	1.05977i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	507.151	+	878.411i	0.834130	+	1.44476i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	170.082	+	1080.88i	0.277911	+	1.76614i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−520.848	+	902.135i	−0.848287	+	1.46928i
$615$		0			0
$616$		0			0
$617$	−597.893	−	1035.58i	−0.969032	−	1.67841i	−0.698368	−	0.715739i	$-0.746090\pi$
$617$	−597.893	−	1035.58i	−0.969032	−	1.67841i	−0.270665	−	0.962674i	$-0.587243\pi$
$618$		0			0
$619$	337.150	−	583.962i	0.544670	−	0.943395i	−0.453958	−	0.891023i	$-0.649988\pi$
$619$	337.150	−	583.962i	0.544670	−	0.943395i	0.998628	−	0.0523724i	$-0.0166783\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$	−30.9388	−	53.5875i	−0.0494230	−	0.0856031i
$627$	−475.616	+	556.308i	−0.758558	+	0.887254i
$628$		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
$641$	−393.893	+	682.242i	−0.614497	+	1.06434i	0.375975	+	0.926630i	$0.377308\pi$
$641$	−393.893	+	682.242i	−0.614497	+	1.06434i	−0.990472	+	0.137711i	$0.956025\pi$
$642$	748.131	−	875.058i	1.16531	−	1.36302i
$643$	119.788	+	207.479i	0.186296	+	0.322674i	0.944012	−	0.329910i	$-0.107018\pi$
$643$	119.788	+	207.479i	0.186296	+	0.322674i	−0.757717	+	0.652584i	$0.773685\pi$
$644$		0			0
$645$		0			0
$646$	−963.393	+	1668.65i	−1.49132	+	2.58304i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−480.686	−	434.563i	−0.741799	−	0.670622i
$649$	−250.033			−0.385258
$650$		0			0
$651$		0			0
$652$	644.000	+	1115.44i	0.987730	+	1.71080i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	1308.60			1.99482
$657$	349.449	+	134.544i	0.531886	+	0.204785i
$658$		0			0
$659$	497.000	−	860.829i	0.754173	−	1.30627i	−0.191611	−	0.981471i	$-0.561371\pi$
$659$	497.000	−	860.829i	0.754173	−	1.30627i	0.945784	−	0.324795i	$-0.105295\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	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	0.135284	−	0.990807i	$-0.456805\pi$
$673$	623.000	−	1079.07i	0.925706	−	1.60337i	0.790422	−	0.612563i	$-0.209861\pi$
$674$	−1301.03			−1.93031
$675$	−593.686	+	321.188i	−0.879535	+	0.475834i
$676$	676.000			1.00000
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$	−578.100	−	107.445i	−0.852655	−	0.158473i
$679$		0			0
$680$		0			0
$681$	577.994	−	676.056i	0.848743	−	0.992740i
$682$		0			0
$683$	−1330.66			−1.94826			−0.974132	−	0.225980i	$-0.927442\pi$
$683$	−1330.66			−1.94826			−0.974132	+	0.225980i	$0.927442\pi$
$684$	−177.373	−	1127.22i	−0.259318	−	1.64798i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	643.878	+	1115.23i	0.935869	+	1.62097i
$689$		0			0
$690$		0			0
$691$	−367.000	+	635.663i	−0.531114	+	0.919917i	0.468226	+	0.883609i	$0.344893\pi$
$691$	−367.000	+	635.663i	−0.531114	+	0.919917i	−0.999341	+	0.0363084i	$0.988440\pi$
$692$		0			0
$693$		0			0
$694$	−1040.12			−1.49873
$695$		0			0
$696$		0			0
$697$	1242.92	+	2152.81i	1.78325	+	3.08868i
$698$		0			0
$699$	−206.555	−	38.3900i	−0.295500	−	0.0549213i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−246.302	+	426.608i	−0.349861	+	0.605977i
$705$		0			0
$706$	490.878	+	850.225i	0.695294	+	1.20428i
$707$		0			0
$708$	253.314	−	296.291i	0.357789	−	0.418491i
$709$		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
$712$	−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
$722$	643.696	−	1114.91i	0.891546	−	1.54420i
$723$	934.040	−	1092.51i	1.29190	−	1.51108i
$724$		0			0
$725$		0			0
$726$	364.304	+	67.7091i	0.501796	+	0.0932632i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$	−1223.12	+	2118.51i	−1.67322	+	2.89810i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	553.245			0.750672
$738$	−1373.87	−	528.962i	−1.86161	−	0.716751i
$739$	1410.24			1.90831			0.954154	−	0.299316i	$-0.0967584\pi$
$739$	1410.24			1.90831			0.954154	+	0.299316i	$0.0967584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1106.00	+	893.783i	−1.48059	+	1.19650i
$748$	−935.759			−1.25101
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	210.398	+	39.1043i	0.279413	+	0.0519313i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−63.7582	+	110.432i	−0.0841137	+	0.145689i
$759$		0			0
$760$		0			0
$761$	−697.000	−	1207.24i	−0.915900	−	1.58639i	−0.805578	−	0.592490i	$-0.798145\pi$
$761$	−697.000	−	1207.24i	−0.915900	−	1.58639i	−0.110322	−	0.993896i	$-0.535188\pi$
$762$		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	0.973341	+	0.229364i	$0.0736647\pi$
$769$	527.000	+	912.791i	0.685306	+	1.18698i	−0.288035	+	0.957620i	$0.593002\pi$
$770$		0			0
$771$	486.939	+	1377.27i	0.631568	+	1.78634i
$772$	−548.665	+	950.316i	−0.710706	+	1.23098i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−225.193	−	1431.12i	−0.290947	−	1.84899i
$775$		0			0
$776$	775.878	−	1343.86i	0.999842	−	1.73178i
$777$		0			0
$778$		0			0
$779$	−1296.21	−	2245.10i	−1.66394	−	2.88203i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$	−365.737	−	67.9753i	−0.465314	−	0.0864826i
$787$	−463.000	−	801.940i	−0.588310	−	1.01898i	−0.994454	−	0.105174i	$-0.966460\pi$
$787$	−463.000	−	801.940i	−0.588310	−	1.01898i	0.406144	−	0.913809i	$-0.366873\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	431.029	−	348.324i	0.544228	−	0.439803i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$	−800.000			−1.00000
$801$	1226.25	+	472.128i	1.53090	+	0.589423i
$802$	−354.486			−0.442002
$803$	−160.120	+	277.336i	−0.199402	+	0.345375i
$804$	−560.506	+	655.601i	−0.697147	+	0.815424i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	497.061			0.614414			0.307207	−	0.951643i	$-0.400606\pi$
$809$	497.061			0.614414			0.307207	+	0.951643i	$0.400606\pi$
$810$		0			0
$811$	−1307.21			−1.61185			−0.805924	−	0.592019i	$-0.798331\pi$
$811$	−1307.21			−1.61185			−0.805924	+	0.592019i	$0.798331\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	948.041	−	1108.88i	1.16181	−	1.35893i
$817$	1275.56	−	2209.33i	1.56127	−	2.70420i
$818$	959.331			1.17278
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	2.84898	+	8.05815i	0.00346592	+	0.00980310i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−192.423	−	544.256i	−0.233241	−	0.659704i
$826$		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
$833$	744.650	+	1289.77i	0.893938	+	1.54835i
$834$	1561.26	+	290.174i	1.87202	+	0.347930i
$835$		0			0
$836$	975.878			1.16732
$837$		0			0
$838$	1028.00			1.22673
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$	463.979	−	542.697i	0.550390	−	0.643769i
$844$	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
$850$	−759.847	−	1316.09i	−0.893938	−	1.54835i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−1535.03			−1.79326
$857$	−601.000	+	1040.96i	−0.701284	+	1.21466i	0.266733	+	0.963771i	$0.414056\pi$
$857$	−601.000	+	1040.96i	−0.701284	+	1.21466i	−0.968016	+	0.250888i	$0.919277\pi$
$858$		0			0
$859$	624.606	+	1081.85i	0.727131	+	1.25943i	0.958091	+	0.286465i	$0.0924801\pi$
$859$	624.606	+	1081.85i	0.727131	+	1.25943i	−0.230960	+	0.972963i	$0.574187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−23.9184	+	863.669i	−0.0276833	+	0.999617i
$865$		0			0
$866$	−847.484	+	1467.88i	−0.978619	+	1.69502i
$867$	1872.30	+	347.983i	2.15952	+	0.401365i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1357.79	+	1097.26i	−1.55531	+	1.25688i
$874$		0			0
$875$		0			0
$876$	−166.424	−	470.720i	−0.189982	−	0.537351i
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		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	$0.803878\pi$
$882$	−823.100	−	316.908i	−0.933220	−	0.359306i
$883$	983.879			1.11425			0.557123	−	0.830430i	$-0.311905\pi$
$883$	983.879			1.11425			0.557123	+	0.830430i	$0.311905\pi$
$884$		0			0
$885$		0			0
$886$	−336.121	−	582.179i	−0.379370	−	0.657087i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−593.324	+	191.466i	−0.665908	+	0.214888i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−638.757	+	1106.36i	−0.711311	+	1.23203i
$899$		0			0
$900$	839.898	+	323.375i	0.933220	+	0.359306i
$901$		0			0
$902$	629.515	−	1090.35i	0.697911	−	1.20882i
$903$		0			0
$904$	392.000	+	678.964i	0.433628	+	0.751066i
$905$		0			0
$906$		0			0
$907$	−171.304	+	296.706i	−0.188868	+	0.327130i	−0.944873	−	0.327436i	$-0.893815\pi$
$907$	−171.304	+	296.706i	−0.188868	+	0.327130i	0.756005	+	0.654566i	$0.227149\pi$
$908$	−1185.94			−1.30610
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	−988.686	+	1156.43i	−1.08409	+	1.26801i
$913$	−608.058	−	1053.19i	−0.666000	−	1.15355i
$914$	883.241	+	1529.82i	0.966347	+	1.67376i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1443.55	+	780.972i	−1.57250	+	0.850732i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−1536.24	−	285.523i	−1.66801	−	0.310014i
$922$		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	0.427193	+	0.904160i	$0.359503\pi$
$929$	−529.000	+	916.255i	−0.569429	+	0.986281i	−0.996622	+	0.0821203i	$0.973831\pi$
$930$		0			0
$931$	−776.575	−	1345.07i	−0.834130	−	1.44476i
$932$	140.061	+	242.593i	0.150280	+	0.260293i
$933$		0			0
$934$	825.332	−	1429.52i	0.883653	−	1.53053i
$935$		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
$939$	60.3148	−	70.5478i	0.0642330	−	0.0751308i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$	−519.755			−0.550588
$945$		0			0
$946$	1238.97			1.30969
$947$	449.605	−	778.738i	0.474767	−	0.822321i	−0.524815	−	0.851216i	$-0.675866\pi$
$947$	449.605	−	778.738i	0.474767	−	0.822321i	0.999582	+	0.0288952i	$0.00919891\pi$
$948$		0			0
$949$		0			0
$950$	792.423	+	1372.52i	0.834130	+	1.44476i
$951$		0			0
$952$		0			0
$953$	1717.06			1.80174			0.900869	−	0.434090i	$-0.142930\pi$
$953$	1717.06			1.80174			0.900869	+	0.434090i	$0.142930\pi$
$954$		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
$963$	1611.58	+	620.487i	1.67350	+	0.644328i
$964$	−1916.48			−1.98805
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	−247.029	−	427.866i	−0.255195	−	0.442010i
$969$	−2841.52	−	528.121i	−2.93242	−	0.545016i
$970$		0			0
$971$	974.000			1.00309			0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000			1.00309			0.501545	+	0.865132i	$0.332765\pi$
$972$	374.222	−	897.074i	0.385003	−	0.922916i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−641.166	−	1110.53i	−0.656260	−	1.13668i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−641.166	−	1110.53i	−0.656260	−	1.13668i	0.325316	−	0.945605i	$-0.394529\pi$
$978$	−1255.47	+	1468.47i	−1.28371	+	1.50151i
$979$	−561.877	+	973.199i	−0.573929	+	0.994074i
$980$		0			0
$981$		0			0
$982$	1810.79			1.84398
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	654.302	+	1850.65i	0.664941	+	1.88074i
$985$		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
$993$	−27.2929	+	31.9233i	−0.0274853	+	0.0321484i
$994$		0			0
$995$		0			0
$996$	1864.08	+	346.455i	1.87156	+	0.347846i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	226.786			0.227240
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.a.43.2	✓	4
3.2	odd	2		216.3.p.a.19.2		4
4.3	odd	2		288.3.t.a.79.1		4
8.3	odd	2	CM	72.3.p.a.43.2	✓	4
8.5	even	2		288.3.t.a.79.1		4
9.2	odd	6		648.3.b.b.163.1		2
9.4	even	3	inner	72.3.p.a.67.2	yes	4
9.5	odd	6		216.3.p.a.91.2		4
9.7	even	3		648.3.b.a.163.2		2
12.11	even	2		864.3.t.a.559.1		4
24.5	odd	2		864.3.t.a.559.1		4
24.11	even	2		216.3.p.a.19.2		4
36.7	odd	6		2592.3.b.b.1135.1		2
36.11	even	6		2592.3.b.a.1135.2		2
36.23	even	6		864.3.t.a.847.1		4
36.31	odd	6		288.3.t.a.175.1		4
72.5	odd	6		864.3.t.a.847.1		4
72.11	even	6		648.3.b.b.163.1		2
72.13	even	6		288.3.t.a.175.1		4
72.29	odd	6		2592.3.b.a.1135.2		2
72.43	odd	6		648.3.b.a.163.2		2
72.59	even	6		216.3.p.a.91.2		4
72.61	even	6		2592.3.b.b.1135.1		2
72.67	odd	6	inner	72.3.p.a.67.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.a.43.2	✓	4	1.1	even	1	trivial
72.3.p.a.43.2	✓	4	8.3	odd	2	CM
72.3.p.a.67.2	yes	4	9.4	even	3	inner
72.3.p.a.67.2	yes	4	72.67	odd	6	inner
216.3.p.a.19.2		4	3.2	odd	2
216.3.p.a.19.2		4	24.11	even	2
216.3.p.a.91.2		4	9.5	odd	6
216.3.p.a.91.2		4	72.59	even	6
288.3.t.a.79.1		4	4.3	odd	2
288.3.t.a.79.1		4	8.5	even	2
288.3.t.a.175.1		4	36.31	odd	6
288.3.t.a.175.1		4	72.13	even	6
648.3.b.a.163.2		2	9.7	even	3
648.3.b.a.163.2		2	72.43	odd	6
648.3.b.b.163.1		2	9.2	odd	6
648.3.b.b.163.1		2	72.11	even	6
864.3.t.a.559.1		4	12.11	even	2
864.3.t.a.559.1		4	24.5	odd	2
864.3.t.a.847.1		4	36.23	even	6
864.3.t.a.847.1		4	72.5	odd	6
2592.3.b.a.1135.2		2	36.11	even	6
2592.3.b.a.1135.2		2	72.29	odd	6
2592.3.b.b.1135.1		2	36.7	odd	6
2592.3.b.b.1135.1		2	72.61	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(2.94949 + 0.548188i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(3.89898 - 4.56048i) q^{6} -8.00000 q^{8} +(8.39898 + 3.23375i) q^{9} +(-3.84847 + 6.66574i) q^{11} +(-4.00000 - 11.3137i) q^{12} +(-8.00000 + 13.8564i) q^{16} -30.3939 q^{17} +(14.0000 - 11.3137i) q^{18} +31.6969 q^{19} +(7.69694 + 13.3315i) q^{22} +(-23.5959 - 4.38551i) q^{24} +(-12.5000 + 21.6506i) q^{25} +(23.0000 + 14.1421i) q^{27} +(16.0000 + 27.7128i) q^{32} +(-15.0051 + 17.5509i) q^{33} +(-30.3939 + 52.6437i) q^{34} +(-5.59592 - 35.5624i) q^{36} +(31.6969 - 54.9007i) q^{38} +(-40.8939 - 70.8303i) q^{41} +(40.2423 - 69.7018i) q^{43} +30.7878 q^{44} +(-31.1918 + 36.4838i) q^{48} +(-24.5000 - 42.4352i) q^{49} +(25.0000 + 43.3013i) q^{50} +(-89.6464 - 16.6616i) q^{51} +(47.4949 - 25.6950i) q^{54} +(93.4898 + 17.3759i) q^{57} +(16.2423 + 28.1326i) q^{59} +64.0000 q^{64} +(15.3939 + 43.5405i) q^{66} +(-35.9393 - 62.2487i) q^{67} +(60.7878 + 105.287i) q^{68} +(-67.1918 - 25.8700i) q^{72} +41.6061 q^{73} +(-48.7372 + 57.0060i) q^{75} +(-63.3939 - 109.801i) q^{76} +(60.0857 + 54.3204i) q^{81} -163.576 q^{82} +(-79.0000 + 136.832i) q^{83} +(-80.4847 - 139.404i) q^{86} +(30.7878 - 53.3260i) q^{88} +146.000 q^{89} +(32.0000 + 90.5097i) q^{96} +(-96.9847 + 167.982i) q^{97} -98.0000 q^{98} +(-53.8786 + 43.5405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 4 q^{6} - 32 q^{8} + 14 q^{9} + 14 q^{11} - 16 q^{12} - 32 q^{16} - 4 q^{17} + 56 q^{18} + 68 q^{19} - 28 q^{22} - 16 q^{24} - 50 q^{25} + 92 q^{27} + 64 q^{32} - 158 q^{33}+ \cdots + 196 q^{99}+O(q^{100}) \) 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

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