LMFDB - Embedded newform 1620.3.t.e.269.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 3]))

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

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

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 81 x^{14} + 4781 x^{12} - 127980 x^{10} + 2502300 x^{8} - 12798000 x^{6} + 47810000 x^{4} + \cdots + 100000000 $ x^16 - 81x^14 + 4781x^12 - 127980x^10 + 2502300x^8 - 12798000x^6 + 47810000x^4 - 81000000*x^2 + 100000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.8
Root			$-4.69001 + 2.70778i$ of defining polynomial
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.e.1349.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.82247 - 1.32053i) q^{5} +(-5.42313 - 3.13104i) q^{7} +O(q^{10})$	$q+(4.82247 - 1.32053i) q^{5} +(-5.42313 - 3.13104i) q^{7} +(0.895275 + 0.516887i) q^{11} +(2.59808 - 1.50000i) q^{13} -15.7776 q^{17} +18.7863 q^{19} +(17.3282 + 30.0134i) q^{23} +(21.5124 - 12.7364i) q^{25} +(2.68583 + 1.55066i) q^{29} +(8.89313 + 15.4034i) q^{31} +(-30.2875 - 7.93797i) q^{35} -43.3104i q^{37} +(40.0960 - 23.1494i) q^{41} +(-17.9596 - 10.3690i) q^{43} +(39.4439 - 68.3188i) q^{47} +(-4.89313 - 8.47515i) q^{49} -44.2313 q^{53} +(5.00000 + 1.31044i) q^{55} +(78.4015 - 45.2651i) q^{59} +(-4.39313 + 7.60913i) q^{61} +(10.5483 - 10.6645i) q^{65} +(16.7233 - 9.65522i) q^{67} -56.6366i q^{71} +109.310i q^{73} +(-3.23679 - 5.60629i) q^{77} +(-19.5000 + 33.7750i) q^{79} +(37.8932 - 65.6330i) q^{83} +(-76.0868 + 20.8347i) q^{85} -90.5302i q^{89} -18.7863 q^{91} +(90.5962 - 24.8078i) q^{95} +(-7.69287 - 4.44148i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 72 q^{19} - 12 q^{25} - 44 q^{31} + 108 q^{49} + 80 q^{55} + 116 q^{61} - 312 q^{79} - 160 q^{85} + 72 q^{91}+O(q^{100}) $ 16 * q - 72 * q^19 - 12 * q^25 - 44 * q^31 + 108 * q^49 + 80 * q^55 + 116 * q^61 - 312 * q^79 - 160 * q^85 + 72 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.82247	−	1.32053i	0.964494	−	0.264106i
$5$	4.82247	−	1.32053i	0.964494	−	0.264106i
$6$		0			0
$7$	−5.42313	−	3.13104i	−0.774732	−	0.447292i	0.0598278	−	0.998209i	$-0.480945\pi$
$7$	−5.42313	−	3.13104i	−0.774732	−	0.447292i	−0.834560	+	0.550917i	$0.814278\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.895275	+	0.516887i	0.0813886	+	0.0469898i	0.540142	−	0.841574i	$-0.318371\pi$
$11$	0.895275	+	0.516887i	0.0813886	+	0.0469898i	−0.458753	+	0.888564i	$0.651704\pi$
$12$		0			0
$13$	2.59808	−	1.50000i	0.199852	−	0.115385i	−0.396734	−	0.917933i	$-0.629857\pi$
$13$	2.59808	−	1.50000i	0.199852	−	0.115385i	0.596586	+	0.802549i	$0.296523\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.7776			−0.928092			−0.464046	−	0.885811i	$-0.653603\pi$
$17$	−15.7776			−0.928092			−0.464046	+	0.885811i	$0.653603\pi$
$18$		0			0
$19$	18.7863			0.988751			0.494375	−	0.869249i	$-0.335397\pi$
$19$	18.7863			0.988751			0.494375	+	0.869249i	$0.335397\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.3282	+	30.0134i	0.753401	+	1.30493i	0.946165	+	0.323683i	$0.104921\pi$
$23$	17.3282	+	30.0134i	0.753401	+	1.30493i	−0.192765	+	0.981245i	$0.561745\pi$
$24$		0			0
$25$	21.5124	−	12.7364i	0.860496	−	0.509457i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.68583	+	1.55066i	0.0926147	+	0.0534711i	0.545592	−	0.838051i	$-0.316305\pi$
$29$	2.68583	+	1.55066i	0.0926147	+	0.0534711i	−0.452977	+	0.891522i	$0.649638\pi$
$30$		0			0
$31$	8.89313	+	15.4034i	0.286875	+	0.496882i	0.973062	−	0.230543i	$-0.0740502\pi$
$31$	8.89313	+	15.4034i	0.286875	+	0.496882i	−0.686187	+	0.727425i	$0.740717\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−30.2875	−	7.93797i	−0.865357	−	0.226799i
$36$		0			0
$37$		−	43.3104i		−	1.17055i	−0.810834	−	0.585276i	$-0.800986\pi$
$37$		−	43.3104i		−	1.17055i	0.810834	−	0.585276i	$-0.199014\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	40.0960	−	23.1494i	0.977952	−	0.564621i	0.0763006	−	0.997085i	$-0.475689\pi$
$41$	40.0960	−	23.1494i	0.977952	−	0.564621i	0.901651	+	0.432464i	$0.142356\pi$
$42$		0			0
$43$	−17.9596	−	10.3690i	−0.417664	−	0.241139i	0.276413	−	0.961039i	$-0.410854\pi$
$43$	−17.9596	−	10.3690i	−0.417664	−	0.241139i	−0.694077	+	0.719900i	$0.744187\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	39.4439	−	68.3188i	0.839232	−	1.45359i	−0.0513061	−	0.998683i	$-0.516338\pi$
$47$	39.4439	−	68.3188i	0.839232	−	1.45359i	0.890538	−	0.454909i	$-0.150328\pi$
$48$		0			0
$49$	−4.89313	−	8.47515i	−0.0998598	−	0.172962i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−44.2313			−0.834554			−0.417277	−	0.908779i	$-0.637015\pi$
$53$	−44.2313			−0.834554			−0.417277	+	0.908779i	$0.637015\pi$
$54$		0			0
$55$	5.00000	+	1.31044i	0.0909091	+	0.0238261i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	78.4015	−	45.2651i	1.32884	−	0.767205i	0.343719	−	0.939073i	$-0.388313\pi$
$59$	78.4015	−	45.2651i	1.32884	−	0.767205i	0.985120	+	0.171867i	$0.0549800\pi$
$60$		0			0
$61$	−4.39313	+	7.60913i	−0.0720185	+	0.124740i	−0.899786	−	0.436332i	$-0.856277\pi$
$61$	−4.39313	+	7.60913i	−0.0720185	+	0.124740i	0.827767	+	0.561072i	$0.189611\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.5483	−	10.6645i	0.162282	−	0.164070i
$66$		0			0
$67$	16.7233	−	9.65522i	0.249602	−	0.144108i	−0.369980	−	0.929040i	$-0.620635\pi$
$67$	16.7233	−	9.65522i	0.249602	−	0.144108i	0.619582	+	0.784932i	$0.287302\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	56.6366i		−	0.797699i	−0.917016	−	0.398850i	$-0.869410\pi$
$71$		−	56.6366i		−	0.797699i	0.917016	−	0.398850i	$-0.130590\pi$
$72$		0			0
$73$			109.310i			1.49740i	0.662907	+	0.748702i	$0.269322\pi$
$73$			109.310i			1.49740i	−0.662907	+	0.748702i	$0.730678\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.23679	−	5.60629i	−0.0420363	−	0.0728090i
$78$		0			0
$79$	−19.5000	+	33.7750i	−0.246835	+	0.427532i	−0.962646	−	0.270763i	$-0.912724\pi$
$79$	−19.5000	+	33.7750i	−0.246835	+	0.427532i	0.715811	+	0.698294i	$0.246057\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	37.8932	−	65.6330i	0.456545	−	0.790759i	−0.542231	−	0.840230i	$-0.682420\pi$
$83$	37.8932	−	65.6330i	0.456545	−	0.790759i	0.998776	+	0.0494706i	$0.0157534\pi$
$84$		0			0
$85$	−76.0868	+	20.8347i	−0.895139	+	0.245114i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	90.5302i		−	1.01719i	−0.861005	−	0.508597i	$-0.830164\pi$
$89$		−	90.5302i		−	1.01719i	0.861005	−	0.508597i	$-0.169836\pi$
$90$		0			0
$91$	−18.7863			−0.206442
$92$		0			0
$93$		0			0
$94$		0			0
$95$	90.5962	−	24.8078i	0.953644	−	0.261135i
$96$		0			0
$97$	−7.69287	−	4.44148i	−0.0793079	−	0.0457885i	0.459822	−	0.888011i	$-0.347913\pi$
$97$	−7.69287	−	4.44148i	−0.0793079	−	0.0457885i	−0.539130	+	0.842223i	$0.681247\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	37.4102	+	21.5988i	0.370398	+	0.213849i	0.673632	−	0.739067i	$-0.264733\pi$
$101$	37.4102	+	21.5988i	0.370398	+	0.213849i	−0.303234	+	0.952916i	$0.598066\pi$
$102$		0			0
$103$	66.4151	−	38.3448i	0.644807	−	0.372279i	−0.141657	−	0.989916i	$-0.545243\pi$
$103$	66.4151	−	38.3448i	0.644807	−	0.372279i	0.786464	+	0.617636i	$0.211910\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−38.0287			−0.355408			−0.177704	−	0.984084i	$-0.556867\pi$
$107$	−38.0287			−0.355408			−0.177704	+	0.984084i	$0.556867\pi$
$108$		0			0
$109$	−146.718			−1.34603			−0.673016	−	0.739628i	$-0.735002\pi$
$109$	−146.718			−1.34603			−0.673016	+	0.739628i	$0.735002\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−93.1147	−	161.279i	−0.824024	−	1.42725i	−0.902663	−	0.430348i	$-0.858391\pi$
$113$	−93.1147	−	161.279i	−0.824024	−	1.42725i	0.0786394	−	0.996903i	$-0.474942\pi$
$114$		0			0
$115$	123.198	+	121.856i	1.07129	+	1.05962i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	85.5637	+	49.4002i	0.719023	+	0.415128i
$120$		0			0
$121$	−59.9657	−	103.864i	−0.495584	−	0.858377i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	86.9241	−	89.8287i	0.695393	−	0.718630i
$126$		0			0
$127$		−	161.242i		−	1.26962i	−0.772668	−	0.634810i	$-0.781078\pi$
$127$		−	161.242i		−	1.26962i	0.772668	−	0.634810i	$-0.218922\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	215.700	−	124.534i	1.64656	−	0.950644i	0.668139	−	0.744036i	$-0.267091\pi$
$131$	215.700	−	124.534i	1.64656	−	0.950644i	0.978424	−	0.206608i	$-0.0662423\pi$
$132$		0			0
$133$	−101.880	−	58.8206i	−0.766017	−	0.442260i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−122.984	+	213.014i	−0.897691	+	1.55485i	−0.0672525	+	0.997736i	$0.521423\pi$
$137$	−122.984	+	213.014i	−0.897691	+	1.55485i	−0.830438	+	0.557110i	$0.811910\pi$
$138$		0			0
$139$	84.9657	+	147.165i	0.611264	+	1.05874i	0.991028	+	0.133656i	$0.0426719\pi$
$139$	84.9657	+	147.165i	0.611264	+	1.05874i	−0.379764	+	0.925083i	$0.623995\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.10132			0.0216876
$144$		0			0
$145$	15.0000	+	3.93131i	0.103448	+	0.0271125i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	140.879	−	81.3368i	0.945499	−	0.545884i	0.0538193	−	0.998551i	$-0.482860\pi$
$149$	140.879	−	81.3368i	0.945499	−	0.545884i	0.891680	+	0.452666i	$0.149527\pi$
$150$		0			0
$151$	55.0725	−	95.3884i	0.364719	−	0.631711i	−0.624012	−	0.781415i	$-0.714498\pi$
$151$	55.0725	−	95.3884i	0.364719	−	0.631711i	0.988731	+	0.149703i	$0.0478318\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	63.2274	+	62.5386i	0.407919	+	0.403475i
$156$		0			0
$157$	−111.944	+	64.6310i	−0.713021	+	0.411663i	−0.812178	−	0.583409i	$-0.801718\pi$
$157$	−111.944	+	64.6310i	−0.713021	+	0.411663i	0.0991578	+	0.995072i	$0.468385\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	217.022i		−	1.34796i
$162$		0			0
$163$			245.621i			1.50688i	0.657519	+	0.753438i	$0.271606\pi$
$163$			245.621i			1.50688i	−0.657519	+	0.753438i	$0.728394\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−41.1300	−	71.2393i	−0.246288	−	0.426583i	0.716205	−	0.697890i	$-0.245877\pi$
$167$	−41.1300	−	71.2393i	−0.246288	−	0.426583i	−0.962493	+	0.271307i	$0.912544\pi$
$168$		0			0
$169$	−80.0000	+	138.564i	−0.473373	+	0.819906i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−110.443	+	191.293i	−0.638398	+	1.10574i	0.347386	+	0.937722i	$0.387069\pi$
$173$	−110.443	+	191.293i	−0.638398	+	1.10574i	−0.985784	+	0.168016i	$0.946264\pi$
$174$		0			0
$175$	−156.543	+	1.71488i	−0.894530	+	0.00979934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	149.234i		−	0.833712i	−0.908973	−	0.416856i	$-0.863132\pi$
$179$		−	149.234i		−	0.833712i	0.908973	−	0.416856i	$-0.136868\pi$
$180$		0			0
$181$	137.931			0.762051			0.381026	−	0.924564i	$-0.375571\pi$
$181$	137.931			0.762051			0.381026	+	0.924564i	$0.375571\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−57.1927	−	208.863i	−0.309149	−	1.12899i
$186$		0			0
$187$	−14.1253	−	8.15522i	−0.0755361	−	0.0436108i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	174.708	+	100.868i	0.914704	+	0.528105i	0.881942	−	0.471359i	$-0.156236\pi$
$191$	174.708	+	100.868i	0.914704	+	0.528105i	0.0327624	+	0.999463i	$0.489570\pi$
$192$		0			0
$193$	319.462	−	184.441i	1.65524	−	0.955655i	0.680378	−	0.732861i	$-0.261816\pi$
$193$	319.462	−	184.441i	1.65524	−	0.955655i	0.974866	−	0.222794i	$-0.0715177\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	170.181			0.863862			0.431931	−	0.901907i	$-0.357832\pi$
$197$	170.181			0.863862			0.431931	+	0.901907i	$0.357832\pi$
$198$		0			0
$199$	136.863			0.687752			0.343876	−	0.939015i	$-0.388260\pi$
$199$	136.863			0.687752			0.343876	+	0.939015i	$0.388260\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.71038	−	16.8189i	−0.0478344	−	0.0828516i
$204$		0			0
$205$	162.792	−	164.585i	0.794109	−	0.802856i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.8189	+	9.71038i	0.0804731	+	0.0464611i
$210$		0			0
$211$	19.9657	+	34.5815i	0.0946240	+	0.163894i	0.909452	−	0.415810i	$-0.136502\pi$
$211$	19.9657	+	34.5815i	0.0946240	+	0.163894i	−0.814828	+	0.579703i	$0.803168\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−100.302	−	26.2879i	−0.466521	−	0.122269i
$216$		0			0
$217$		−	111.379i		−	0.513268i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−40.9913	+	23.6663i	−0.185481	+	0.107087i
$222$		0			0
$223$	−268.181	−	154.835i	−1.20261	−	0.694326i	−0.241473	−	0.970408i	$-0.577631\pi$
$223$	−268.181	−	154.835i	−1.20261	−	0.694326i	−0.961134	+	0.276082i	$0.910964\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	53.6708	−	92.9605i	0.236435	−	0.409518i	−0.723254	−	0.690583i	$-0.757354\pi$
$227$	53.6708	−	92.9605i	0.236435	−	0.409518i	0.959689	+	0.281065i	$0.0906876\pi$
$228$		0			0
$229$	44.4275	+	76.9506i	0.194006	+	0.336029i	0.946574	−	0.322486i	$-0.104518\pi$
$229$	44.4275	+	76.9506i	0.194006	+	0.336029i	−0.752568	+	0.658515i	$0.771185\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	245.696			1.05449			0.527245	−	0.849713i	$-0.323225\pi$
$233$	245.696			1.05449			0.527245	+	0.849713i	$0.323225\pi$
$234$		0			0
$235$	100.000	−	381.552i	0.425532	−	1.62363i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	205.852	−	118.849i	0.861304	−	0.497274i	−0.00314447	−	0.999995i	$-0.501001\pi$
$239$	205.852	−	118.849i	0.861304	−	0.497274i	0.864449	+	0.502721i	$0.167668\pi$
$240$		0			0
$241$	80.5000	−	139.430i	0.334025	−	0.578548i	−0.649272	−	0.760556i	$-0.724926\pi$
$241$	80.5000	−	139.430i	0.334025	−	0.578548i	0.983297	+	0.182008i	$0.0582597\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−34.7886	−	34.4096i	−0.141994	−	0.140447i
$246$		0			0
$247$	48.8081	−	28.1794i	0.197604	−	0.114087i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			91.5640i			0.364797i	0.983225	+	0.182398i	$0.0583861\pi$
$251$			91.5640i			0.364797i	−0.983225	+	0.182398i	$0.941614\pi$
$252$		0			0
$253$			35.8269i			0.141608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−124.534	−	215.700i	−0.484569	−	0.839299i	0.515274	−	0.857026i	$-0.327690\pi$
$257$	−124.534	−	215.700i	−0.484569	−	0.839299i	−0.999843	+	0.0177270i	$0.994357\pi$
$258$		0			0
$259$	−135.607	+	234.878i	−0.523579	+	0.906865i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	98.0376	−	169.806i	0.372767	−	0.645651i	−0.617223	−	0.786788i	$-0.711743\pi$
$263$	98.0376	−	169.806i	0.372767	−	0.645651i	0.989990	+	0.141137i	$0.0450759\pi$
$264$		0			0
$265$	−213.304	+	58.4087i	−0.804922	+	0.220410i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	166.809i		−	0.620106i	−0.950719	−	0.310053i	$-0.899653\pi$
$269$		−	166.809i		−	0.620106i	0.950719	−	0.310053i	$-0.100347\pi$
$270$		0			0
$271$	−89.9313			−0.331850			−0.165925	−	0.986138i	$-0.553061\pi$
$271$	−89.9313			−0.331850			−0.165925	+	0.986138i	$0.553061\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	25.8428	−	0.283101i	0.0939739	−	0.00102946i
$276$		0			0
$277$	364.436	+	210.407i	1.31565	+	0.759593i	0.983026	−	0.183466i	$-0.0587317\pi$
$277$	364.436	+	210.407i	1.31565	+	0.759593i	0.332627	+	0.943059i	$0.392065\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−267.434	−	154.403i	−0.951724	−	0.549478i	−0.0581078	−	0.998310i	$-0.518507\pi$
$281$	−267.434	−	154.403i	−0.951724	−	0.549478i	−0.893616	+	0.448832i	$0.851840\pi$
$282$		0			0
$283$	51.2563	−	29.5929i	0.181118	−	0.104568i	−0.406700	−	0.913562i	$-0.633321\pi$
$283$	51.2563	−	29.5929i	0.181118	−	0.104568i	0.587818	+	0.808993i	$0.299987\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−289.928			−1.01020
$288$		0			0
$289$	−40.0687			−0.138646
$290$		0			0
$291$		0			0
$292$		0			0
$293$	113.680	+	196.899i	0.387985	+	0.672010i	0.992178	−	0.124828i	$-0.0398378\pi$
$293$	113.680	+	196.899i	0.387985	+	0.672010i	−0.604193	+	0.796838i	$0.706504\pi$
$294$		0			0
$295$	318.315	−	321.821i	1.07903	−	1.09092i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	90.0401	+	51.9847i	0.301137	+	0.173862i
$300$		0			0
$301$	64.9313	+	112.464i	0.215719	+	0.373636i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.1377	+	42.4960i	−0.0365170	+	0.139331i
$306$		0			0
$307$			301.221i			0.981177i	0.871391	+	0.490589i	$0.163218\pi$
$307$			301.221i			0.981177i	−0.871391	+	0.490589i	$0.836782\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	153.222	−	88.4627i	0.492675	−	0.284446i	−0.233009	−	0.972475i	$-0.574857\pi$
$311$	153.222	−	88.4627i	0.492675	−	0.284446i	0.725683	+	0.688029i	$0.241524\pi$
$312$		0			0
$313$	−302.512	−	174.655i	−0.966491	−	0.558004i	−0.0683264	−	0.997663i	$-0.521766\pi$
$313$	−302.512	−	174.655i	−0.966491	−	0.558004i	−0.898165	+	0.439659i	$0.855099\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	185.958	−	322.089i	0.586620	−	1.01606i	−0.408052	−	0.912959i	$-0.633792\pi$
$317$	185.958	−	322.089i	0.586620	−	1.01606i	0.994671	−	0.103096i	$-0.0328750\pi$
$318$		0			0
$319$	1.60303	+	2.77654i	0.00502519	+	0.00870388i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−296.401			−0.917651
$324$		0			0
$325$	36.7863	−	65.3588i	0.113188	−	0.201104i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−427.818	+	247.001i	−1.30036	+	0.750763i
$330$		0			0
$331$	−97.8970	+	169.563i	−0.295761	+	0.512273i	−0.975162	−	0.221494i	$-0.928907\pi$
$331$	−97.8970	+	169.563i	−0.295761	+	0.512273i	0.679400	+	0.733768i	$0.262240\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	67.8977	−	68.6456i	0.202680	−	0.204912i
$336$		0			0
$337$	−408.048	+	235.587i	−1.21082	+	0.699070i	−0.962939	−	0.269720i	$-0.913069\pi$
$337$	−408.048	+	235.587i	−1.21082	+	0.699070i	−0.247886	+	0.968789i	$0.579736\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			18.3870i			0.0539208i
$342$		0			0
$343$			368.125i			1.07325i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	176.519	+	305.740i	0.508700	+	0.881095i	0.999949	+	0.0100755i	$0.00320718\pi$
$347$	176.519	+	305.740i	0.508700	+	0.881095i	−0.491249	+	0.871019i	$0.663459\pi$
$348$		0			0
$349$	−308.966	+	535.144i	−0.885288	+	1.53336i	−0.0399057	+	0.999203i	$0.512706\pi$
$349$	−308.966	+	535.144i	−0.885288	+	1.53336i	−0.845383	+	0.534161i	$0.820628\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	26.4967	−	45.8937i	0.0750615	−	0.130010i	−0.826051	−	0.563595i	$-0.809418\pi$
$353$	26.4967	−	45.8937i	0.0750615	−	0.130010i	0.901113	+	0.433584i	$0.142751\pi$
$354$		0			0
$355$	−74.7903	−	273.128i	−0.210677	−	0.769376i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			618.426i			1.72264i	0.508067	+	0.861318i	$0.330360\pi$
$359$			618.426i			1.72264i	−0.508067	+	0.861318i	$0.669640\pi$
$360$		0			0
$361$	−8.07636			−0.0223722
$362$		0			0
$363$		0			0
$364$		0			0
$365$	144.348	+	527.146i	0.395473	+	1.44424i
$366$		0			0
$367$	18.3364	+	10.5865i	0.0499630	+	0.0288461i	0.524773	−	0.851242i	$-0.324150\pi$
$367$	18.3364	+	10.5865i	0.0499630	+	0.0288461i	−0.474810	+	0.880088i	$0.657483\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	239.872	+	138.490i	0.646556	+	0.373289i
$372$		0			0
$373$	500.821	−	289.149i	1.34268	−	0.775198i	0.355482	−	0.934683i	$-0.384317\pi$
$373$	500.821	−	289.149i	1.34268	−	0.775198i	0.987200	+	0.159485i	$0.0509834\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.30397			0.0246790
$378$		0			0
$379$	−675.657			−1.78273			−0.891367	−	0.453281i	$-0.850253\pi$
$379$	−675.657			−1.78273			−0.891367	+	0.453281i	$0.850253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	83.9462	+	145.399i	0.219181	+	0.379632i	0.954558	−	0.298026i	$-0.0963283\pi$
$383$	83.9462	+	145.399i	0.219181	+	0.379632i	−0.735377	+	0.677658i	$0.762995\pi$
$384$		0			0
$385$	−23.0126	−	22.7619i	−0.0597730	−	0.0591218i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−586.221	−	338.455i	−1.50699	−	0.870063i	−0.999967	−	0.00813318i	$-0.997411\pi$
$389$	−586.221	−	338.455i	−1.50699	−	0.870063i	−0.507027	−	0.861930i	$-0.669256\pi$
$390$		0			0
$391$	−273.397	−	473.537i	−0.699225	−	1.21109i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−49.4373	+	188.629i	−0.125158	+	0.477542i
$396$		0			0
$397$			63.8423i			0.160812i	0.996762	+	0.0804059i	$0.0256216\pi$
$397$			63.8423i			0.160812i	−0.996762	+	0.0804059i	$0.974378\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−471.496	+	272.218i	−1.17580	+	0.678848i	−0.955039	−	0.296480i	$-0.904187\pi$
$401$	−471.496	+	272.218i	−1.17580	+	0.678848i	−0.220760	+	0.975328i	$0.570854\pi$
$402$		0			0
$403$	46.2101	+	26.6794i	0.114665	+	0.0662020i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	22.3866	−	38.7748i	0.0550040	−	0.0952697i
$408$		0			0
$409$	373.363	+	646.683i	0.912867	+	1.58113i	0.809994	+	0.586438i	$0.199470\pi$
$409$	373.363	+	646.683i	0.912867	+	1.58113i	0.102873	+	0.994695i	$0.467197\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−566.908			−1.37266
$414$		0			0
$415$	96.0687	−	366.552i	0.231491	−	0.883258i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	33.8291	−	19.5312i	0.0807377	−	0.0466139i	−0.459088	−	0.888391i	$-0.651824\pi$
$419$	33.8291	−	19.5312i	0.0807377	−	0.0466139i	0.539825	+	0.841777i	$0.318490\pi$
$420$		0			0
$421$	179.966	−	311.710i	0.427472	−	0.740403i	−0.569176	−	0.822216i	$-0.692738\pi$
$421$	179.966	−	311.710i	0.427472	−	0.740403i	0.996648	+	0.0818128i	$0.0260710\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−339.413	+	200.949i	−0.798619	+	0.472822i
$426$		0			0
$427$	47.6490	−	27.5102i	0.111590	−	0.0644266i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			431.163i			1.00038i	0.865916	+	0.500189i	$0.166736\pi$
$431$			431.163i			1.00038i	−0.865916	+	0.500189i	$0.833264\pi$
$432$		0			0
$433$		−	6.05602i		−	0.0139862i	−0.999976	−	0.00699309i	$-0.997774\pi$
$433$		−	6.05602i		−	0.0139862i	0.999976	−	0.00699309i	$-0.00222599\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	325.532	+	563.839i	0.744926	+	1.29025i
$438$		0			0
$439$	−246.641	+	427.195i	−0.561825	+	0.973110i	0.435512	+	0.900183i	$0.356567\pi$
$439$	−246.641	+	427.195i	−0.561825	+	0.973110i	−0.997337	+	0.0729268i	$0.976766\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.4053	−	21.4866i	0.0280029	−	0.0485025i	−0.851684	−	0.524055i	$-0.824419\pi$
$443$	12.4053	−	21.4866i	0.0280029	−	0.0485025i	0.879687	+	0.475553i	$0.157752\pi$
$444$		0			0
$445$	−119.548	−	436.579i	−0.268647	−	0.981077i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			131.881i			0.293722i	0.989157	+	0.146861i	$0.0469170\pi$
$449$			131.881i			0.293722i	−0.989157	+	0.146861i	$0.953083\pi$
$450$		0			0
$451$	47.8626			0.106126
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−90.5962	+	24.8078i	−0.199112	+	0.0545226i
$456$		0			0
$457$	262.734	+	151.690i	0.574910	+	0.331925i	0.759108	−	0.650965i	$-0.225635\pi$
$457$	262.734	+	151.690i	0.574910	+	0.331925i	−0.184198	+	0.982889i	$0.558969\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−563.135	−	325.126i	−1.22155	−	0.705263i	−0.256302	−	0.966597i	$-0.582504\pi$
$461$	−563.135	−	325.126i	−1.22155	−	0.705263i	−0.965248	+	0.261334i	$0.915838\pi$
$462$		0			0
$463$	−624.973	+	360.828i	−1.34983	+	0.779327i	−0.988226	−	0.153004i	$-0.951105\pi$
$463$	−624.973	+	360.828i	−1.34983	+	0.779327i	−0.361608	+	0.932330i	$0.617772\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	341.445			0.731147			0.365573	−	0.930783i	$-0.380873\pi$
$467$	341.445			0.731147			0.365573	+	0.930783i	$0.380873\pi$
$468$		0			0
$469$	−120.924			−0.257833
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.7192	−	18.5661i	−0.0226621	−	0.0392519i
$474$		0			0
$475$	404.138	−	239.270i	0.850816	−	0.503725i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	553.478	+	319.551i	1.15549	+	0.667121i	0.950218	−	0.311585i	$-0.100860\pi$
$479$	553.478	+	319.551i	1.15549	+	0.667121i	0.205269	+	0.978706i	$0.434193\pi$
$480$		0			0
$481$	−64.9657	−	112.524i	−0.135064	−	0.233937i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−42.9637	−	11.2602i	−0.0885850	−	0.0232170i
$486$		0			0
$487$			411.931i			0.845855i	0.906164	+	0.422927i	$0.138997\pi$
$487$			411.931i			0.845855i	−0.906164	+	0.422927i	$0.861003\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−508.714	+	293.706i	−1.03608	+	0.598180i	−0.918720	−	0.394910i	$-0.870776\pi$
$491$	−508.714	+	293.706i	−1.03608	+	0.598180i	−0.117358	+	0.993090i	$0.537443\pi$
$492$		0			0
$493$	−42.3758	−	24.4657i	−0.0859549	−	0.0496261i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−177.332	+	307.148i	−0.356804	+	0.618003i
$498$		0			0
$499$	387.290	+	670.806i	0.776132	+	1.34430i	0.934156	+	0.356865i	$0.116155\pi$
$499$	387.290	+	670.806i	0.776132	+	1.34430i	−0.158023	+	0.987435i	$0.550512\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	546.283			1.08605			0.543025	−	0.839717i	$-0.317279\pi$
$503$	546.283			1.08605			0.543025	+	0.839717i	$0.317279\pi$
$504$		0			0
$505$	208.931	+	54.7583i	0.413725	+	0.108432i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−603.422	+	348.386i	−1.18551	+	0.684452i	−0.957282	−	0.289157i	$-0.906625\pi$
$509$	−603.422	+	348.386i	−1.18551	+	0.684452i	−0.228224	+	0.973609i	$0.573292\pi$
$510$		0			0
$511$	342.256	−	592.804i	0.669776	−	1.16009i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	269.649	−	272.620i	0.523591	−	0.529358i
$516$		0			0
$517$	70.6263	−	40.7761i	0.136608	−	0.0788706i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			558.909i			1.07276i	0.843976	+	0.536381i	$0.180209\pi$
$521$			558.909i			1.07276i	−0.843976	+	0.536381i	$0.819791\pi$
$522$		0			0
$523$		−	402.669i		−	0.769922i	−0.922933	−	0.384961i	$-0.874215\pi$
$523$		−	402.669i		−	0.769922i	0.922933	−	0.384961i	$-0.125785\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−140.312	−	243.027i	−0.266246	−	0.461152i
$528$		0			0
$529$	−336.034	+	582.029i	−0.635226	+	1.10024i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	69.4483	−	120.288i	0.130297	−	0.225681i
$534$		0			0
$535$	−183.392	+	50.2180i	−0.342789	+	0.0938654i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	10.1168i		−	0.0187696i
$540$		0			0
$541$	636.649			1.17680			0.588400	−	0.808570i	$-0.299758\pi$
$541$	636.649			1.17680			0.588400	+	0.808570i	$0.299758\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−707.541	+	193.745i	−1.29824	+	0.355495i
$546$		0			0
$547$	−257.866	−	148.879i	−0.471419	−	0.272174i	0.245415	−	0.969418i	$-0.421076\pi$
$547$	−257.866	−	148.879i	−0.471419	−	0.272174i	−0.716834	+	0.697244i	$0.754409\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	50.4566	+	29.1311i	0.0915728	+	0.0528696i
$552$		0			0
$553$	211.502	−	122.111i	0.382463	−	0.220815i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−391.067			−0.702095			−0.351047	−	0.936358i	$-0.614174\pi$
$557$	−391.067			−0.702095			−0.351047	+	0.936358i	$0.614174\pi$
$558$		0			0
$559$	−62.2137			−0.111295
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−205.108	−	355.258i	−0.364313	−	0.631009i	0.624353	−	0.781143i	$-0.285363\pi$
$563$	−205.108	−	355.258i	−0.364313	−	0.631009i	−0.988666	+	0.150134i	$0.952030\pi$
$564$		0			0
$565$	−662.017	−	654.804i	−1.17171	−	1.15895i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−538.450	−	310.874i	−0.946309	−	0.546352i	−0.0543766	−	0.998520i	$-0.517317\pi$
$569$	−538.450	−	310.874i	−0.946309	−	0.546352i	−0.891932	+	0.452169i	$0.850650\pi$
$570$		0			0
$571$	−198.324	−	343.508i	−0.347328	−	0.601590i	0.638446	−	0.769667i	$-0.279578\pi$
$571$	−198.324	−	343.508i	−0.347328	−	0.601590i	−0.985774	+	0.168077i	$0.946244\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	755.034	+	424.960i	1.31310	+	0.739061i
$576$		0			0
$577$		−	571.601i		−	0.990642i	−0.868710	−	0.495321i	$-0.835050\pi$
$577$		−	571.601i		−	0.990642i	0.868710	−	0.495321i	$-0.164950\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−411.000	+	237.291i	−0.707400	+	0.408418i
$582$		0			0
$583$	−39.5992	−	22.8626i	−0.0679232	−	0.0392155i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−384.458	+	665.900i	−0.654953	+	1.13441i	0.326952	+	0.945041i	$0.393978\pi$
$587$	−384.458	+	665.900i	−0.654953	+	1.13441i	−0.981905	+	0.189372i	$0.939355\pi$
$588$		0			0
$589$	167.069	+	289.371i	0.283648	+	0.491293i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	917.116			1.54657			0.773285	−	0.634059i	$-0.218612\pi$
$593$	917.116			1.54657			0.773285	+	0.634059i	$0.218612\pi$
$594$		0			0
$595$	477.863	+	125.242i	0.803130	+	0.210490i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−765.405	+	441.907i	−1.27781	+	0.737741i	−0.976444	−	0.215769i	$-0.930774\pi$
$599$	−765.405	+	441.907i	−1.27781	+	0.737741i	−0.301361	+	0.953510i	$0.597441\pi$
$600$		0			0
$601$	337.824	−	585.129i	0.562104	−	0.973593i	−0.435209	−	0.900330i	$-0.643325\pi$
$601$	337.824	−	585.129i	0.562104	−	0.973593i	0.997313	−	0.0732629i	$-0.0233412\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−426.337	−	421.692i	−0.704690	−	0.697012i
$606$		0			0
$607$	595.109	−	343.587i	0.980411	−	0.566040i	0.0780166	−	0.996952i	$-0.475141\pi$
$607$	595.109	−	343.587i	0.980411	−	0.566040i	0.902394	+	0.430912i	$0.141808\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	236.663i		−	0.387338i
$612$		0			0
$613$		−	193.896i		−	0.316306i	−0.987415	−	0.158153i	$-0.949446\pi$
$613$		−	193.896i		−	0.316306i	0.987415	−	0.158153i	$-0.0505539\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−278.802	−	482.900i	−0.451867	−	0.782657i	0.546635	−	0.837371i	$-0.315909\pi$
$617$	−278.802	−	482.900i	−0.451867	−	0.782657i	−0.998502	+	0.0547139i	$0.982575\pi$
$618$		0			0
$619$	−513.324	+	889.104i	−0.829280	+	1.43636i	0.0693235	+	0.997594i	$0.477916\pi$
$619$	−513.324	+	889.104i	−0.829280	+	1.43636i	−0.898604	+	0.438761i	$0.855417\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−283.454	+	490.957i	−0.454983	+	0.788053i
$624$		0			0
$625$	300.568	−	547.982i	0.480908	−	0.876771i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			683.333i			1.08638i
$630$		0			0
$631$	145.794			0.231052			0.115526	−	0.993304i	$-0.463145\pi$
$631$	145.794			0.231052			0.115526	+	0.993304i	$0.463145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−212.924	−	777.583i	−0.335314	−	1.22454i
$636$		0			0
$637$	−25.4255	−	14.6794i	−0.0399144	−	0.0230446i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	74.8204	+	43.1976i	0.116724	+	0.0673909i	0.557226	−	0.830361i	$-0.311866\pi$
$641$	74.8204	+	43.1976i	0.116724	+	0.0673909i	−0.440501	+	0.897752i	$0.645199\pi$
$642$		0			0
$643$	−163.857	+	94.6030i	−0.254832	+	0.147128i	−0.621975	−	0.783037i	$-0.713669\pi$
$643$	−163.857	+	94.6030i	−0.254832	+	0.147128i	0.367143	+	0.930165i	$0.380336\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	801.554			1.23888			0.619439	−	0.785045i	$-0.287360\pi$
$647$	801.554			1.23888			0.619439	+	0.785045i	$0.287360\pi$
$648$		0			0
$649$	93.5879			0.144203
$650$		0			0
$651$		0			0
$652$		0			0
$653$	416.961	+	722.198i	0.638531	+	1.10597i	0.985755	+	0.168187i	$0.0537911\pi$
$653$	416.961	+	722.198i	0.638531	+	1.10597i	−0.347224	+	0.937782i	$0.612876\pi$
$654$		0			0
$655$	875.754	−	885.401i	1.33703	−	1.35176i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−132.118	−	76.2784i	−0.200483	−	0.115749i	0.396398	−	0.918079i	$-0.370260\pi$
$659$	−132.118	−	76.2784i	−0.200483	−	0.115749i	−0.596881	+	0.802330i	$0.703593\pi$
$660$		0			0
$661$	171.760	+	297.496i	0.259848	+	0.450070i	0.966201	−	0.257790i	$-0.0829942\pi$
$661$	171.760	+	297.496i	0.259848	+	0.450070i	−0.706353	+	0.707860i	$0.749661\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−568.989	−	149.125i	−0.855622	−	0.224248i
$666$		0			0
$667$			107.481i			0.161141i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.86612	+	4.54151i	−0.0117230	+	0.00676827i
$672$		0			0
$673$	924.380	+	533.691i	1.37352	+	0.793003i	0.991370	−	0.131097i	$-0.0418499\pi$
$673$	924.380	+	533.691i	1.37352	+	0.793003i	0.382152	+	0.924100i	$0.375183\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	62.5684	−	108.372i	0.0924200	−	0.160076i	−0.816109	−	0.577898i	$-0.803873\pi$
$677$	62.5684	−	108.372i	0.0924200	−	0.160076i	0.908529	+	0.417822i	$0.137206\pi$
$678$		0			0
$679$	27.8129	+	48.1734i	0.0409616	+	0.0709476i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	675.334			0.988775			0.494388	−	0.869241i	$-0.335392\pi$
$683$	675.334			0.988775			0.494388	+	0.869241i	$0.335392\pi$
$684$		0			0
$685$	−311.794	+	1189.66i	−0.455174	+	1.73672i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−114.916	+	66.3470i	−0.166787	+	0.0962947i
$690$		0			0
$691$	−213.863	+	370.421i	−0.309497	+	0.536065i	−0.978252	−	0.207418i	$-0.933494\pi$
$691$	−213.863	+	370.421i	−0.309497	+	0.536065i	0.668755	+	0.743483i	$0.266827\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	604.080	+	597.498i	0.869179	+	0.859710i
$696$		0			0
$697$	−632.617	+	365.242i	−0.907629	+	0.524020i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	410.708i		−	0.585889i	−0.956129	−	0.292945i	$-0.905365\pi$
$701$		−	410.708i		−	0.585889i	0.956129	−	0.292945i	$-0.0946352\pi$
$702$		0			0
$703$		−	813.641i		−	1.15738i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−135.253	−	234.266i	−0.191306	−	0.331352i
$708$		0			0
$709$	−395.462	+	684.960i	−0.557774	+	0.966093i	0.439908	+	0.898043i	$0.355011\pi$
$709$	−395.462	+	684.960i	−0.557774	+	0.966093i	−0.997682	+	0.0680501i	$0.978322\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−308.204	+	533.825i	−0.432264	+	0.748703i
$714$		0			0
$715$	14.9560	−	4.09539i	0.0209175	−	0.00572781i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			79.3797i			0.110403i	0.998475	+	0.0552014i	$0.0175801\pi$
$719$			79.3797i			0.110403i	−0.998475	+	0.0552014i	$0.982420\pi$
$720$		0			0
$721$	−480.237			−0.666070
$722$		0			0
$723$		0			0
$724$		0			0
$725$	77.5284	−	0.849303i	0.106936	−	0.00117145i
$726$		0			0
$727$	−1070.30	−	617.939i	−1.47222	−	0.849985i	−0.472705	−	0.881221i	$-0.656722\pi$
$727$	−1070.30	−	617.939i	−1.47222	−	0.849985i	−0.999512	+	0.0312360i	$0.990056\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	283.358	+	163.597i	0.387631	+	0.223799i
$732$		0			0
$733$	−1017.43	+	587.415i	−1.38804	+	0.801384i	−0.993094	−	0.117321i	$-0.962570\pi$
$733$	−1017.43	+	587.415i	−1.38804	+	0.801384i	−0.394944	+	0.918705i	$0.629236\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19.9626			0.0270863
$738$		0			0
$739$	−425.863			−0.576269			−0.288134	−	0.957590i	$-0.593035\pi$
$739$	−425.863			−0.576269			−0.288134	+	0.957590i	$0.593035\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−330.184	−	571.896i	−0.444394	−	0.769712i	0.553616	−	0.832772i	$-0.313247\pi$
$743$	−330.184	−	571.896i	−0.444394	−	0.769712i	−0.998010	+	0.0630597i	$0.979914\pi$
$744$		0			0
$745$	571.979	−	578.279i	0.767757	−	0.776214i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	206.234	+	119.070i	0.275346	+	0.158971i
$750$		0			0
$751$	38.3626	+	66.4460i	0.0510821	+	0.0884767i	0.890436	−	0.455109i	$-0.150400\pi$
$751$	38.3626	+	66.4460i	0.0510821	+	0.0884767i	−0.839354	+	0.543586i	$0.817066\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	139.622	−	532.732i	0.184930	−	0.705606i
$756$		0			0
$757$		−	530.532i		−	0.700835i	−0.936594	−	0.350417i	$-0.886040\pi$
$757$		−	530.532i		−	0.700835i	0.936594	−	0.350417i	$-0.113960\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−635.078	+	366.662i	−0.834531	+	0.481817i	−0.855401	−	0.517966i	$-0.826690\pi$
$761$	−635.078	+	366.662i	−0.834531	+	0.481817i	0.0208705	+	0.999782i	$0.493356\pi$
$762$		0			0
$763$	795.668	+	459.379i	1.04282	+	0.602070i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	135.795	−	235.204i	0.177047	−	0.306655i
$768$		0			0
$769$	122.637	+	212.414i	0.159476	+	0.276221i	0.934680	−	0.355490i	$-0.115686\pi$
$769$	122.637	+	212.414i	0.159476	+	0.276221i	−0.775204	+	0.631712i	$0.782353\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1211.23			−1.56692			−0.783460	−	0.621443i	$-0.786547\pi$
$773$	−1211.23			−1.56692			−0.783460	+	0.621443i	$0.786547\pi$
$774$		0			0
$775$	387.496	+	218.097i	0.499995	+	0.281415i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	753.254	−	434.892i	0.966950	−	0.558269i
$780$		0			0
$781$	29.2748	−	50.7054i	0.0374837	−	0.0649236i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−454.501	+	459.507i	−0.578982	+	0.585359i
$786$		0			0
$787$	1041.88	−	601.528i	1.32386	−	0.764330i	0.339517	−	0.940600i	$-0.389736\pi$
$787$	1041.88	−	601.528i	1.32386	−	0.764330i	0.984342	+	0.176269i	$0.0564030\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1166.18i			1.47432i
$792$		0			0
$793$			26.3588i			0.0332393i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	507.170	+	878.445i	0.636349	+	1.10219i	0.986228	+	0.165394i	$0.0528896\pi$
$797$	507.170	+	878.445i	0.636349	+	1.10219i	−0.349878	+	0.936795i	$0.613777\pi$
$798$		0			0
$799$	−622.328	+	1077.90i	−0.778884	+	1.34907i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−56.5012	+	97.8629i	−0.0703626	+	0.121872i
$804$		0			0
$805$	−286.583	−	1046.58i	−0.356004	−	1.30010i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	951.230i		−	1.17581i	−0.808930	−	0.587905i	$-0.799953\pi$
$809$		−	951.230i		−	1.17581i	0.808930	−	0.587905i	$-0.200047\pi$
$810$		0			0
$811$	−404.580			−0.498866			−0.249433	−	0.968392i	$-0.580244\pi$
$811$	−404.580			−0.498866			−0.249433	+	0.968392i	$0.580244\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	324.349	+	1184.50i	0.397975	+	1.45337i
$816$		0			0
$817$	−337.393	−	194.794i	−0.412966	−	0.238426i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1026.38	−	592.582i	−1.25016	−	0.721780i	−0.279019	−	0.960286i	$-0.590009\pi$
$821$	−1026.38	−	592.582i	−1.25016	−	0.721780i	−0.971141	+	0.238505i	$0.923343\pi$
$822$		0			0
$823$	−578.564	+	334.034i	−0.702994	+	0.405874i	−0.808462	−	0.588549i	$-0.799700\pi$
$823$	−578.564	+	334.034i	−0.702994	+	0.405874i	0.105467	+	0.994423i	$0.466366\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	427.199			0.516564			0.258282	−	0.966070i	$-0.416844\pi$
$827$	427.199			0.516564			0.258282	+	0.966070i	$0.416844\pi$
$828$		0			0
$829$	−1143.52			−1.37940			−0.689698	−	0.724097i	$-0.742257\pi$
$829$	−1143.52			−1.37940			−0.689698	+	0.724097i	$0.742257\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	77.2017	+	133.717i	0.0926791	+	0.160525i
$834$		0			0
$835$	−292.422	−	289.236i	−0.350206	−	0.346390i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	627.212	+	362.121i	0.747571	+	0.431610i	0.824816	−	0.565402i	$-0.191279\pi$
$839$	627.212	+	362.121i	0.747571	+	0.431610i	−0.0772447	+	0.997012i	$0.524612\pi$
$840$		0			0
$841$	−415.691	−	719.998i	−0.494282	−	0.856121i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−202.820	+	773.863i	−0.240023	+	0.915814i
$846$		0			0
$847$			751.020i			0.886683i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1299.89	−	750.493i	1.52749	−	0.881895i
$852$		0			0
$853$	423.187	+	244.327i	0.496116	+	0.286433i	0.727108	−	0.686523i	$-0.240864\pi$
$853$	423.187	+	244.327i	0.496116	+	0.286433i	−0.230992	+	0.972956i	$0.574197\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	67.1598	−	116.324i	0.0783662	−	0.135734i	−0.824179	−	0.566329i	$-0.808363\pi$
$857$	67.1598	−	116.324i	0.0783662	−	0.135734i	0.902545	+	0.430595i	$0.141696\pi$
$858$		0			0
$859$	660.691	+	1144.35i	0.769140	+	1.33219i	0.938030	+	0.346554i	$0.112648\pi$
$859$	660.691	+	1144.35i	0.769140	+	1.33219i	−0.168890	+	0.985635i	$0.554018\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−948.942			−1.09959			−0.549793	−	0.835301i	$-0.685293\pi$
$863$	−948.942			−1.09959			−0.549793	+	0.835301i	$0.685293\pi$
$864$		0			0
$865$	−280.000	+	1068.35i	−0.323699	+	1.23508i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−34.9157	+	20.1586i	−0.0401792	+	0.0231975i
$870$		0			0
$871$	28.9657	−	50.1700i	0.0332556	−	0.0576004i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−752.658	+	214.989i	−0.860181	+	0.245702i
$876$		0			0
$877$	−47.0452	+	27.1616i	−0.0536433	+	0.0309710i	−0.526582	−	0.850124i	$-0.676527\pi$
$877$	−47.0452	+	27.1616i	−0.0536433	+	0.0309710i	0.472938	+	0.881095i	$0.343193\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	762.713i		−	0.865735i	−0.901458	−	0.432868i	$-0.857502\pi$
$881$		−	762.713i		−	0.865735i	0.901458	−	0.432868i	$-0.142498\pi$
$882$		0			0
$883$			95.0917i			0.107692i	0.998549	+	0.0538458i	$0.0171479\pi$
$883$			95.0917i			0.107692i	−0.998549	+	0.0538458i	$0.982852\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−336.387	−	582.640i	−0.379241	−	0.656865i	0.611711	−	0.791082i	$-0.290482\pi$
$887$	−336.387	−	582.640i	−0.379241	−	0.656865i	−0.990952	+	0.134216i	$0.957148\pi$
$888$		0			0
$889$	−504.855	+	874.434i	−0.567891	+	0.983616i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	741.003	−	1283.46i	0.829791	−	1.43724i
$894$		0			0
$895$	−197.068	−	719.678i	−0.220188	−	0.804110i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			55.1610i			0.0613581i
$900$		0			0
$901$	697.863			0.774542
$902$		0			0
$903$		0			0
$904$		0			0
$905$	665.169	−	182.142i	0.734994	−	0.201262i
$906$		0			0
$907$	−118.883	−	68.6374i	−0.131073	−	0.0756752i	0.433030	−	0.901380i	$-0.357445\pi$
$907$	−118.883	−	68.6374i	−0.131073	−	0.0756752i	−0.564103	+	0.825705i	$0.690778\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1479.65	+	854.277i	1.62420	+	0.937735i	0.985778	+	0.168052i	$0.0537477\pi$
$911$	1479.65	+	854.277i	1.62420	+	0.937735i	0.638427	+	0.769683i	$0.279586\pi$
$912$		0			0
$913$	67.8497	−	39.1731i	0.0743151	−	0.0429059i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1559.69			−1.70086
$918$		0			0
$919$	475.802			0.517738			0.258869	−	0.965912i	$-0.416650\pi$
$919$	475.802			0.517738			0.258869	+	0.965912i	$0.416650\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−84.9550	−	147.146i	−0.0920422	−	0.159422i
$924$		0			0
$925$	−551.620	−	931.712i	−0.596346	−	1.00726i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−851.352	−	491.528i	−0.916418	−	0.529094i	−0.0339271	−	0.999424i	$-0.510801\pi$
$929$	−851.352	−	491.528i	−0.916418	−	0.529094i	−0.882490	+	0.470330i	$0.844135\pi$
$930$		0			0
$931$	−91.9236	−	159.216i	−0.0987365	−	0.171017i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−78.8878	−	20.6755i	−0.0843720	−	0.0221128i
$936$		0			0
$937$		−	614.705i		−	0.656035i	−0.944672	−	0.328018i	$-0.893619\pi$
$937$		−	614.705i		−	0.656035i	0.944672	−	0.328018i	$-0.106381\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1000.42	−	577.592i	1.06314	−	0.613807i	0.136844	−	0.990593i	$-0.456304\pi$
$941$	1000.42	−	577.592i	1.06314	−	0.613807i	0.926300	+	0.376786i	$0.122971\pi$
$942$		0			0
$943$	1389.59	+	802.277i	1.47358	+	0.850771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	706.076	−	1222.96i	0.745592	−	1.29140i	−0.204325	−	0.978903i	$-0.565500\pi$
$947$	706.076	−	1222.96i	0.745592	−	1.29140i	0.949918	−	0.312501i	$-0.101167\pi$
$948$		0			0
$949$	163.966	+	283.997i	0.172777	+	0.299259i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	243.408			0.255412			0.127706	−	0.991812i	$-0.459239\pi$
$953$	243.408			0.255412			0.127706	+	0.991812i	$0.459239\pi$
$954$		0			0
$955$	975.725	+	255.725i	1.02170	+	0.267775i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1333.91	−	770.134i	1.39094	−	0.803060i
$960$		0			0
$961$	322.324	−	558.282i	0.335405	−	0.580939i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1297.04	−	1311.32i	1.34408	−	1.35888i
$966$		0			0
$967$	763.356	−	440.724i	0.789407	−	0.455764i	−0.0503470	−	0.998732i	$-0.516033\pi$
$967$	763.356	−	440.724i	0.789407	−	0.455764i	0.839754	+	0.542968i	$0.182699\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1537.17i		−	1.58308i	−0.611119	−	0.791538i	$-0.709280\pi$
$971$		−	1537.17i		−	1.58308i	0.611119	−	0.791538i	$-0.290720\pi$
$972$		0			0
$973$		−	1064.12i		−	1.09365i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	197.084	+	341.359i	0.201724	+	0.349396i	0.949084	−	0.315023i	$-0.102012\pi$
$977$	197.084	+	341.359i	0.201724	+	0.349396i	−0.747360	+	0.664419i	$0.768679\pi$
$978$		0			0
$979$	46.7939	−	81.0495i	0.0477977	−	0.0827880i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	364.434	−	631.219i	0.370737	−	0.642135i	−0.618942	−	0.785437i	$-0.712438\pi$
$983$	364.434	−	631.219i	0.370737	−	0.642135i	0.989679	+	0.143301i	$0.0457718\pi$
$984$		0			0
$985$	820.692	−	224.729i	0.833190	−	0.228151i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	718.702i		−	0.726696i
$990$		0			0
$991$	−732.282			−0.738933			−0.369466	−	0.929244i	$-0.620460\pi$
$991$	−732.282			−0.738933			−0.369466	+	0.929244i	$0.620460\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	660.016	−	180.731i	0.663332	−	0.181639i
$996$		0			0
$997$	691.538	+	399.260i	0.693619	+	0.400461i	0.804966	−	0.593321i	$-0.202183\pi$
$997$	691.538	+	399.260i	0.693619	+	0.400461i	−0.111348	+	0.993782i	$0.535517\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.e.269.8		16
3.2	odd	2	inner	1620.3.t.e.269.1		16
5.4	even	2	inner	1620.3.t.e.269.7		16
9.2	odd	6		540.3.b.c.269.5	yes	8
9.4	even	3	inner	1620.3.t.e.1349.2		16
9.5	odd	6	inner	1620.3.t.e.1349.7		16
9.7	even	3		540.3.b.c.269.4	yes	8
15.14	odd	2	inner	1620.3.t.e.269.2		16
36.7	odd	6		2160.3.c.n.1889.4		8
36.11	even	6		2160.3.c.n.1889.5		8
45.2	even	12		2700.3.g.p.701.2		4
45.4	even	6	inner	1620.3.t.e.1349.1		16
45.7	odd	12		2700.3.g.p.701.1		4
45.14	odd	6	inner	1620.3.t.e.1349.8		16
45.29	odd	6		540.3.b.c.269.3	✓	8
45.34	even	6		540.3.b.c.269.6	yes	8
45.38	even	12		2700.3.g.o.701.4		4
45.43	odd	12		2700.3.g.o.701.3		4
180.79	odd	6		2160.3.c.n.1889.6		8
180.119	even	6		2160.3.c.n.1889.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.c.269.3	✓	8	45.29	odd	6
540.3.b.c.269.4	yes	8	9.7	even	3
540.3.b.c.269.5	yes	8	9.2	odd	6
540.3.b.c.269.6	yes	8	45.34	even	6
1620.3.t.e.269.1		16	3.2	odd	2	inner
1620.3.t.e.269.2		16	15.14	odd	2	inner
1620.3.t.e.269.7		16	5.4	even	2	inner
1620.3.t.e.269.8		16	1.1	even	1	trivial
1620.3.t.e.1349.1		16	45.4	even	6	inner
1620.3.t.e.1349.2		16	9.4	even	3	inner
1620.3.t.e.1349.7		16	9.5	odd	6	inner
1620.3.t.e.1349.8		16	45.14	odd	6	inner
2160.3.c.n.1889.3		8	180.119	even	6
2160.3.c.n.1889.4		8	36.7	odd	6
2160.3.c.n.1889.5		8	36.11	even	6
2160.3.c.n.1889.6		8	180.79	odd	6
2700.3.g.o.701.3		4	45.43	odd	12
2700.3.g.o.701.4		4	45.38	even	12
2700.3.g.p.701.1		4	45.7	odd	12
2700.3.g.p.701.2		4	45.2	even	12

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.82247 - 1.32053i) q^{5} +(-5.42313 - 3.13104i) q^{7} +O(q^{10})\)	\(q+(4.82247 - 1.32053i) q^{5} +(-5.42313 - 3.13104i) q^{7} +(0.895275 + 0.516887i) q^{11} +(2.59808 - 1.50000i) q^{13} -15.7776 q^{17} +18.7863 q^{19} +(17.3282 + 30.0134i) q^{23} +(21.5124 - 12.7364i) q^{25} +(2.68583 + 1.55066i) q^{29} +(8.89313 + 15.4034i) q^{31} +(-30.2875 - 7.93797i) q^{35} -43.3104i q^{37} +(40.0960 - 23.1494i) q^{41} +(-17.9596 - 10.3690i) q^{43} +(39.4439 - 68.3188i) q^{47} +(-4.89313 - 8.47515i) q^{49} -44.2313 q^{53} +(5.00000 + 1.31044i) q^{55} +(78.4015 - 45.2651i) q^{59} +(-4.39313 + 7.60913i) q^{61} +(10.5483 - 10.6645i) q^{65} +(16.7233 - 9.65522i) q^{67} -56.6366i q^{71} +109.310i q^{73} +(-3.23679 - 5.60629i) q^{77} +(-19.5000 + 33.7750i) q^{79} +(37.8932 - 65.6330i) q^{83} +(-76.0868 + 20.8347i) q^{85} -90.5302i q^{89} -18.7863 q^{91} +(90.5962 - 24.8078i) q^{95} +(-7.69287 - 4.44148i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 72 q^{19} - 12 q^{25} - 44 q^{31} + 108 q^{49} + 80 q^{55} + 116 q^{61} - 312 q^{79} - 160 q^{85} + 72 q^{91}+O(q^{100}) \) 16 * q - 72 * q^19 - 12 * q^25 - 44 * q^31 + 108 * q^49 + 80 * q^55 + 116 * q^61 - 312 * q^79 - 160 * q^85 + 72 * q^91

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