LMFDB - Embedded newform 1620.3.t.a.269.4

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.154550410641.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 221x^{4} - 60x^{2} + 16 $ x^8 - 15x^6 + 221x^4 - 60*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.4
Root			$-0.451318 - 0.260569i$ of defining polynomial
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.a.1349.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.98136 + 0.431312i) q^{5} +(6.19588 + 3.57719i) q^{7} +O(q^{10})$	$q+(4.98136 + 0.431312i) q^{5} +(6.19588 + 3.57719i) q^{7} +(-4.39061 - 2.53492i) q^{11} +(-2.70791 + 1.56341i) q^{13} +8.72842 q^{17} -20.1852 q^{19} +(7.36421 + 12.7552i) q^{23} +(24.6279 + 4.29704i) q^{25} +(34.4674 + 19.8997i) q^{29} +(19.6852 + 34.0959i) q^{31} +(29.3210 + 20.4917i) q^{35} -34.8712i q^{37} +(-11.4891 + 6.63325i) q^{41} +(57.6908 + 33.3078i) q^{43} +(8.45683 - 14.6477i) q^{47} +(1.09262 + 1.89248i) q^{49} -4.62950 q^{53} +(-20.7779 - 14.5211i) q^{55} +(-22.3208 + 12.8869i) q^{59} +(6.09262 - 10.5527i) q^{61} +(-14.1634 + 6.61997i) q^{65} +(-92.1581 + 53.2075i) q^{67} +101.487i q^{71} -23.2646i q^{73} +(-18.1358 - 31.4121i) q^{77} +(33.2779 - 57.6390i) q^{79} +(72.1421 - 124.954i) q^{83} +(43.4794 + 3.76467i) q^{85} +154.553i q^{89} -22.3705 q^{91} +(-100.550 - 8.70613i) q^{95} +(-152.189 - 87.8664i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{5}+O(q^{10}) $ 8 * q - 3 * q^5	$ 8 q - 3 q^{5} + 12 q^{17} + 12 q^{19} + 30 q^{23} - 9 q^{25} - 16 q^{31} + 90 q^{35} - 48 q^{47} - 78 q^{49} - 384 q^{53} + 94 q^{55} - 38 q^{61} - 138 q^{65} - 174 q^{77} + 6 q^{79} + 288 q^{83} + 100 q^{85} + 168 q^{91} - 318 q^{95}+O(q^{100}) $ 8 * q - 3 * q^5 + 12 * q^17 + 12 * q^19 + 30 * q^23 - 9 * q^25 - 16 * q^31 + 90 * q^35 - 48 * q^47 - 78 * q^49 - 384 * q^53 + 94 * q^55 - 38 * q^61 - 138 * q^65 - 174 * q^77 + 6 * q^79 + 288 * q^83 + 100 * q^85 + 168 * q^91 - 318 * q^95

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
$5$	4.98136	+	0.431312i	0.996272	+	0.0862624i
$5$	4.98136	+	0.431312i	0.996272	+	0.0862624i
$6$		0			0
$7$	6.19588	+	3.57719i	0.885126	+	0.511028i	0.872345	−	0.488891i	$-0.162598\pi$
$7$	6.19588	+	3.57719i	0.885126	+	0.511028i	0.0127808	+	0.999918i	$0.495932\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.39061	−	2.53492i	−0.399146	−	0.230447i	0.286969	−	0.957940i	$-0.407352\pi$
$11$	−4.39061	−	2.53492i	−0.399146	−	0.230447i	−0.686116	+	0.727493i	$0.740686\pi$
$12$		0			0
$13$	−2.70791	+	1.56341i	−0.208301	+	0.120262i	−0.600521	−	0.799609i	$-0.705040\pi$
$13$	−2.70791	+	1.56341i	−0.208301	+	0.120262i	0.392221	+	0.919871i	$0.371707\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.72842			0.513436			0.256718	−	0.966486i	$-0.417359\pi$
$17$	8.72842			0.513436			0.256718	+	0.966486i	$0.417359\pi$
$18$		0			0
$19$	−20.1852			−1.06238			−0.531191	−	0.847252i	$-0.678255\pi$
$19$	−20.1852			−1.06238			−0.531191	+	0.847252i	$0.678255\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.36421	+	12.7552i	0.320183	+	0.554573i	0.980526	−	0.196391i	$-0.0629223\pi$
$23$	7.36421	+	12.7552i	0.320183	+	0.554573i	−0.660343	+	0.750964i	$0.729589\pi$
$24$		0			0
$25$	24.6279	+	4.29704i	0.985118	+	0.171882i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.4674	+	19.8997i	1.18853	+	0.686198i	0.957972	−	0.286860i	$-0.0926116\pi$
$29$	34.4674	+	19.8997i	1.18853	+	0.686198i	0.230558	+	0.973059i	$0.425945\pi$
$30$		0			0
$31$	19.6852	+	34.0959i	0.635008	+	1.09987i	0.986513	+	0.163680i	$0.0523365\pi$
$31$	19.6852	+	34.0959i	0.635008	+	1.09987i	−0.351505	+	0.936186i	$0.614330\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	29.3210	+	20.4917i	0.837744	+	0.585476i
$36$		0			0
$37$		−	34.8712i		−	0.942465i	−0.882009	−	0.471232i	$-0.843809\pi$
$37$		−	34.8712i		−	0.942465i	0.882009	−	0.471232i	$-0.156191\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.4891	+	6.63325i	−0.280223	+	0.161787i	−0.633524	−	0.773723i	$-0.718392\pi$
$41$	−11.4891	+	6.63325i	−0.280223	+	0.161787i	0.353302	+	0.935509i	$0.385059\pi$
$42$		0			0
$43$	57.6908	+	33.3078i	1.34165	+	0.774600i	0.987049	−	0.160420i	$-0.0512848\pi$
$43$	57.6908	+	33.3078i	1.34165	+	0.774600i	0.354597	+	0.935019i	$0.384618\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.45683	−	14.6477i	0.179933	−	0.311652i	−0.761925	−	0.647666i	$-0.775745\pi$
$47$	8.45683	−	14.6477i	0.179933	−	0.311652i	0.941857	+	0.336013i	$0.109079\pi$
$48$		0			0
$49$	1.09262	+	1.89248i	0.0222985	+	0.0386221i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.62950			−0.0873491			−0.0436746	−	0.999046i	$-0.513906\pi$
$53$	−4.62950			−0.0873491			−0.0436746	+	0.999046i	$0.513906\pi$
$54$		0			0
$55$	−20.7779	−	14.5211i	−0.377780	−	0.264019i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−22.3208	+	12.8869i	−0.378318	+	0.218422i	−0.677086	−	0.735904i	$-0.736758\pi$
$59$	−22.3208	+	12.8869i	−0.378318	+	0.218422i	0.298768	+	0.954326i	$0.403424\pi$
$60$		0			0
$61$	6.09262	−	10.5527i	0.0998791	−	0.172996i	−0.811755	−	0.583998i	$-0.801488\pi$
$61$	6.09262	−	10.5527i	0.0998791	−	0.172996i	0.911634	+	0.411002i	$0.134821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.1634	+	6.61997i	−0.217898	+	0.101846i
$66$		0			0
$67$	−92.1581	+	53.2075i	−1.37549	+	0.794142i	−0.991613	−	0.129240i	$-0.958746\pi$
$67$	−92.1581	+	53.2075i	−1.37549	+	0.794142i	−0.383881	+	0.923382i	$0.625413\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			101.487i			1.42939i	0.699436	+	0.714695i	$0.253435\pi$
$71$			101.487i			1.42939i	−0.699436	+	0.714695i	$0.746565\pi$
$72$		0			0
$73$		−	23.2646i		−	0.318694i	−0.987223	−	0.159347i	$-0.949061\pi$
$73$		−	23.2646i		−	0.318694i	0.987223	−	0.159347i	$-0.0509388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−18.1358	−	31.4121i	−0.235530	−	0.407950i
$78$		0			0
$79$	33.2779	−	57.6390i	0.421239	−	0.729607i	−0.574822	−	0.818278i	$-0.694929\pi$
$79$	33.2779	−	57.6390i	0.421239	−	0.729607i	0.996061	+	0.0886713i	$0.0282621\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	72.1421	−	124.954i	0.869182	−	1.50547i	0.00634748	−	0.999980i	$-0.497980\pi$
$83$	72.1421	−	124.954i	0.869182	−	1.50547i	0.862834	−	0.505487i	$-0.168687\pi$
$84$		0			0
$85$	43.4794	+	3.76467i	0.511522	+	0.0442902i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			154.553i			1.73655i	0.496085	+	0.868274i	$0.334770\pi$
$89$			154.553i			1.73655i	−0.496085	+	0.868274i	$0.665230\pi$
$90$		0			0
$91$	−22.3705			−0.245830
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−100.550	−	8.70613i	−1.05842	−	0.0916435i
$96$		0			0
$97$	−152.189	−	87.8664i	−1.56896	−	0.905839i	−0.996291	−	0.0860529i	$-0.972575\pi$
$97$	−152.189	−	87.8664i	−1.56896	−	0.905839i	−0.572669	−	0.819786i	$-0.694092\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	129.456	+	74.7415i	1.28174	+	0.740014i	0.977167	−	0.212474i	$-0.0681520\pi$
$101$	129.456	+	74.7415i	1.28174	+	0.740014i	0.304576	+	0.952488i	$0.401485\pi$
$102$		0			0
$103$	−24.7835	+	14.3088i	−0.240617	+	0.138920i	−0.615460	−	0.788168i	$-0.711030\pi$
$103$	−24.7835	+	14.3088i	−0.240617	+	0.138920i	0.374843	+	0.927088i	$0.377696\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	156.297			1.46072			0.730359	−	0.683064i	$-0.239353\pi$
$107$	156.297			1.46072			0.730359	+	0.683064i	$0.239353\pi$
$108$		0			0
$109$	8.55575			0.0784931			0.0392465	−	0.999230i	$-0.487504\pi$
$109$	8.55575			0.0784931			0.0392465	+	0.999230i	$0.487504\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	92.1852	+	159.670i	0.815799	+	1.41300i	0.908753	+	0.417334i	$0.137036\pi$
$113$	92.1852	+	159.670i	0.815799	+	1.41300i	−0.0929544	+	0.995670i	$0.529631\pi$
$114$		0			0
$115$	31.1823	+	66.7145i	0.271151	+	0.580126i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	54.0802	+	31.2232i	0.454456	+	0.262380i
$120$		0			0
$121$	−47.6484	−	82.5294i	−0.393788	−	0.682061i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	120.827	+	32.0274i	0.966619	+	0.256219i
$126$		0			0
$127$		−	40.7002i		−	0.320474i	−0.987079	−	0.160237i	$-0.948774\pi$
$127$		−	40.7002i		−	0.320474i	0.987079	−	0.160237i	$-0.0512259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−69.3025	+	40.0118i	−0.529026	+	0.305434i	−0.740620	−	0.671924i	$-0.765468\pi$
$131$	−69.3025	+	40.0118i	−0.529026	+	0.305434i	0.211593	+	0.977358i	$0.432135\pi$
$132$		0			0
$133$	−125.065	−	72.2065i	−0.940341	−	0.542906i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.53688	+	11.3222i	−0.0477144	+	0.0826438i	−0.888896	−	0.458109i	$-0.848527\pi$
$137$	−6.53688	+	11.3222i	−0.0477144	+	0.0826438i	0.841182	+	0.540752i	$0.181860\pi$
$138$		0			0
$139$	−72.1852	−	125.029i	−0.519318	−	0.899486i	−0.999748	−	0.0224524i	$-0.992853\pi$
$139$	−72.1852	−	125.029i	−0.519318	−	0.899486i	0.480430	−	0.877033i	$-0.340481\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.8525			0.110857
$144$		0			0
$145$	163.111	+	113.994i	1.12491	+	0.786166i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	46.9817	−	27.1249i	0.315313	−	0.182046i	−0.333988	−	0.942577i	$-0.608395\pi$
$149$	46.9817	−	27.1249i	0.315313	−	0.182046i	0.649302	+	0.760531i	$0.275061\pi$
$150$		0			0
$151$	−10.5926	+	18.3470i	−0.0701498	+	0.121503i	−0.898967	−	0.438017i	$-0.855681\pi$
$151$	−10.5926	+	18.3470i	−0.0701498	+	0.121503i	0.828817	+	0.559520i	$0.189014\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	83.3534	+	178.334i	0.537764	+	1.15054i
$156$		0			0
$157$	91.7458	−	52.9695i	0.584368	−	0.337385i	−0.178499	−	0.983940i	$-0.557124\pi$
$157$	91.7458	−	52.9695i	0.584368	−	0.337385i	0.762867	+	0.646555i	$0.223791\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			105.373i			0.654489i
$162$		0			0
$163$			154.746i			0.949361i	0.880158	+	0.474680i	$0.157436\pi$
$163$			154.746i			0.949361i	−0.880158	+	0.474680i	$0.842564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	75.8336	+	131.348i	0.454094	+	0.786513i	0.998636	−	0.0522205i	$-0.0166299\pi$
$167$	75.8336	+	131.348i	0.454094	+	0.786513i	−0.544542	+	0.838734i	$0.683297\pi$
$168$		0			0
$169$	−79.6115	+	137.891i	−0.471074	+	0.815924i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	155.599	−	269.505i	0.899416	−	1.55783i	0.0711731	−	0.997464i	$-0.477326\pi$
$173$	155.599	−	269.505i	0.899416	−	1.55783i	0.828243	−	0.560370i	$-0.189341\pi$
$174$		0			0
$175$	137.220	+	114.723i	0.784117	+	0.655559i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			211.170i			1.17972i	0.807505	+	0.589861i	$0.200817\pi$
$179$			211.170i			1.17972i	−0.807505	+	0.589861i	$0.799183\pi$
$180$		0			0
$181$	149.297			0.824844			0.412422	−	0.910993i	$-0.364683\pi$
$181$	149.297			0.824844			0.412422	+	0.910993i	$0.364683\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0404	−	173.706i	0.0812992	−	0.938952i
$186$		0			0
$187$	−38.3231	−	22.1258i	−0.204936	−	0.118320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.9278	−	12.0827i	−0.109570	−	0.0632602i	0.444214	−	0.895921i	$-0.353483\pi$
$191$	−20.9278	−	12.0827i	−0.109570	−	0.0632602i	−0.553783	+	0.832661i	$0.686816\pi$
$192$		0			0
$193$	288.086	−	166.327i	1.49267	−	0.861796i	0.492709	−	0.870194i	$-0.336007\pi$
$193$	288.086	−	166.327i	1.49267	−	0.861796i	0.999965	+	0.00839799i	$0.00267319\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	279.568			1.41913			0.709564	−	0.704641i	$-0.248892\pi$
$197$	279.568			1.41913			0.709564	+	0.704641i	$0.248892\pi$
$198$		0			0
$199$	−161.185			−0.809976			−0.404988	−	0.914322i	$-0.632724\pi$
$199$	−161.185			−0.809976			−0.404988	+	0.914322i	$0.632724\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	142.370	+	246.593i	0.701332	+	1.21474i
$204$		0			0
$205$	−60.0925	+	28.0872i	−0.293134	+	0.137011i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	88.6255	+	51.1680i	0.424046	+	0.244823i
$210$		0			0
$211$	46.0926	+	79.8348i	0.218448	+	0.378364i	0.954334	−	0.298742i	$-0.0965671\pi$
$211$	46.0926	+	79.8348i	0.218448	+	0.378364i	−0.735885	+	0.677106i	$0.763234\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	273.013	+	190.801i	1.26983	+	0.887446i
$216$		0			0
$217$			281.672i			1.29803i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−23.6357	+	13.6461i	−0.106949	+	0.0617471i
$222$		0			0
$223$	171.189	+	98.8361i	0.767664	+	0.443211i	0.832041	−	0.554715i	$-0.187173\pi$
$223$	171.189	+	98.8361i	0.767664	+	0.443211i	−0.0643766	+	0.997926i	$0.520506\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.3642	+	28.3436i	−0.0720890	+	0.124862i	−0.899817	−	0.436268i	$-0.856300\pi$
$227$	−16.3642	+	28.3436i	−0.0720890	+	0.124862i	0.827728	+	0.561130i	$0.189633\pi$
$228$		0			0
$229$	−175.463	−	303.911i	−0.766215	−	1.32712i	−0.939602	−	0.342269i	$-0.888805\pi$
$229$	−175.463	−	303.911i	−0.766215	−	1.32712i	0.173388	−	0.984854i	$-0.444529\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−399.827			−1.71600			−0.857999	−	0.513652i	$-0.828292\pi$
$233$	−399.827			−1.71600			−0.857999	+	0.513652i	$0.828292\pi$
$234$		0			0
$235$	48.4443	−	69.3178i	0.206146	−	0.294969i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−314.965	+	181.845i	−1.31784	+	0.760858i	−0.983381	−	0.181551i	$-0.941888\pi$
$239$	−314.965	+	181.845i	−1.31784	+	0.760858i	−0.334463	+	0.942409i	$0.608555\pi$
$240$		0			0
$241$	−113.834	+	197.166i	−0.472339	+	0.818115i	−0.999499	−	0.0316513i	$-0.989923\pi$
$241$	−113.834	+	197.166i	−0.472339	+	0.818115i	0.527160	+	0.849766i	$0.323257\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.62651	+	9.89839i	0.0188837	+	0.0404016i
$246$		0			0
$247$	54.6598	−	31.5578i	0.221295	−	0.127765i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			227.988i			0.908319i	0.890920	+	0.454160i	$0.150060\pi$
$251$			227.988i			0.908319i	−0.890920	+	0.454160i	$0.849940\pi$
$252$		0			0
$253$		−	74.6707i		−	0.295141i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−228.019	−	394.940i	−0.887233	−	1.53673i	−0.843133	−	0.537705i	$-0.819291\pi$
$257$	−228.019	−	394.940i	−0.887233	−	1.53673i	−0.0441002	−	0.999027i	$-0.514042\pi$
$258$		0			0
$259$	124.741	−	216.058i	0.481625	−	0.834200i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	93.2716	−	161.551i	0.354645	−	0.614263i	−0.632412	−	0.774632i	$-0.717935\pi$
$263$	93.2716	−	161.551i	0.354645	−	0.614263i	0.987057	+	0.160369i	$0.0512685\pi$
$264$		0			0
$265$	−23.0612	−	1.99676i	−0.0870235	−	0.00753494i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	337.157i		−	1.25337i	−0.779272	−	0.626686i	$-0.784411\pi$
$269$		−	337.157i		−	1.25337i	0.779272	−	0.626686i	$-0.215589\pi$
$270$		0			0
$271$	58.2590			0.214978			0.107489	−	0.994206i	$-0.465719\pi$
$271$	58.2590			0.214978			0.107489	+	0.994206i	$0.465719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−97.2390	−	81.2965i	−0.353596	−	0.295623i
$276$		0			0
$277$	287.718	+	166.114i	1.03869	+	0.599691i	0.919463	−	0.393176i	$-0.128624\pi$
$277$	287.718	+	166.114i	1.03869	+	0.599691i	0.119231	+	0.992866i	$0.461957\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	97.2509	+	56.1478i	0.346089	+	0.199814i	0.662961	−	0.748654i	$-0.269299\pi$
$281$	97.2509	+	56.1478i	0.346089	+	0.199814i	−0.316873	+	0.948468i	$0.602633\pi$
$282$		0			0
$283$	−282.303	+	162.987i	−0.997536	+	0.575927i	−0.907518	−	0.420013i	$-0.862026\pi$
$283$	−282.303	+	162.987i	−0.997536	+	0.575927i	−0.0900175	+	0.995940i	$0.528692\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−94.9137			−0.330710
$288$		0			0
$289$	−212.815			−0.736383
$290$		0			0
$291$		0			0
$292$		0			0
$293$	35.9811	+	62.3212i	0.122803	+	0.212700i	0.920872	−	0.389865i	$-0.127478\pi$
$293$	35.9811	+	62.3212i	0.122803	+	0.212700i	−0.798069	+	0.602566i	$0.794145\pi$
$294$		0			0
$295$	−116.746	+	54.5671i	−0.395749	+	0.184973i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−39.8832	−	23.0266i	−0.133389	−	0.0770119i
$300$		0			0
$301$	238.297	+	412.742i	0.791684	+	1.37124i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	34.9011	−	49.9392i	0.114430	−	0.163735i
$306$		0			0
$307$		−	245.900i		−	0.800977i	−0.916302	−	0.400488i	$-0.868841\pi$
$307$		−	245.900i		−	0.800977i	0.916302	−	0.400488i	$-0.131159\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	60.0756	−	34.6847i	0.193169	−	0.111526i	−0.400296	−	0.916386i	$-0.631093\pi$
$311$	60.0756	−	34.6847i	0.193169	−	0.111526i	0.593465	+	0.804860i	$0.297759\pi$
$312$		0			0
$313$	−189.364	−	109.330i	−0.604998	−	0.349296i	0.166007	−	0.986125i	$-0.446912\pi$
$313$	−189.364	−	109.330i	−0.604998	−	0.349296i	−0.771005	+	0.636829i	$0.780246\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−37.2158	+	64.4597i	−0.117400	+	0.203343i	−0.918737	−	0.394871i	$-0.870789\pi$
$317$	−37.2158	+	64.4597i	−0.117400	+	0.203343i	0.801337	+	0.598214i	$0.204123\pi$
$318$		0			0
$319$	−100.889	−	174.744i	−0.316265	−	0.547787i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−176.185			−0.545465
$324$		0			0
$325$	−73.4082	+	26.8676i	−0.225871	+	0.0826696i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	104.795	−	60.5034i	0.318526	−	0.183901i
$330$		0			0
$331$	110.111	−	190.719i	0.332663	−	0.576189i	−0.650370	−	0.759618i	$-0.725386\pi$
$331$	110.111	−	190.719i	0.332663	−	0.576189i	0.983033	+	0.183428i	$0.0587195\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−482.022	+	225.297i	−1.43887	+	0.672529i
$336$		0			0
$337$	−21.3402	+	12.3208i	−0.0633240	+	0.0365601i	−0.531328	−	0.847166i	$-0.678307\pi$
$337$	−21.3402	+	12.3208i	−0.0633240	+	0.0365601i	0.468004	+	0.883726i	$0.344973\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	199.602i		−	0.585343i
$342$		0			0
$343$		−	334.931i		−	0.976475i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	73.7779	+	127.787i	0.212616	+	0.368262i	0.952533	−	0.304437i	$-0.0984682\pi$
$347$	73.7779	+	127.787i	0.212616	+	0.368262i	−0.739916	+	0.672699i	$0.765135\pi$
$348$		0			0
$349$	251.575	−	435.740i	0.720844	−	1.24854i	−0.239818	−	0.970818i	$-0.577088\pi$
$349$	251.575	−	435.740i	0.720844	−	1.24854i	0.960662	−	0.277721i	$-0.0895790\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	118.358	−	205.002i	0.335292	−	0.580742i	−0.648249	−	0.761428i	$-0.724499\pi$
$353$	118.358	−	205.002i	0.335292	−	0.580742i	0.983541	+	0.180686i	$0.0578318\pi$
$354$		0			0
$355$	−43.7724	+	505.542i	−0.123303	+	1.42406i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			195.871i			0.545601i	0.962071	+	0.272800i	$0.0879499\pi$
$359$			195.871i			0.545601i	−0.962071	+	0.272800i	$0.912050\pi$
$360$		0			0
$361$	46.4443			0.128654
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.0343	−	115.890i	0.0274913	−	0.317506i
$366$		0			0
$367$	294.237	+	169.878i	0.801737	+	0.462883i	0.844078	−	0.536220i	$-0.180148\pi$
$367$	294.237	+	169.878i	0.801737	+	0.462883i	−0.0423414	+	0.999103i	$0.513482\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−28.6838	−	16.5606i	−0.0773150	−	0.0446378i
$372$		0			0
$373$	−468.502	+	270.490i	−1.25604	+	0.725174i	−0.972302	−	0.233729i	$-0.924907\pi$
$373$	−468.502	+	270.490i	−1.25604	+	0.725174i	−0.283736	+	0.958902i	$0.591574\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−124.446			−0.330095
$378$		0			0
$379$	−432.926			−1.14229			−0.571143	−	0.820851i	$-0.693500\pi$
$379$	−432.926			−1.14229			−0.571143	+	0.820851i	$0.693500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	239.488	+	414.806i	0.625296	+	1.08304i	0.988484	+	0.151328i	$0.0483550\pi$
$383$	239.488	+	414.806i	0.625296	+	1.08304i	−0.363188	+	0.931716i	$0.618312\pi$
$384$		0			0
$385$	−76.7925	−	164.297i	−0.199461	−	0.426746i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−293.747	−	169.595i	−0.755134	−	0.435977i	0.0724120	−	0.997375i	$-0.476930\pi$
$389$	−293.747	−	169.595i	−0.755134	−	0.435977i	−0.827546	+	0.561398i	$0.810264\pi$
$390$		0			0
$391$	64.2779	+	111.333i	0.164394	+	0.284738i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	190.630	−	272.767i	0.482606	−	0.690550i
$396$		0			0
$397$		−	113.042i		−	0.284740i	−0.989814	−	0.142370i	$-0.954528\pi$
$397$		−	113.042i		−	0.284740i	0.989814	−	0.142370i	$-0.0454723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	413.764	−	238.887i	1.03183	−	0.595728i	0.114323	−	0.993444i	$-0.463530\pi$
$401$	413.764	−	238.887i	1.03183	−	0.595728i	0.917509	+	0.397716i	$0.130197\pi$
$402$		0			0
$403$	−106.612	−	61.5523i	−0.264545	−	0.152735i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−88.3957	+	153.106i	−0.217188	+	0.376181i
$408$		0			0
$409$	−338.685	−	586.620i	−0.828081	−	1.43428i	−0.899541	−	0.436835i	$-0.856099\pi$
$409$	−338.685	−	586.620i	−0.828081	−	1.43428i	0.0714602	−	0.997443i	$-0.477234\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−184.396			−0.446479
$414$		0			0
$415$	413.260	−	591.324i	0.995807	−	1.42488i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	298.060	−	172.085i	0.711360	−	0.410704i	−0.100205	−	0.994967i	$-0.531950\pi$
$419$	298.060	−	172.085i	0.711360	−	0.410704i	0.811564	+	0.584263i	$0.198616\pi$
$420$		0			0
$421$	−14.7959	+	25.6272i	−0.0351446	+	0.0608723i	−0.883063	−	0.469255i	$-0.844523\pi$
$421$	−14.7959	+	25.6272i	−0.0351446	+	0.0608723i	0.847918	+	0.530127i	$0.177856\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	214.963	+	37.5064i	0.505795	+	0.0882502i
$426$		0			0
$427$	75.4983	−	43.5890i	0.176811	−	0.102082i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	724.253i		−	1.68040i	−0.542276	−	0.840201i	$-0.682437\pi$
$431$		−	724.253i		−	1.68040i	0.542276	−	0.840201i	$-0.317563\pi$
$432$		0			0
$433$		−	96.6100i		−	0.223118i	−0.993758	−	0.111559i	$-0.964416\pi$
$433$		−	96.6100i		−	0.223118i	0.993758	−	0.111559i	$-0.0355844\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−148.648	−	257.467i	−0.340156	−	0.589168i
$438$		0			0
$439$	−50.6115	+	87.6617i	−0.115288	+	0.199685i	−0.917895	−	0.396824i	$-0.870112\pi$
$439$	−50.6115	+	87.6617i	−0.115288	+	0.199685i	0.802607	+	0.596508i	$0.203446\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−188.760	+	326.942i	−0.426094	+	0.738017i	−0.996522	−	0.0833307i	$-0.973444\pi$
$443$	−188.760	+	326.942i	−0.426094	+	0.738017i	0.570427	+	0.821348i	$0.306778\pi$
$444$		0			0
$445$	−66.6604	+	769.883i	−0.149799	+	1.73007i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	654.974i		−	1.45874i	−0.684120	−	0.729369i	$-0.739814\pi$
$449$		−	654.974i		−	1.45874i	0.684120	−	0.729369i	$-0.260186\pi$
$450$		0			0
$451$	67.2590			0.149133
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−111.436	−	9.64866i	−0.244913	−	0.0212058i
$456$		0			0
$457$	278.492	+	160.787i	0.609391	+	0.351832i	0.772727	−	0.634739i	$-0.218892\pi$
$457$	278.492	+	160.787i	0.609391	+	0.351832i	−0.163336	+	0.986570i	$0.552226\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−403.145	−	232.756i	−0.874500	−	0.504893i	−0.00565911	−	0.999984i	$-0.501801\pi$
$461$	−403.145	−	232.756i	−0.874500	−	0.504893i	−0.868841	+	0.495091i	$0.835135\pi$
$462$		0			0
$463$	217.268	−	125.440i	0.469262	−	0.270928i	−0.246669	−	0.969100i	$-0.579336\pi$
$463$	217.268	−	125.440i	0.469262	−	0.270928i	0.715931	+	0.698171i	$0.246003\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−246.259			−0.527321			−0.263661	−	0.964616i	$-0.584930\pi$
$467$	−246.259			−0.527321			−0.263661	+	0.964616i	$0.584930\pi$
$468$		0			0
$469$	−761.334			−1.62331
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−168.865	−	292.483i	−0.357009	−	0.618357i
$474$		0			0
$475$	−497.121	−	86.7368i	−1.04657	−	0.182604i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−151.565	−	87.5060i	−0.316419	−	0.182685i	0.333376	−	0.942794i	$-0.391812\pi$
$479$	−151.565	−	87.5060i	−0.316419	−	0.182685i	−0.649795	+	0.760109i	$0.725145\pi$
$480$		0			0
$481$	54.5180	+	94.4280i	0.113343	+	0.196316i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−720.211	−	503.335i	−1.48497	−	1.03781i
$486$		0			0
$487$			28.9906i			0.0595289i	0.999557	+	0.0297645i	$0.00947573\pi$
$487$			28.9906i			0.0595289i	−0.999557	+	0.0297645i	$0.990524\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−843.832	+	487.187i	−1.71860	+	0.992234i	−0.797106	+	0.603840i	$0.793637\pi$
$491$	−843.832	+	487.187i	−1.71860	+	0.992234i	−0.921493	+	0.388394i	$0.873030\pi$
$492$		0			0
$493$	300.846	+	173.693i	0.610234	+	0.352319i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−363.038	+	628.800i	−0.730458	+	1.26519i
$498$		0			0
$499$	−313.093	−	542.292i	−0.627440	−	1.08676i	−0.988064	−	0.154047i	$-0.950769\pi$
$499$	−313.093	−	542.292i	−0.627440	−	1.08676i	0.360623	−	0.932711i	$-0.382564\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−71.5197			−0.142186			−0.0710932	−	0.997470i	$-0.522649\pi$
$503$	−71.5197			−0.142186			−0.0710932	+	0.997470i	$0.522649\pi$
$504$		0			0
$505$	612.630	+	428.150i	1.21313	+	0.847822i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	807.392	−	466.148i	1.58623	−	0.915812i	0.592312	−	0.805708i	$-0.298215\pi$
$509$	807.392	−	466.148i	1.58623	−	0.915812i	0.993920	−	0.110103i	$-0.0351182\pi$
$510$		0			0
$511$	83.2221	−	144.145i	0.162861	−	0.282084i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−129.627	+	60.5878i	−0.251703	+	0.117646i
$516$		0			0
$517$	−74.2613	+	42.8748i	−0.143639	+	0.0829299i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	373.070i		−	0.716066i	−0.933709	−	0.358033i	$-0.883448\pi$
$521$		−	373.070i		−	0.716066i	0.933709	−	0.358033i	$-0.116552\pi$
$522$		0			0
$523$			828.480i			1.58409i	0.610461	+	0.792046i	$0.290984\pi$
$523$			828.480i			1.58409i	−0.610461	+	0.792046i	$0.709016\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	171.821	+	297.603i	0.326036	+	0.564711i
$528$		0			0
$529$	156.037	−	270.264i	0.294966	−	0.510896i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.7410	−	35.9245i	0.0389137	−	0.0674005i
$534$		0			0
$535$	778.571	+	67.4126i	1.45527	+	0.126005i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	11.0789i		−	0.0205545i
$540$		0			0
$541$	−50.5935			−0.0935184			−0.0467592	−	0.998906i	$-0.514889\pi$
$541$	−50.5935			−0.0935184			−0.0467592	+	0.998906i	$0.514889\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	42.6193	+	3.69019i	0.0782005	+	0.00677100i
$546$		0			0
$547$	−420.083	−	242.535i	−0.767976	−	0.443391i	0.0641762	−	0.997939i	$-0.479558\pi$
$547$	−420.083	−	242.535i	−0.767976	−	0.443391i	−0.832152	+	0.554547i	$0.812891\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−695.733	−	401.681i	−1.26267	−	0.729004i
$552$		0			0
$553$	412.371	−	238.083i	0.745699	−	0.430529i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−198.372			−0.356144			−0.178072	−	0.984017i	$-0.556986\pi$
$557$	−198.372			−0.356144			−0.178072	+	0.984017i	$0.556986\pi$
$558$		0			0
$559$	−208.295			−0.372621
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−330.420	−	572.304i	−0.586892	−	1.01653i	−0.994637	−	0.103430i	$-0.967018\pi$
$563$	−330.420	−	572.304i	−0.586892	−	1.01653i	0.407745	−	0.913096i	$-0.366315\pi$
$564$		0			0
$565$	390.341	+	835.132i	0.690869	+	1.47811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	141.157	+	81.4970i	0.248079	+	0.143229i	0.618884	−	0.785482i	$-0.287585\pi$
$569$	141.157	+	81.4970i	0.248079	+	0.143229i	−0.370805	+	0.928711i	$0.620918\pi$
$570$		0			0
$571$	−131.316	−	227.445i	−0.229975	−	0.398328i	0.727826	−	0.685762i	$-0.240531\pi$
$571$	−131.316	−	227.445i	−0.229975	−	0.398328i	−0.957800	+	0.287434i	$0.907198\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	126.556	+	345.778i	0.220097	+	0.601353i
$576$		0			0
$577$			532.551i			0.922966i	0.887149	+	0.461483i	$0.152682\pi$
$577$			532.551i			0.922966i	−0.887149	+	0.461483i	$0.847318\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	893.967	−	516.132i	1.53867	−	0.888352i
$582$		0			0
$583$	20.3263	+	11.7354i	0.0348651	+	0.0201294i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	463.797	−	803.320i	0.790114	−	1.36852i	−0.135782	−	0.990739i	$-0.543355\pi$
$587$	463.797	−	803.320i	0.790114	−	1.36852i	0.925896	−	0.377778i	$-0.123312\pi$
$588$		0			0
$589$	−397.352	−	688.233i	−0.674621	−	1.16848i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−890.336			−1.50141			−0.750705	−	0.660638i	$-0.770286\pi$
$593$	−890.336			−1.50141			−0.750705	+	0.660638i	$0.770286\pi$
$594$		0			0
$595$	255.926	+	178.860i	0.430128	+	0.300604i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−897.856	+	518.378i	−1.49893	+	0.865405i	−0.999999	−	0.00123933i	$-0.999606\pi$
$599$	−897.856	+	518.378i	−1.49893	+	0.865405i	−0.498926	+	0.866644i	$0.666272\pi$
$600$		0			0
$601$	−432.908	+	749.819i	−0.720313	+	1.24762i	0.240561	+	0.970634i	$0.422669\pi$
$601$	−432.908	+	749.819i	−0.720313	+	1.24762i	−0.960874	+	0.276985i	$0.910665\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−201.758	−	431.660i	−0.333484	−	0.713488i
$606$		0			0
$607$	613.671	−	354.303i	1.01099	−	0.583695i	0.0995083	−	0.995037i	$-0.468273\pi$
$607$	613.671	−	354.303i	1.01099	−	0.583695i	0.911481	+	0.411342i	$0.134940\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			52.8860i			0.0865565i
$612$		0			0
$613$		−	677.814i		−	1.10573i	−0.833270	−	0.552866i	$-0.813534\pi$
$613$		−	677.814i		−	1.10573i	0.833270	−	0.552866i	$-0.186466\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−237.031	−	410.551i	−0.384168	−	0.665398i	0.607486	−	0.794331i	$-0.292178\pi$
$617$	−237.031	−	410.551i	−0.384168	−	0.665398i	−0.991653	+	0.128933i	$0.958845\pi$
$618$		0			0
$619$	−161.593	+	279.888i	−0.261056	+	0.452162i	−0.966523	−	0.256581i	$-0.917404\pi$
$619$	−161.593	+	279.888i	−0.261056	+	0.452162i	0.705467	+	0.708743i	$0.250737\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−552.865	+	957.590i	−0.887424	+	1.53706i
$624$		0			0
$625$	588.071	+	211.654i	0.940913	+	0.338647i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	304.370i		−	0.483896i
$630$		0			0
$631$	977.669			1.54940			0.774698	−	0.632331i	$-0.217902\pi$
$631$	977.669			1.54940			0.774698	+	0.632331i	$0.217902\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.5545	−	202.743i	0.0276449	−	0.319280i
$636$		0			0
$637$	−5.91745	−	3.41644i	−0.00928956	−	0.00536333i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	212.376	+	122.615i	0.331320	+	0.191288i	0.656427	−	0.754390i	$-0.272067\pi$
$641$	212.376	+	122.615i	0.331320	+	0.191288i	−0.325107	+	0.945677i	$0.605400\pi$
$642$		0			0
$643$	211.718	−	122.236i	0.329267	−	0.190102i	−0.326249	−	0.945284i	$-0.605785\pi$
$643$	211.718	−	122.236i	0.329267	−	0.190102i	0.655515	+	0.755182i	$0.272451\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−866.779			−1.33969			−0.669844	−	0.742501i	$-0.733639\pi$
$647$	−866.779			−1.33969			−0.669844	+	0.742501i	$0.733639\pi$
$648$		0			0
$649$	130.669			0.201339
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−320.599	−	555.294i	−0.490963	−	0.850373i	0.508983	−	0.860777i	$-0.330022\pi$
$653$	−320.599	−	555.294i	−0.490963	−	0.850373i	−0.999946	+	0.0104037i	$0.996688\pi$
$654$		0			0
$655$	−362.478	+	169.422i	−0.553402	+	0.258660i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−668.130	−	385.745i	−1.01385	−	0.585349i	−0.101536	−	0.994832i	$-0.532376\pi$
$659$	−668.130	−	385.745i	−1.01385	−	0.585349i	−0.912318	+	0.409483i	$0.865709\pi$
$660$		0			0
$661$	398.038	+	689.422i	0.602175	+	1.04300i	0.992491	+	0.122317i	$0.0390324\pi$
$661$	398.038	+	689.422i	0.602175	+	1.04300i	−0.390316	+	0.920681i	$0.627634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−591.852	−	413.629i	−0.890004	−	0.621999i
$666$		0			0
$667$			586.184i			0.878836i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−53.5007	+	30.8886i	−0.0797327	+	0.0460337i
$672$		0			0
$673$	−604.767	−	349.162i	−0.898613	−	0.518815i	−0.0218633	−	0.999761i	$-0.506960\pi$
$673$	−604.767	−	349.162i	−0.898613	−	0.518815i	−0.876750	+	0.480946i	$0.840293\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−216.827	+	375.556i	−0.320277	+	0.554736i	−0.980545	−	0.196294i	$-0.937109\pi$
$677$	−216.827	+	375.556i	−0.320277	+	0.554736i	0.660268	+	0.751030i	$0.270443\pi$
$678$		0			0
$679$	−628.630	−	1088.82i	−0.925818	−	1.60356i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1112.83			1.62933			0.814663	−	0.579935i	$-0.196922\pi$
$683$	1112.83			1.62933			0.814663	+	0.579935i	$0.196922\pi$
$684$		0			0
$685$	−37.4460	+	53.5806i	−0.0546656	+	0.0782198i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.5363	−	7.23782i	0.0181949	−	0.0105048i
$690$		0			0
$691$	−248.871	+	431.058i	−0.360161	+	0.623817i	−0.987987	−	0.154536i	$-0.950612\pi$
$691$	−248.871	+	431.058i	−0.360161	+	0.623817i	0.627826	+	0.778354i	$0.283945\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−305.655	−	653.947i	−0.439791	−	0.940930i
$696$		0			0
$697$	−100.282	+	57.8978i	−0.143876	+	0.0830671i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	660.713i		−	0.942529i	−0.881992	−	0.471264i	$-0.843798\pi$
$701$		−	660.713i		−	0.942529i	0.881992	−	0.471264i	$-0.156202\pi$
$702$		0			0
$703$			703.884i			1.00126i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	534.729	+	926.178i	0.756336	+	1.31001i
$708$		0			0
$709$	506.779	−	877.767i	0.714780	−	1.23803i	−0.248265	−	0.968692i	$-0.579860\pi$
$709$	506.779	−	877.767i	0.714780	−	1.23803i	0.963044	−	0.269342i	$-0.0868063\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−289.933	+	502.178i	−0.406637	+	0.704317i
$714$		0			0
$715$	78.9670	+	6.83736i	0.110443	+	0.00956275i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1067.57i		−	1.48480i	−0.669955	−	0.742402i	$-0.733687\pi$
$719$		−	1067.57i		−	1.48480i	0.669955	−	0.742402i	$-0.266313\pi$
$720$		0			0
$721$	−204.741			−0.283968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	763.350	+	638.198i	1.05290	+	0.880272i
$726$		0			0
$727$	−760.856	−	439.281i	−1.04657	−	0.604237i	−0.124883	−	0.992172i	$-0.539855\pi$
$727$	−760.856	−	439.281i	−1.04657	−	0.604237i	−0.921687	+	0.387934i	$0.873189\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	503.549	+	290.724i	0.688850	+	0.397707i
$732$		0			0
$733$	−66.6392	+	38.4742i	−0.0909130	+	0.0524886i	−0.544767	−	0.838587i	$-0.683382\pi$
$733$	−66.6392	+	38.4742i	−0.0909130	+	0.0524886i	0.453854	+	0.891076i	$0.350049\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	539.507			0.732031
$738$		0			0
$739$	−613.815			−0.830602			−0.415301	−	0.909684i	$-0.636324\pi$
$739$	−613.815			−0.830602			−0.415301	+	0.909684i	$0.636324\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−165.518	−	286.686i	−0.222770	−	0.385849i	0.732878	−	0.680360i	$-0.238177\pi$
$743$	−165.518	−	286.686i	−0.222770	−	0.385849i	−0.955648	+	0.294511i	$0.904843\pi$
$744$		0			0
$745$	245.732	−	114.855i	0.329842	−	0.154168i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	968.396	+	559.104i	1.29292	+	0.746467i
$750$		0			0
$751$	248.426	+	430.287i	0.330794	+	0.572952i	0.982668	−	0.185376i	$-0.0593502\pi$
$751$	248.426	+	430.287i	0.330794	+	0.572952i	−0.651874	+	0.758327i	$0.726017\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−60.6790	+	86.8241i	−0.0803695	+	0.114999i
$756$		0			0
$757$		−	1234.26i		−	1.63046i	−0.579135	−	0.815232i	$-0.696610\pi$
$757$		−	1234.26i		−	1.63046i	0.579135	−	0.815232i	$-0.303390\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	60.0756	−	34.6847i	0.0789430	−	0.0455777i	−0.460009	−	0.887914i	$-0.652154\pi$
$761$	60.0756	−	34.6847i	0.0789430	−	0.0455777i	0.538952	+	0.842337i	$0.318820\pi$
$762$		0			0
$763$	53.0104	+	30.6056i	0.0694762	+	0.0401121i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	40.2950	−	69.7930i	0.0525359	−	0.0909948i
$768$		0			0
$769$	−555.426	−	962.026i	−0.722271	−	1.25101i	−0.960088	−	0.279700i	$-0.909765\pi$
$769$	−555.426	−	962.026i	−0.722271	−	1.25101i	0.237817	−	0.971310i	$-0.423568\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	733.074			0.948349			0.474174	−	0.880431i	$-0.342747\pi$
$773$	733.074			0.948349			0.474174	+	0.880431i	$0.342747\pi$
$774$		0			0
$775$	338.296	+	924.299i	0.436511	+	1.19264i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	231.911	−	133.894i	0.297703	−	0.171879i
$780$		0			0
$781$	257.261	−	445.589i	0.329399	−	0.570536i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	479.865	−	224.289i	0.611293	−	0.285718i
$786$		0			0
$787$	−870.220	+	502.422i	−1.10574	+	0.638402i	−0.937724	−	0.347382i	$-0.887071\pi$
$787$	−870.220	+	502.422i	−1.10574	+	0.638402i	−0.168020	+	0.985784i	$0.553737\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1319.06i			1.66758i
$792$		0			0
$793$			38.1011i			0.0480468i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−310.550	−	537.889i	−0.389649	−	0.674892i	0.602753	−	0.797928i	$-0.294070\pi$
$797$	−310.550	−	537.889i	−0.389649	−	0.674892i	−0.992402	+	0.123036i	$0.960737\pi$
$798$		0			0
$799$	73.8148	−	127.851i	0.0923839	−	0.160014i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−58.9740	+	102.146i	−0.0734421	+	0.127205i
$804$		0			0
$805$	−45.4485	+	524.900i	−0.0564578	+	0.652050i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			276.049i			0.341222i	0.985338	+	0.170611i	$0.0545742\pi$
$809$			276.049i			0.341222i	−0.985338	+	0.170611i	$0.945426\pi$
$810$		0			0
$811$	1024.08			1.26273			0.631366	−	0.775485i	$-0.282495\pi$
$811$	1024.08			1.26273			0.631366	+	0.775485i	$0.282495\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−66.7437	+	770.845i	−0.0818941	+	0.945822i
$816$		0			0
$817$	−1164.50	−	672.326i	−1.42534	−	0.822920i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.57167	−	3.21681i	−0.00678645	−	0.00391816i	0.496603	−	0.867978i	$-0.334581\pi$
$821$	−5.57167	−	3.21681i	−0.00678645	−	0.00391816i	−0.503389	+	0.864060i	$0.667914\pi$
$822$		0			0
$823$	244.849	−	141.364i	0.297508	−	0.171766i	−0.343815	−	0.939037i	$-0.611719\pi$
$823$	244.849	−	141.364i	0.297508	−	0.171766i	0.641323	+	0.767271i	$0.278386\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−223.567			−0.270334			−0.135167	−	0.990823i	$-0.543157\pi$
$827$	−223.567			−0.270334			−0.135167	+	0.990823i	$0.543157\pi$
$828$		0			0
$829$	−1355.63			−1.63526			−0.817631	−	0.575742i	$-0.804713\pi$
$829$	−1355.63			−1.63526			−0.817631	+	0.575742i	$0.804713\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.53688	+	16.5184i	0.0114488	+	0.0198300i
$834$		0			0
$835$	321.103	+	686.998i	0.384554	+	0.822752i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−557.473	−	321.857i	−0.664450	−	0.383620i	0.129521	−	0.991577i	$-0.458656\pi$
$839$	−557.473	−	321.857i	−0.664450	−	0.383620i	−0.793970	+	0.607956i	$0.791990\pi$
$840$		0			0
$841$	371.500	+	643.457i	0.441736	+	0.765109i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−456.048	+	652.548i	−0.539701	+	0.772247i
$846$		0			0
$847$		−	681.790i		−	0.804947i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	444.788	−	256.799i	0.522666	−	0.301761i
$852$		0			0
$853$	835.485	+	482.368i	0.979467	+	0.565495i	0.902109	−	0.431508i	$-0.142018\pi$
$853$	835.485	+	482.368i	0.979467	+	0.565495i	0.0773577	+	0.997003i	$0.475352\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	227.539	−	394.108i	0.265506	−	0.459870i	−0.702190	−	0.711989i	$-0.747794\pi$
$857$	227.539	−	394.108i	0.265506	−	0.459870i	0.967696	+	0.252120i	$0.0811277\pi$
$858$		0			0
$859$	−543.389	−	941.178i	−0.632584	−	1.09567i	−0.987022	−	0.160588i	$-0.948661\pi$
$859$	−543.389	−	941.178i	−0.632584	−	1.09567i	0.354438	−	0.935080i	$-0.384672\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1273.67			1.47586			0.737930	−	0.674877i	$-0.235803\pi$
$863$	1273.67			1.47586			0.737930	+	0.674877i	$0.235803\pi$
$864$		0			0
$865$	891.335	−	1275.39i	1.03045	−	1.47444i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−292.220	+	168.713i	−0.336272	+	0.194147i
$870$		0			0
$871$	166.370	−	288.162i	0.191011	−	0.330841i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	634.063	+	630.661i	0.724644	+	0.720755i
$876$		0			0
$877$	−1012.41	+	584.517i	−1.15441	+	0.666496i	−0.949957	−	0.312381i	$-0.898873\pi$
$877$	−1012.41	+	584.517i	−1.15441	+	0.666496i	−0.204448	+	0.978877i	$0.565540\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	547.812i		−	0.621808i	−0.950441	−	0.310904i	$-0.899368\pi$
$881$		−	547.812i		−	0.621808i	0.950441	−	0.310904i	$-0.100632\pi$
$882$		0			0
$883$		−	494.077i		−	0.559544i	−0.960066	−	0.279772i	$-0.909741\pi$
$883$		−	494.077i		−	0.559544i	0.960066	−	0.279772i	$-0.0902589\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	459.834	+	796.455i	0.518414	+	0.897920i	0.999771	+	0.0213953i	$0.00681087\pi$
$887$	459.834	+	796.455i	0.518414	+	0.897920i	−0.481357	+	0.876525i	$0.659856\pi$
$888$		0			0
$889$	145.593	−	252.174i	0.163771	−	0.283660i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−170.703	+	295.667i	−0.191157	+	0.331094i
$894$		0			0
$895$	−91.0802	+	1051.92i	−0.101766	+	1.17532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1566.93i			1.74297i
$900$		0			0
$901$	−40.4082			−0.0448482
$902$		0			0
$903$		0			0
$904$		0			0
$905$	743.701	+	64.3934i	0.821769	+	0.0711530i
$906$		0			0
$907$	123.093	+	71.0677i	0.135714	+	0.0783547i	0.566320	−	0.824185i	$-0.308367\pi$
$907$	123.093	+	71.0677i	0.135714	+	0.0783547i	−0.430606	+	0.902540i	$0.641700\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.5395	+	7.81706i	0.0148623	+	0.00858074i	0.507413	−	0.861703i	$-0.330602\pi$
$911$	13.5395	+	7.81706i	0.0148623	+	0.00858074i	−0.492550	+	0.870284i	$0.663935\pi$
$912$		0			0
$913$	−633.495	+	365.749i	−0.693861	+	0.400601i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−572.520			−0.624340
$918$		0			0
$919$	−1549.48			−1.68605			−0.843027	−	0.537871i	$-0.819229\pi$
$919$	−1549.48			−1.68605			−0.843027	+	0.537871i	$0.819229\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−158.666	−	274.817i	−0.171902	−	0.297743i
$924$		0			0
$925$	149.843	−	858.806i	0.161992	−	0.928439i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−850.853	−	491.240i	−0.915880	−	0.528784i	−0.0335618	−	0.999437i	$-0.510685\pi$
$929$	−850.853	−	491.240i	−0.915880	−	0.528784i	−0.882318	+	0.470653i	$0.844018\pi$
$930$		0			0
$931$	−22.0549	−	38.2002i	−0.0236895	−	0.0410314i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−181.358	−	126.746i	−0.193966	−	0.135557i
$936$		0			0
$937$			329.526i			0.351682i	0.984419	+	0.175841i	$0.0562645\pi$
$937$			329.526i			0.351682i	−0.984419	+	0.175841i	$0.943735\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−813.967	+	469.944i	−0.865002	+	0.499409i	−0.865684	−	0.500590i	$-0.833116\pi$
$941$	−813.967	+	469.944i	−0.865002	+	0.499409i	0.000681863		1.00000i	$0.499783\pi$
$942$		0			0
$943$	−169.217	−	97.6973i	−0.179445	−	0.103603i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	728.390	−	1261.61i	0.769155	−	1.33222i	−0.168866	−	0.985639i	$-0.554010\pi$
$947$	728.390	−	1261.61i	0.769155	−	1.33222i	0.938021	−	0.346577i	$-0.112656\pi$
$948$		0			0
$949$	36.3722	+	62.9985i	0.0383269	+	0.0663841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1391.14			1.45975			0.729874	−	0.683582i	$-0.239579\pi$
$953$	1391.14			1.45975			0.729874	+	0.683582i	$0.239579\pi$
$954$		0			0
$955$	−99.0377	−	69.2147i	−0.103704	−	0.0724761i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−81.0034	+	46.7674i	−0.0844666	+	0.0487668i
$960$		0			0
$961$	−294.518	+	510.120i	−0.306470	+	0.530822i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1506.80	−	704.278i	1.56145	−	0.729822i
$966$		0			0
$967$	10.3746	−	5.98980i	0.0107287	−	0.00619421i	−0.494626	−	0.869106i	$-0.664695\pi$
$967$	10.3746	−	5.98980i	0.0107287	−	0.00619421i	0.505355	+	0.862912i	$0.331362\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			504.732i			0.519806i	0.965635	+	0.259903i	$0.0836906\pi$
$971$			504.732i			0.519806i	−0.965635	+	0.259903i	$0.916309\pi$
$972$		0			0
$973$		−	1032.88i		−	1.06154i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−264.777	−	458.607i	−0.271010	−	0.469404i	0.698111	−	0.715990i	$-0.254024\pi$
$977$	−264.777	−	458.607i	−0.271010	−	0.469404i	−0.969121	+	0.246586i	$0.920691\pi$
$978$		0			0
$979$	391.779	−	678.581i	0.400183	−	0.693137i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−555.328	+	961.857i	−0.564932	+	0.978491i	0.432124	+	0.901814i	$0.357764\pi$
$983$	−555.328	+	961.857i	−0.564932	+	0.978491i	−0.997056	+	0.0766768i	$0.975569\pi$
$984$		0			0
$985$	1392.63	+	120.581i	1.41384	+	0.122417i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			981.142i			0.992054i
$990$		0			0
$991$	−1048.11			−1.05763			−0.528817	−	0.848736i	$-0.677364\pi$
$991$	−1048.11			−1.05763			−0.528817	+	0.848736i	$0.677364\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−802.922	−	69.5211i	−0.806957	−	0.0698704i
$996$		0			0
$997$	1382.87	+	798.403i	1.38703	+	0.800805i	0.992980	−	0.118282i	$-0.0377387\pi$
$997$	1382.87	+	798.403i	1.38703	+	0.800805i	0.394055	+	0.919087i	$0.371072\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.a.269.4		8
3.2	odd	2		1620.3.t.d.269.1		8
5.4	even	2		1620.3.t.d.269.3		8
9.2	odd	6		540.3.b.a.269.3	✓	4
9.4	even	3	inner	1620.3.t.a.1349.2		8
9.5	odd	6		1620.3.t.d.1349.3		8
9.7	even	3		540.3.b.b.269.2	yes	4
15.14	odd	2	inner	1620.3.t.a.269.2		8
36.7	odd	6		2160.3.c.l.1889.2		4
36.11	even	6		2160.3.c.h.1889.3		4
45.2	even	12		2700.3.g.s.701.7		8
45.4	even	6		1620.3.t.d.1349.1		8
45.7	odd	12		2700.3.g.s.701.8		8
45.14	odd	6	inner	1620.3.t.a.1349.4		8
45.29	odd	6		540.3.b.b.269.1	yes	4
45.34	even	6		540.3.b.a.269.4	yes	4
45.38	even	12		2700.3.g.s.701.1		8
45.43	odd	12		2700.3.g.s.701.2		8
180.79	odd	6		2160.3.c.h.1889.4		4
180.119	even	6		2160.3.c.l.1889.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.a.269.3	✓	4	9.2	odd	6
540.3.b.a.269.4	yes	4	45.34	even	6
540.3.b.b.269.1	yes	4	45.29	odd	6
540.3.b.b.269.2	yes	4	9.7	even	3
1620.3.t.a.269.2		8	15.14	odd	2	inner
1620.3.t.a.269.4		8	1.1	even	1	trivial
1620.3.t.a.1349.2		8	9.4	even	3	inner
1620.3.t.a.1349.4		8	45.14	odd	6	inner
1620.3.t.d.269.1		8	3.2	odd	2
1620.3.t.d.269.3		8	5.4	even	2
1620.3.t.d.1349.1		8	45.4	even	6
1620.3.t.d.1349.3		8	9.5	odd	6
2160.3.c.h.1889.3		4	36.11	even	6
2160.3.c.h.1889.4		4	180.79	odd	6
2160.3.c.l.1889.1		4	180.119	even	6
2160.3.c.l.1889.2		4	36.7	odd	6
2700.3.g.s.701.1		8	45.38	even	12
2700.3.g.s.701.2		8	45.43	odd	12
2700.3.g.s.701.7		8	45.2	even	12
2700.3.g.s.701.8		8	45.7	odd	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)

\(f(q)\)	\(=\)	\(q+(4.98136 + 0.431312i) q^{5} +(6.19588 + 3.57719i) q^{7} +O(q^{10})\)	\(q+(4.98136 + 0.431312i) q^{5} +(6.19588 + 3.57719i) q^{7} +(-4.39061 - 2.53492i) q^{11} +(-2.70791 + 1.56341i) q^{13} +8.72842 q^{17} -20.1852 q^{19} +(7.36421 + 12.7552i) q^{23} +(24.6279 + 4.29704i) q^{25} +(34.4674 + 19.8997i) q^{29} +(19.6852 + 34.0959i) q^{31} +(29.3210 + 20.4917i) q^{35} -34.8712i q^{37} +(-11.4891 + 6.63325i) q^{41} +(57.6908 + 33.3078i) q^{43} +(8.45683 - 14.6477i) q^{47} +(1.09262 + 1.89248i) q^{49} -4.62950 q^{53} +(-20.7779 - 14.5211i) q^{55} +(-22.3208 + 12.8869i) q^{59} +(6.09262 - 10.5527i) q^{61} +(-14.1634 + 6.61997i) q^{65} +(-92.1581 + 53.2075i) q^{67} +101.487i q^{71} -23.2646i q^{73} +(-18.1358 - 31.4121i) q^{77} +(33.2779 - 57.6390i) q^{79} +(72.1421 - 124.954i) q^{83} +(43.4794 + 3.76467i) q^{85} +154.553i q^{89} -22.3705 q^{91} +(-100.550 - 8.70613i) q^{95} +(-152.189 - 87.8664i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{5}+O(q^{10}) \) 8 * q - 3 * q^5	\( 8 q - 3 q^{5} + 12 q^{17} + 12 q^{19} + 30 q^{23} - 9 q^{25} - 16 q^{31} + 90 q^{35} - 48 q^{47} - 78 q^{49} - 384 q^{53} + 94 q^{55} - 38 q^{61} - 138 q^{65} - 174 q^{77} + 6 q^{79} + 288 q^{83} + 100 q^{85} + 168 q^{91} - 318 q^{95}+O(q^{100}) \) 8 * q - 3 * q^5 + 12 * q^17 + 12 * q^19 + 30 * q^23 - 9 * q^25 - 16 * q^31 + 90 * q^35 - 48 * q^47 - 78 * q^49 - 384 * q^53 + 94 * q^55 - 38 * q^61 - 138 * q^65 - 174 * q^77 + 6 * q^79 + 288 * q^83 + 100 * q^85 + 168 * q^91 - 318 * q^95

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