LMFDB - Embedded newform 2736.2.dc.e.449.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(449,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + \cdots + 414864 $ x^20 - 8x^19 - 18x^18 + 232x^17 + 208x^16 - 3152x^15 - 2732x^14 + 24552x^13 + 30728x^12 - 110512x^11 - 211368x^10 + 231024x^9 + 827916x^8 + 179664x^7 - 1551960x^6 - 1894880x^5 + 305564x^4 + 2772832x^3 + 3072904x^2 + 1645728*x + 414864
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.9
Root			$-1.67971 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.e.1889.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.66570 - 1.53904i) q^{5} -1.99260 q^{7} +O(q^{10})$	$q+(2.66570 - 1.53904i) q^{5} -1.99260 q^{7} -1.69919i q^{11} +(-3.28196 - 1.89484i) q^{13} +(4.26035 - 2.45972i) q^{17} +(-4.35882 - 0.0268439i) q^{19} +(-3.60711 - 2.08257i) q^{23} +(2.23732 - 3.87514i) q^{25} +(-2.81633 + 4.87803i) q^{29} +2.51260i q^{31} +(-5.31169 + 3.06671i) q^{35} -5.47497i q^{37} +(-2.32605 - 4.02884i) q^{41} +(-0.0197780 - 0.0342565i) q^{43} +(-4.34785 - 2.51023i) q^{47} -3.02953 q^{49} +(-5.90513 + 10.2280i) q^{53} +(-2.61513 - 4.52954i) q^{55} +(-1.80192 - 3.12101i) q^{59} +(4.19220 - 7.26110i) q^{61} -11.6650 q^{65} +(9.33239 + 5.38806i) q^{67} +(1.22385 + 2.11976i) q^{71} +(-4.06832 - 7.04654i) q^{73} +3.38582i q^{77} +(-13.2735 + 7.66347i) q^{79} -9.21882i q^{83} +(7.57122 - 13.1137i) q^{85} +(3.32271 - 5.75510i) q^{89} +(6.53964 + 3.77567i) q^{91} +(-11.6606 + 6.63685i) q^{95} +(-1.63733 + 0.945311i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{7}+O(q^{10}) $ 20 * q + 4 * q^7	$ 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) $ 20 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 4 * q^19 + 14 * q^25 - 12 * q^35 - 8 * q^41 + 2 * q^43 - 36 * q^47 + 32 * q^49 + 8 * q^53 - 12 * q^55 + 8 * q^59 - 2 * q^61 - 8 * q^65 - 30 * q^67 - 4 * q^71 - 22 * q^73 - 54 * q^79 - 4 * q^85 + 32 * q^89 - 18 * q^91 - 32 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.66570	−	1.53904i	1.19214	−	0.688282i	0.233348	−	0.972393i	$-0.425032\pi$
$5$	2.66570	−	1.53904i	1.19214	−	0.688282i	0.958791	+	0.284112i	$0.0916987\pi$
$6$		0			0
$7$	−1.99260			−0.753134			−0.376567	−	0.926389i	$-0.622895\pi$
$7$	−1.99260			−0.753134			−0.376567	+	0.926389i	$0.622895\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.69919i		−	0.512326i	−0.966634	−	0.256163i	$-0.917542\pi$
$11$		−	1.69919i		−	0.512326i	0.966634	−	0.256163i	$-0.0824583\pi$
$12$		0			0
$13$	−3.28196	−	1.89484i	−0.910251	−	0.525534i	−0.0297394	−	0.999558i	$-0.509468\pi$
$13$	−3.28196	−	1.89484i	−0.910251	−	0.525534i	−0.880512	+	0.474024i	$0.842801\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.26035	−	2.45972i	1.03329	−	0.596569i	0.115362	−	0.993323i	$-0.463197\pi$
$17$	4.26035	−	2.45972i	1.03329	−	0.596569i	0.917925	+	0.396755i	$0.129864\pi$
$18$		0			0
$19$	−4.35882	−	0.0268439i	−0.999981	−	0.00615842i
$19$	−4.35882	−	0.0268439i	−0.999981	−	0.00615842i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.60711	−	2.08257i	−0.752135	−	0.434245i	0.0743299	−	0.997234i	$-0.476318\pi$
$23$	−3.60711	−	2.08257i	−0.752135	−	0.434245i	−0.826465	+	0.562988i	$0.809652\pi$
$24$		0			0
$25$	2.23732	−	3.87514i	0.447463	−	0.775029i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.81633	+	4.87803i	−0.522980	+	0.905827i	0.476663	+	0.879086i	$0.341846\pi$
$29$	−2.81633	+	4.87803i	−0.522980	+	0.905827i	−0.999642	+	0.0267410i	$0.991487\pi$
$30$		0			0
$31$			2.51260i			0.451276i	0.974211	+	0.225638i	$0.0724467\pi$
$31$			2.51260i			0.451276i	−0.974211	+	0.225638i	$0.927553\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.31169	+	3.06671i	−0.897840	+	0.518368i
$36$		0			0
$37$		−	5.47497i		−	0.900079i	−0.893009	−	0.450040i	$-0.851410\pi$
$37$		−	5.47497i		−	0.900079i	0.893009	−	0.450040i	$-0.148590\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.32605	−	4.02884i	−0.363269	−	0.629200i	0.625228	−	0.780442i	$-0.285006\pi$
$41$	−2.32605	−	4.02884i	−0.363269	−	0.629200i	−0.988497	+	0.151242i	$0.951673\pi$
$42$		0			0
$43$	−0.0197780	−	0.0342565i	−0.00301611	−	0.00522406i	0.864513	−	0.502610i	$-0.167627\pi$
$43$	−0.0197780	−	0.0342565i	−0.00301611	−	0.00522406i	−0.867530	+	0.497386i	$0.834293\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.34785	−	2.51023i	−0.634199	−	0.366155i	0.148177	−	0.988961i	$-0.452659\pi$
$47$	−4.34785	−	2.51023i	−0.634199	−	0.366155i	−0.782376	+	0.622806i	$0.785993\pi$
$48$		0			0
$49$	−3.02953			−0.432790
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.90513	+	10.2280i	−0.811132	+	1.40492i	0.100940	+	0.994893i	$0.467815\pi$
$53$	−5.90513	+	10.2280i	−0.811132	+	1.40492i	−0.912072	+	0.410030i	$0.865518\pi$
$54$		0			0
$55$	−2.61513	−	4.52954i	−0.352624	−	0.610763i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.80192	−	3.12101i	−0.234590	−	0.406321i	0.724564	−	0.689208i	$-0.242041\pi$
$59$	−1.80192	−	3.12101i	−0.234590	−	0.406321i	−0.959153	+	0.282887i	$0.908708\pi$
$60$		0			0
$61$	4.19220	−	7.26110i	0.536756	−	0.929689i	−0.462320	−	0.886713i	$-0.652983\pi$
$61$	4.19220	−	7.26110i	0.536756	−	0.929689i	0.999076	−	0.0429757i	$-0.0136838\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−11.6650			−1.44686
$66$		0			0
$67$	9.33239	+	5.38806i	1.14013	+	0.658256i	0.946464	−	0.322811i	$-0.104628\pi$
$67$	9.33239	+	5.38806i	1.14013	+	0.658256i	0.193670	+	0.981067i	$0.437961\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.22385	+	2.11976i	0.145244	+	0.251570i	0.929464	−	0.368913i	$-0.120270\pi$
$71$	1.22385	+	2.11976i	0.145244	+	0.251570i	−0.784220	+	0.620483i	$0.786937\pi$
$72$		0			0
$73$	−4.06832	−	7.04654i	−0.476161	−	0.824735i	0.523466	−	0.852047i	$-0.324639\pi$
$73$	−4.06832	−	7.04654i	−0.476161	−	0.824735i	−0.999627	+	0.0273116i	$0.991305\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.38582i			0.385850i
$78$		0			0
$79$	−13.2735	+	7.66347i	−1.49339	+	0.862208i	−0.999971	−	0.00758495i	$-0.997586\pi$
$79$	−13.2735	+	7.66347i	−1.49339	+	0.862208i	−0.493417	+	0.869793i	$0.664252\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	9.21882i		−	1.01190i	−0.862564	−	0.505949i	$-0.831143\pi$
$83$		−	9.21882i		−	1.01190i	0.862564	−	0.505949i	$-0.168857\pi$
$84$		0			0
$85$	7.57122	−	13.1137i	0.821214	−	1.42238i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.32271	−	5.75510i	0.352206	−	0.610039i	−0.634429	−	0.772981i	$-0.718765\pi$
$89$	3.32271	−	5.75510i	0.352206	−	0.610039i	0.986636	+	0.162941i	$0.0520982\pi$
$90$		0			0
$91$	6.53964	+	3.77567i	0.685541	+	0.395797i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.6606	+	6.63685i	−1.19635	+	0.680927i
$96$		0			0
$97$	−1.63733	+	0.945311i	−0.166245	+	0.0959817i	−0.580814	−	0.814036i	$-0.697266\pi$
$97$	−1.63733	+	0.945311i	−0.166245	+	0.0959817i	0.414569	+	0.910018i	$0.363932\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.4015	+	7.16001i	1.23400	+	0.712448i	0.967860	−	0.251488i	$-0.0809200\pi$
$101$	12.4015	+	7.16001i	1.23400	+	0.712448i	0.266135	+	0.963936i	$0.414253\pi$
$102$		0			0
$103$		−	10.2832i		−	1.01323i	−0.862172	−	0.506615i	$-0.830896\pi$
$103$		−	10.2832i		−	1.01323i	0.862172	−	0.506615i	$-0.169104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.9921			−1.35266			−0.676331	−	0.736597i	$-0.736431\pi$
$107$	−13.9921			−1.35266			−0.676331	+	0.736597i	$0.736431\pi$
$108$		0			0
$109$	3.55880	−	2.05468i	0.340872	−	0.196802i	−0.319786	−	0.947490i	$-0.603611\pi$
$109$	3.55880	−	2.05468i	0.340872	−	0.196802i	0.660657	+	0.750688i	$0.270278\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.151418			0.0142442			0.00712211	−	0.999975i	$-0.497733\pi$
$113$	0.151418			0.0142442			0.00712211	+	0.999975i	$0.497733\pi$
$114$		0			0
$115$	−12.8207			−1.19553
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.48919	+	4.90124i	−0.778203	+	0.449296i
$120$		0			0
$121$	8.11275			0.737522
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.61713i			0.144641i
$126$		0			0
$127$	17.0756	+	9.85863i	1.51522	+	0.874812i	0.999841	+	0.0178489i	$0.00568179\pi$
$127$	17.0756	+	9.85863i	1.51522	+	0.874812i	0.515378	+	0.856963i	$0.327652\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.4457	+	6.03080i	−0.912642	+	0.526914i	−0.881280	−	0.472594i	$-0.843318\pi$
$131$	−10.4457	+	6.03080i	−0.912642	+	0.526914i	−0.0313614	+	0.999508i	$0.509984\pi$
$132$		0			0
$133$	8.68540	+	0.0534893i	0.753119	+	0.00463811i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.6665	−	8.46769i	−1.25304	−	0.723444i	−0.281329	−	0.959611i	$-0.590775\pi$
$137$	−14.6665	−	8.46769i	−1.25304	−	0.723444i	−0.971712	+	0.236168i	$0.924108\pi$
$138$		0			0
$139$	5.77539	−	10.0033i	0.489862	−	0.848467i	−0.510070	−	0.860133i	$-0.670380\pi$
$139$	5.77539	−	10.0033i	0.489862	−	0.848467i	0.999932	+	0.0116666i	$0.00371367\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.21970	+	5.57668i	−0.269245	+	0.466345i
$144$		0			0
$145$			17.3378i			1.43983i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.1747	+	9.33846i	−1.32508	+	0.765037i	−0.984535	−	0.175190i	$-0.943946\pi$
$149$	−16.1747	+	9.33846i	−1.32508	+	0.765037i	−0.340548	+	0.940227i	$0.610613\pi$
$150$		0			0
$151$		−	22.8345i		−	1.85824i	−0.369773	−	0.929122i	$-0.620565\pi$
$151$		−	22.8345i		−	1.85824i	0.369773	−	0.929122i	$-0.379435\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.86700	+	6.69784i	0.310605	+	0.537984i
$156$		0			0
$157$	−1.76719	−	3.06087i	−0.141037	−	0.244284i	0.786850	−	0.617144i	$-0.211710\pi$
$157$	−1.76719	−	3.06087i	−0.141037	−	0.244284i	−0.927888	+	0.372860i	$0.878377\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.18755	+	4.14973i	0.566458	+	0.327045i
$162$		0			0
$163$	1.05334			0.0825038			0.0412519	−	0.999149i	$-0.486865\pi$
$163$	1.05334			0.0825038			0.0412519	+	0.999149i	$0.486865\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.48242	−	6.03173i	0.269478	−	0.466749i	−0.699249	−	0.714878i	$-0.746482\pi$
$167$	3.48242	−	6.03173i	0.269478	−	0.466749i	0.968727	+	0.248129i	$0.0798156\pi$
$168$		0			0
$169$	0.680833	+	1.17924i	0.0523718	+	0.0907105i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.534578	+	0.925915i	0.0406432	+	0.0703960i	0.885631	−	0.464389i	$-0.153726\pi$
$173$	0.534578	+	0.925915i	0.0406432	+	0.0703960i	−0.844988	+	0.534785i	$0.820393\pi$
$174$		0			0
$175$	−4.45808	+	7.72163i	−0.337000	+	0.583700i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.91235			0.740884			0.370442	−	0.928856i	$-0.379206\pi$
$179$	9.91235			0.740884			0.370442	+	0.928856i	$0.379206\pi$
$180$		0			0
$181$	3.60095	+	2.07901i	0.267657	+	0.154532i	0.627822	−	0.778357i	$-0.283946\pi$
$181$	3.60095	+	2.07901i	0.267657	+	0.154532i	−0.360165	+	0.932888i	$0.617280\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.42622	−	14.5946i	−0.619508	−	1.07302i
$186$		0			0
$187$	−4.17953	−	7.23916i	−0.305637	−	0.529379i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			16.0696i			1.16276i	0.813633	+	0.581378i	$0.197486\pi$
$191$			16.0696i			1.16276i	−0.813633	+	0.581378i	$0.802514\pi$
$192$		0			0
$193$	−6.18003	+	3.56804i	−0.444849	+	0.256833i	−0.705652	−	0.708558i	$-0.749346\pi$
$193$	−6.18003	+	3.56804i	−0.444849	+	0.256833i	0.260804	+	0.965392i	$0.416013\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.05313i		−	0.217527i	−0.994068	−	0.108763i	$-0.965311\pi$
$197$		−	3.05313i		−	0.217527i	0.994068	−	0.108763i	$-0.0346890\pi$
$198$		0			0
$199$	8.37119	−	14.4993i	0.593418	−	1.02783i	−0.400350	−	0.916362i	$-0.631111\pi$
$199$	8.37119	−	14.4993i	0.593418	−	1.02783i	0.993768	−	0.111468i	$-0.0355553\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.61183	−	9.71998i	0.393874	−	0.682209i
$204$		0			0
$205$	−12.4011	−	7.15980i	−0.866133	−	0.500062i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.0456130	+	7.40647i	−0.00315512	+	0.512316i
$210$		0			0
$211$	7.63524	−	4.40821i	0.525631	−	0.303473i	−0.213604	−	0.976920i	$-0.568520\pi$
$211$	7.63524	−	4.40821i	0.525631	−	0.303473i	0.739236	+	0.673447i	$0.235187\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.105444	−	0.0608784i	−0.00719125	−	0.00415187i
$216$		0			0
$217$		−	5.00661i		−	0.339871i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.6431			−1.25407
$222$		0			0
$223$	−9.43800	+	5.44903i	−0.632015	+	0.364894i	−0.781532	−	0.623865i	$-0.785562\pi$
$223$	−9.43800	+	5.44903i	−0.632015	+	0.364894i	0.149517	+	0.988759i	$0.452228\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.53821			0.102094			0.0510472	−	0.998696i	$-0.483744\pi$
$227$	1.53821			0.102094			0.0510472	+	0.998696i	$0.483744\pi$
$228$		0			0
$229$	−15.9880			−1.05652			−0.528259	−	0.849083i	$-0.677155\pi$
$229$	−15.9880			−1.05652			−0.528259	+	0.849083i	$0.677155\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.9776	−	8.06999i	0.915706	−	0.528683i	0.0334432	−	0.999441i	$-0.489353\pi$
$233$	13.9776	−	8.06999i	0.915706	−	0.528683i	0.882263	+	0.470758i	$0.156019\pi$
$234$		0			0
$235$	−15.4534			−1.00807
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	21.2980i		−	1.37765i	−0.724927	−	0.688826i	$-0.758127\pi$
$239$		−	21.2980i		−	1.37765i	0.724927	−	0.688826i	$-0.241873\pi$
$240$		0			0
$241$	−9.30003	−	5.36937i	−0.599067	−	0.345872i	0.169607	−	0.985512i	$-0.445750\pi$
$241$	−9.30003	−	5.36937i	−0.599067	−	0.345872i	−0.768675	+	0.639640i	$0.779083\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.07582	+	4.66258i	−0.515945	+	0.297881i
$246$		0			0
$247$	14.2546	+	8.34736i	0.906998	+	0.531130i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.0224	+	5.78644i	0.632609	+	0.365237i	0.781762	−	0.623577i	$-0.214321\pi$
$251$	10.0224	+	5.78644i	0.632609	+	0.365237i	−0.149153	+	0.988814i	$0.547655\pi$
$252$		0			0
$253$	−3.53868	+	6.12918i	−0.222475	+	0.385338i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.53936	−	13.0585i	0.470292	−	0.814570i	−0.529131	−	0.848540i	$-0.677482\pi$
$257$	7.53936	−	13.0585i	0.470292	−	0.814570i	0.999423	+	0.0339705i	$0.0108152\pi$
$258$		0			0
$259$			10.9094i			0.677880i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.9126	−	10.9192i	1.16620	−	0.673308i	0.213421	−	0.976960i	$-0.431539\pi$
$263$	18.9126	−	10.9192i	1.16620	−	0.673308i	0.952783	+	0.303652i	$0.0982061\pi$
$264$		0			0
$265$			36.3530i			2.23315i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.544537	+	0.943165i	0.0332010	+	0.0575058i	0.882148	−	0.470971i	$-0.156097\pi$
$269$	0.544537	+	0.943165i	0.0332010	+	0.0575058i	−0.848947	+	0.528477i	$0.822763\pi$
$270$		0			0
$271$	0.978115	+	1.69414i	0.0594162	+	0.102912i	0.894204	−	0.447661i	$-0.147743\pi$
$271$	0.978115	+	1.69414i	0.0594162	+	0.102912i	−0.834787	+	0.550573i	$0.814409\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.58461	−	3.80163i	−0.397067	−	0.229247i
$276$		0			0
$277$	−22.1790			−1.33261			−0.666304	−	0.745680i	$-0.732125\pi$
$277$	−22.1790			−1.33261			−0.666304	+	0.745680i	$0.732125\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.51462	+	16.4798i	−0.567595	+	0.983103i	0.429209	+	0.903205i	$0.358793\pi$
$281$	−9.51462	+	16.4798i	−0.567595	+	0.983103i	−0.996803	+	0.0798973i	$0.974541\pi$
$282$		0			0
$283$	−16.1263	−	27.9316i	−0.958611	−	1.66036i	−0.725880	−	0.687822i	$-0.758567\pi$
$283$	−16.1263	−	27.9316i	−0.958611	−	1.66036i	−0.232731	−	0.972541i	$-0.574766\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.63491	+	8.02789i	0.273590	+	0.473872i
$288$		0			0
$289$	3.60040	−	6.23607i	0.211788	−	0.366828i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−25.7934			−1.50687			−0.753435	−	0.657523i	$-0.771604\pi$
$293$	−25.7934			−1.50687			−0.753435	+	0.657523i	$0.771604\pi$
$294$		0			0
$295$	−9.60675	−	5.54646i	−0.559327	−	0.322927i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.89226	+	13.6698i	0.456421	+	0.790545i
$300$		0			0
$301$	0.0394097	+	0.0682596i	0.00227154	+	0.00393442i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	25.8079i		−	1.47776i
$306$		0			0
$307$	10.8935	−	6.28939i	0.621727	−	0.358954i	−0.155814	−	0.987786i	$-0.549800\pi$
$307$	10.8935	−	6.28939i	0.621727	−	0.358954i	0.777541	+	0.628832i	$0.216467\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	26.1395i		−	1.48224i	−0.671374	−	0.741119i	$-0.734296\pi$
$311$		−	26.1395i		−	1.48224i	0.671374	−	0.741119i	$-0.265704\pi$
$312$		0			0
$313$	8.19557	−	14.1951i	0.463241	−	0.802357i	−0.535879	−	0.844295i	$-0.680020\pi$
$313$	8.19557	−	14.1951i	0.463241	−	0.802357i	0.999120	+	0.0419376i	$0.0133531\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.565432	−	0.979356i	0.0317578	−	0.0550061i	−0.849710	−	0.527251i	$-0.823223\pi$
$317$	0.565432	−	0.979356i	0.0317578	−	0.0550061i	0.881467	+	0.472245i	$0.156556\pi$
$318$		0			0
$319$	8.28871	+	4.78549i	0.464079	+	0.267936i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.6361	+	10.6071i	−1.03694	+	0.590194i
$324$		0			0
$325$	−14.6856	+	8.47871i	−0.814608	+	0.470314i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.66354	+	5.00190i	0.477637	+	0.275764i
$330$		0			0
$331$		−	32.8398i		−	1.80504i	−0.430651	−	0.902519i	$-0.641716\pi$
$331$		−	32.8398i		−	1.80504i	0.430651	−	0.902519i	$-0.358284\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	33.1699			1.81226
$336$		0			0
$337$	25.7677	−	14.8770i	1.40366	−	0.810401i	0.408890	−	0.912584i	$-0.365916\pi$
$337$	25.7677	−	14.8770i	1.40366	−	0.810401i	0.994766	+	0.102183i	$0.0325826\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.26939			0.231200
$342$		0			0
$343$	19.9849			1.07908
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.8009	−	12.5867i	1.17033	−	0.675692i	0.216574	−	0.976266i	$-0.430512\pi$
$347$	21.8009	−	12.5867i	1.17033	−	0.675692i	0.953758	+	0.300575i	$0.0971784\pi$
$348$		0			0
$349$	−29.6760			−1.58852			−0.794259	−	0.607579i	$-0.792141\pi$
$349$	−29.6760			−1.58852			−0.794259	+	0.607579i	$0.792141\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5.48254i		−	0.291806i	−0.989299	−	0.145903i	$-0.953391\pi$
$353$		−	5.48254i		−	0.291806i	0.989299	−	0.145903i	$-0.0466087\pi$
$354$		0			0
$355$	6.52482	+	3.76711i	0.346302	+	0.199937i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.4217	+	12.3678i	−1.13059	+	0.652749i	−0.944084	−	0.329705i	$-0.893051\pi$
$359$	−21.4217	+	12.3678i	−1.13059	+	0.652749i	−0.186509	+	0.982453i	$0.559718\pi$
$360$		0			0
$361$	18.9986	+	0.234015i	0.999924	+	0.0123166i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.6899	−	12.5227i	−1.13530	−	0.655466i
$366$		0			0
$367$	0.674890	−	1.16894i	0.0352290	−	0.0610184i	−0.847873	−	0.530199i	$-0.822117\pi$
$367$	0.674890	−	1.16894i	0.0352290	−	0.0610184i	0.883102	+	0.469180i	$0.155451\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.7666	−	20.3803i	0.610891	−	1.05809i
$372$		0			0
$373$			20.4342i			1.05804i	0.848608	+	0.529022i	$0.177441\pi$
$373$			20.4342i			1.05804i	−0.848608	+	0.529022i	$0.822559\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.4862	−	10.6730i	0.952086	−	0.549687i
$378$		0			0
$379$			21.3835i			1.09840i	0.835692	+	0.549198i	$0.185067\pi$
$379$			21.3835i			1.09840i	−0.835692	+	0.549198i	$0.814933\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.82045	+	3.15312i	0.0930209	+	0.161117i	0.908781	−	0.417274i	$-0.137014\pi$
$383$	1.82045	+	3.15312i	0.0930209	+	0.161117i	−0.815760	+	0.578391i	$0.803681\pi$
$384$		0			0
$385$	5.21092	+	9.02558i	0.265573	+	0.459986i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	29.4944	+	17.0286i	1.49543	+	0.863386i	0.999986	−	0.00525546i	$-0.00167287\pi$
$389$	29.4944	+	17.0286i	1.49543	+	0.863386i	0.495442	+	0.868641i	$0.335006\pi$
$390$		0			0
$391$	−20.4901			−1.03623
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.5888	+	40.8571i	−1.18688	+	2.05574i
$396$		0			0
$397$	14.0335	+	24.3067i	0.704321	+	1.21992i	0.966936	+	0.255019i	$0.0820818\pi$
$397$	14.0335	+	24.3067i	0.704321	+	1.21992i	−0.262615	+	0.964901i	$0.584585\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.60679	−	2.78305i	−0.0802394	−	0.138979i	0.823113	−	0.567877i	$-0.192235\pi$
$401$	−1.60679	−	2.78305i	−0.0802394	−	0.138979i	−0.903353	+	0.428898i	$0.858902\pi$
$402$		0			0
$403$	4.76097	−	8.24624i	0.237161	−	0.410775i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.30302			−0.461134
$408$		0			0
$409$	15.4161	+	8.90051i	0.762279	+	0.440102i	0.830113	−	0.557595i	$-0.188276\pi$
$409$	15.4161	+	8.90051i	0.762279	+	0.440102i	−0.0678347	+	0.997697i	$0.521609\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.59051	+	6.21894i	0.176677	+	0.306014i
$414$		0			0
$415$	−14.1882	−	24.5746i	−0.696470	−	1.20632i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			31.0864i			1.51867i	0.650701	+	0.759334i	$0.274475\pi$
$419$			31.0864i			1.51867i	−0.650701	+	0.759334i	$0.725525\pi$
$420$		0			0
$421$	−26.5471	+	15.3270i	−1.29382	+	0.746990i	−0.979330	−	0.202269i	$-0.935168\pi$
$421$	−26.5471	+	15.3270i	−1.29382	+	0.746990i	−0.314495	+	0.949259i	$0.601835\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	22.0126i		−	1.06777i
$426$		0			0
$427$	−8.35339	+	14.4685i	−0.404249	+	0.700180i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.61118	+	6.25474i	−0.173944	+	0.301280i	−0.939795	−	0.341738i	$-0.888985\pi$
$431$	−3.61118	+	6.25474i	−0.173944	+	0.301280i	0.765851	+	0.643018i	$0.222318\pi$
$432$		0			0
$433$	11.2525	+	6.49662i	0.540760	+	0.312208i	0.745387	−	0.666632i	$-0.232265\pi$
$433$	11.2525	+	6.49662i	0.540760	+	0.312208i	−0.204627	+	0.978840i	$0.565598\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.6668	+	9.17436i	0.749446	+	0.438869i
$438$		0			0
$439$	26.3040	−	15.1866i	1.25542	−	0.724819i	0.283241	−	0.959049i	$-0.408590\pi$
$439$	26.3040	−	15.1866i	1.25542	−	0.724819i	0.972181	+	0.234230i	$0.0752569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.9178	+	6.30337i	0.518718	+	0.299482i	0.736410	−	0.676536i	$-0.236519\pi$
$443$	10.9178	+	6.30337i	0.518718	+	0.299482i	−0.217692	+	0.976018i	$0.569853\pi$
$444$		0			0
$445$		−	20.4552i		−	0.969669i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.9409			0.893874			0.446937	−	0.894565i	$-0.352515\pi$
$449$	18.9409			0.893874			0.446937	+	0.894565i	$0.352515\pi$
$450$		0			0
$451$	−6.84578	+	3.95241i	−0.322355	+	0.186112i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	23.2437			1.08968
$456$		0			0
$457$	−9.84685			−0.460616			−0.230308	−	0.973118i	$-0.573973\pi$
$457$	−9.84685			−0.460616			−0.230308	+	0.973118i	$0.573973\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.0131	+	14.4413i	−1.16498	+	0.672600i	−0.952492	−	0.304564i	$-0.901489\pi$
$461$	−25.0131	+	14.4413i	−1.16498	+	0.672600i	−0.212486	+	0.977164i	$0.568156\pi$
$462$		0			0
$463$	9.04717			0.420458			0.210229	−	0.977652i	$-0.432579\pi$
$463$	9.04717			0.420458			0.210229	+	0.977652i	$0.432579\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	29.2536i		−	1.35370i	−0.736123	−	0.676848i	$-0.763346\pi$
$467$		−	29.2536i		−	1.35370i	0.736123	−	0.676848i	$-0.236654\pi$
$468$		0			0
$469$	−18.5958	−	10.7363i	−0.858673	−	0.495755i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.0582083	+	0.0336066i	−0.00267642	+	0.00154523i
$474$		0			0
$475$	−9.85607	+	16.8310i	−0.452228	+	0.772258i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.628028	+	0.362592i	0.0286953	+	0.0165672i	0.514279	−	0.857623i	$-0.328060\pi$
$479$	0.628028	+	0.362592i	0.0286953	+	0.0165672i	−0.485584	+	0.874190i	$0.661393\pi$
$480$		0			0
$481$	−10.3742	+	17.9686i	−0.473022	+	0.819298i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.90975	+	5.03983i	−0.132125	+	0.228847i
$486$		0			0
$487$			10.6245i			0.481440i	0.970595	+	0.240720i	$0.0773836\pi$
$487$			10.6245i			0.481440i	−0.970595	+	0.240720i	$0.922616\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.66767	−	5.58163i	0.436296	−	0.251895i	−0.265729	−	0.964048i	$-0.585613\pi$
$491$	9.66767	−	5.58163i	0.436296	−	0.251895i	0.702025	+	0.712152i	$0.252279\pi$
$492$		0			0
$493$			27.7095i			1.24797i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.43864	−	4.22385i	−0.109388	−	0.189466i
$498$		0			0
$499$	9.24877	+	16.0193i	0.414032	+	0.717124i	0.995326	−	0.0965690i	$-0.0307868\pi$
$499$	9.24877	+	16.0193i	0.414032	+	0.717124i	−0.581294	+	0.813693i	$0.697453\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.5452	+	6.66564i	0.514776	+	0.297206i	0.734795	−	0.678289i	$-0.237278\pi$
$503$	11.5452	+	6.66564i	0.514776	+	0.297206i	−0.220018	+	0.975496i	$0.570612\pi$
$504$		0			0
$505$	44.0783			1.96146
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.10049	+	3.63816i	−0.0931028	+	0.161259i	−0.908815	−	0.417199i	$-0.863012\pi$
$509$	−2.10049	+	3.63816i	−0.0931028	+	0.161259i	0.815712	+	0.578458i	$0.196345\pi$
$510$		0			0
$511$	8.10655	+	14.0410i	0.358613	+	0.621136i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.8263	−	27.4119i	−0.697388	−	1.20791i
$516$		0			0
$517$	−4.26537	+	7.38783i	−0.187591	+	0.324916i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	25.1752			1.10295			0.551474	−	0.834192i	$-0.314066\pi$
$521$	25.1752			1.10295			0.551474	+	0.834192i	$0.314066\pi$
$522$		0			0
$523$	5.97759	+	3.45116i	0.261382	+	0.150909i	0.624965	−	0.780653i	$-0.285113\pi$
$523$	5.97759	+	3.45116i	0.261382	+	0.150909i	−0.363583	+	0.931562i	$0.618447\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.18028	+	10.7046i	0.269217	+	0.466298i
$528$		0			0
$529$	−2.82583	−	4.89448i	−0.122862	−	0.212804i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			17.6300i			0.763640i
$534$		0			0
$535$	−37.2987	+	21.5344i	−1.61256	+	0.931013i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			5.14775i			0.221729i
$540$		0			0
$541$	7.27614	−	12.6026i	0.312826	−	0.541830i	−0.666147	−	0.745820i	$-0.732058\pi$
$541$	7.27614	−	12.6026i	0.312826	−	0.541830i	0.978973	+	0.203990i	$0.0653911\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.32448	−	10.9543i	0.270911	−	0.469231i
$546$		0			0
$547$	37.5739	+	21.6933i	1.60655	+	0.927539i	0.990135	+	0.140114i	$0.0447469\pi$
$547$	37.5739	+	21.6933i	1.60655	+	0.927539i	0.616410	+	0.787425i	$0.288586\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.4068	−	21.1868i	0.528548	−	0.902589i
$552$		0			0
$553$	26.4489	−	15.2703i	1.12472	−	0.649358i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.1990	+	11.6619i	0.855861	+	0.494131i	0.862624	−	0.505846i	$-0.168819\pi$
$557$	20.1990	+	11.6619i	0.855861	+	0.494131i	−0.00676316	+	0.999977i	$0.502153\pi$
$558$		0			0
$559$			0.149904i			0.00634028i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.6564			−0.954854			−0.477427	−	0.878671i	$-0.658430\pi$
$563$	−22.6564			−0.954854			−0.477427	+	0.878671i	$0.658430\pi$
$564$		0			0
$565$	0.403636	−	0.233039i	0.0169811	−	0.00980403i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.9710			−1.04684			−0.523420	−	0.852075i	$-0.675344\pi$
$569$	−24.9710			−1.04684			−0.523420	+	0.852075i	$0.675344\pi$
$570$		0			0
$571$	29.4002			1.23036			0.615179	−	0.788387i	$-0.289084\pi$
$571$	29.4002			1.23036			0.615179	+	0.788387i	$0.289084\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.1405	+	9.31872i	−0.673105	+	0.388617i
$576$		0			0
$577$	30.8631			1.28485			0.642424	−	0.766349i	$-0.277929\pi$
$577$	30.8631			1.28485			0.642424	+	0.766349i	$0.277929\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			18.3695i			0.762094i
$582$		0			0
$583$	17.3793	+	10.0340i	0.719778	+	0.415564i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.4415	−	14.1113i	1.00881	−	0.582437i	0.0979673	−	0.995190i	$-0.468766\pi$
$587$	24.4415	−	14.1113i	1.00881	−	0.582437i	0.910843	+	0.412753i	$0.135433\pi$
$588$		0			0
$589$	0.0674480	−	10.9520i	0.00277915	−	0.451267i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20.5510	+	11.8651i	0.843927	+	0.487241i	0.858597	−	0.512651i	$-0.171337\pi$
$593$	20.5510	+	11.8651i	0.843927	+	0.487241i	−0.0146705	+	0.999892i	$0.504670\pi$
$594$		0			0
$595$	−15.0864	+	26.1305i	−0.618484	+	1.07125i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.74748	−	15.1511i	0.357412	−	0.619056i	−0.630115	−	0.776501i	$-0.716992\pi$
$599$	8.74748	−	15.1511i	0.357412	−	0.619056i	0.987528	+	0.157445i	$0.0503258\pi$
$600$		0			0
$601$		−	5.56507i		−	0.227004i	−0.993538	−	0.113502i	$-0.963793\pi$
$601$		−	5.56507i		−	0.227004i	0.993538	−	0.113502i	$-0.0362068\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.6262	−	12.4859i	0.879229	−	0.507623i
$606$		0			0
$607$			28.5045i			1.15696i	0.815695	+	0.578482i	$0.196355\pi$
$607$			28.5045i			1.15696i	−0.815695	+	0.578482i	$0.803645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.51297	+	16.4770i	0.384854	+	0.666586i
$612$		0			0
$613$	−12.9752	−	22.4736i	−0.524062	−	0.907701i	−0.999608	−	0.0280104i	$-0.991083\pi$
$613$	−12.9752	−	22.4736i	−0.524062	−	0.907701i	0.475546	−	0.879691i	$-0.342250\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.3443	+	19.8287i	1.38265	+	0.798273i	0.992473	−	0.122468i	$-0.0390808\pi$
$617$	34.3443	+	19.8287i	1.38265	+	0.798273i	0.390176	+	0.920740i	$0.372414\pi$
$618$		0			0
$619$	−29.0421			−1.16730			−0.583651	−	0.812005i	$-0.698376\pi$
$619$	−29.0421			−1.16730			−0.583651	+	0.812005i	$0.698376\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.62084	+	11.4676i	−0.265259	+	0.459441i
$624$		0			0
$625$	13.6754	+	23.6865i	0.547017	+	0.947461i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.4669	−	23.3253i	−0.536959	−	0.930040i
$630$		0			0
$631$	−3.89172	+	6.74065i	−0.154927	+	0.268341i	−0.933032	−	0.359792i	$-0.882848\pi$
$631$	−3.89172	+	6.74065i	−0.154927	+	0.268341i	0.778106	+	0.628134i	$0.216181\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	60.6915			2.40847
$636$		0			0
$637$	9.94278	+	5.74047i	0.393947	+	0.227446i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.8762	+	22.3022i	0.508579	+	0.880884i	0.999951	+	0.00993439i	$0.00316227\pi$
$641$	12.8762	+	22.3022i	0.508579	+	0.880884i	−0.491372	+	0.870950i	$0.663504\pi$
$642$		0			0
$643$	20.2168	+	35.0166i	0.797273	+	1.38092i	0.921386	+	0.388650i	$0.127058\pi$
$643$	20.2168	+	35.0166i	0.797273	+	1.38092i	−0.124112	+	0.992268i	$0.539608\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.50801i			0.295170i	0.989049	+	0.147585i	$0.0471501\pi$
$647$			7.50801i			0.295170i	−0.989049	+	0.147585i	$0.952850\pi$
$648$		0			0
$649$	−5.30320	+	3.06180i	−0.208169	+	0.120186i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.73149i			0.0677584i	0.999426	+	0.0338792i	$0.0107861\pi$
$653$			1.73149i			0.0677584i	−0.999426	+	0.0338792i	$0.989214\pi$
$654$		0			0
$655$	−18.5634	+	32.1527i	−0.725330	+	1.25631i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.54026	+	2.66781i	−0.0600001	+	0.103923i	−0.894465	−	0.447138i	$-0.852443\pi$
$659$	−1.54026	+	2.66781i	−0.0600001	+	0.103923i	0.834465	+	0.551061i	$0.185777\pi$
$660$		0			0
$661$	17.9794	+	10.3804i	0.699316	+	0.403750i	0.807093	−	0.590425i	$-0.201040\pi$
$661$	17.9794	+	10.3804i	0.699316	+	0.403750i	−0.107776	+	0.994175i	$0.534373\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.2350	−	13.2246i	0.901015	−	0.512829i
$666$		0			0
$667$	20.3176	−	11.7304i	0.786702	−	0.454203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.3380	−	7.12335i	−0.476304	−	0.274994i
$672$		0			0
$673$		−	30.7209i		−	1.18421i	−0.805863	−	0.592103i	$-0.798298\pi$
$673$		−	30.7209i		−	1.18421i	0.805863	−	0.592103i	$-0.201702\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.55398			0.0981573			0.0490786	−	0.998795i	$-0.484372\pi$
$677$	2.55398			0.0981573			0.0490786	+	0.998795i	$0.484372\pi$
$678$		0			0
$679$	3.26254	−	1.88363i	0.125205	−	0.0722871i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	23.7994			0.910657			0.455329	−	0.890323i	$-0.349522\pi$
$683$	23.7994			0.910657			0.455329	+	0.890323i	$0.349522\pi$
$684$		0			0
$685$	−52.1286			−1.99173
$686$		0			0
$687$		0			0
$688$		0			0
$689$	38.7608	−	22.3786i	1.47667	−	0.852555i
$690$		0			0
$691$	−24.7626			−0.942014			−0.471007	−	0.882130i	$-0.656109\pi$
$691$	−24.7626			−0.942014			−0.471007	+	0.882130i	$0.656109\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	35.5543i		−	1.34865i
$696$		0			0
$697$	−19.8196	−	11.4429i	−0.750722	−	0.433429i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−40.2721	+	23.2511i	−1.52106	+	0.878183i	−0.521367	+	0.853333i	$0.674578\pi$
$701$	−40.2721	+	23.2511i	−1.52106	+	0.878183i	−0.999691	+	0.0248507i	$0.992089\pi$
$702$		0			0
$703$	−0.146970	+	23.8644i	−0.00554306	+	0.900062i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.7113	−	14.2671i	−0.929363	−	0.536568i
$708$		0			0
$709$	13.4733	−	23.3364i	0.506000	−	0.876417i	−0.493976	−	0.869475i	$-0.664457\pi$
$709$	13.4733	−	23.3364i	0.506000	−	0.876417i	0.999976	−	0.00694154i	$-0.00220958\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.23265	−	9.06322i	0.195964	−	0.339420i
$714$		0			0
$715$			19.8210i			0.741264i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.34462	+	4.81777i	−0.311202	+	0.179672i	−0.647464	−	0.762096i	$-0.724170\pi$
$719$	−8.34462	+	4.81777i	−0.311202	+	0.179672i	0.336262	+	0.941768i	$0.390837\pi$
$720$		0			0
$721$			20.4903i			0.763098i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.6020	+	21.8274i	0.468028	+	0.810649i
$726$		0			0
$727$	−3.12517	−	5.41296i	−0.115906	−	0.200756i	0.802235	−	0.597008i	$-0.203644\pi$
$727$	−3.12517	−	5.41296i	−0.115906	−	0.200756i	−0.918142	+	0.396252i	$0.870311\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.168522	−	0.0972964i	−0.00623302	−	0.00359864i
$732$		0			0
$733$	30.5368			1.12790			0.563950	−	0.825809i	$-0.309281\pi$
$733$	30.5368			1.12790			0.563950	+	0.825809i	$0.309281\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.15535	−	15.8575i	0.337242	−	0.584120i
$738$		0			0
$739$	−18.1823	−	31.4926i	−0.668846	−	1.15848i	−0.978227	−	0.207537i	$-0.933455\pi$
$739$	−18.1823	−	31.4926i	−0.668846	−	1.15848i	0.309381	−	0.950938i	$-0.399878\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.59055	+	16.6113i	0.351843	+	0.609410i	0.986572	−	0.163324i	$-0.0522217\pi$
$743$	9.59055	+	16.6113i	0.351843	+	0.609410i	−0.634729	+	0.772735i	$0.718888\pi$
$744$		0			0
$745$	−28.7446	+	49.7871i	−1.05312	+	1.82406i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	27.8806			1.01874
$750$		0			0
$751$	−30.7207	−	17.7366i	−1.12101	−	0.647218i	−0.179355	−	0.983784i	$-0.557401\pi$
$751$	−30.7207	−	17.7366i	−1.12101	−	0.647218i	−0.941660	+	0.336566i	$0.890734\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−35.1433	−	60.8700i	−1.27900	−	2.21528i
$756$		0			0
$757$	−8.20314	−	14.2083i	−0.298148	−	0.516408i	0.677564	−	0.735464i	$-0.263036\pi$
$757$	−8.20314	−	14.2083i	−0.298148	−	0.516408i	−0.975712	+	0.219056i	$0.929702\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			14.3019i			0.518445i	0.965818	+	0.259223i	$0.0834663\pi$
$761$			14.3019i			0.518445i	−0.965818	+	0.259223i	$0.916534\pi$
$762$		0			0
$763$	−7.09129	+	4.09416i	−0.256722	+	0.148218i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			13.6574i			0.493139i
$768$		0			0
$769$	18.7306	−	32.4423i	0.675441	−	1.16990i	−0.300899	−	0.953656i	$-0.597287\pi$
$769$	18.7306	−	32.4423i	0.675441	−	1.16990i	0.976340	−	0.216242i	$-0.0693800\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.1435	−	38.3537i	0.796448	−	1.37949i	−0.125468	−	0.992098i	$-0.540043\pi$
$773$	22.1435	−	38.3537i	0.796448	−	1.37949i	0.921916	−	0.387390i	$-0.126623\pi$
$774$		0			0
$775$	9.73668	+	5.62148i	0.349752	+	0.201929i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.0307	+	17.6234i	0.359387	+	0.631425i
$780$		0			0
$781$	3.60189	−	2.07955i	0.128886	−	0.0744122i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.42163	−	5.43958i	−0.336272	−	0.194147i
$786$		0			0
$787$		−	28.5111i		−	1.01631i	−0.861265	−	0.508156i	$-0.830327\pi$
$787$		−	28.5111i		−	1.01631i	0.861265	−	0.508156i	$-0.169673\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.301716			−0.0107278
$792$		0			0
$793$	−27.5172	+	15.8871i	−0.977166	+	0.564167i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.4739			1.25655			0.628276	−	0.777991i	$-0.283761\pi$
$797$	35.4739			1.25655			0.628276	+	0.777991i	$0.283761\pi$
$798$		0			0
$799$	−24.6978			−0.873746
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.9734	+	6.91286i	−0.422533	+	0.243949i
$804$		0			0
$805$	25.5465			0.900395
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	21.1812i		−	0.744692i	−0.928094	−	0.372346i	$-0.878553\pi$
$809$		−	21.1812i		−	0.744692i	0.928094	−	0.372346i	$-0.121447\pi$
$810$		0			0
$811$	−31.4474	−	18.1562i	−1.10427	−	0.637549i	−0.166929	−	0.985969i	$-0.553385\pi$
$811$	−31.4474	−	18.1562i	−1.10427	−	0.637549i	−0.937339	+	0.348420i	$0.886718\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.80789	−	1.62113i	0.0983560	−	0.0567859i
$816$		0			0
$817$	0.0852890	+	0.149849i	0.00298388	+	0.00524254i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32.3149	−	18.6570i	−1.12780	−	0.651134i	−0.184417	−	0.982848i	$-0.559040\pi$
$821$	−32.3149	−	18.6570i	−1.12780	−	0.651134i	−0.943380	+	0.331714i	$0.892373\pi$
$822$		0			0
$823$	9.05698	−	15.6872i	0.315707	−	0.546820i	−0.663881	−	0.747838i	$-0.731092\pi$
$823$	9.05698	−	15.6872i	0.315707	−	0.546820i	0.979587	+	0.201019i	$0.0644252\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.87917	−	4.98686i	0.100118	−	0.173410i	−0.811615	−	0.584193i	$-0.801411\pi$
$827$	2.87917	−	4.98686i	0.100118	−	0.173410i	0.911733	+	0.410783i	$0.134745\pi$
$828$		0			0
$829$		−	23.4978i		−	0.816114i	−0.912957	−	0.408057i	$-0.866207\pi$
$829$		−	23.4978i		−	0.816114i	0.912957	−	0.408057i	$-0.133793\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.9069	+	7.45178i	−0.447196	+	0.258189i
$834$		0			0
$835$		−	21.4384i		−	0.741906i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.2221	−	17.7051i	−0.352905	−	0.611249i	0.633852	−	0.773454i	$-0.281473\pi$
$839$	−10.2221	−	17.7051i	−0.352905	−	0.611249i	−0.986757	+	0.162205i	$0.948139\pi$
$840$		0			0
$841$	−1.36345	−	2.36156i	−0.0470154	−	0.0814330i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.62980	+	2.09566i	0.124869	+	0.0720930i
$846$		0			0
$847$	−16.1655			−0.555453
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.4020	+	19.7488i	−0.390855	+	0.676981i
$852$		0			0
$853$	−25.2111	−	43.6670i	−0.863213	−	1.49513i	−0.868811	−	0.495144i	$-0.835115\pi$
$853$	−25.2111	−	43.6670i	−0.863213	−	1.49513i	0.00559855	−	0.999984i	$-0.498218\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.8719	−	36.1512i	−0.712971	−	1.23490i	−0.963737	−	0.266855i	$-0.914016\pi$
$857$	−20.8719	−	36.1512i	−0.712971	−	1.23490i	0.250766	−	0.968048i	$-0.419318\pi$
$858$		0			0
$859$	5.03760	−	8.72538i	0.171881	−	0.297706i	−0.767197	−	0.641412i	$-0.778349\pi$
$859$	5.03760	−	8.72538i	0.171881	−	0.297706i	0.939077	+	0.343706i	$0.111682\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−34.5534			−1.17621			−0.588107	−	0.808783i	$-0.700126\pi$
$863$	−34.5534			−1.17621			−0.588107	+	0.808783i	$0.700126\pi$
$864$		0			0
$865$	2.85005	+	1.64548i	0.0969046	+	0.0559479i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.0217	+	22.5543i	0.441731	+	0.765101i
$870$		0			0
$871$	−20.4190	−	35.3668i	−0.691872	−	1.19836i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	3.22230i		−	0.108934i
$876$		0			0
$877$	26.0094	−	15.0165i	0.878276	−	0.507073i	0.00818631	−	0.999966i	$-0.497394\pi$
$877$	26.0094	−	15.0165i	0.878276	−	0.507073i	0.870090	+	0.492894i	$0.164061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	31.0504i		−	1.04611i	−0.852298	−	0.523057i	$-0.824792\pi$
$881$		−	31.0504i		−	1.04611i	0.852298	−	0.523057i	$-0.175208\pi$
$882$		0			0
$883$	−22.3288	+	38.6747i	−0.751425	+	1.30151i	0.195706	+	0.980663i	$0.437300\pi$
$883$	−22.3288	+	38.6747i	−0.751425	+	1.30151i	−0.947132	+	0.320845i	$0.896033\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.1523	+	36.6369i	−0.710225	+	1.23015i	0.254548	+	0.967060i	$0.418073\pi$
$887$	−21.1523	+	36.6369i	−0.710225	+	1.23015i	−0.964773	+	0.263085i	$0.915260\pi$
$888$		0			0
$889$	−34.0250	−	19.6443i	−1.14116	−	0.658850i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.8841	+	11.0584i	0.631932	+	0.370054i
$894$		0			0
$895$	26.4234	−	15.2555i	0.883236	−	0.509937i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.2565	−	7.07631i	−0.408778	−	0.236008i
$900$		0			0
$901$			58.0998i			1.93558i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.7988			0.425445
$906$		0			0
$907$	−23.6001	+	13.6255i	−0.783629	+	0.452429i	−0.837715	−	0.546108i	$-0.816109\pi$
$907$	−23.6001	+	13.6255i	−0.783629	+	0.452429i	0.0540855	+	0.998536i	$0.482776\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	36.1256			1.19689			0.598447	−	0.801162i	$-0.295785\pi$
$911$	36.1256			1.19689			0.598447	+	0.801162i	$0.295785\pi$
$912$		0			0
$913$	−15.6645			−0.518421
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.8141	−	12.0170i	0.687341	−	0.396837i
$918$		0			0
$919$	27.4494			0.905474			0.452737	−	0.891644i	$-0.350448\pi$
$919$	27.4494			0.905474			0.452737	+	0.891644i	$0.350448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	9.27597i		−	0.305322i
$924$		0			0
$925$	−21.2163	−	12.2492i	−0.697587	−	0.402752i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−43.7728	+	25.2723i	−1.43614	+	0.829156i	−0.997579	−	0.0695451i	$-0.977845\pi$
$929$	−43.7728	+	25.2723i	−1.43614	+	0.829156i	−0.438562	+	0.898701i	$0.644512\pi$
$930$		0			0
$931$	13.2052	+	0.0813244i	0.432782	+	0.00266530i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−22.2828	−	12.8650i	−0.728724	−	0.420729i
$936$		0			0
$937$	29.1604	−	50.5074i	0.952630	−	1.65000i	0.212929	−	0.977068i	$-0.431700\pi$
$937$	29.1604	−	50.5074i	0.952630	−	1.65000i	0.739701	−	0.672936i	$-0.234967\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−24.8845	+	43.1013i	−0.811212	+	1.40506i	0.100804	+	0.994906i	$0.467859\pi$
$941$	−24.8845	+	43.1013i	−0.811212	+	1.40506i	−0.912016	+	0.410155i	$0.865475\pi$
$942$		0			0
$943$			19.3767i			0.630991i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−32.4108	+	18.7124i	−1.05321	+	0.608071i	−0.923546	−	0.383488i	$-0.874723\pi$
$947$	−32.4108	+	18.7124i	−1.05321	+	0.608071i	−0.129663	+	0.991558i	$0.541390\pi$
$948$		0			0
$949$			30.8353i			1.00095i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	27.6613	+	47.9108i	0.896038	+	1.55198i	0.832514	+	0.554003i	$0.186901\pi$
$953$	27.6613	+	47.9108i	0.896038	+	1.55198i	0.0635239	+	0.997980i	$0.479766\pi$
$954$		0			0
$955$	24.7319	+	42.8368i	0.800304	+	1.38617i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	29.2245	+	16.8728i	0.943707	+	0.544850i
$960$		0			0
$961$	24.6868			0.796350
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.9828	+	19.0227i	−0.353547	+	0.612362i
$966$		0			0
$967$	24.7733	+	42.9086i	0.796655	+	1.37985i	0.921783	+	0.387707i	$0.126733\pi$
$967$	24.7733	+	42.9086i	0.796655	+	1.37985i	−0.125127	+	0.992141i	$0.539934\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.8414	+	29.1702i	0.540467	+	0.936116i	0.998877	+	0.0473755i	$0.0150857\pi$
$971$	16.8414	+	29.1702i	0.540467	+	0.936116i	−0.458410	+	0.888741i	$0.651581\pi$
$972$		0			0
$973$	−11.5081	+	19.9326i	−0.368932	+	0.639009i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−16.3104			−0.521817			−0.260908	−	0.965364i	$-0.584022\pi$
$977$	−16.3104			−0.521817			−0.260908	+	0.965364i	$0.584022\pi$
$978$		0			0
$979$	−9.77902	−	5.64592i	−0.312539	−	0.180444i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.256152	−	0.443669i	−0.00816999	−	0.0141508i	0.861912	−	0.507059i	$-0.169267\pi$
$983$	−0.256152	−	0.443669i	−0.00816999	−	0.0141508i	−0.870081	+	0.492908i	$0.835934\pi$
$984$		0			0
$985$	−4.69890	−	8.13874i	−0.149720	−	0.259322i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.164756i			0.00523893i
$990$		0			0
$991$	−49.2049	+	28.4084i	−1.56304	+	0.902424i	−0.566097	+	0.824339i	$0.691547\pi$
$991$	−49.2049	+	28.4084i	−1.56304	+	0.902424i	−0.996947	+	0.0780851i	$0.975119\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	51.5346i		−	1.63376i
$996$		0			0
$997$	21.1094	−	36.5626i	0.668542	−	1.15795i	−0.309769	−	0.950812i	$-0.600252\pi$
$997$	21.1094	−	36.5626i	0.668542	−	1.15795i	0.978312	−	0.207138i	$-0.0664148\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.e.449.9		20
3.2	odd	2		2736.2.dc.f.449.2		20
4.3	odd	2		1368.2.cu.a.449.9	✓	20
12.11	even	2		1368.2.cu.b.449.2	yes	20
19.8	odd	6		2736.2.dc.f.1889.2		20
57.8	even	6	inner	2736.2.dc.e.1889.9		20
76.27	even	6		1368.2.cu.b.521.2	yes	20
228.179	odd	6		1368.2.cu.a.521.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.cu.a.449.9	✓	20	4.3	odd	2
1368.2.cu.a.521.9	yes	20	228.179	odd	6
1368.2.cu.b.449.2	yes	20	12.11	even	2
1368.2.cu.b.521.2	yes	20	76.27	even	6
2736.2.dc.e.449.9		20	1.1	even	1	trivial
2736.2.dc.e.1889.9		20	57.8	even	6	inner
2736.2.dc.f.449.2		20	3.2	odd	2
2736.2.dc.f.1889.2		20	19.8	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.66570 - 1.53904i) q^{5} -1.99260 q^{7} +O(q^{10})\)	\(q+(2.66570 - 1.53904i) q^{5} -1.99260 q^{7} -1.69919i q^{11} +(-3.28196 - 1.89484i) q^{13} +(4.26035 - 2.45972i) q^{17} +(-4.35882 - 0.0268439i) q^{19} +(-3.60711 - 2.08257i) q^{23} +(2.23732 - 3.87514i) q^{25} +(-2.81633 + 4.87803i) q^{29} +2.51260i q^{31} +(-5.31169 + 3.06671i) q^{35} -5.47497i q^{37} +(-2.32605 - 4.02884i) q^{41} +(-0.0197780 - 0.0342565i) q^{43} +(-4.34785 - 2.51023i) q^{47} -3.02953 q^{49} +(-5.90513 + 10.2280i) q^{53} +(-2.61513 - 4.52954i) q^{55} +(-1.80192 - 3.12101i) q^{59} +(4.19220 - 7.26110i) q^{61} -11.6650 q^{65} +(9.33239 + 5.38806i) q^{67} +(1.22385 + 2.11976i) q^{71} +(-4.06832 - 7.04654i) q^{73} +3.38582i q^{77} +(-13.2735 + 7.66347i) q^{79} -9.21882i q^{83} +(7.57122 - 13.1137i) q^{85} +(3.32271 - 5.75510i) q^{89} +(6.53964 + 3.77567i) q^{91} +(-11.6606 + 6.63685i) q^{95} +(-1.63733 + 0.945311i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{7}+O(q^{10}) \) 20 * q + 4 * q^7	\( 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) \) 20 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 4 * q^19 + 14 * q^25 - 12 * q^35 - 8 * q^41 + 2 * q^43 - 36 * q^47 + 32 * q^49 + 8 * q^53 - 12 * q^55 + 8 * q^59 - 2 * q^61 - 8 * q^65 - 30 * q^67 - 4 * q^71 - 22 * q^73 - 54 * q^79 - 4 * q^85 + 32 * q^89 - 18 * q^91 - 32 * q^95 + 12 * q^97

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