LMFDB - Embedded newform 480.2.b.a.431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(431,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.b (of order $2$, degree $1$, 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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1649659456.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 $ x^8 - x^7 - 2x^5 + 4x^4 - 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.5
Root			$1.40014 + 0.199044i$ of defining polynomial
Character	$\chi$	$=$	480.431
Dual form			480.2.b.a.431.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.520627 - 1.65195i) q^{3} -1.00000 q^{5} +1.92736i q^{7} +(-2.45790 - 1.72010i) q^{9} +O(q^{10})$	\(q+(0.520627 - 1.65195i) q^{3} -1.00000 q^{5} +1.92736i q^{7} +(-2.45790 - 1.72010i) q^{9} -4.02057i q^{11} -4.81675i q^{13} +(-0.520627 + 1.65195i) q^{15} -5.23126i q^{17} +0.684753 q^{19} +(3.18390 + 1.00343i) q^{21} +1.72601 q^{23} +1.00000 q^{25} +(-4.12117 + 3.16480i) q^{27} -6.99830 q^{29} -4.23638i q^{31} +(-6.64180 - 2.09322i) q^{33} -1.92736i q^{35} +9.83221i q^{37} +(-7.95705 - 2.50773i) q^{39} +3.44020i q^{41} -1.04125 q^{43} +(2.45790 + 1.72010i) q^{45} +7.55759 q^{47} +3.28530 q^{49} +(-8.64180 - 2.72353i) q^{51} +4.08251 q^{53} +4.02057i q^{55} +(0.356500 - 1.13118i) q^{57} +0.994883i q^{59} -3.16761i q^{61} +(3.31525 - 4.73724i) q^{63} +4.81675i q^{65} +14.8728 q^{67} +(0.898604 - 2.85128i) q^{69} -9.28360 q^{71} +11.2836 q^{73} +(0.520627 - 1.65195i) q^{75} +7.74908 q^{77} -9.25184i q^{79} +(3.08251 + 8.45566i) q^{81} +7.15862i q^{83} +5.23126i q^{85} +(-3.64350 + 11.5609i) q^{87} +0.829022i q^{89} +9.28360 q^{91} +(-6.99830 - 2.20557i) q^{93} -0.684753 q^{95} -1.45201 q^{97} +(-6.91579 + 9.88215i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 4 q^{19} - 4 q^{21} + 4 q^{23} + 8 q^{25} + 12 q^{27} - 4 q^{33} - 16 q^{39} - 28 q^{47} - 16 q^{49} - 20 q^{51} + 16 q^{53} - 4 q^{57} + 28 q^{63} + 32 q^{67} + 20 q^{69} + 24 q^{71} - 8 q^{73} + 8 q^{81} - 36 q^{87} - 24 q^{91} - 4 q^{95} + 8 q^{97} - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 4 * q^19 - 4 * q^21 + 4 * q^23 + 8 * q^25 + 12 * q^27 - 4 * q^33 - 16 * q^39 - 28 * q^47 - 16 * q^49 - 20 * q^51 + 16 * q^53 - 4 * q^57 + 28 * q^63 + 32 * q^67 + 20 * q^69 + 24 * q^71 - 8 * q^73 + 8 * q^81 - 36 * q^87 - 24 * q^91 - 4 * q^95 + 8 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	0.520627	−	1.65195i	0.300584	−	0.953755i
$3$	0.520627	−	1.65195i	0.300584	−	0.953755i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.92736i			0.728472i	0.931307	+	0.364236i	$0.118670\pi$
$7$			1.92736i			0.728472i	−0.931307	+	0.364236i	$0.881330\pi$
$8$		0			0
$9$	−2.45790	−	1.72010i	−0.819299	−	0.573367i
$10$		0			0
$11$		−	4.02057i		−	1.21225i	−0.795370	−	0.606124i	$-0.792723\pi$
$11$		−	4.02057i		−	1.21225i	0.795370	−	0.606124i	$-0.207277\pi$
$12$		0			0
$13$		−	4.81675i		−	1.33593i	−0.744194	−	0.667963i	$-0.767166\pi$
$13$		−	4.81675i		−	1.33593i	0.744194	−	0.667963i	$-0.232834\pi$
$14$		0			0
$15$	−0.520627	+	1.65195i	−0.134425	+	0.426532i
$16$		0			0
$17$		−	5.23126i		−	1.26877i	−0.773018	−	0.634384i	$-0.781254\pi$
$17$		−	5.23126i		−	1.26877i	0.773018	−	0.634384i	$-0.218746\pi$
$18$		0			0
$19$	0.684753			0.157093			0.0785465	−	0.996910i	$-0.474972\pi$
$19$	0.684753			0.157093			0.0785465	+	0.996910i	$0.474972\pi$
$20$		0			0
$21$	3.18390	+	1.00343i	0.694784	+	0.218967i
$22$		0			0
$23$	1.72601			0.359897			0.179949	−	0.983676i	$-0.442407\pi$
$23$	1.72601			0.359897			0.179949	+	0.983676i	$0.442407\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−4.12117	+	3.16480i	−0.793120	+	0.609066i
$28$		0			0
$29$	−6.99830			−1.29955			−0.649776	−	0.760126i	$-0.725137\pi$
$29$	−6.99830			−1.29955			−0.649776	+	0.760126i	$0.725137\pi$
$30$		0			0
$31$		−	4.23638i		−	0.760876i	−0.924806	−	0.380438i	$-0.875773\pi$
$31$		−	4.23638i		−	0.760876i	0.924806	−	0.380438i	$-0.124227\pi$
$32$		0			0
$33$	−6.64180	−	2.09322i	−1.15619	−	0.364382i
$34$		0			0
$35$		−	1.92736i		−	0.325783i
$36$		0			0
$37$			9.83221i			1.61640i	0.588905	+	0.808202i	$0.299559\pi$
$37$			9.83221i			1.61640i	−0.588905	+	0.808202i	$0.700441\pi$
$38$		0			0
$39$	−7.95705	−	2.50773i	−1.27415	−	0.401558i
$40$		0			0
$41$			3.44020i			0.537269i	0.963242	+	0.268635i	$0.0865724\pi$
$41$			3.44020i			0.537269i	−0.963242	+	0.268635i	$0.913428\pi$
$42$		0			0
$43$	−1.04125			−0.158790			−0.0793948	−	0.996843i	$-0.525299\pi$
$43$	−1.04125			−0.158790			−0.0793948	+	0.996843i	$0.525299\pi$
$44$		0			0
$45$	2.45790	+	1.72010i	0.366402	+	0.256417i
$46$		0			0
$47$	7.55759			1.10239			0.551194	−	0.834377i	$-0.314172\pi$
$47$	7.55759			1.10239			0.551194	+	0.834377i	$0.314172\pi$
$48$		0			0
$49$	3.28530			0.469328
$50$		0			0
$51$	−8.64180	−	2.72353i	−1.21009	−	0.381371i
$52$		0			0
$53$	4.08251			0.560775			0.280388	−	0.959887i	$-0.409537\pi$
$53$	4.08251			0.560775			0.280388	+	0.959887i	$0.409537\pi$
$54$		0			0
$55$			4.02057i			0.542134i
$56$		0			0
$57$	0.356500	−	1.13118i	0.0472196	−	0.149828i
$58$		0			0
$59$			0.994883i			0.129523i	0.997901	+	0.0647614i	$0.0206286\pi$
$59$			0.994883i			0.129523i	−0.997901	+	0.0647614i	$0.979371\pi$
$60$		0			0
$61$		−	3.16761i		−	0.405571i	−0.979223	−	0.202785i	$-0.935001\pi$
$61$		−	3.16761i		−	0.405571i	0.979223	−	0.202785i	$-0.0649994\pi$
$62$		0			0
$63$	3.31525	−	4.73724i	0.417682	−	0.596836i
$64$		0			0
$65$			4.81675i			0.597444i
$66$		0			0
$67$	14.8728			1.81701			0.908503	−	0.417878i	$-0.137226\pi$
$67$	14.8728			1.81701			0.908503	+	0.417878i	$0.137226\pi$
$68$		0			0
$69$	0.898604	−	2.85128i	0.108179	−	0.343254i
$70$		0			0
$71$	−9.28360			−1.10176			−0.550880	−	0.834584i	$-0.685708\pi$
$71$	−9.28360			−1.10176			−0.550880	+	0.834584i	$0.685708\pi$
$72$		0			0
$73$	11.2836			1.32064			0.660322	−	0.750982i	$-0.270420\pi$
$73$	11.2836			1.32064			0.660322	+	0.750982i	$0.270420\pi$
$74$		0			0
$75$	0.520627	−	1.65195i	0.0601168	−	0.190751i
$76$		0			0
$77$	7.74908			0.883089
$78$		0			0
$79$		−	9.25184i		−	1.04091i	−0.853888	−	0.520456i	$-0.825762\pi$
$79$		−	9.25184i		−	1.04091i	0.853888	−	0.520456i	$-0.174238\pi$
$80$		0			0
$81$	3.08251	+	8.45566i	0.342501	+	0.939518i
$82$		0			0
$83$			7.15862i			0.785760i	0.919590	+	0.392880i	$0.128521\pi$
$83$			7.15862i			0.785760i	−0.919590	+	0.392880i	$0.871479\pi$
$84$		0			0
$85$			5.23126i			0.567410i
$86$		0			0
$87$	−3.64350	+	11.5609i	−0.390624	+	1.23945i
$88$		0			0
$89$			0.829022i			0.0878762i	0.999034	+	0.0439381i	$0.0139904\pi$
$89$			0.829022i			0.0878762i	−0.999034	+	0.0439381i	$0.986010\pi$
$90$		0			0
$91$	9.28360			0.973185
$92$		0			0
$93$	−6.99830	−	2.20557i	−0.725690	−	0.228707i
$94$		0			0
$95$	−0.684753			−0.0702541
$96$		0			0
$97$	−1.45201			−0.147429			−0.0737147	−	0.997279i	$-0.523485\pi$
$97$	−1.45201			−0.147429			−0.0737147	+	0.997279i	$0.523485\pi$
$98$		0			0
$99$	−6.91579	+	9.88215i	−0.695063	+	0.993194i
$100$		0			0
$101$	−4.20279			−0.418193			−0.209097	−	0.977895i	$-0.567052\pi$
$101$	−4.20279			−0.418193			−0.209097	+	0.977895i	$0.567052\pi$
$102$		0			0
$103$		−	7.10183i		−	0.699764i	−0.936794	−	0.349882i	$-0.886222\pi$
$103$		−	7.10183i		−	0.699764i	0.936794	−	0.349882i	$-0.113778\pi$
$104$		0			0
$105$	−3.18390	−	1.00343i	−0.310717	−	0.0979250i
$106$		0			0
$107$			7.76293i			0.750471i	0.926930	+	0.375235i	$0.122438\pi$
$107$			7.76293i			0.750471i	−0.926930	+	0.375235i	$0.877562\pi$
$108$		0			0
$109$		−	20.5105i		−	1.96455i	−0.187437	−	0.982277i	$-0.560018\pi$
$109$		−	20.5105i		−	1.96455i	0.187437	−	0.982277i	$-0.439982\pi$
$110$		0			0
$111$	16.2423	+	5.11891i	1.54165	+	0.485865i
$112$		0			0
$113$		−	0.215805i		−	0.0203013i	−0.999948	−	0.0101506i	$-0.996769\pi$
$113$		−	0.215805i		−	0.0203013i	0.999948	−	0.0101506i	$-0.00323110\pi$
$114$		0			0
$115$	−1.72601			−0.160951
$116$		0			0
$117$	−8.28530	+	11.8391i	−0.765976	+	1.09452i
$118$		0			0
$119$	10.0825			0.924262
$120$		0			0
$121$	−5.16501			−0.469547
$122$		0			0
$123$	5.68305	+	1.79106i	0.512423	+	0.161494i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			16.5763i			1.47091i	0.677574	+	0.735455i	$0.263032\pi$
$127$			16.5763i			1.47091i	−0.677574	+	0.735455i	$0.736968\pi$
$128$		0			0
$129$	−0.542104	+	1.72010i	−0.0477296	+	0.151446i
$130$		0			0
$131$			5.61293i			0.490404i	0.969472	+	0.245202i	$0.0788543\pi$
$131$			5.61293i			0.490404i	−0.969472	+	0.245202i	$0.921146\pi$
$132$		0			0
$133$			1.31976i			0.114438i
$134$		0			0
$135$	4.12117	−	3.16480i	0.354694	−	0.272382i
$136$		0			0
$137$			9.41770i			0.804608i	0.915506	+	0.402304i	$0.131791\pi$
$137$			9.41770i			0.804608i	−0.915506	+	0.402304i	$0.868209\pi$
$138$		0			0
$139$	6.51634			0.552708			0.276354	−	0.961056i	$-0.410874\pi$
$139$	6.51634			0.552708			0.276354	+	0.961056i	$0.410874\pi$
$140$		0			0
$141$	3.93468	−	12.4848i	0.331360	−	1.05141i
$142$		0			0
$143$	−19.3661			−1.61947
$144$		0			0
$145$	6.99830			0.581177
$146$		0			0
$147$	1.71041	−	5.42716i	0.141072	−	0.447624i
$148$		0			0
$149$	7.53452			0.617252			0.308626	−	0.951184i	$-0.400131\pi$
$149$	7.53452			0.617252			0.308626	+	0.951184i	$0.400131\pi$
$150$		0			0
$151$			9.41085i			0.765844i	0.923781	+	0.382922i	$0.125082\pi$
$151$			9.41085i			0.765844i	−0.923781	+	0.382922i	$0.874918\pi$
$152$		0			0
$153$	−8.99830	+	12.8579i	−0.727469	+	1.03950i
$154$		0			0
$155$			4.23638i			0.340274i
$156$		0			0
$157$			3.49699i			0.279090i	0.990216	+	0.139545i	$0.0445640\pi$
$157$			3.49699i			0.279090i	−0.990216	+	0.139545i	$0.955436\pi$
$158$		0			0
$159$	2.12546	−	6.74411i	0.168560	−	0.534842i
$160$		0			0
$161$			3.32663i			0.262175i
$162$		0			0
$163$	−16.9553			−1.32804			−0.664022	−	0.747713i	$-0.731152\pi$
$163$	−16.9553			−1.32804			−0.664022	+	0.747713i	$0.731152\pi$
$164$		0			0
$165$	6.64180	+	2.09322i	0.517063	+	0.162957i
$166$		0			0
$167$	11.3926			0.881584			0.440792	−	0.897609i	$-0.354698\pi$
$167$	11.3926			0.881584			0.440792	+	0.897609i	$0.354698\pi$
$168$		0			0
$169$	−10.2011			−0.784699
$170$		0			0
$171$	−1.68305	−	1.17784i	−0.128706	−	0.0900720i
$172$		0			0
$173$	−2.16501			−0.164603			−0.0823014	−	0.996607i	$-0.526227\pi$
$173$	−2.16501			−0.164603			−0.0823014	+	0.996607i	$0.526227\pi$
$174$		0			0
$175$			1.92736i			0.145694i
$176$		0			0
$177$	1.64350	+	0.517962i	0.123533	+	0.0389324i
$178$		0			0
$179$		−	5.34034i		−	0.399155i	−0.979882	−	0.199578i	$-0.936043\pi$
$179$		−	5.34034i		−	0.399155i	0.979882	−	0.199578i	$-0.0639570\pi$
$180$		0			0
$181$			10.7942i			0.802330i	0.916006	+	0.401165i	$0.131395\pi$
$181$			10.7942i			0.802330i	−0.916006	+	0.401165i	$0.868605\pi$
$182$		0			0
$183$	−5.23274	−	1.64914i	−0.386815	−	0.121908i
$184$		0			0
$185$		−	9.83221i		−	0.722878i
$186$		0			0
$187$	−21.0327			−1.53806
$188$		0			0
$189$	−6.09969	−	7.94297i	−0.443687	−	0.577766i
$190$		0			0
$191$	−19.7491			−1.42899			−0.714497	−	0.699639i	$-0.753344\pi$
$191$	−19.7491			−1.42899			−0.714497	+	0.699639i	$0.753344\pi$
$192$		0			0
$193$	5.45201			0.392444			0.196222	−	0.980559i	$-0.437133\pi$
$193$	5.45201			0.392444			0.196222	+	0.980559i	$0.437133\pi$
$194$		0			0
$195$	7.95705	+	2.50773i	0.569816	+	0.179582i
$196$		0			0
$197$	22.6497			1.61372			0.806862	−	0.590740i	$-0.201164\pi$
$197$	22.6497			1.61372			0.806862	+	0.590740i	$0.201164\pi$
$198$		0			0
$199$		−	18.8853i		−	1.33875i	−0.742926	−	0.669373i	$-0.766563\pi$
$199$		−	18.8853i		−	1.33875i	0.742926	−	0.669373i	$-0.233437\pi$
$200$		0			0
$201$	7.74319	−	24.5692i	0.546163	−	1.73298i
$202$		0			0
$203$		−	13.4882i		−	0.946687i
$204$		0			0
$205$		−	3.44020i		−	0.240274i
$206$		0			0
$207$	−4.24234	−	2.96890i	−0.294863	−	0.206353i
$208$		0			0
$209$		−	2.75310i		−	0.190436i
$210$		0			0
$211$	10.7673			0.741249			0.370624	−	0.928783i	$-0.379144\pi$
$211$	10.7673			0.741249			0.370624	+	0.928783i	$0.379144\pi$
$212$		0			0
$213$	−4.83329	+	15.3361i	−0.331171	+	1.05081i
$214$		0			0
$215$	1.04125			0.0710129
$216$		0			0
$217$	8.16501			0.554277
$218$		0			0
$219$	5.87454	−	18.6400i	0.396965	−	1.25957i
$220$		0			0
$221$	−25.1977			−1.69498
$222$		0			0
$223$			6.54540i			0.438313i	0.975690	+	0.219156i	$0.0703304\pi$
$223$			6.54540i			0.438313i	−0.975690	+	0.219156i	$0.929670\pi$
$224$		0			0
$225$	−2.45790	−	1.72010i	−0.163860	−	0.114673i
$226$		0			0
$227$		−	22.5118i		−	1.49416i	−0.664735	−	0.747080i	$-0.731455\pi$
$227$		−	22.5118i		−	1.49416i	0.664735	−	0.747080i	$-0.268545\pi$
$228$		0			0
$229$			12.8839i			0.851392i	0.904866	+	0.425696i	$0.139971\pi$
$229$			12.8839i			0.851392i	−0.904866	+	0.425696i	$0.860029\pi$
$230$		0			0
$231$	4.03438	−	12.8011i	0.265442	−	0.842251i
$232$		0			0
$233$		−	10.8510i		−	0.710875i	−0.934700	−	0.355437i	$-0.884332\pi$
$233$		−	10.8510i		−	0.710875i	0.934700	−	0.355437i	$-0.115668\pi$
$234$		0			0
$235$	−7.55759			−0.493003
$236$		0			0
$237$	−15.2836	−	4.81675i	−0.992776	−	0.312882i
$238$		0			0
$239$	−6.63049			−0.428891			−0.214446	−	0.976736i	$-0.568794\pi$
$239$	−6.63049			−0.428891			−0.214446	+	0.976736i	$0.568794\pi$
$240$		0			0
$241$	−15.9519			−1.02755			−0.513775	−	0.857925i	$-0.671753\pi$
$241$	−15.9519			−1.02755			−0.513775	+	0.857925i	$0.671753\pi$
$242$		0			0
$243$	15.5732	−	0.689915i	0.999020	−	0.0442580i
$244$		0			0
$245$	−3.28530			−0.209890
$246$		0			0
$247$		−	3.29828i		−	0.209865i
$248$		0			0
$249$	11.8257	+	3.72697i	0.749423	+	0.236187i
$250$		0			0
$251$			15.2464i			0.962346i	0.876626	+	0.481173i	$0.159789\pi$
$251$			15.2464i			0.962346i	−0.876626	+	0.481173i	$0.840211\pi$
$252$		0			0
$253$		−	6.93953i		−	0.436285i
$254$		0			0
$255$	8.64180	+	2.72353i	0.541170	+	0.170554i
$256$		0			0
$257$		−	16.1845i		−	1.00956i	−0.863247	−	0.504782i	$-0.831573\pi$
$257$		−	16.1845i		−	1.00956i	0.863247	−	0.504782i	$-0.168427\pi$
$258$		0			0
$259$	−18.9502			−1.17751
$260$		0			0
$261$	17.2011	+	12.0378i	1.06472	+	0.745120i
$262$		0			0
$263$	21.4751			1.32421			0.662105	−	0.749411i	$-0.269663\pi$
$263$	21.4751			1.32421			0.662105	+	0.749411i	$0.269663\pi$
$264$		0			0
$265$	−4.08251			−0.250786
$266$		0			0
$267$	1.36951	+	0.431611i	0.0838124	+	0.0264142i
$268$		0			0
$269$	23.0327			1.40433			0.702163	−	0.712016i	$-0.252218\pi$
$269$	23.0327			1.40433			0.702163	+	0.712016i	$0.252218\pi$
$270$		0			0
$271$		−	16.1914i		−	0.983556i	−0.870721	−	0.491778i	$-0.836347\pi$
$271$		−	16.1914i		−	0.983556i	0.870721	−	0.491778i	$-0.163653\pi$
$272$		0			0
$273$	4.83329	−	15.3361i	0.292524	−	0.928181i
$274$		0			0
$275$		−	4.02057i		−	0.242450i
$276$		0			0
$277$			3.81503i			0.229223i	0.993410	+	0.114611i	$0.0365623\pi$
$277$			3.81503i			0.229223i	−0.993410	+	0.114611i	$0.963438\pi$
$278$		0			0
$279$	−7.28700	+	10.4126i	−0.436261	+	0.623385i
$280$		0			0
$281$		−	27.9474i		−	1.66720i	−0.552368	−	0.833600i	$-0.686276\pi$
$281$		−	27.9474i		−	1.66720i	0.552368	−	0.833600i	$-0.313724\pi$
$282$		0			0
$283$	5.58924			0.332246			0.166123	−	0.986105i	$-0.446875\pi$
$283$	5.58924			0.332246			0.166123	+	0.986105i	$0.446875\pi$
$284$		0			0
$285$	−0.356500	+	1.13118i	−0.0211173	+	0.0670053i
$286$		0			0
$287$	−6.63049			−0.391386
$288$		0			0
$289$	−10.3661			−0.609771
$290$		0			0
$291$	−0.755956	+	2.39865i	−0.0443149	+	0.140612i
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		−	0.994883i		−	0.0579243i
$296$		0			0
$297$	12.7243	+	16.5695i	0.738339	+	0.961458i
$298$		0			0
$299$		−	8.31374i		−	0.480796i
$300$		0			0
$301$		−	2.00687i		−	0.115674i
$302$		0			0
$303$	−2.18808	+	6.94281i	−0.125702	+	0.398854i
$304$		0			0
$305$			3.16761i			0.181377i
$306$		0			0
$307$	4.79033			0.273399			0.136699	−	0.990613i	$-0.456351\pi$
$307$	4.79033			0.273399			0.136699	+	0.990613i	$0.456351\pi$
$308$		0			0
$309$	−11.7319	−	3.69740i	−0.667404	−	0.210338i
$310$		0			0
$311$	12.8780			0.730245			0.365123	−	0.930959i	$-0.381027\pi$
$311$	12.8780			0.730245			0.365123	+	0.930959i	$0.381027\pi$
$312$		0			0
$313$	−20.4022			−1.15320			−0.576600	−	0.817027i	$-0.695621\pi$
$313$	−20.4022			−1.15320			−0.576600	+	0.817027i	$0.695621\pi$
$314$		0			0
$315$	−3.31525	+	4.73724i	−0.186793	+	0.266913i
$316$		0			0
$317$	−5.34350			−0.300121			−0.150060	−	0.988677i	$-0.547947\pi$
$317$	−5.34350			−0.300121			−0.150060	+	0.988677i	$0.547947\pi$
$318$		0			0
$319$			28.1372i			1.57538i
$320$		0			0
$321$	12.8240	+	4.04159i	0.715766	+	0.225579i
$322$		0			0
$323$		−	3.58212i		−	0.199315i
$324$		0			0
$325$		−	4.81675i		−	0.267185i
$326$		0			0
$327$	−33.8824	−	10.6783i	−1.87370	−	0.590513i
$328$		0			0
$329$			14.5662i			0.803059i
$330$		0			0
$331$	−25.1694			−1.38344			−0.691719	−	0.722167i	$-0.743146\pi$
$331$	−25.1694			−1.38344			−0.691719	+	0.722167i	$0.743146\pi$
$332$		0			0
$333$	16.9124	−	24.1665i	0.926793	−	1.32432i
$334$		0			0
$335$	−14.8728			−0.812590
$336$		0			0
$337$	−20.1616			−1.09827			−0.549136	−	0.835733i	$-0.685043\pi$
$337$	−20.1616			−1.09827			−0.549136	+	0.835733i	$0.685043\pi$
$338$		0			0
$339$	−0.356500	−	0.112354i	−0.0193624	−	0.00610223i
$340$		0			0
$341$	−17.0327			−0.922371
$342$		0			0
$343$			19.8234i			1.07036i
$344$		0			0
$345$	−0.898604	+	2.85128i	−0.0483792	+	0.153508i
$346$		0			0
$347$		−	9.41442i		−	0.505392i	−0.967546	−	0.252696i	$-0.918683\pi$
$347$		−	9.41442i		−	0.505392i	0.967546	−	0.252696i	$-0.0813173\pi$
$348$		0			0
$349$			10.3968i			0.556530i	0.960504	+	0.278265i	$0.0897593\pi$
$349$			10.3968i			0.556530i	−0.960504	+	0.278265i	$0.910241\pi$
$350$		0			0
$351$	15.2440	+	19.8507i	0.813667	+	1.05955i
$352$		0			0
$353$			20.4254i			1.08713i	0.839366	+	0.543567i	$0.182927\pi$
$353$			20.4254i			1.08713i	−0.839366	+	0.543567i	$0.817073\pi$
$354$		0			0
$355$	9.28360			0.492722
$356$		0			0
$357$	5.24922	−	16.6558i	0.277818	−	0.881520i
$358$		0			0
$359$	6.87107			0.362641			0.181320	−	0.983424i	$-0.441963\pi$
$359$	6.87107			0.362641			0.181320	+	0.983424i	$0.441963\pi$
$360$		0			0
$361$	−18.5311			−0.975322
$362$		0			0
$363$	−2.68904	+	8.53236i	−0.141138	+	0.447833i
$364$		0			0
$365$	−11.2836			−0.590610
$366$		0			0
$367$		−	15.8130i		−	0.825431i	−0.910860	−	0.412715i	$-0.864580\pi$
$367$		−	15.8130i		−	0.825431i	0.910860	−	0.412715i	$-0.135420\pi$
$368$		0			0
$369$	5.91749	−	8.45566i	0.308052	−	0.440184i
$370$		0			0
$371$			7.86844i			0.408509i
$372$		0			0
$373$		−	7.51072i		−	0.388890i	−0.980913	−	0.194445i	$-0.937709\pi$
$373$		−	7.51072i		−	0.388890i	0.980913	−	0.194445i	$-0.0622906\pi$
$374$		0			0
$375$	−0.520627	+	1.65195i	−0.0268850	+	0.0853065i
$376$		0			0
$377$			33.7091i			1.73610i
$378$		0			0
$379$	13.1468			0.675307			0.337654	−	0.941270i	$-0.390367\pi$
$379$	13.1468			0.675307			0.337654	+	0.941270i	$0.390367\pi$
$380$		0			0
$381$	27.3833	+	8.63007i	1.40289	+	0.442132i
$382$		0			0
$383$	23.0887			1.17978			0.589889	−	0.807484i	$-0.299172\pi$
$383$	23.0887			1.17978			0.589889	+	0.807484i	$0.299172\pi$
$384$		0			0
$385$	−7.74908			−0.394930
$386$		0			0
$387$	2.55929	+	1.79106i	0.130096	+	0.0910447i
$388$		0			0
$389$	−12.9040			−0.654260			−0.327130	−	0.944979i	$-0.606081\pi$
$389$	−12.9040			−0.654260			−0.327130	+	0.944979i	$0.606081\pi$
$390$		0			0
$391$		−	9.02919i		−	0.456626i
$392$		0			0
$393$	9.27229	+	2.92224i	0.467725	+	0.147407i
$394$		0			0
$395$			9.25184i			0.465510i
$396$		0			0
$397$		−	18.1459i		−	0.910719i	−0.890308	−	0.455359i	$-0.849511\pi$
$397$		−	18.1459i		−	0.910719i	0.890308	−	0.455359i	$-0.150489\pi$
$398$		0			0
$399$	2.18019	+	0.687103i	0.109146	+	0.0343982i
$400$		0			0
$401$			33.7433i			1.68506i	0.538651	+	0.842529i	$0.318934\pi$
$401$			33.7433i			1.68506i	−0.538651	+	0.842529i	$0.681066\pi$
$402$		0			0
$403$	−20.4056			−1.01647
$404$		0			0
$405$	−3.08251	−	8.45566i	−0.153171	−	0.420165i
$406$		0			0
$407$	39.5311			1.95948
$408$		0			0
$409$	31.6480			1.56489			0.782446	−	0.622718i	$-0.213972\pi$
$409$	31.6480			1.56489			0.782446	+	0.622718i	$0.213972\pi$
$410$		0			0
$411$	15.5576	+	4.90310i	0.767399	+	0.241852i
$412$		0			0
$413$	−1.91749			−0.0943537
$414$		0			0
$415$		−	7.15862i		−	0.351403i
$416$		0			0
$417$	3.39258	−	10.7647i	0.166135	−	0.527149i
$418$		0			0
$419$			29.8954i			1.46049i	0.683188	+	0.730243i	$0.260593\pi$
$419$			29.8954i			1.46049i	−0.683188	+	0.730243i	$0.739407\pi$
$420$		0			0
$421$		−	6.86330i		−	0.334497i	−0.985915	−	0.167248i	$-0.946512\pi$
$421$		−	6.86330i		−	0.334497i	0.985915	−	0.167248i	$-0.0534882\pi$
$422$		0			0
$423$	−18.5758	−	12.9998i	−0.903185	−	0.632073i
$424$		0			0
$425$		−	5.23126i		−	0.253753i
$426$		0			0
$427$	6.10511			0.295447
$428$		0			0
$429$	−10.0825	+	31.9919i	−0.486788	+	1.54458i
$430$		0			0
$431$	−20.0226			−0.964455			−0.482228	−	0.876046i	$-0.660172\pi$
$431$	−20.0226			−0.964455			−0.482228	+	0.876046i	$0.660172\pi$
$432$		0			0
$433$	10.2112			0.490717			0.245358	−	0.969432i	$-0.421094\pi$
$433$	10.2112			0.490717			0.245358	+	0.969432i	$0.421094\pi$
$434$		0			0
$435$	3.64350	−	11.5609i	0.174692	−	0.554301i
$436$		0			0
$437$	1.18189			0.0565373
$438$		0			0
$439$			33.6933i			1.60809i	0.594566	+	0.804047i	$0.297324\pi$
$439$			33.6933i			1.60809i	−0.594566	+	0.804047i	$0.702676\pi$
$440$		0			0
$441$	−8.07492	−	5.65104i	−0.384520	−	0.269097i
$442$		0			0
$443$			4.46465i			0.212122i	0.994360	+	0.106061i	$0.0338239\pi$
$443$			4.46465i			0.212122i	−0.994360	+	0.106061i	$0.966176\pi$
$444$		0			0
$445$		−	0.829022i		−	0.0392994i
$446$		0			0
$447$	3.92267	−	12.4467i	0.185536	−	0.588707i
$448$		0			0
$449$		−	27.5500i		−	1.30016i	−0.759865	−	0.650081i	$-0.774735\pi$
$449$		−	27.5500i		−	1.30016i	0.759865	−	0.650081i	$-0.225265\pi$
$450$		0			0
$451$	13.8316			0.651304
$452$		0			0
$453$	15.5463	+	4.89954i	0.730428	+	0.230200i
$454$		0			0
$455$	−9.28360			−0.435222
$456$		0			0
$457$	−11.5016			−0.538020			−0.269010	−	0.963137i	$-0.586697\pi$
$457$	−11.5016			−0.538020			−0.269010	+	0.963137i	$0.586697\pi$
$458$		0			0
$459$	16.5559	+	21.5589i	0.772763	+	1.00628i
$460$		0			0
$461$	8.25929			0.384673			0.192337	−	0.981329i	$-0.438393\pi$
$461$	8.25929			0.384673			0.192337	+	0.981329i	$0.438393\pi$
$462$		0			0
$463$		−	11.7199i		−	0.544669i	−0.962203	−	0.272334i	$-0.912204\pi$
$463$		−	11.7199i		−	0.544669i	0.962203	−	0.272334i	$-0.0877957\pi$
$464$		0			0
$465$	6.99830	+	2.20557i	0.324538	+	0.102281i
$466$		0			0
$467$			17.1895i			0.795437i	0.917508	+	0.397718i	$0.130198\pi$
$467$			17.1895i			0.795437i	−0.917508	+	0.397718i	$0.869802\pi$
$468$		0			0
$469$			28.6653i			1.32364i
$470$		0			0
$471$	5.77686	+	1.82062i	0.266184	+	0.0838900i
$472$		0			0
$473$			4.18643i			0.192492i
$474$		0			0
$475$	0.684753			0.0314186
$476$		0			0
$477$	−10.0344	−	7.02232i	−0.459442	−	0.321530i
$478$		0			0
$479$	−11.5379			−0.527181			−0.263591	−	0.964635i	$-0.584907\pi$
$479$	−11.5379			−0.527181			−0.263591	+	0.964635i	$0.584907\pi$
$480$		0			0
$481$	47.3593			2.15940
$482$		0			0
$483$	5.49543	+	1.73193i	0.250051	+	0.0788056i
$484$		0			0
$485$	1.45201			0.0659324
$486$		0			0
$487$		−	33.1015i		−	1.49997i	−0.661455	−	0.749985i	$-0.730061\pi$
$487$		−	33.1015i		−	1.49997i	0.661455	−	0.749985i	$-0.269939\pi$
$488$		0			0
$489$	−8.82740	+	28.0094i	−0.399189	+	1.26663i
$490$		0			0
$491$		−	21.3635i		−	0.964121i	−0.876138	−	0.482061i	$-0.839888\pi$
$491$		−	21.3635i		−	0.964121i	0.876138	−	0.482061i	$-0.160112\pi$
$492$		0			0
$493$			36.6099i			1.64883i
$494$		0			0
$495$	6.91579	−	9.88215i	0.310842	−	0.444170i
$496$		0			0
$497$		−	17.8928i		−	0.802602i
$498$		0			0
$499$	−14.8464			−0.664614			−0.332307	−	0.943171i	$-0.607827\pi$
$499$	−14.8464			−0.664614			−0.332307	+	0.943171i	$0.607827\pi$
$500$		0			0
$501$	5.93128	−	18.8200i	0.264990	−	0.840816i
$502$		0			0
$503$	1.86841			0.0833084			0.0416542	−	0.999132i	$-0.486737\pi$
$503$	1.86841			0.0833084			0.0416542	+	0.999132i	$0.486737\pi$
$504$		0			0
$505$	4.20279			0.187022
$506$		0			0
$507$	−5.31096	+	16.8517i	−0.235868	+	0.748411i
$508$		0			0
$509$	1.62879			0.0721950			0.0360975	−	0.999348i	$-0.488507\pi$
$509$	1.62879			0.0721950			0.0360975	+	0.999348i	$0.488507\pi$
$510$		0			0
$511$			21.7475i			0.962053i
$512$		0			0
$513$	−2.82198	+	2.16710i	−0.124594	+	0.0956800i
$514$		0			0
$515$			7.10183i			0.312944i
$516$		0			0
$517$		−	30.3858i		−	1.33637i
$518$		0			0
$519$	−1.12716	+	3.57650i	−0.0494770	+	0.156991i
$520$		0			0
$521$		−	7.82768i		−	0.342937i	−0.985190	−	0.171469i	$-0.945149\pi$
$521$		−	7.82768i		−	0.342937i	0.985190	−	0.171469i	$-0.0548512\pi$
$522$		0			0
$523$	32.2423			1.40986			0.704930	−	0.709277i	$-0.250979\pi$
$523$	32.2423			1.40986			0.704930	+	0.709277i	$0.250979\pi$
$524$		0			0
$525$	3.18390	+	1.00343i	0.138957	+	0.0437934i
$526$		0			0
$527$	−22.1616			−0.965375
$528$		0			0
$529$	−20.0209			−0.870474
$530$		0			0
$531$	1.71130	−	2.44532i	0.0742640	−	0.106118i
$532$		0			0
$533$	16.5706			0.717752
$534$		0			0
$535$		−	7.76293i		−	0.335621i
$536$		0			0
$537$	−8.82198	−	2.78032i	−0.380697	−	0.119980i
$538$		0			0
$539$		−	13.2088i		−	0.568942i
$540$		0			0
$541$			3.25040i			0.139746i	0.997556	+	0.0698728i	$0.0222593\pi$
$541$			3.25040i			0.139746i	−0.997556	+	0.0698728i	$0.977741\pi$
$542$		0			0
$543$	17.8316	+	5.61977i	0.765227	+	0.241167i
$544$		0			0
$545$			20.5105i			0.878575i
$546$		0			0
$547$	−10.3248			−0.441459			−0.220729	−	0.975335i	$-0.570844\pi$
$547$	−10.3248			−0.441459			−0.220729	+	0.975335i	$0.570844\pi$
$548$		0			0
$549$	−5.44861	+	7.78565i	−0.232541	+	0.332284i
$550$		0			0
$551$	−4.79210			−0.204150
$552$		0			0
$553$	17.8316			0.758276
$554$		0			0
$555$	−16.2423	−	5.11891i	−0.689449	−	0.217286i
$556$		0			0
$557$	8.33343			0.353099			0.176549	−	0.984292i	$-0.443506\pi$
$557$	8.33343			0.353099			0.176549	+	0.984292i	$0.443506\pi$
$558$		0			0
$559$			5.01546i			0.212131i
$560$		0			0
$561$	−10.9502	+	34.7450i	−0.462316	+	1.46693i
$562$		0			0
$563$			14.0982i			0.594166i	0.954852	+	0.297083i	$0.0960138\pi$
$563$			14.0982i			0.594166i	−0.954852	+	0.297083i	$0.903986\pi$
$564$		0			0
$565$			0.215805i			0.00907900i
$566$		0			0
$567$	−16.2971	+	5.94109i	−0.684412	+	0.249502i
$568$		0			0
$569$			9.37801i			0.393147i	0.980489	+	0.196573i	$0.0629814\pi$
$569$			9.37801i			0.393147i	−0.980489	+	0.196573i	$0.937019\pi$
$570$		0			0
$571$	−10.1967			−0.426717			−0.213359	−	0.976974i	$-0.568440\pi$
$571$	−10.1967			−0.426717			−0.213359	+	0.976974i	$0.568440\pi$
$572$		0			0
$573$	−10.2819	+	32.6245i	−0.429532	+	1.36291i
$574$		0			0
$575$	1.72601			0.0719794
$576$		0			0
$577$	14.9762			0.623466			0.311733	−	0.950170i	$-0.399090\pi$
$577$	14.9762			0.623466			0.311733	+	0.950170i	$0.399090\pi$
$578$		0			0
$579$	2.83846	−	9.00647i	0.117962	−	0.374296i
$580$		0			0
$581$	−13.7972			−0.572405
$582$		0			0
$583$		−	16.4140i		−	0.679799i
$584$		0			0
$585$	8.28530	−	11.8391i	0.342555	−	0.489485i
$586$		0			0
$587$			35.1368i			1.45025i	0.688617	+	0.725125i	$0.258218\pi$
$587$			35.1368i			1.45025i	−0.688617	+	0.725125i	$0.741782\pi$
$588$		0			0
$589$		−	2.90087i		−	0.119528i
$590$		0			0
$591$	11.7920	−	37.4162i	0.485059	−	1.53910i
$592$		0			0
$593$		−	11.6209i		−	0.477214i	−0.971116	−	0.238607i	$-0.923309\pi$
$593$		−	11.6209i		−	0.477214i	0.971116	−	0.238607i	$-0.0766908\pi$
$594$		0			0
$595$	−10.0825			−0.413342
$596$		0			0
$597$	−31.1977	−	9.83221i	−1.27684	−	0.402405i
$598$		0			0
$599$	9.69953			0.396312			0.198156	−	0.980170i	$-0.436505\pi$
$599$	9.69953			0.396312			0.198156	+	0.980170i	$0.436505\pi$
$600$		0			0
$601$	0.585768			0.0238940			0.0119470	−	0.999929i	$-0.496197\pi$
$601$	0.585768			0.0238940			0.0119470	+	0.999929i	$0.496197\pi$
$602$		0			0
$603$	−36.5559	−	25.5828i	−1.48867	−	1.04181i
$604$		0			0
$605$	5.16501			0.209988
$606$		0			0
$607$		−	22.9594i		−	0.931894i	−0.884813	−	0.465947i	$-0.845714\pi$
$607$		−	22.9594i		−	0.931894i	0.884813	−	0.465947i	$-0.154286\pi$
$608$		0			0
$609$	−22.2819	−	7.02232i	−0.902908	−	0.284559i
$610$		0			0
$611$		−	36.4030i		−	1.47271i
$612$		0			0
$613$			4.10130i			0.165650i	0.996564	+	0.0828250i	$0.0263943\pi$
$613$			4.10130i			0.165650i	−0.996564	+	0.0828250i	$0.973606\pi$
$614$		0			0
$615$	−5.68305	−	1.79106i	−0.229163	−	0.0722225i
$616$		0			0
$617$		−	14.1493i		−	0.569630i	−0.958583	−	0.284815i	$-0.908068\pi$
$617$		−	14.1493i		−	0.569630i	0.958583	−	0.284815i	$-0.0919322\pi$
$618$		0			0
$619$	10.1108			0.406386			0.203193	−	0.979139i	$-0.434868\pi$
$619$	10.1108			0.406386			0.203193	+	0.979139i	$0.434868\pi$
$620$		0			0
$621$	−7.11317	+	5.46246i	−0.285441	+	0.219201i
$622$		0			0
$623$	−1.59782			−0.0640153
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−4.54799	−	1.43334i	−0.181629	−	0.0572419i
$628$		0			0
$629$	51.4349			2.05084
$630$		0			0
$631$			28.7572i			1.14481i	0.819972	+	0.572404i	$0.193989\pi$
$631$			28.7572i			1.14481i	−0.819972	+	0.572404i	$0.806011\pi$
$632$		0			0
$633$	5.60572	−	17.7870i	0.222807	−	0.706970i
$634$		0			0
$635$		−	16.5763i		−	0.657811i
$636$		0			0
$637$		−	15.8245i		−	0.626988i
$638$		0			0
$639$	22.8181	+	15.9687i	0.902671	+	0.631713i
$640$		0			0
$641$			35.7751i			1.41303i	0.707698	+	0.706515i	$0.249734\pi$
$641$			35.7751i			1.41303i	−0.707698	+	0.706515i	$0.750266\pi$
$642$		0			0
$643$	4.64793			0.183296			0.0916481	−	0.995791i	$-0.470786\pi$
$643$	4.64793			0.183296			0.0916481	+	0.995791i	$0.470786\pi$
$644$		0			0
$645$	0.542104	−	1.72010i	0.0213453	−	0.0677289i
$646$		0			0
$647$	−6.90109			−0.271310			−0.135655	−	0.990756i	$-0.543314\pi$
$647$	−6.90109			−0.271310			−0.135655	+	0.990756i	$0.543314\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	4.25092	−	13.4882i	0.166607	−	0.528645i
$652$		0			0
$653$	16.8181			0.658144			0.329072	−	0.944305i	$-0.393264\pi$
$653$	16.8181			0.658144			0.329072	+	0.944305i	$0.393264\pi$
$654$		0			0
$655$		−	5.61293i		−	0.219315i
$656$		0			0
$657$	−27.7339	−	19.4089i	−1.08200	−	0.757214i
$658$		0			0
$659$		−	15.3712i		−	0.598779i	−0.954131	−	0.299389i	$-0.903217\pi$
$659$		−	15.3712i		−	0.598779i	0.954131	−	0.299389i	$-0.0967830\pi$
$660$		0			0
$661$		−	14.7252i		−	0.572743i	−0.958119	−	0.286372i	$-0.907551\pi$
$661$		−	14.7252i		−	0.572743i	0.958119	−	0.286372i	$-0.0924492\pi$
$662$		0			0
$663$	−13.1186	+	41.6254i	−0.509484	+	1.61660i
$664$		0			0
$665$		−	1.31976i		−	0.0511782i
$666$		0			0
$667$	−12.0791			−0.467705
$668$		0			0
$669$	10.8127	+	3.40771i	0.418043	+	0.131750i
$670$		0			0
$671$	−12.7356			−0.491653
$672$		0			0
$673$	−44.9434			−1.73244			−0.866220	−	0.499663i	$-0.833457\pi$
$673$	−44.9434			−1.73244			−0.866220	+	0.499663i	$0.833457\pi$
$674$		0			0
$675$	−4.12117	+	3.16480i	−0.158624	+	0.121813i
$676$		0			0
$677$	−31.3401			−1.20450			−0.602249	−	0.798308i	$-0.705728\pi$
$677$	−31.3401			−1.20450			−0.602249	+	0.798308i	$0.705728\pi$
$678$		0			0
$679$		−	2.79854i		−	0.107398i
$680$		0			0
$681$	−37.1884	−	11.7202i	−1.42506	−	0.449120i
$682$		0			0
$683$			6.76121i			0.258710i	0.991598	+	0.129355i	$0.0412907\pi$
$683$			6.76121i			0.258710i	−0.991598	+	0.129355i	$0.958709\pi$
$684$		0			0
$685$		−	9.41770i		−	0.359832i
$686$		0			0
$687$	21.2836	+	6.70770i	0.812020	+	0.255915i
$688$		0			0
$689$		−	19.6644i		−	0.749155i
$690$		0			0
$691$	−14.4304			−0.548959			−0.274480	−	0.961593i	$-0.588506\pi$
$691$	−14.4304			−0.548959			−0.274480	+	0.961593i	$0.588506\pi$
$692$		0			0
$693$	−19.0464	−	13.3292i	−0.723514	−	0.506334i
$694$		0			0
$695$	−6.51634			−0.247179
$696$		0			0
$697$	17.9966			0.681670
$698$		0			0
$699$	−17.9254	−	5.64934i	−0.678001	−	0.213677i
$700$		0			0
$701$	−9.83499			−0.371462			−0.185731	−	0.982601i	$-0.559465\pi$
$701$	−9.83499			−0.371462			−0.185731	+	0.982601i	$0.559465\pi$
$702$		0			0
$703$			6.73263i			0.253926i
$704$		0			0
$705$	−3.93468	+	12.4848i	−0.148189	+	0.470204i
$706$		0			0
$707$		−	8.10028i		−	0.304642i
$708$		0			0
$709$		−	20.4277i		−	0.767180i	−0.923503	−	0.383590i	$-0.874688\pi$
$709$		−	20.4277i		−	0.767180i	0.923503	−	0.383590i	$-0.125312\pi$
$710$		0			0
$711$	−15.9141	+	22.7401i	−0.596825	+	0.852819i
$712$		0			0
$713$		−	7.31201i		−	0.273837i
$714$		0			0
$715$	19.3661			0.724251
$716$		0			0
$717$	−3.45201	+	10.9533i	−0.128918	+	0.409057i
$718$		0			0
$719$	28.2338			1.05294			0.526471	−	0.850193i	$-0.323515\pi$
$719$	28.2338			1.05294			0.526471	+	0.850193i	$0.323515\pi$
$720$		0			0
$721$	13.6878			0.509759
$722$		0			0
$723$	−8.30497	+	26.3517i	−0.308865	+	0.980032i
$724$		0			0
$725$	−6.99830			−0.259910
$726$		0			0
$727$			28.1339i			1.04343i	0.853120	+	0.521714i	$0.174707\pi$
$727$			28.1339i			1.04343i	−0.853120	+	0.521714i	$0.825293\pi$
$728$		0			0
$729$	6.96811	−	26.0854i	0.258078	−	0.966124i
$730$		0			0
$731$			5.44707i			0.201467i
$732$		0			0
$733$			10.2296i			0.377840i	0.981993	+	0.188920i	$0.0604986\pi$
$733$			10.2296i			0.377840i	−0.981993	+	0.188920i	$0.939501\pi$
$734$		0			0
$735$	−1.71041	+	5.42716i	−0.0630895	+	0.200184i
$736$		0			0
$737$		−	59.7973i		−	2.20266i
$738$		0			0
$739$	48.0440			1.76733			0.883664	−	0.468121i	$-0.155069\pi$
$739$	48.0440			1.76733			0.883664	+	0.468121i	$0.155069\pi$
$740$		0			0
$741$	−5.44861	−	1.71717i	−0.200160	−	0.0630819i
$742$		0			0
$743$	−35.9667			−1.31949			−0.659745	−	0.751489i	$-0.729336\pi$
$743$	−35.9667			−1.31949			−0.659745	+	0.751489i	$0.729336\pi$
$744$		0			0
$745$	−7.53452			−0.276043
$746$		0			0
$747$	12.3135	−	17.5951i	0.450529	−	0.643772i
$748$		0			0
$749$	−14.9619			−0.546697
$750$		0			0
$751$			26.4357i			0.964654i	0.875991	+	0.482327i	$0.160208\pi$
$751$			26.4357i			0.964654i	−0.875991	+	0.482327i	$0.839792\pi$
$752$		0			0
$753$	25.1864	+	7.93770i	0.917843	+	0.289266i
$754$		0			0
$755$		−	9.41085i		−	0.342496i
$756$		0			0
$757$		−	24.6881i		−	0.897303i	−0.893707	−	0.448652i	$-0.851904\pi$
$757$		−	24.6881i		−	0.897303i	0.893707	−	0.448652i	$-0.148096\pi$
$758$		0			0
$759$	−11.4638	−	3.61290i	−0.416109	−	0.131140i
$760$		0			0
$761$		−	46.5273i		−	1.68661i	−0.537434	−	0.843306i	$-0.680606\pi$
$761$		−	46.5273i		−	1.68661i	0.537434	−	0.843306i	$-0.319394\pi$
$762$		0			0
$763$	39.5311			1.43112
$764$		0			0
$765$	8.99830	−	12.8579i	0.325334	−	0.464878i
$766$		0			0
$767$	4.79210			0.173033
$768$		0			0
$769$	36.8643			1.32936			0.664680	−	0.747129i	$-0.268568\pi$
$769$	36.8643			1.32936			0.664680	+	0.747129i	$0.268568\pi$
$770$		0			0
$771$	−26.7361	−	8.42609i	−0.962876	−	0.303458i
$772$		0			0
$773$	−41.5537			−1.49458			−0.747292	−	0.664496i	$-0.768646\pi$
$773$	−41.5537			−1.49458			−0.747292	+	0.664496i	$0.768646\pi$
$774$		0			0
$775$		−	4.23638i		−	0.152175i
$776$		0			0
$777$	−9.86596	+	31.3048i	−0.353939	+	1.12305i
$778$		0			0
$779$			2.35569i			0.0844013i
$780$		0			0
$781$			37.3254i			1.33561i
$782$		0			0
$783$	28.8412	−	22.1482i	1.03070	−	0.791512i
$784$		0			0
$785$		−	3.49699i		−	0.124813i
$786$		0			0
$787$	48.4300			1.72634			0.863171	−	0.504912i	$-0.168475\pi$
$787$	48.4300			1.72634			0.863171	+	0.504912i	$0.168475\pi$
$788$		0			0
$789$	11.1805	−	35.4758i	0.398036	−	1.26297i
$790$		0			0
$791$	0.415934			0.0147889
$792$		0			0
$793$	−15.2576			−0.541813
$794$		0			0
$795$	−2.12546	+	6.74411i	−0.0753823	+	0.239189i
$796$		0			0
$797$	32.0757			1.13618			0.568090	−	0.822967i	$-0.307683\pi$
$797$	32.0757			1.13618			0.568090	+	0.822967i	$0.307683\pi$
$798$		0			0
$799$		−	39.5357i		−	1.39867i
$800$		0			0
$801$	1.42600	−	2.03765i	0.0503853	−	0.0719968i
$802$		0			0
$803$		−	45.3665i		−	1.60095i
$804$		0			0
$805$		−	3.32663i		−	0.117248i
$806$		0			0
$807$	11.9914	−	38.0489i	0.422118	−	1.33938i
$808$		0			0
$809$		−	1.09505i		−	0.0385001i	−0.999815	−	0.0192500i	$-0.993872\pi$
$809$		−	1.09505i		−	0.0385001i	0.999815	−	0.0192500i	$-0.00612785\pi$
$810$		0			0
$811$	−3.56617			−0.125225			−0.0626126	−	0.998038i	$-0.519943\pi$
$811$	−3.56617			−0.125225			−0.0626126	+	0.998038i	$0.519943\pi$
$812$		0			0
$813$	−26.7474	−	8.42966i	−0.938071	−	0.295641i
$814$		0			0
$815$	16.9553			0.593920
$816$		0			0
$817$	−0.713001			−0.0249447
$818$		0			0
$819$	−22.8181	−	15.9687i	−0.797329	−	0.557992i
$820$		0			0
$821$	47.0327			1.64145			0.820726	−	0.571322i	$-0.193569\pi$
$821$	47.0327			1.64145			0.820726	+	0.571322i	$0.193569\pi$
$822$		0			0
$823$			30.2235i			1.05353i	0.850012	+	0.526763i	$0.176595\pi$
$823$			30.2235i			1.05353i	−0.850012	+	0.526763i	$0.823405\pi$
$824$		0			0
$825$	−6.64180	−	2.09322i	−0.231238	−	0.0728765i
$826$		0			0
$827$		−	31.2229i		−	1.08573i	−0.839821	−	0.542864i	$-0.817340\pi$
$827$		−	31.2229i		−	1.08573i	0.839821	−	0.542864i	$-0.182660\pi$
$828$		0			0
$829$			42.0990i			1.46216i	0.682292	+	0.731080i	$0.260983\pi$
$829$			42.0990i			1.46216i	−0.682292	+	0.731080i	$0.739017\pi$
$830$		0			0
$831$	6.30224	+	1.98620i	0.218622	+	0.0689006i
$832$		0			0
$833$		−	17.1863i		−	0.595468i
$834$		0			0
$835$	−11.3926			−0.394256
$836$		0			0
$837$	13.4073	+	17.4588i	0.463424	+	0.603466i
$838$		0			0
$839$	14.0599			0.485402			0.242701	−	0.970101i	$-0.421967\pi$
$839$	14.0599			0.485402			0.242701	+	0.970101i	$0.421967\pi$
$840$		0			0
$841$	19.9762			0.688834
$842$		0			0
$843$	−46.1677	−	14.5501i	−1.59010	−	0.501134i
$844$		0			0
$845$	10.2011			0.350928
$846$		0			0
$847$		−	9.95482i		−	0.342052i
$848$		0			0
$849$	2.90991	−	9.23316i	0.0998678	−	0.316881i
$850$		0			0
$851$			16.9704i			0.581739i
$852$		0			0
$853$		−	25.1966i		−	0.862716i	−0.902181	−	0.431358i	$-0.858035\pi$
$853$		−	25.1966i		−	0.862716i	0.902181	−	0.431358i	$-0.141965\pi$
$854$		0			0
$855$	1.68305	+	1.17784i	0.0575591	+	0.0402814i
$856$		0			0
$857$			46.7431i			1.59671i	0.602185	+	0.798357i	$0.294297\pi$
$857$			46.7431i			1.59671i	−0.602185	+	0.798357i	$0.705703\pi$
$858$		0			0
$859$	−40.8430			−1.39354			−0.696772	−	0.717293i	$-0.745381\pi$
$859$	−40.8430			−1.39354			−0.696772	+	0.717293i	$0.745381\pi$
$860$		0			0
$861$	−3.45201	+	10.9533i	−0.117644	+	0.373286i
$862$		0			0
$863$	−39.3858			−1.34071			−0.670354	−	0.742042i	$-0.733858\pi$
$863$	−39.3858			−1.34071			−0.670354	+	0.742042i	$0.733858\pi$
$864$		0			0
$865$	2.16501			0.0736126
$866$		0			0
$867$	−5.39687	+	17.1243i	−0.183287	+	0.581572i
$868$		0			0
$869$	−37.1977			−1.26185
$870$		0			0
$871$		−	71.6388i		−	2.42739i
$872$		0			0
$873$	3.56889	+	2.49761i	0.120789	+	0.0845312i
$874$		0			0
$875$		−	1.92736i		−	0.0651565i
$876$		0			0
$877$			57.4498i			1.93994i	0.243218	+	0.969972i	$0.421797\pi$
$877$			57.4498i			1.93994i	−0.243218	+	0.969972i	$0.578203\pi$
$878$		0			0
$879$	−3.12376	+	9.91172i	−0.105362	+	0.334314i
$880$		0			0
$881$			44.2957i			1.49236i	0.665744	+	0.746180i	$0.268114\pi$
$881$			44.2957i			1.49236i	−0.665744	+	0.746180i	$0.731886\pi$
$882$		0			0
$883$	25.7701			0.867231			0.433616	−	0.901098i	$-0.357238\pi$
$883$	25.7701			0.867231			0.433616	+	0.901098i	$0.357238\pi$
$884$		0			0
$885$	−1.64350	−	0.517962i	−0.0552456	−	0.0174111i
$886$		0			0
$887$	−21.4751			−0.721063			−0.360531	−	0.932747i	$-0.617405\pi$
$887$	−21.4751			−0.721063			−0.360531	+	0.932747i	$0.617405\pi$
$888$		0			0
$889$	−31.9485			−1.07152
$890$		0			0
$891$	33.9966	−	12.3934i	1.13893	−	0.415196i
$892$		0			0
$893$	5.17508			0.173177
$894$		0			0
$895$			5.34034i			0.178508i
$896$		0			0
$897$	−13.7339	−	4.32835i	−0.458562	−	0.144520i
$898$		0			0
$899$			29.6474i			0.988798i
$900$		0			0
$901$		−	21.3567i		−	0.711493i
$902$		0			0
$903$	−3.31525	−	1.04483i	−0.110325	−	0.0347697i
$904$		0			0
$905$		−	10.7942i		−	0.358813i
$906$		0			0
$907$	−34.5984			−1.14882			−0.574410	−	0.818568i	$-0.694769\pi$
$907$	−34.5984			−1.14882			−0.574410	+	0.818568i	$0.694769\pi$
$908$		0			0
$909$	10.3300	+	7.22923i	0.342625	+	0.239778i
$910$		0			0
$911$	−20.3074			−0.672815			−0.336407	−	0.941717i	$-0.609212\pi$
$911$	−20.3074			−0.672815			−0.336407	+	0.941717i	$0.609212\pi$
$912$		0			0
$913$	28.7818			0.952537
$914$		0			0
$915$	5.23274	+	1.64914i	0.172989	+	0.0545189i
$916$		0			0
$917$	−10.8181			−0.357246
$918$		0			0
$919$			11.1280i			0.367080i	0.983012	+	0.183540i	$0.0587556\pi$
$919$			11.1280i			0.367080i	−0.983012	+	0.183540i	$0.941244\pi$
$920$		0			0
$921$	2.49397	−	7.91340i	0.0821792	−	0.260755i
$922$		0			0
$923$			44.7168i			1.47187i
$924$		0			0
$925$			9.83221i			0.323281i
$926$		0			0
$927$	−12.2159	+	17.4556i	−0.401222	+	0.573316i
$928$		0			0
$929$			20.3857i			0.668834i	0.942425	+	0.334417i	$0.108539\pi$
$929$			20.3857i			0.668834i	−0.942425	+	0.334417i	$0.891461\pi$
$930$		0			0
$931$	2.24962			0.0737282
$932$		0			0
$933$	6.70464	−	21.2739i	0.219500	−	0.696475i
$934$		0			0
$935$	21.0327			0.687842
$936$		0			0
$937$	35.4418			1.15783			0.578917	−	0.815387i	$-0.303476\pi$
$937$	35.4418			1.15783			0.578917	+	0.815387i	$0.303476\pi$
$938$		0			0
$939$	−10.6219	+	33.7034i	−0.346633	+	1.09987i
$940$		0			0
$941$	13.4402			0.438139			0.219070	−	0.975709i	$-0.429698\pi$
$941$	13.4402			0.438139			0.219070	+	0.975709i	$0.429698\pi$
$942$		0			0
$943$			5.93781i			0.193362i
$944$		0			0
$945$	6.09969	+	7.94297i	0.198423	+	0.258385i
$946$		0			0
$947$		−	4.96195i		−	0.161242i	−0.996745	−	0.0806208i	$-0.974310\pi$
$947$		−	4.96195i		−	0.161242i	0.996745	−	0.0806208i	$-0.0256903\pi$
$948$		0			0
$949$		−	54.3503i		−	1.76428i
$950$		0			0
$951$	−2.78197	+	8.82720i	−0.0902114	+	0.286242i
$952$		0			0
$953$			51.4156i			1.66551i	0.553639	+	0.832757i	$0.313239\pi$
$953$			51.4156i			1.66551i	−0.553639	+	0.832757i	$0.686761\pi$
$954$		0			0
$955$	19.7491			0.639065
$956$		0			0
$957$	46.4813	+	14.6490i	1.50253	+	0.473534i
$958$		0			0
$959$	−18.1513			−0.586135
$960$		0			0
$961$	13.0531			0.421068
$962$		0			0
$963$	13.3530	−	19.0805i	0.430295	−	0.614860i
$964$		0			0
$965$	−5.45201			−0.175506
$966$		0			0
$967$			20.9875i			0.674911i	0.941341	+	0.337456i	$0.109566\pi$
$967$			20.9875i			0.674911i	−0.941341	+	0.337456i	$0.890434\pi$
$968$		0			0
$969$	−5.91749	−	1.86495i	−0.190097	−	0.0599107i
$970$		0			0
$971$			25.4429i			0.816502i	0.912870	+	0.408251i	$0.133861\pi$
$971$			25.4429i			0.816502i	−0.912870	+	0.408251i	$0.866139\pi$
$972$		0			0
$973$			12.5593i			0.402633i
$974$		0			0
$975$	−7.95705	−	2.50773i	−0.254829	−	0.0803116i
$976$		0			0
$977$			26.6495i			0.852593i	0.904584	+	0.426296i	$0.140182\pi$
$977$			26.6495i			0.852593i	−0.904584	+	0.426296i	$0.859818\pi$
$978$		0			0
$979$	3.33314			0.106528
$980$		0			0
$981$	−35.2802	+	50.4128i	−1.12641	+	1.60956i
$982$		0			0
$983$	47.0887			1.50190			0.750948	−	0.660361i	$-0.229597\pi$
$983$	47.0887			1.50190			0.750948	+	0.660361i	$0.229597\pi$
$984$		0			0
$985$	−22.6497			−0.721679
$986$		0			0
$987$	24.0626	+	7.58353i	0.765922	+	0.241387i
$988$		0			0
$989$	−1.79721			−0.0571479
$990$		0			0
$991$		−	26.3879i		−	0.838238i	−0.907931	−	0.419119i	$-0.862339\pi$
$991$		−	26.3879i		−	0.838238i	0.907931	−	0.419119i	$-0.137661\pi$
$992$		0			0
$993$	−13.1039	+	41.5787i	−0.415839	+	1.31946i
$994$		0			0
$995$			18.8853i			0.598705i
$996$		0			0
$997$			10.2047i			0.323186i	0.986858	+	0.161593i	$0.0516631\pi$
$997$			10.2047i			0.323186i	−0.986858	+	0.161593i	$0.948337\pi$
$998$		0			0
$999$	−31.1170	−	40.5202i	−0.984497	−	1.28200i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.b.a.431.5		8
3.2	odd	2		480.2.b.b.431.6		8
4.3	odd	2		120.2.b.b.11.8	yes	8
5.2	odd	4		2400.2.m.c.1199.1		16
5.3	odd	4		2400.2.m.c.1199.16		16
5.4	even	2		2400.2.b.e.2351.4		8
8.3	odd	2		480.2.b.b.431.5		8
8.5	even	2		120.2.b.a.11.2	yes	8
12.11	even	2		120.2.b.a.11.1	✓	8
15.2	even	4		2400.2.m.d.1199.15		16
15.8	even	4		2400.2.m.d.1199.2		16
15.14	odd	2		2400.2.b.f.2351.3		8
20.3	even	4		600.2.m.d.299.9		16
20.7	even	4		600.2.m.d.299.8		16
20.19	odd	2		600.2.b.e.251.1		8
24.5	odd	2		120.2.b.b.11.7	yes	8
24.11	even	2	inner	480.2.b.a.431.6		8
40.3	even	4		2400.2.m.d.1199.16		16
40.13	odd	4		600.2.m.c.299.10		16
40.19	odd	2		2400.2.b.f.2351.4		8
40.27	even	4		2400.2.m.d.1199.1		16
40.29	even	2		600.2.b.f.251.7		8
40.37	odd	4		600.2.m.c.299.7		16
60.23	odd	4		600.2.m.c.299.8		16
60.47	odd	4		600.2.m.c.299.9		16
60.59	even	2		600.2.b.f.251.8		8
120.29	odd	2		600.2.b.e.251.2		8
120.53	even	4		600.2.m.d.299.7		16
120.59	even	2		2400.2.b.e.2351.3		8
120.77	even	4		600.2.m.d.299.10		16
120.83	odd	4		2400.2.m.c.1199.2		16
120.107	odd	4		2400.2.m.c.1199.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.b.a.11.1	✓	8	12.11	even	2
120.2.b.a.11.2	yes	8	8.5	even	2
120.2.b.b.11.7	yes	8	24.5	odd	2
120.2.b.b.11.8	yes	8	4.3	odd	2
480.2.b.a.431.5		8	1.1	even	1	trivial
480.2.b.a.431.6		8	24.11	even	2	inner
480.2.b.b.431.5		8	8.3	odd	2
480.2.b.b.431.6		8	3.2	odd	2
600.2.b.e.251.1		8	20.19	odd	2
600.2.b.e.251.2		8	120.29	odd	2
600.2.b.f.251.7		8	40.29	even	2
600.2.b.f.251.8		8	60.59	even	2
600.2.m.c.299.7		16	40.37	odd	4
600.2.m.c.299.8		16	60.23	odd	4
600.2.m.c.299.9		16	60.47	odd	4
600.2.m.c.299.10		16	40.13	odd	4
600.2.m.d.299.7		16	120.53	even	4
600.2.m.d.299.8		16	20.7	even	4
600.2.m.d.299.9		16	20.3	even	4
600.2.m.d.299.10		16	120.77	even	4
2400.2.b.e.2351.3		8	120.59	even	2
2400.2.b.e.2351.4		8	5.4	even	2
2400.2.b.f.2351.3		8	15.14	odd	2
2400.2.b.f.2351.4		8	40.19	odd	2
2400.2.m.c.1199.1		16	5.2	odd	4
2400.2.m.c.1199.2		16	120.83	odd	4
2400.2.m.c.1199.15		16	120.107	odd	4
2400.2.m.c.1199.16		16	5.3	odd	4
2400.2.m.d.1199.1		16	40.27	even	4
2400.2.m.d.1199.2		16	15.8	even	4
2400.2.m.d.1199.15		16	15.2	even	4
2400.2.m.d.1199.16		16	40.3	even	4

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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.520627 - 1.65195i) q^{3} -1.00000 q^{5} +1.92736i q^{7} +(-2.45790 - 1.72010i) q^{9} +O(q^{10})\)	\(q+(0.520627 - 1.65195i) q^{3} -1.00000 q^{5} +1.92736i q^{7} +(-2.45790 - 1.72010i) q^{9} -4.02057i q^{11} -4.81675i q^{13} +(-0.520627 + 1.65195i) q^{15} -5.23126i q^{17} +0.684753 q^{19} +(3.18390 + 1.00343i) q^{21} +1.72601 q^{23} +1.00000 q^{25} +(-4.12117 + 3.16480i) q^{27} -6.99830 q^{29} -4.23638i q^{31} +(-6.64180 - 2.09322i) q^{33} -1.92736i q^{35} +9.83221i q^{37} +(-7.95705 - 2.50773i) q^{39} +3.44020i q^{41} -1.04125 q^{43} +(2.45790 + 1.72010i) q^{45} +7.55759 q^{47} +3.28530 q^{49} +(-8.64180 - 2.72353i) q^{51} +4.08251 q^{53} +4.02057i q^{55} +(0.356500 - 1.13118i) q^{57} +0.994883i q^{59} -3.16761i q^{61} +(3.31525 - 4.73724i) q^{63} +4.81675i q^{65} +14.8728 q^{67} +(0.898604 - 2.85128i) q^{69} -9.28360 q^{71} +11.2836 q^{73} +(0.520627 - 1.65195i) q^{75} +7.74908 q^{77} -9.25184i q^{79} +(3.08251 + 8.45566i) q^{81} +7.15862i q^{83} +5.23126i q^{85} +(-3.64350 + 11.5609i) q^{87} +0.829022i q^{89} +9.28360 q^{91} +(-6.99830 - 2.20557i) q^{93} -0.684753 q^{95} -1.45201 q^{97} +(-6.91579 + 9.88215i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 4 q^{19} - 4 q^{21} + 4 q^{23} + 8 q^{25} + 12 q^{27} - 4 q^{33} - 16 q^{39} - 28 q^{47} - 16 q^{49} - 20 q^{51} + 16 q^{53} - 4 q^{57} + 28 q^{63} + 32 q^{67} + 20 q^{69} + 24 q^{71} - 8 q^{73} + 8 q^{81} - 36 q^{87} - 24 q^{91} - 4 q^{95} + 8 q^{97} - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 4 * q^19 - 4 * q^21 + 4 * q^23 + 8 * q^25 + 12 * q^27 - 4 * q^33 - 16 * q^39 - 28 * q^47 - 16 * q^49 - 20 * q^51 + 16 * q^53 - 4 * q^57 + 28 * q^63 + 32 * q^67 + 20 * q^69 + 24 * q^71 - 8 * q^73 + 8 * q^81 - 36 * q^87 - 24 * q^91 - 4 * q^95 + 8 * q^97 - 16 * q^99

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