LMFDB - Embedded newform 2340.2.fo.a.1241.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(1241,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.1241"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.fo (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$10$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			1241.7
Character	$\chi$	$=$	2340.1241
Dual form			2340.2.fo.a.1961.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(-0.420712 + 1.57012i) q^{7} +(-0.0781861 - 0.291794i) q^{11} +(0.635992 - 3.54902i) q^{13} +(-1.26811 - 2.19643i) q^{17} +(-4.60232 - 1.23319i) q^{19} +(2.25419 - 3.90437i) q^{23} +1.00000i q^{25} +(-8.42481 - 4.86407i) q^{29} +(6.01033 - 6.01033i) q^{31} +(-1.40773 + 0.812752i) q^{35} +(9.00020 - 2.41160i) q^{37} +(3.76339 - 1.00840i) q^{41} +(-2.10913 + 1.21771i) q^{43} +(-4.45078 + 4.45078i) q^{47} +(3.77391 + 2.17887i) q^{49} -5.50999i q^{53} +(0.151044 - 0.261616i) q^{55} +(6.04535 + 1.61985i) q^{59} +(-1.78181 - 3.08619i) q^{61} +(2.95925 - 2.05982i) q^{65} +(-0.318095 - 1.18715i) q^{67} +(-1.47669 + 5.51107i) q^{71} +(1.50185 + 1.50185i) q^{73} +0.491045 q^{77} +13.6526 q^{79} +(-5.60322 - 5.60322i) q^{83} +(0.656423 - 2.44980i) q^{85} +(-3.34186 - 12.4720i) q^{89} +(5.30480 + 2.49169i) q^{91} +(-2.38234 - 4.12633i) q^{95} +(12.5306 + 3.35757i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) $ 40 * q - 12 * q^13 - 20 * q^19 - 28 * q^31 + 12 * q^49 + 16 * q^55 + 48 * q^61 + 16 * q^67 - 20 * q^73 + 80 * q^79 + 20 * q^85 + 4 * q^91 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{5}{12}\right)$	$1$	$-1$

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
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	−0.420712	+	1.57012i	−0.159014	+	0.593448i	0.839714	+	0.543029i	$0.182723\pi$
$7$	−0.420712	+	1.57012i	−0.159014	+	0.593448i	−0.998728	+	0.0504196i	$0.983944\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.0781861	−	0.291794i	−0.0235740	−	0.0879793i	0.953137	−	0.302540i	$-0.0978346\pi$
$11$	−0.0781861	−	0.291794i	−0.0235740	−	0.0879793i	−0.976711	+	0.214561i	$0.931168\pi$
$12$		0			0
$13$	0.635992	−	3.54902i	0.176392	−	0.984320i
$13$	0.635992	−	3.54902i	0.176392	−	0.984320i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.26811	−	2.19643i	−0.307562	−	0.532713i	0.670266	−	0.742121i	$-0.266180\pi$
$17$	−1.26811	−	2.19643i	−0.307562	−	0.532713i	−0.977828	+	0.209407i	$0.932847\pi$
$18$		0			0
$19$	−4.60232	−	1.23319i	−1.05584	−	0.282913i	−0.311180	−	0.950351i	$-0.600724\pi$
$19$	−4.60232	−	1.23319i	−1.05584	−	0.282913i	−0.744665	+	0.667438i	$0.767391\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.25419	−	3.90437i	0.470031	−	0.814117i	−0.529382	−	0.848383i	$-0.677576\pi$
$23$	2.25419	−	3.90437i	0.470031	−	0.814117i	0.999413	+	0.0342667i	$0.0109096\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.42481	−	4.86407i	−1.56445	−	0.903235i	−0.996798	−	0.0799649i	$-0.974519\pi$
$29$	−8.42481	−	4.86407i	−1.56445	−	0.903235i	−0.567651	−	0.823270i	$-0.692148\pi$
$30$		0			0
$31$	6.01033	−	6.01033i	1.07949	−	1.07949i	0.0829327	−	0.996555i	$-0.473571\pi$
$31$	6.01033	−	6.01033i	1.07949	−	1.07949i	0.996555	−	0.0829327i	$-0.0264287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.40773	+	0.812752i	−0.237949	+	0.137380i
$36$		0			0
$37$	9.00020	−	2.41160i	1.47962	−	0.396464i	0.573404	−	0.819272i	$-0.305622\pi$
$37$	9.00020	−	2.41160i	1.47962	−	0.396464i	0.906219	+	0.422809i	$0.138956\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.76339	−	1.00840i	0.587743	−	0.157485i	0.0473216	−	0.998880i	$-0.484931\pi$
$41$	3.76339	−	1.00840i	0.587743	−	0.157485i	0.540422	+	0.841394i	$0.318265\pi$
$42$		0			0
$43$	−2.10913	+	1.21771i	−0.321640	+	0.185699i	−0.652123	−	0.758113i	$-0.726122\pi$
$43$	−2.10913	+	1.21771i	−0.321640	+	0.185699i	0.330483	+	0.943812i	$0.392788\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.45078	+	4.45078i	−0.649213	+	0.649213i	−0.952803	−	0.303590i	$-0.901815\pi$
$47$	−4.45078	+	4.45078i	−0.649213	+	0.649213i	0.303590	+	0.952803i	$0.401815\pi$
$48$		0			0
$49$	3.77391	+	2.17887i	0.539130	+	0.311267i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.50999i		−	0.756855i	−0.925631	−	0.378428i	$-0.876465\pi$
$53$		−	5.50999i		−	0.756855i	0.925631	−	0.378428i	$-0.123535\pi$
$54$		0			0
$55$	0.151044	−	0.261616i	0.0203668	−	0.0352763i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.04535	+	1.61985i	0.787038	+	0.210886i	0.629885	−	0.776688i	$-0.283102\pi$
$59$	6.04535	+	1.61985i	0.787038	+	0.210886i	0.157153	+	0.987574i	$0.449769\pi$
$60$		0			0
$61$	−1.78181	−	3.08619i	−0.228138	−	0.395146i	0.729118	−	0.684388i	$-0.239930\pi$
$61$	−1.78181	−	3.08619i	−0.228138	−	0.395146i	−0.957256	+	0.289241i	$0.906597\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.95925	−	2.05982i	0.367049	−	0.255489i
$66$		0			0
$67$	−0.318095	−	1.18715i	−0.0388614	−	0.145033i	0.943769	−	0.330605i	$-0.107253\pi$
$67$	−0.318095	−	1.18715i	−0.0388614	−	0.145033i	−0.982631	+	0.185572i	$0.940586\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.47669	+	5.51107i	−0.175250	+	0.654043i	0.821258	+	0.570556i	$0.193272\pi$
$71$	−1.47669	+	5.51107i	−0.175250	+	0.654043i	−0.996509	+	0.0834870i	$0.973394\pi$
$72$		0			0
$73$	1.50185	+	1.50185i	0.175778	+	0.175778i	0.789513	−	0.613734i	$-0.210333\pi$
$73$	1.50185	+	1.50185i	0.175778	+	0.175778i	−0.613734	+	0.789513i	$0.710333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.491045			0.0559598
$78$		0			0
$79$	13.6526			1.53604			0.768019	−	0.640427i	$-0.221243\pi$
$79$	13.6526			1.53604			0.768019	+	0.640427i	$0.221243\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.60322	−	5.60322i	−0.615033	−	0.615033i	0.329220	−	0.944253i	$-0.393214\pi$
$83$	−5.60322	−	5.60322i	−0.615033	−	0.615033i	−0.944253	+	0.329220i	$0.893214\pi$
$84$		0			0
$85$	0.656423	−	2.44980i	0.0711990	−	0.265718i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.34186	−	12.4720i	−0.354236	−	1.32203i	−0.881443	−	0.472290i	$-0.843427\pi$
$89$	−3.34186	−	12.4720i	−0.354236	−	1.32203i	0.527207	−	0.849737i	$-0.323239\pi$
$90$		0			0
$91$	5.30480	+	2.49169i	0.556094	+	0.261200i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.38234	−	4.12633i	−0.244423	−	0.423352i
$96$		0			0
$97$	12.5306	+	3.35757i	1.27229	+	0.340909i	0.830908	−	0.556409i	$-0.187821\pi$
$97$	12.5306	+	3.35757i	1.27229	+	0.340909i	0.441383	+	0.897319i	$0.354488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.14827	−	8.91706i	0.512272	−	0.887280i	−0.487627	−	0.873052i	$-0.662137\pi$
$101$	5.14827	−	8.91706i	0.512272	−	0.887280i	0.999899	−	0.0142285i	$-0.00452921\pi$
$102$		0			0
$103$			7.50125i			0.739120i	0.929207	+	0.369560i	$0.120492\pi$
$103$			7.50125i			0.739120i	−0.929207	+	0.369560i	$0.879508\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.759642	−	0.438579i	−0.0734373	−	0.0423991i	0.462832	−	0.886446i	$-0.346833\pi$
$107$	−0.759642	−	0.438579i	−0.0734373	−	0.0423991i	−0.536269	+	0.844047i	$0.680167\pi$
$108$		0			0
$109$	8.00205	−	8.00205i	0.766458	−	0.766458i	−0.211023	−	0.977481i	$-0.567680\pi$
$109$	8.00205	−	8.00205i	0.766458	−	0.766458i	0.977481	+	0.211023i	$0.0676796\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.58293	+	4.37801i	−0.713342	+	0.411848i	−0.812297	−	0.583244i	$-0.801783\pi$
$113$	−7.58293	+	4.37801i	−0.713342	+	0.411848i	0.0989551	+	0.995092i	$0.468450\pi$
$114$		0			0
$115$	4.35476	−	1.16685i	0.406083	−	0.108810i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.98217	−	1.06702i	0.365045	−	0.0978134i
$120$		0			0
$121$	9.44725	−	5.45437i	0.858841	−	0.495852i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	−1.03801	−	0.599294i	−0.0921083	−	0.0531788i	0.453238	−	0.891389i	$-0.350269\pi$
$127$	−1.03801	−	0.599294i	−0.0921083	−	0.0531788i	−0.545347	+	0.838211i	$0.683602\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			9.59841i			0.838617i	0.907844	+	0.419309i	$0.137727\pi$
$131$			9.59841i			0.838617i	−0.907844	+	0.419309i	$0.862273\pi$
$132$		0			0
$133$	3.87250	−	6.70736i	0.335788	−	0.581602i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.6056	−	4.98537i	−1.58959	−	0.425929i	−0.647711	−	0.761886i	$-0.724273\pi$
$137$	−18.6056	−	4.98537i	−1.58959	−	0.425929i	−0.941877	+	0.335957i	$0.890940\pi$
$138$		0			0
$139$	2.45541	+	4.25289i	0.208265	+	0.360726i	0.951168	−	0.308673i	$-0.0998850\pi$
$139$	2.45541	+	4.25289i	0.208265	+	0.360726i	−0.742903	+	0.669399i	$0.766552\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.08531	+	0.0919048i	−0.0907581	+	0.00768547i
$144$		0			0
$145$	−2.51783	−	9.39666i	−0.209094	−	0.780350i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.09098	+	4.07160i	−0.0893767	+	0.333558i	−0.996107	−	0.0881526i	$-0.971904\pi$
$149$	−1.09098	+	4.07160i	−0.0893767	+	0.333558i	0.906730	+	0.421711i	$0.138570\pi$
$150$		0			0
$151$	−0.324656	−	0.324656i	−0.0264202	−	0.0264202i	0.693773	−	0.720193i	$-0.255947\pi$
$151$	−0.324656	−	0.324656i	−0.0264202	−	0.0264202i	−0.720193	+	0.693773i	$0.755947\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.49990			0.682728
$156$		0			0
$157$	−24.2736			−1.93725			−0.968623	−	0.248535i	$-0.920051\pi$
$157$	−24.2736			−1.93725			−0.968623	+	0.248535i	$0.920051\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.18195	+	5.18195i	0.408395	+	0.408395i
$162$		0			0
$163$	4.58873	−	17.1254i	0.359417	−	1.34136i	−0.515418	−	0.856939i	$-0.672363\pi$
$163$	4.58873	−	17.1254i	0.359417	−	1.34136i	0.874834	−	0.484422i	$-0.160970\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.01928	−	22.4643i	−0.465786	−	1.73834i	−0.654270	−	0.756261i	$-0.727024\pi$
$167$	−6.01928	−	22.4643i	−0.465786	−	1.73834i	0.188484	−	0.982076i	$-0.439643\pi$
$168$		0			0
$169$	−12.1910	−	4.51429i	−0.937771	−	0.347253i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.800773	−	1.38698i	−0.0608816	−	0.105450i	0.833978	−	0.551798i	$-0.186058\pi$
$173$	−0.800773	−	1.38698i	−0.0608816	−	0.105450i	−0.894860	+	0.446347i	$0.852725\pi$
$174$		0			0
$175$	−1.57012	−	0.420712i	−0.118690	−	0.0318028i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.30134	+	16.1104i	−0.695215	+	1.20415i	0.274893	+	0.961475i	$0.411358\pi$
$179$	−9.30134	+	16.1104i	−0.695215	+	1.20415i	−0.970108	+	0.242673i	$0.921976\pi$
$180$		0			0
$181$			0.370426i			0.0275335i	0.999905	+	0.0137668i	$0.00438224\pi$
$181$			0.370426i			0.0275335i	−0.999905	+	0.0137668i	$0.995618\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.06936	+	4.65885i	0.593271	+	0.342525i
$186$		0			0
$187$	−0.541758	+	0.541758i	−0.0396173	+	0.0396173i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.0379	−	9.83683i	1.23282	−	0.711768i	0.265202	−	0.964193i	$-0.414561\pi$
$191$	17.0379	−	9.83683i	1.23282	−	0.711768i	0.967617	+	0.252424i	$0.0812279\pi$
$192$		0			0
$193$	15.0040	−	4.02031i	1.08001	−	0.289388i	0.325411	−	0.945573i	$-0.394497\pi$
$193$	15.0040	−	4.02031i	1.08001	−	0.289388i	0.754600	+	0.656185i	$0.227831\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.3449	+	3.84371i	−1.02203	+	0.273853i	−0.730649	−	0.682754i	$-0.760782\pi$
$197$	−14.3449	+	3.84371i	−1.02203	+	0.273853i	−0.291383	+	0.956606i	$0.594116\pi$
$198$		0			0
$199$	−11.8297	+	6.82985i	−0.838582	+	0.484156i	−0.856782	−	0.515679i	$-0.827540\pi$
$199$	−11.8297	+	6.82985i	−0.838582	+	0.484156i	0.0181999	+	0.999834i	$0.494206\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11.1816	−	11.1816i	0.784792	−	0.784792i
$204$		0			0
$205$	3.37417	+	1.94808i	0.235662	+	0.136059i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.43935i			0.0995619i
$210$		0			0
$211$	8.58519	−	14.8700i	0.591029	−	1.02369i	−0.403065	−	0.915171i	$-0.632055\pi$
$211$	8.58519	−	14.8700i	0.591029	−	1.02369i	0.994094	−	0.108521i	$-0.0346115\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.35243	−	0.630333i	−0.160435	−	0.0429883i
$216$		0			0
$217$	6.90831	+	11.9655i	0.468967	+	0.812274i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.60168	+	3.10363i	−0.578612	+	0.208773i
$222$		0			0
$223$	6.36076	+	23.7387i	0.425948	+	1.58966i	0.761843	+	0.647761i	$0.224295\pi$
$223$	6.36076	+	23.7387i	0.425948	+	1.58966i	−0.335895	+	0.941899i	$0.609039\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.59352	−	5.94710i	0.105766	−	0.394723i	−0.892665	−	0.450720i	$-0.851167\pi$
$227$	1.59352	−	5.94710i	0.105766	−	0.394723i	0.998431	+	0.0559975i	$0.0178339\pi$
$228$		0			0
$229$	2.54915	+	2.54915i	0.168453	+	0.168453i	0.786299	−	0.617846i	$-0.211995\pi$
$229$	2.54915	+	2.54915i	0.168453	+	0.168453i	−0.617846	+	0.786299i	$0.711995\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−27.0259			−1.77053			−0.885263	−	0.465091i	$-0.846022\pi$
$233$	−27.0259			−1.77053			−0.885263	+	0.465091i	$0.846022\pi$
$234$		0			0
$235$	−6.29435			−0.410598
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.65404	−	8.65404i	−0.559783	−	0.559783i	0.369462	−	0.929246i	$-0.379542\pi$
$239$	−8.65404	−	8.65404i	−0.559783	−	0.559783i	−0.929246	+	0.369462i	$0.879542\pi$
$240$		0			0
$241$	7.39942	−	27.6150i	0.476639	−	1.77884i	−0.138436	−	0.990371i	$-0.544208\pi$
$241$	7.39942	−	27.6150i	0.476639	−	1.77884i	0.615075	−	0.788469i	$-0.289126\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.12786	+	4.20925i	0.0720566	+	0.268919i
$246$		0			0
$247$	−7.30364	+	15.5494i	−0.464719	+	0.989385i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.3346	+	19.6321i	0.715433	+	1.23917i	0.962792	+	0.270242i	$0.0871038\pi$
$251$	11.3346	+	19.6321i	0.715433	+	1.23917i	−0.247360	+	0.968924i	$0.579563\pi$
$252$		0			0
$253$	−1.31552	−	0.352492i	−0.0827059	−	0.0221610i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.2115	−	21.1510i	0.761733	−	1.31936i	−0.180224	−	0.983626i	$-0.557682\pi$
$257$	12.2115	−	21.1510i	0.761733	−	1.31936i	0.941957	−	0.335735i	$-0.108985\pi$
$258$		0			0
$259$			15.1459i			0.941123i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11.9156	+	6.87949i	0.734749	+	0.424207i	0.820157	−	0.572139i	$-0.193886\pi$
$263$	11.9156	+	6.87949i	0.734749	+	0.424207i	−0.0854082	+	0.996346i	$0.527219\pi$
$264$		0			0
$265$	3.89615	−	3.89615i	0.239339	−	0.239339i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.79742	+	1.61509i	−0.170562	+	0.0984739i	−0.582851	−	0.812579i	$-0.698063\pi$
$269$	−2.79742	+	1.61509i	−0.170562	+	0.0984739i	0.412289	+	0.911053i	$0.364729\pi$
$270$		0			0
$271$	−9.32074	+	2.49749i	−0.566195	+	0.151711i	−0.530551	−	0.847653i	$-0.678015\pi$
$271$	−9.32074	+	2.49749i	−0.566195	+	0.151711i	−0.0356439	+	0.999365i	$0.511348\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.291794	−	0.0781861i	0.0175959	−	0.00471480i
$276$		0			0
$277$	−3.39445	+	1.95978i	−0.203952	+	0.117752i	−0.598498	−	0.801124i	$-0.704235\pi$
$277$	−3.39445	+	1.95978i	−0.203952	+	0.117752i	0.394545	+	0.918877i	$0.370902\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.0443	−	15.0443i	0.897466	−	0.897466i	−0.0977457	−	0.995211i	$-0.531163\pi$
$281$	15.0443	−	15.0443i	0.897466	−	0.897466i	0.995211	+	0.0977457i	$0.0311632\pi$
$282$		0			0
$283$	−12.3539	−	7.13255i	−0.734365	−	0.423986i	0.0856516	−	0.996325i	$-0.472703\pi$
$283$	−12.3539	−	7.13255i	−0.734365	−	0.423986i	−0.820017	+	0.572339i	$0.806036\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			6.33321i			0.373838i
$288$		0			0
$289$	5.28379	−	9.15179i	0.310811	−	0.538341i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.99230	+	2.67743i	0.583756	+	0.156417i	0.538598	−	0.842563i	$-0.318954\pi$
$293$	9.99230	+	2.67743i	0.583756	+	0.156417i	0.0451582	+	0.998980i	$0.485621\pi$
$294$		0			0
$295$	3.12931	+	5.42012i	0.182195	+	0.315571i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.4230	−	10.4833i	−0.718442	−	0.606264i
$300$		0			0
$301$	−1.02461	−	3.82389i	−0.0590575	−	0.220405i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.922335	−	3.44220i	0.0528127	−	0.197100i
$306$		0			0
$307$	7.31436	+	7.31436i	0.417452	+	0.417452i	0.884325	−	0.466872i	$-0.154619\pi$
$307$	7.31436	+	7.31436i	0.417452	+	0.417452i	−0.466872	+	0.884325i	$0.654619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	31.4363			1.78259			0.891296	−	0.453422i	$-0.149797\pi$
$311$	31.4363			1.78259			0.891296	+	0.453422i	$0.149797\pi$
$312$		0			0
$313$	−8.29498			−0.468860			−0.234430	−	0.972133i	$-0.575322\pi$
$313$	−8.29498			−0.468860			−0.234430	+	0.972133i	$0.575322\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.133879	+	0.133879i	0.00751938	+	0.00751938i	0.710856	−	0.703337i	$-0.248308\pi$
$317$	0.133879	+	0.133879i	0.00751938	+	0.00751938i	−0.703337	+	0.710856i	$0.748308\pi$
$318$		0			0
$319$	−0.760605	+	2.83862i	−0.0425857	+	0.158932i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.12764	+	11.6725i	0.174026	+	0.649476i
$324$		0			0
$325$	3.54902	+	0.635992i	0.196864	+	0.0352785i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.11575	−	8.86073i	−0.282040	−	0.488508i
$330$		0			0
$331$	−12.3195	−	3.30101i	−0.677143	−	0.181440i	−0.0961728	−	0.995365i	$-0.530660\pi$
$331$	−12.3195	−	3.30101i	−0.677143	−	0.181440i	−0.580970	+	0.813925i	$0.697327\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.614512	−	1.06437i	0.0335744	−	0.0581525i
$336$		0			0
$337$		−	12.4924i		−	0.680504i	−0.940334	−	0.340252i	$-0.889488\pi$
$337$		−	12.4924i		−	0.680504i	0.940334	−	0.340252i	$-0.110512\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.22371	−	1.28386i	−0.120420	−	0.0695248i
$342$		0			0
$343$	−13.0546	+	13.0546i	−0.704884	+	0.704884i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.72792	+	5.61642i	−0.522222	+	0.301505i	−0.737843	−	0.674972i	$-0.764156\pi$
$347$	−9.72792	+	5.61642i	−0.522222	+	0.301505i	0.215621	+	0.976477i	$0.430822\pi$
$348$		0			0
$349$	3.03532	−	0.813312i	0.162477	−	0.0435356i	−0.176664	−	0.984271i	$-0.556530\pi$
$349$	3.03532	−	0.813312i	0.162477	−	0.0435356i	0.339141	+	0.940736i	$0.389864\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.19720	−	2.19643i	0.436293	−	0.116904i	−0.0339857	−	0.999422i	$-0.510820\pi$
$353$	8.19720	−	2.19643i	0.436293	−	0.116904i	0.470279	+	0.882518i	$0.344153\pi$
$354$		0			0
$355$	−4.94109	+	2.85274i	−0.262246	+	0.151408i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.9016	+	12.9016i	−0.680920	+	0.680920i	−0.960208	−	0.279287i	$-0.909902\pi$
$359$	−12.9016	+	12.9016i	−0.680920	+	0.680920i	0.279287	+	0.960208i	$0.409902\pi$
$360$		0			0
$361$	3.20611	+	1.85105i	0.168743	+	0.0974236i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			2.12394i			0.111172i
$366$		0			0
$367$	−16.0090	+	27.7284i	−0.835663	+	1.44741i	0.0578258	+	0.998327i	$0.481583\pi$
$367$	−16.0090	+	27.7284i	−0.835663	+	1.44741i	−0.893489	+	0.449085i	$0.851750\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.65133	+	2.31812i	0.449155	+	0.120351i
$372$		0			0
$373$	6.99741	+	12.1199i	0.362312	+	0.627543i	0.988341	−	0.152257i	$-0.0486541\pi$
$373$	6.99741	+	12.1199i	0.362312	+	0.627543i	−0.626029	+	0.779800i	$0.715321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.6208	+	26.8063i	−1.16503	+	1.38059i
$378$		0			0
$379$	7.46212	+	27.8490i	0.383303	+	1.43051i	0.840824	+	0.541309i	$0.182071\pi$
$379$	7.46212	+	27.8490i	0.383303	+	1.43051i	−0.457521	+	0.889199i	$0.651262\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.79864	+	29.1049i	−0.398492	+	1.48719i	0.417258	+	0.908788i	$0.362991\pi$
$383$	−7.79864	+	29.1049i	−0.398492	+	1.48719i	−0.815750	+	0.578405i	$0.803675\pi$
$384$		0			0
$385$	0.347221	+	0.347221i	0.0176960	+	0.0176960i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.4861			0.937285			0.468642	−	0.883388i	$-0.344743\pi$
$389$	18.4861			0.937285			0.468642	+	0.883388i	$0.344743\pi$
$390$		0			0
$391$	−11.4342			−0.578254
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.65385	+	9.65385i	0.485738	+	0.485738i
$396$		0			0
$397$	−2.13821	+	7.97990i	−0.107314	+	0.400499i	−0.998597	−	0.0529463i	$-0.983139\pi$
$397$	−2.13821	+	7.97990i	−0.107314	+	0.400499i	0.891284	+	0.453446i	$0.149805\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.18126	+	4.40851i	0.0589892	+	0.220151i	0.989128	−	0.147058i	$-0.0469803\pi$
$401$	1.18126	+	4.40851i	0.0589892	+	0.220151i	−0.930139	+	0.367208i	$0.880314\pi$
$402$		0			0
$403$	−17.5082	−	25.1533i	−0.872148	−	1.25297i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.40738	−	2.43765i	−0.0697613	−	0.120830i
$408$		0			0
$409$	−22.6073	−	6.05760i	−1.11786	−	0.299529i	−0.347842	−	0.937553i	$-0.613085\pi$
$409$	−22.6073	−	6.05760i	−1.11786	−	0.299529i	−0.770016	+	0.638024i	$0.779752\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.08670	+	8.81042i	−0.250300	+	0.433533i
$414$		0			0
$415$		−	7.92415i		−	0.388981i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.3201	−	5.95834i	−0.504172	−	0.291084i	0.226263	−	0.974066i	$-0.427349\pi$
$419$	−10.3201	−	5.95834i	−0.504172	−	0.291084i	−0.730435	+	0.682982i	$0.760683\pi$
$420$		0			0
$421$	−2.93825	+	2.93825i	−0.143202	+	0.143202i	−0.775073	−	0.631871i	$-0.782287\pi$
$421$	−2.93825	+	2.93825i	−0.143202	+	0.143202i	0.631871	+	0.775073i	$0.282287\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.19643	−	1.26811i	0.106543	−	0.0615124i
$426$		0			0
$427$	5.59531	−	1.49926i	0.270776	−	0.0725542i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.5055	−	5.49445i	0.987718	−	0.264658i	0.271426	−	0.962459i	$-0.412505\pi$
$431$	20.5055	−	5.49445i	0.987718	−	0.264658i	0.716291	+	0.697801i	$0.245838\pi$
$432$		0			0
$433$	−9.34080	+	5.39291i	−0.448890	+	0.259167i	−0.707361	−	0.706852i	$-0.750115\pi$
$433$	−9.34080	+	5.39291i	−0.448890	+	0.259167i	0.258471	+	0.966019i	$0.416781\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.1893	+	15.1893i	−0.726603	+	0.726603i
$438$		0			0
$439$	−9.89837	−	5.71482i	−0.472423	−	0.272754i	0.244830	−	0.969566i	$-0.421268\pi$
$439$	−9.89837	−	5.71482i	−0.472423	−	0.272754i	−0.717254	+	0.696812i	$0.754601\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			17.1103i			0.812933i	0.913666	+	0.406467i	$0.133239\pi$
$443$			17.1103i			0.812933i	−0.913666	+	0.406467i	$0.866761\pi$
$444$		0			0
$445$	6.45597	−	11.1821i	0.306042	−	0.530081i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.50000	+	1.74167i	0.306754	+	0.0821945i	0.408912	−	0.912574i	$-0.365908\pi$
$449$	6.50000	+	1.74167i	0.306754	+	0.0821945i	−0.102158	+	0.994768i	$0.532575\pi$
$450$		0			0
$451$	−0.588490	−	1.01929i	−0.0277109	−	0.0479967i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.98917	+	5.51295i	0.0932536	+	0.258451i
$456$		0			0
$457$	2.76840	+	10.3318i	0.129500	+	0.483301i	0.999960	−	0.00893821i	$-0.00284516\pi$
$457$	2.76840	+	10.3318i	0.129500	+	0.483301i	−0.870460	+	0.492239i	$0.836178\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.12588	+	19.1300i	−0.238736	+	0.890975i	0.737693	+	0.675136i	$0.235915\pi$
$461$	−5.12588	+	19.1300i	−0.238736	+	0.890975i	−0.976429	+	0.215838i	$0.930752\pi$
$462$		0			0
$463$	8.24595	+	8.24595i	0.383222	+	0.383222i	0.872262	−	0.489040i	$-0.162653\pi$
$463$	8.24595	+	8.24595i	0.383222	+	0.383222i	−0.489040	+	0.872262i	$0.662653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.7618			−0.729369			−0.364685	−	0.931131i	$-0.618823\pi$
$467$	−15.7618			−0.729369			−0.364685	+	0.931131i	$0.618823\pi$
$468$		0			0
$469$	1.99778			0.0922490
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.520226	+	0.520226i	0.0239200	+	0.0239200i
$474$		0			0
$475$	1.23319	−	4.60232i	0.0565825	−	0.211169i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.49788	−	5.59016i	−0.0684398	−	0.255421i	0.923226	−	0.384257i	$-0.125542\pi$
$479$	−1.49788	−	5.59016i	−0.0684398	−	0.255421i	−0.991666	+	0.128836i	$0.958876\pi$
$480$		0			0
$481$	−2.83474	−	33.4756i	−0.129253	−	1.52636i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.48632	+	11.2346i	0.294529	+	0.510139i
$486$		0			0
$487$	−34.8084	−	9.32688i	−1.57732	−	0.422641i	−0.639224	−	0.769021i	$-0.720744\pi$
$487$	−34.8084	−	9.32688i	−1.57732	−	0.422641i	−0.938094	+	0.346380i	$0.887411\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.2965	−	19.5662i	0.509805	−	0.883008i	−0.490130	−	0.871649i	$-0.663051\pi$
$491$	11.2965	−	19.5662i	0.509805	−	0.883008i	0.999935	−	0.0113593i	$-0.00361585\pi$
$492$		0			0
$493$			24.6727i			1.11120i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.03176	−	4.63714i	−0.360274	−	0.208004i
$498$		0			0
$499$	−23.9443	+	23.9443i	−1.07189	+	1.07189i	−0.0746867	+	0.997207i	$0.523796\pi$
$499$	−23.9443	+	23.9443i	−1.07189	+	1.07189i	−0.997207	+	0.0746867i	$0.976204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.4162	−	16.9834i	1.31160	−	0.757254i	0.329241	−	0.944246i	$-0.393207\pi$
$503$	29.4162	−	16.9834i	1.31160	−	0.757254i	0.982362	+	0.186991i	$0.0598737\pi$
$504$		0			0
$505$	9.94569	−	2.66494i	0.442577	−	0.118588i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.54032	−	2.55632i	0.422867	−	0.113307i	−0.0411089	−	0.999155i	$-0.513089\pi$
$509$	9.54032	−	2.55632i	0.422867	−	0.113307i	0.463976	+	0.885848i	$0.346422\pi$
$510$		0			0
$511$	−2.98993	+	1.72623i	−0.132267	+	0.0763641i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.30418	+	5.30418i	−0.233730	+	0.233730i
$516$		0			0
$517$	1.64670	+	0.950723i	0.0724218	+	0.0418128i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			25.7050i			1.12616i	0.826404	+	0.563078i	$0.190383\pi$
$521$			25.7050i			1.12616i	−0.826404	+	0.563078i	$0.809617\pi$
$522$		0			0
$523$	3.70723	−	6.42111i	0.162106	−	0.280775i	−0.773518	−	0.633774i	$-0.781505\pi$
$523$	3.70723	−	6.42111i	0.162106	−	0.280775i	0.935624	+	0.352999i	$0.114838\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.8231	−	5.57952i	−0.907067	−	0.243048i
$528$		0			0
$529$	1.33728	+	2.31624i	0.0581426	+	0.100706i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.18533	−	13.9977i	−0.0513425	−	0.606307i
$534$		0			0
$535$	−0.227025	−	0.847270i	−0.00981516	−	0.0366307i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.340714	−	1.27156i	0.0146756	−	0.0547701i
$540$		0			0
$541$	−4.25112	−	4.25112i	−0.182770	−	0.182770i	0.609792	−	0.792562i	$-0.291253\pi$
$541$	−4.25112	−	4.25112i	−0.182770	−	0.182770i	−0.792562	+	0.609792i	$0.791253\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.3166			0.484750
$546$		0			0
$547$	−9.95059			−0.425457			−0.212728	−	0.977111i	$-0.568235\pi$
$547$	−9.95059			−0.425457			−0.212728	+	0.977111i	$0.568235\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	32.7754	+	32.7754i	1.39628	+	1.39628i
$552$		0			0
$553$	−5.74381	+	21.4362i	−0.244252	+	0.911559i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.88622	−	10.7715i	−0.122293	−	0.456404i	0.877436	−	0.479694i	$-0.159252\pi$
$557$	−2.88622	−	10.7715i	−0.122293	−	0.456404i	−0.999729	+	0.0232903i	$0.992586\pi$
$558$		0			0
$559$	2.98028	+	8.25980i	0.126052	+	0.349353i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.61327	−	13.1866i	−0.320861	−	0.555747i	0.659805	−	0.751437i	$-0.270639\pi$
$563$	−7.61327	−	13.1866i	−0.320861	−	0.555747i	−0.980666	+	0.195690i	$0.937306\pi$
$564$		0			0
$565$	−8.45766	−	2.26622i	−0.355816	−	0.0953407i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.358823	−	0.621499i	0.0150426	−	0.0260546i	−0.858406	−	0.512971i	$-0.828545\pi$
$569$	0.358823	−	0.621499i	0.0150426	−	0.0260546i	0.873449	+	0.486916i	$0.161878\pi$
$570$		0			0
$571$			16.1777i			0.677017i	0.940963	+	0.338508i	$0.109922\pi$
$571$			16.1777i			0.677017i	−0.940963	+	0.338508i	$0.890078\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.90437	+	2.25419i	0.162823	+	0.0940061i
$576$		0			0
$577$	−13.9025	+	13.9025i	−0.578767	+	0.578767i	−0.934563	−	0.355797i	$-0.884210\pi$
$577$	−13.9025	+	13.9025i	−0.578767	+	0.578767i	0.355797	+	0.934563i	$0.384210\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.1550	−	6.44037i	0.462789	−	0.267192i
$582$		0			0
$583$	−1.60778	+	0.430804i	−0.0665876	+	0.0178421i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.63709	−	1.77840i	0.273942	−	0.0734026i	−0.119233	−	0.992866i	$-0.538043\pi$
$587$	6.63709	−	1.77840i	0.273942	−	0.0734026i	0.393175	+	0.919464i	$0.371377\pi$
$588$		0			0
$589$	−35.0733	+	20.2496i	−1.44517	+	0.834371i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.4449	+	24.4449i	−1.00383	+	1.00383i	−0.00384061	+	0.999993i	$0.501223\pi$
$593$	−24.4449	+	24.4449i	−1.00383	+	1.00383i	−0.999993	+	0.00384061i	$0.998777\pi$
$594$		0			0
$595$	3.57031	+	2.06132i	0.146369	+	0.0845059i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.8655i			1.50628i	0.657859	+	0.753141i	$0.271462\pi$
$599$			36.8655i			1.50628i	−0.657859	+	0.753141i	$0.728538\pi$
$600$		0			0
$601$	−3.70418	+	6.41582i	−0.151097	+	0.261707i	−0.931631	−	0.363406i	$-0.881614\pi$
$601$	−3.70418	+	6.41582i	−0.151097	+	0.261707i	0.780534	+	0.625113i	$0.214947\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.5370	+	2.82339i	0.428391	+	0.114787i
$606$		0			0
$607$	23.8138	+	41.2467i	0.966571	+	1.67415i	0.705333	+	0.708876i	$0.250797\pi$
$607$	23.8138	+	41.2467i	0.966571	+	1.67415i	0.261238	+	0.965274i	$0.415869\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.9652	+	18.6265i	0.524517	+	0.753549i
$612$		0			0
$613$	−7.08397	−	26.4377i	−0.286119	−	1.06781i	−0.948018	−	0.318217i	$-0.896916\pi$
$613$	−7.08397	−	26.4377i	−0.286119	−	1.06781i	0.661899	−	0.749593i	$-0.269751\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.364815	+	1.36151i	−0.0146869	+	0.0548122i	−0.972880	−	0.231309i	$-0.925699\pi$
$617$	−0.364815	+	1.36151i	−0.0146869	+	0.0548122i	0.958194	+	0.286121i	$0.0923659\pi$
$618$		0			0
$619$	−21.4931	−	21.4931i	−0.863882	−	0.863882i	0.127905	−	0.991786i	$-0.459175\pi$
$619$	−21.4931	−	21.4931i	−0.863882	−	0.863882i	−0.991786	+	0.127905i	$0.959175\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20.9884			0.840884
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.7102	−	16.7102i	−0.666278	−	0.666278i
$630$		0			0
$631$	3.38949	−	12.6498i	0.134933	−	0.503579i	−0.865065	−	0.501660i	$-0.832723\pi$
$631$	3.38949	−	12.6498i	0.134933	−	0.503579i	0.999998	−	0.00191829i	$-0.000610612\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.310218	−	1.15775i	−0.0123106	−	0.0459438i
$636$		0			0
$637$	10.1330	−	12.0079i	0.401485	−	0.475771i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.54227	+	11.3315i	0.258404	+	0.447569i	0.965815	−	0.259234i	$-0.0834701\pi$
$641$	6.54227	+	11.3315i	0.258404	+	0.447569i	−0.707411	+	0.706803i	$0.750137\pi$
$642$		0			0
$643$	13.8580	+	3.71325i	0.546508	+	0.146436i	0.521500	−	0.853251i	$-0.325372\pi$
$643$	13.8580	+	3.71325i	0.546508	+	0.146436i	0.0250072	+	0.999687i	$0.492039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.3014	+	26.5028i	−0.601561	+	1.04193i	0.391024	+	0.920381i	$0.372121\pi$
$647$	−15.3014	+	26.5028i	−0.601561	+	1.04193i	−0.992585	+	0.121554i	$0.961212\pi$
$648$		0			0
$649$		−	1.89065i		−	0.0742145i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0233	−	17.3340i	−1.17490	−	0.678331i	−0.220073	−	0.975483i	$-0.570630\pi$
$653$	−30.0233	−	17.3340i	−1.17490	−	0.678331i	−0.954830	+	0.297153i	$0.903963\pi$
$654$		0			0
$655$	−6.78710	+	6.78710i	−0.265194	+	0.265194i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.8484	−	13.1916i	0.890049	−	0.513870i	0.0160904	−	0.999871i	$-0.494878\pi$
$659$	22.8484	−	13.1916i	0.890049	−	0.513870i	0.873958	+	0.486001i	$0.161545\pi$
$660$		0			0
$661$	30.6023	−	8.19987i	1.19029	−	0.318938i	0.391291	−	0.920267i	$-0.372028\pi$
$661$	30.6023	−	8.19987i	1.19029	−	0.318938i	0.799002	+	0.601329i	$0.205362\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.48109	−	2.00455i	0.290104	−	0.0777332i
$666$		0			0
$667$	−37.9822	+	21.9290i	−1.47068	+	0.849096i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.761221	+	0.761221i	−0.0293866	+	0.0293866i
$672$		0			0
$673$	14.1791	+	8.18631i	0.546564	+	0.315559i	0.747735	−	0.663997i	$-0.231141\pi$
$673$	14.1791	+	8.18631i	0.546564	+	0.315559i	−0.201171	+	0.979556i	$0.564475\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	30.6337i		−	1.17735i	−0.808370	−	0.588675i	$-0.799650\pi$
$677$		−	30.6337i		−	1.17735i	0.808370	−	0.588675i	$-0.200350\pi$
$678$		0			0
$679$	−10.5435	+	18.2620i	−0.404624	+	0.700830i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.5382	−	9.52243i	−1.35983	−	0.364365i	−0.496076	−	0.868279i	$-0.665226\pi$
$683$	−35.5382	−	9.52243i	−1.35983	−	0.364365i	−0.863754	+	0.503913i	$0.831893\pi$
$684$		0			0
$685$	−9.63099	−	16.6814i	−0.367981	−	0.637362i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.5550	−	3.50431i	−0.744988	−	0.133503i
$690$		0			0
$691$	−6.32813	−	23.6169i	−0.240733	−	0.898430i	−0.975480	−	0.220088i	$-0.929366\pi$
$691$	−6.32813	−	23.6169i	−0.240733	−	0.898430i	0.734747	−	0.678342i	$-0.237301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.27101	+	4.74348i	−0.0482123	+	0.179931i
$696$		0			0
$697$	−6.98728	−	6.98728i	−0.264662	−	0.264662i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	16.4448			0.621113			0.310556	−	0.950555i	$-0.399485\pi$
$701$	16.4448			0.621113			0.310556	+	0.950555i	$0.399485\pi$
$702$		0			0
$703$	−44.3957			−1.67442
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.8349	+	11.8349i	0.445097	+	0.445097i
$708$		0			0
$709$	6.24975	−	23.3244i	0.234714	−	0.875966i	−0.743563	−	0.668666i	$-0.766866\pi$
$709$	6.24975	−	23.3244i	0.234714	−	0.875966i	0.978277	−	0.207300i	$-0.0664677\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.91813	−	37.0150i	−0.371437	−	1.38622i
$714$		0			0
$715$	−0.832416	−	0.702443i	−0.0311306	−	0.0262699i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.0640	+	33.0198i	0.710967	+	1.23143i	0.964495	+	0.264103i	$0.0850758\pi$
$719$	19.0640	+	33.0198i	0.710967	+	1.23143i	−0.253528	+	0.967328i	$0.581591\pi$
$720$		0			0
$721$	−11.7778	−	3.15586i	−0.438629	−	0.117530i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.86407	−	8.42481i	0.180647	−	0.312890i
$726$		0			0
$727$			42.9583i			1.59324i	0.604483	+	0.796618i	$0.293380\pi$
$727$			42.9583i			1.59324i	−0.604483	+	0.796618i	$0.706620\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.34923	+	3.08838i	0.197849	+	0.114228i
$732$		0			0
$733$	8.28937	−	8.28937i	0.306175	−	0.306175i	−0.537249	−	0.843424i	$-0.680536\pi$
$733$	8.28937	−	8.28937i	0.306175	−	0.306175i	0.843424	+	0.537249i	$0.180536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.321532	+	0.185636i	−0.0118438	+	0.00683801i
$738$		0			0
$739$	36.0666	−	9.66403i	1.32673	−	0.355497i	0.475237	−	0.879858i	$-0.342362\pi$
$739$	36.0666	−	9.66403i	1.32673	−	0.355497i	0.851496	+	0.524361i	$0.175696\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.9775	−	4.54912i	0.622846	−	0.166891i	0.0664246	−	0.997791i	$-0.478841\pi$
$743$	16.9775	−	4.54912i	0.622846	−	0.166891i	0.556421	+	0.830900i	$0.312174\pi$
$744$		0			0
$745$	−3.65050	+	2.10762i	−0.133744	+	0.0772171i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.00821	−	1.00821i	0.0368392	−	0.0368392i
$750$		0			0
$751$	−16.3665	−	9.44923i	−0.597224	−	0.344807i	0.170725	−	0.985319i	$-0.445389\pi$
$751$	−16.3665	−	9.44923i	−0.597224	−	0.344807i	−0.767949	+	0.640512i	$0.778722\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	0.459133i		−	0.0167096i
$756$		0			0
$757$	10.4334	−	18.0713i	0.379210	−	0.656811i	−0.611738	−	0.791061i	$-0.709529\pi$
$757$	10.4334	−	18.0713i	0.379210	−	0.656811i	0.990948	+	0.134250i	$0.0428625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	39.7865	+	10.6608i	1.44226	+	0.386453i	0.893325	−	0.449412i	$-0.148366\pi$
$761$	39.7865	+	10.6608i	1.44226	+	0.386453i	0.548936	+	0.835864i	$0.315033\pi$
$762$		0			0
$763$	9.19760	+	15.9307i	0.332976	+	0.576731i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.59366	−	20.4248i	0.346407	−	0.737498i
$768$		0			0
$769$	−4.60484	−	17.1855i	−0.166055	−	0.619725i	−0.997903	−	0.0647223i	$-0.979384\pi$
$769$	−4.60484	−	17.1855i	−0.166055	−	0.619725i	0.831848	−	0.555003i	$-0.187283\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.66423	+	28.6033i	−0.275663	+	1.02879i	0.679732	+	0.733460i	$0.262096\pi$
$773$	−7.66423	+	28.6033i	−0.275663	+	1.02879i	−0.955396	+	0.295329i	$0.904571\pi$
$774$		0			0
$775$	6.01033	+	6.01033i	0.215898	+	0.215898i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.5639			−0.665120
$780$		0			0
$781$	1.72355			0.0616736
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.1640	−	17.1640i	−0.612611	−	0.612611i
$786$		0			0
$787$	2.96313	−	11.0585i	0.105624	−	0.394194i	−0.892791	−	0.450471i	$-0.851256\pi$
$787$	2.96313	−	11.0585i	0.105624	−	0.394194i	0.998415	+	0.0562765i	$0.0179229\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.68376	−	13.7480i	−0.130979	−	0.488821i
$792$		0			0
$793$	−12.0862	+	4.36089i	−0.429192	+	0.154860i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.04417	+	1.80855i	0.0369864	+	0.0640622i	0.883926	−	0.467627i	$-0.154891\pi$
$797$	1.04417	+	1.80855i	0.0369864	+	0.0640622i	−0.846940	+	0.531689i	$0.821558\pi$
$798$		0			0
$799$	15.4199	+	4.13175i	0.545517	+	0.146171i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.320808	−	0.555655i	0.0113211	−	0.0196087i
$804$		0			0
$805$			7.32838i			0.258292i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.8915	+	8.02025i	0.488399	+	0.281977i	0.723910	−	0.689895i	$-0.242343\pi$
$809$	13.8915	+	8.02025i	0.488399	+	0.281977i	−0.235511	+	0.971872i	$0.575676\pi$
$810$		0			0
$811$	18.6035	−	18.6035i	0.653256	−	0.653256i	−0.300519	−	0.953776i	$-0.597160\pi$
$811$	18.6035	−	18.6035i	0.653256	−	0.653256i	0.953776	+	0.300519i	$0.0971600\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.3542	−	8.86474i	0.537833	−	0.310518i
$816$		0			0
$817$	11.2086	−	3.00333i	0.392138	−	0.105073i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.2462	+	4.08522i	−0.532098	+	0.142575i	−0.514857	−	0.857276i	$-0.672155\pi$
$821$	−15.2462	+	4.08522i	−0.532098	+	0.142575i	−0.0172409	+	0.999851i	$0.505488\pi$
$822$		0			0
$823$	8.10520	−	4.67954i	0.282530	−	0.163119i	−0.352038	−	0.935986i	$-0.614511\pi$
$823$	8.10520	−	4.67954i	0.282530	−	0.163119i	0.634568	+	0.772867i	$0.281178\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.91823	−	4.91823i	0.171023	−	0.171023i	−0.616405	−	0.787429i	$-0.711412\pi$
$827$	4.91823	−	4.91823i	0.171023	−	0.171023i	0.787429	+	0.616405i	$0.211412\pi$
$828$		0			0
$829$	40.8552	+	23.5878i	1.41896	+	0.819238i	0.996208	−	0.0870069i	$-0.0277302\pi$
$829$	40.8552	+	23.5878i	1.41896	+	0.819238i	0.422754	+	0.906245i	$0.361064\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.0522i		−	0.382936i
$834$		0			0
$835$	11.6284	−	20.1409i	0.402416	−	0.697005i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.0792	−	7.25585i	−0.934878	−	0.250500i	−0.240944	−	0.970539i	$-0.577457\pi$
$839$	−27.0792	−	7.25585i	−0.934878	−	0.250500i	−0.693933	+	0.720039i	$0.744124\pi$
$840$		0			0
$841$	32.8183	+	56.8430i	1.13167	+	1.96010i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.42827	−	11.8124i	−0.186738	−	0.406360i
$846$		0			0
$847$	4.58943	+	17.1280i	0.157695	+	0.588525i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.8724	−	40.5763i	0.372700	−	1.39094i
$852$		0			0
$853$	−20.8964	−	20.8964i	−0.715479	−	0.715479i	0.252197	−	0.967676i	$-0.418847\pi$
$853$	−20.8964	−	20.8964i	−0.715479	−	0.715479i	−0.967676	+	0.252197i	$0.918847\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.02510			−0.205813			−0.102907	−	0.994691i	$-0.532814\pi$
$857$	−6.02510			−0.205813			−0.102907	+	0.994691i	$0.532814\pi$
$858$		0			0
$859$	−16.2184			−0.553365			−0.276682	−	0.960961i	$-0.589235\pi$
$859$	−16.2184			−0.553365			−0.276682	+	0.960961i	$0.589235\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37.4312	+	37.4312i	1.27417	+	1.27417i	0.943878	+	0.330293i	$0.107148\pi$
$863$	37.4312	+	37.4312i	1.27417	+	1.27417i	0.330293	+	0.943878i	$0.392852\pi$
$864$		0			0
$865$	0.414510	−	1.54697i	0.0140938	−	0.0525987i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.06744	−	3.98375i	−0.0362105	−	0.135140i
$870$		0			0
$871$	−4.41550	+	0.373908i	−0.149614	+	0.0126694i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.812752	−	1.40773i	−0.0274760	−	0.0475899i
$876$		0			0
$877$	−20.9684	−	5.61847i	−0.708052	−	0.189722i	−0.113218	−	0.993570i	$-0.536116\pi$
$877$	−20.9684	−	5.61847i	−0.708052	−	0.189722i	−0.594835	+	0.803848i	$0.702782\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−19.2171	+	33.2850i	−0.647441	+	1.12140i	0.336291	+	0.941758i	$0.390827\pi$
$881$	−19.2171	+	33.2850i	−0.647441	+	1.12140i	−0.983732	+	0.179643i	$0.942506\pi$
$882$		0			0
$883$		−	1.98458i		−	0.0667865i	−0.999442	−	0.0333933i	$-0.989369\pi$
$883$		−	1.98458i		−	0.0667865i	0.999442	−	0.0333933i	$-0.0106314\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.5671	−	19.3799i	−1.12707	−	0.650715i	−0.183875	−	0.982950i	$-0.558864\pi$
$887$	−33.5671	−	19.3799i	−1.12707	−	0.650715i	−0.943197	+	0.332235i	$0.892197\pi$
$888$		0			0
$889$	1.37766	−	1.37766i	0.0462054	−	0.0462054i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	25.9725	−	14.9953i	0.869138	−	0.501797i
$894$		0			0
$895$	−17.9688	+	4.81473i	−0.600631	+	0.160939i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−79.8706	+	21.4013i	−2.66383	+	0.713772i
$900$		0			0
$901$	−12.1023	+	6.98728i	−0.403187	+	0.232780i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.261931	+	0.261931i	−0.00870687	+	0.00870687i
$906$		0			0
$907$	−9.03950	−	5.21896i	−0.300152	−	0.173293i	0.342359	−	0.939569i	$-0.388774\pi$
$907$	−9.03950	−	5.21896i	−0.300152	−	0.173293i	−0.642511	+	0.766276i	$0.722107\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	29.0318i		−	0.961867i	−0.876757	−	0.480934i	$-0.840298\pi$
$911$		−	29.0318i		−	0.961867i	0.876757	−	0.480934i	$-0.159702\pi$
$912$		0			0
$913$	−1.19689	+	2.07308i	−0.0396114	+	0.0686090i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.0706	−	4.03816i	−0.497676	−	0.133352i
$918$		0			0
$919$	26.0364	+	45.0963i	0.858861	+	1.48759i	0.873016	+	0.487691i	$0.162161\pi$
$919$	26.0364	+	45.0963i	0.858861	+	1.48759i	−0.0141548	+	0.999900i	$0.504506\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	18.6197	+	8.74578i	0.612875	+	0.287871i
$924$		0			0
$925$	2.41160	+	9.00020i	0.0792928	+	0.295925i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.00843	−	18.6917i	0.164321	−	0.613255i	−0.833804	−	0.552060i	$-0.813842\pi$
$929$	5.00843	−	18.6917i	0.164321	−	0.613255i	0.998126	−	0.0611956i	$-0.0194913\pi$
$930$		0			0
$931$	−14.6818	−	14.6818i	−0.481176	−	0.481176i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.766162			−0.0250562
$936$		0			0
$937$	10.6914			0.349274			0.174637	−	0.984633i	$-0.444125\pi$
$937$	10.6914			0.349274			0.174637	+	0.984633i	$0.444125\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.7460	+	11.7460i	0.382908	+	0.382908i	0.872149	−	0.489241i	$-0.162726\pi$
$941$	11.7460	+	11.7460i	0.382908	+	0.382908i	−0.489241	+	0.872149i	$0.662726\pi$
$942$		0			0
$943$	4.54624	−	16.9668i	0.148046	−	0.552515i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.83997	−	14.3310i	−0.124782	−	0.465693i	0.875050	−	0.484033i	$-0.160829\pi$
$947$	−3.83997	−	14.3310i	−0.124782	−	0.465693i	−0.999832	+	0.0183398i	$0.994162\pi$
$948$		0			0
$949$	6.28526	−	4.37493i	0.204028	−	0.142016i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.8452	+	32.6409i	0.610457	+	1.05734i	0.991163	+	0.132646i	$0.0423475\pi$
$953$	18.8452	+	32.6409i	0.610457	+	1.05734i	−0.380706	+	0.924696i	$0.624319\pi$
$954$		0			0
$955$	19.0033	+	5.09192i	0.614933	+	0.164771i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.6552	−	27.1156i	0.505533	−	0.875610i
$960$		0			0
$961$		−	41.2482i		−	1.33059i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	13.4522	+	7.76664i	0.433042	+	0.250017i
$966$		0			0
$967$	2.05564	−	2.05564i	0.0661048	−	0.0661048i	−0.673281	−	0.739386i	$-0.735116\pi$
$967$	2.05564	−	2.05564i	0.0661048	−	0.0661048i	0.739386	+	0.673281i	$0.235116\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.8798	+	21.2926i	−1.18353	+	0.683311i	−0.956829	−	0.290653i	$-0.906128\pi$
$971$	−36.8798	+	21.2926i	−1.18353	+	0.683311i	−0.226702	+	0.973964i	$0.572794\pi$
$972$		0			0
$973$	−7.71055	+	2.06604i	−0.247189	+	0.0662341i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.11542	−	0.834774i	0.0996711	−	0.0267068i	−0.208639	−	0.977993i	$-0.566903\pi$
$977$	3.11542	−	0.834774i	0.0996711	−	0.0267068i	0.308310	+	0.951286i	$0.400237\pi$
$978$		0			0
$979$	−3.37797	+	1.95027i	−0.107960	+	0.0623309i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.2041	−	21.2041i	0.676305	−	0.676305i	−0.282857	−	0.959162i	$-0.591282\pi$
$983$	21.2041	−	21.2041i	0.676305	−	0.676305i	0.959162	+	0.282857i	$0.0912823\pi$
$984$		0			0
$985$	−12.8613	−	7.42547i	−0.409795	−	0.236595i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			10.9798i			0.349137i
$990$		0			0
$991$	13.4162	−	23.2376i	0.426180	−	0.738165i	−0.570350	−	0.821402i	$-0.693192\pi$
$991$	13.4162	−	23.2376i	0.426180	−	0.738165i	0.996530	+	0.0832365i	$0.0265257\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.1943	−	3.53539i	−0.418286	−	0.112079i
$996$		0			0
$997$	9.30538	+	16.1174i	0.294704	+	0.510443i	0.974916	−	0.222573i	$-0.0714455\pi$
$997$	9.30538	+	16.1174i	0.294704	+	0.510443i	−0.680212	+	0.733016i	$0.738112\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.fo.a.1241.7	yes	40
3.2	odd	2	inner	2340.2.fo.a.1241.3	✓	40
13.11	odd	12	inner	2340.2.fo.a.1961.3	yes	40
39.11	even	12	inner	2340.2.fo.a.1961.7	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.a.1241.3	✓	40	3.2	odd	2	inner
2340.2.fo.a.1241.7	yes	40	1.1	even	1	trivial
2340.2.fo.a.1961.3	yes	40	13.11	odd	12	inner
2340.2.fo.a.1961.7	yes	40	39.11	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.fo (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(-0.420712 + 1.57012i) q^{7} +(-0.0781861 - 0.291794i) q^{11} +(0.635992 - 3.54902i) q^{13} +(-1.26811 - 2.19643i) q^{17} +(-4.60232 - 1.23319i) q^{19} +(2.25419 - 3.90437i) q^{23} +1.00000i q^{25} +(-8.42481 - 4.86407i) q^{29} +(6.01033 - 6.01033i) q^{31} +(-1.40773 + 0.812752i) q^{35} +(9.00020 - 2.41160i) q^{37} +(3.76339 - 1.00840i) q^{41} +(-2.10913 + 1.21771i) q^{43} +(-4.45078 + 4.45078i) q^{47} +(3.77391 + 2.17887i) q^{49} -5.50999i q^{53} +(0.151044 - 0.261616i) q^{55} +(6.04535 + 1.61985i) q^{59} +(-1.78181 - 3.08619i) q^{61} +(2.95925 - 2.05982i) q^{65} +(-0.318095 - 1.18715i) q^{67} +(-1.47669 + 5.51107i) q^{71} +(1.50185 + 1.50185i) q^{73} +0.491045 q^{77} +13.6526 q^{79} +(-5.60322 - 5.60322i) q^{83} +(0.656423 - 2.44980i) q^{85} +(-3.34186 - 12.4720i) q^{89} +(5.30480 + 2.49169i) q^{91} +(-2.38234 - 4.12633i) q^{95} +(12.5306 + 3.35757i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 40 * q - 12 * q^13 - 20 * q^19 - 28 * q^31 + 12 * q^49 + 16 * q^55 + 48 * q^61 + 16 * q^67 - 20 * q^73 + 80 * q^79 + 20 * q^85 + 4 * q^91 - 40 * q^97

Introduction

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