LMFDB - Embedded newform 2548.2.j.f.1353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(1145,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("2548.1145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2548.j (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1353.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2548.1353
Dual form			2548.2.j.f.1145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(-4.00000 + 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +2.00000 q^{29} +(5.00000 + 8.66025i) q^{31} +(3.00000 - 5.19615i) q^{37} +6.00000 q^{41} +4.00000 q^{43} +(-3.00000 - 5.19615i) q^{45} +(-1.00000 + 1.73205i) q^{47} +(-3.00000 - 5.19615i) q^{53} +4.00000 q^{55} +(-5.00000 - 8.66025i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.0000 q^{71} +(1.00000 + 1.73205i) q^{73} +(2.00000 - 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +6.00000 q^{83} +12.0000 q^{85} +(-3.00000 + 5.19615i) q^{89} +(6.00000 + 10.3923i) q^{95} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 + 3 * q^9	$ 2 q + 2 q^{5} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{17} - 6 q^{19} - 8 q^{23} + q^{25} + 4 q^{29} + 10 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 6 q^{45} - 2 q^{47} - 6 q^{53} + 8 q^{55} - 10 q^{59} - 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} + 2 q^{73} + 4 q^{79} - 9 q^{81} + 12 q^{83} + 24 q^{85} - 6 q^{89} + 12 q^{95} - 4 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^17 - 6 * q^19 - 8 * q^23 + q^25 + 4 * q^29 + 10 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 6 * q^45 - 2 * q^47 - 6 * q^53 + 8 * q^55 - 10 * q^59 - 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 + 2 * q^73 + 4 * q^79 - 9 * q^81 + 12 * q^83 + 24 * q^85 - 6 * q^89 + 12 * q^95 - 4 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$.

$n$	$197$	$885$	$1275$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	0.284157	+	0.958778i	$0.408286\pi$
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	−0.972404	+	0.233301i	$0.925047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	0.0607377	+	0.998154i	$0.480655\pi$
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	−0.894795	+	0.446476i	$0.852679\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	−3.00000	−	5.19615i	−0.447214	−	0.774597i
$46$		0			0
$47$	−1.00000	+	1.73205i	−0.145865	+	0.252646i	−0.929695	−	0.368329i	$-0.879930\pi$
$47$	−1.00000	+	1.73205i	−0.145865	+	0.252646i	0.783830	+	0.620975i	$0.213263\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	−	1.73205i	0.124035	−	0.214834i
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	0.918594	−	0.395203i	$-0.129326\pi$
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	+	10.3923i	0.615587	+	1.06623i
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	0.811976	−	0.583691i	$-0.198392\pi$
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	−0.911479	+	0.411346i	$0.865059\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	−0.162306	−	0.986740i	$-0.551893\pi$
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	0.935695	−	0.352809i	$-0.114773\pi$
$108$		0			0
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	0.977769	+	0.209687i	$0.0672444\pi$
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	−0.307290	+	0.951616i	$0.599422\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$	8.00000	+	13.8564i	0.746004	+	1.29212i
$116$		0			0
$117$	1.50000	−	2.59808i	0.138675	−	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	0.269863	+	0.962899i	$0.413022\pi$
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	−0.968826	+	0.247741i	$0.920312\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$	2.00000	−	3.46410i	0.166091	−	0.287678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	0.216394	+	0.976306i	$0.430570\pi$
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	−0.953703	+	0.300750i	$0.902763\pi$
$150$		0			0
$151$	−3.00000	−	5.19615i	−0.244137	−	0.422857i	0.717752	−	0.696299i	$-0.245171\pi$
$151$	−3.00000	−	5.19615i	−0.244137	−	0.422857i	−0.961888	+	0.273442i	$0.911838\pi$
$152$		0			0
$153$	18.0000			1.45521
$154$		0			0
$155$	20.0000			1.60644
$156$		0			0
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	0.903167	−	0.429289i	$-0.141236\pi$
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	−0.823359	+	0.567521i	$0.807902\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.00000	−	8.66025i	0.391630	−	0.678323i	−0.601035	−	0.799223i	$-0.705245\pi$
$163$	5.00000	−	8.66025i	0.391630	−	0.678323i	0.992665	+	0.120900i	$0.0385779\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	9.00000	+	15.5885i	0.688247	+	1.19208i
$172$		0			0
$173$	−5.00000	+	8.66025i	−0.380143	+	0.658427i	−0.991082	−	0.133250i	$-0.957459\pi$
$173$	−5.00000	+	8.66025i	−0.380143	+	0.658427i	0.610939	+	0.791677i	$0.290792\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	0.549825	−	0.835280i	$-0.314694\pi$
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	−0.998286	+	0.0585225i	$0.981361\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	−	10.3923i	−0.441129	−	0.764057i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000	−	10.3923i	0.419058	−	0.725830i
$206$		0			0
$207$	12.0000	+	20.7846i	0.834058	+	1.44463i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	−	6.92820i	0.272798	−	0.472500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	6.00000			0.401790			0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000			0.401790			0.200895	+	0.979613i	$0.435615\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$		0			0
$227$	9.00000	+	15.5885i	0.597351	+	1.03464i	0.993210	+	0.116331i	$0.0371134\pi$
$227$	9.00000	+	15.5885i	0.597351	+	1.03464i	−0.395860	+	0.918311i	$0.629553\pi$
$228$		0			0
$229$	9.00000	−	15.5885i	0.594737	−	1.03011i	−0.398847	−	0.917017i	$-0.630590\pi$
$229$	9.00000	−	15.5885i	0.594737	−	1.03011i	0.993584	−	0.113097i	$-0.0360770\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	−0.982027	−	0.188739i	$-0.939560\pi$
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	0.654466	+	0.756091i	$0.272893\pi$
$234$		0			0
$235$	2.00000	+	3.46410i	0.130466	+	0.225973i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	18.0000			1.16432			0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000			1.16432			0.582162	+	0.813073i	$0.302207\pi$
$240$		0			0
$241$	−3.00000	−	5.19615i	−0.193247	−	0.334714i	0.753077	−	0.657932i	$-0.228569\pi$
$241$	−3.00000	−	5.19615i	−0.193247	−	0.334714i	−0.946324	+	0.323218i	$0.895235\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.00000	+	5.19615i	−0.190885	+	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	−0.435970	−	0.899961i	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.00000	−	5.19615i	0.185695	−	0.321634i
$262$		0			0
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	0.998361	+	0.0572259i	$0.0182255\pi$
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	−0.449622	+	0.893219i	$0.648441\pi$
$270$		0			0
$271$	−1.00000	+	1.73205i	−0.0607457	+	0.105215i	−0.894799	−	0.446469i	$-0.852681\pi$
$271$	−1.00000	+	1.73205i	−0.0607457	+	0.105215i	0.834053	+	0.551684i	$0.186015\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	+	1.73205i	−0.0603023	+	0.104447i
$276$		0			0
$277$	−15.0000	−	25.9808i	−0.901263	−	1.56103i	−0.825857	−	0.563880i	$-0.809308\pi$
$277$	−15.0000	−	25.9808i	−0.901263	−	1.56103i	−0.0754058	−	0.997153i	$-0.524025\pi$
$278$		0			0
$279$	30.0000			1.79605
$280$		0			0
$281$	−14.0000			−0.835170			−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000			−0.835170			−0.417585	+	0.908638i	$0.637123\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$	−20.0000			−1.16445
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.00000	+	6.92820i	−0.231326	+	0.400668i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.00000	+	3.46410i	0.114520	+	0.198354i
$306$		0			0
$307$	34.0000			1.94048			0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000			1.94048			0.970241	+	0.242140i	$0.0778494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	2.00000	+	3.46410i	0.111979	+	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.0000			−2.00309
$324$		0			0
$325$	0.500000	+	0.866025i	0.0277350	+	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	−0.387498	−	0.921871i	$-0.626660\pi$
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	0.992112	−	0.125353i	$-0.0400062\pi$
$332$		0			0
$333$	−9.00000	−	15.5885i	−0.493197	−	0.854242i
$334$		0			0
$335$	−20.0000			−1.09272
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.0000	+	17.3205i	−0.541530	+	0.937958i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	0.617388	−	0.786659i	$-0.288191\pi$
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	−0.989960	+	0.141344i	$0.954858\pi$
$354$		0			0
$355$	10.0000	−	17.3205i	0.530745	−	0.919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.0000	+	25.9808i	−0.791670	+	1.37121i	0.133263	+	0.991081i	$0.457455\pi$
$359$	−15.0000	+	25.9808i	−0.791670	+	1.37121i	−0.924932	+	0.380131i	$0.875879\pi$
$360$		0			0
$361$	−8.50000	−	14.7224i	−0.447368	−	0.774865i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000			0.209370
$366$		0			0
$367$	−2.00000	−	3.46410i	−0.104399	−	0.180825i	0.809093	−	0.587680i	$-0.199959\pi$
$367$	−2.00000	−	3.46410i	−0.104399	−	0.180825i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$	9.00000	−	15.5885i	0.468521	−	0.811503i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	0.996610	−	0.0822766i	$-0.0262191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.00000			0.103005
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	−0.539076	−	0.842257i	$-0.681226\pi$
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	0.998954	+	0.0457244i	$0.0145596\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.00000	−	10.3923i	0.304997	−	0.528271i
$388$		0			0
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$	−48.0000			−2.42746
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000	−	6.92820i	−0.201262	−	0.348596i
$396$		0			0
$397$	1.00000	−	1.73205i	0.0501886	−	0.0869291i	−0.839840	−	0.542834i	$-0.817351\pi$
$397$	1.00000	−	1.73205i	0.0501886	−	0.0869291i	0.890028	+	0.455905i	$0.150684\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$	5.00000	+	8.66025i	0.249068	+	0.431398i
$404$		0			0
$405$	−18.0000			−0.894427
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	3.00000	+	5.19615i	0.145865	+	0.252646i
$424$		0			0
$425$	−3.00000	+	5.19615i	−0.145521	+	0.252050i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	0.959985	+	0.280052i	$0.0903517\pi$
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	−0.237460	+	0.971397i	$0.576315\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.0000	−	41.5692i	−1.14808	−	1.98853i
$438$		0			0
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	−0.991322	−	0.131453i	$-0.958036\pi$
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	0.609503	+	0.792784i	$0.291369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	−0.426414	−	0.904528i	$-0.640223\pi$
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	0.996551	−	0.0829786i	$-0.0264433\pi$
$444$		0			0
$445$	6.00000	+	10.3923i	0.284427	+	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$		0			0
$451$	6.00000	+	10.3923i	0.282529	+	0.489355i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	0.127469	+	0.991843i	$0.459315\pi$
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	−0.922695	+	0.385530i	$0.874019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$	22.0000			1.02243			0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000			1.02243			0.511213	+	0.859454i	$0.329196\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.0000	−	17.3205i	0.462745	−	0.801498i	−0.536352	−	0.843995i	$-0.680198\pi$
$467$	10.0000	−	17.3205i	0.462745	−	0.801498i	0.999097	+	0.0424970i	$0.0135313\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	−6.00000			−0.275299
$476$		0			0
$477$	−18.0000			−0.824163
$478$		0			0
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	0.583803	−	0.811895i	$-0.301564\pi$
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	−0.995023	+	0.0996406i	$0.968231\pi$
$480$		0			0
$481$	3.00000	−	5.19615i	0.136788	−	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000	+	3.46410i	−0.0908153	+	0.157297i
$486$		0			0
$487$	−1.00000	−	1.73205i	−0.0453143	−	0.0784867i	0.842479	−	0.538730i	$-0.181096\pi$
$487$	−1.00000	−	1.73205i	−0.0453143	−	0.0784867i	−0.887793	+	0.460243i	$0.847762\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	6.00000	+	10.3923i	0.270226	+	0.468046i
$494$		0			0
$495$	6.00000	−	10.3923i	0.269680	−	0.467099i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7.00000	−	12.1244i	0.313363	−	0.542761i	−0.665725	−	0.746197i	$-0.731878\pi$
$499$	7.00000	−	12.1244i	0.313363	−	0.542761i	0.979088	+	0.203436i	$0.0652110\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.0000			−1.24846			−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000			−1.24846			−0.624229	+	0.781241i	$0.714587\pi$
$504$		0			0
$505$	−4.00000			−0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	−0.978421	−	0.206623i	$-0.933753\pi$
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	0.668151	+	0.744026i	$0.267086\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	13.8564i	0.352522	+	0.610586i
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	0.924229	−	0.381839i	$-0.124709\pi$
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	12.0000	−	20.7846i	0.524723	−	0.908848i	−0.474862	−	0.880060i	$-0.657502\pi$
$523$	12.0000	−	20.7846i	0.524723	−	0.908848i	0.999586	−	0.0287874i	$-0.00916457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−30.0000	+	51.9615i	−1.30682	+	2.26348i
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$	−30.0000			−1.30189
$532$		0			0
$533$	6.00000			0.259889
$534$		0			0
$535$	−16.0000	−	27.7128i	−0.691740	−	1.19813i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.0000	+	22.5167i	−0.558914	+	0.968067i	0.438674	+	0.898646i	$0.355448\pi$
$541$	−13.0000	+	22.5167i	−0.558914	+	0.968067i	−0.997587	+	0.0694205i	$0.977885\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	28.0000			1.19939
$546$		0			0
$547$	−32.0000			−1.36822			−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000			−1.36822			−0.684111	+	0.729378i	$0.739809\pi$
$548$		0			0
$549$	3.00000	+	5.19615i	0.128037	+	0.221766i
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.840113	−	0.542411i	$-0.817511\pi$
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.0496855	−	0.998765i	$-0.515822\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	0.820798	−	0.571218i	$-0.193529\pi$
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	−0.905088	+	0.425223i	$0.860196\pi$
$564$		0			0
$565$	14.0000	−	24.2487i	0.588984	−	1.02015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	−0.951592	−	0.307364i	$-0.900553\pi$
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	0.741981	+	0.670421i	$0.233886\pi$
$570$		0			0
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	0.963827	−	0.266529i	$-0.0858769\pi$
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	−0.712735	+	0.701434i	$0.752544\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$	−3.00000	−	5.19615i	−0.124892	−	0.216319i	0.796799	−	0.604245i	$-0.206525\pi$
$577$	−3.00000	−	5.19615i	−0.124892	−	0.216319i	−0.921691	+	0.387926i	$0.873192\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$	−3.00000	−	5.19615i	−0.124035	−	0.214834i
$586$		0			0
$587$	−26.0000			−1.07313			−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000			−1.07313			−0.536567	+	0.843857i	$0.680279\pi$
$588$		0			0
$589$	−60.0000			−2.47226
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.0000	−	22.5167i	0.533846	−	0.924648i	−0.465372	−	0.885115i	$-0.654080\pi$
$593$	13.0000	−	22.5167i	0.533846	−	0.924648i	0.999218	−	0.0395334i	$-0.0125871\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.00000	−	3.46410i	−0.0817178	−	0.141539i	0.822270	−	0.569097i	$-0.192707\pi$
$599$	−2.00000	−	3.46410i	−0.0817178	−	0.141539i	−0.903988	+	0.427558i	$0.859374\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$	−30.0000			−1.22169
$604$		0			0
$605$	−7.00000	−	12.1244i	−0.284590	−	0.492925i
$606$		0			0
$607$	−6.00000	+	10.3923i	−0.243532	+	0.421811i	−0.961718	−	0.274041i	$-0.911640\pi$
$607$	−6.00000	+	10.3923i	−0.243532	+	0.421811i	0.718186	+	0.695852i	$0.244973\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.00000	+	1.73205i	−0.0404557	+	0.0700713i
$612$		0			0
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	−0.999574	−	0.0291886i	$-0.990708\pi$
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	0.474509	−	0.880251i	$-0.342626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.0000			1.36879			0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000			1.36879			0.684394	+	0.729112i	$0.260067\pi$
$618$		0			0
$619$	−1.00000	−	1.73205i	−0.0401934	−	0.0696170i	0.845229	−	0.534404i	$-0.179464\pi$
$619$	−1.00000	−	1.73205i	−0.0401934	−	0.0696170i	−0.885422	+	0.464787i	$0.846131\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	36.0000			1.43541
$630$		0			0
$631$	22.0000			0.875806			0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000			0.875806			0.437903	+	0.899022i	$0.355721\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000	+	13.8564i	−0.317470	+	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	15.0000	−	25.9808i	0.593391	−	1.02778i
$640$		0			0
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	0.999878	+	0.0156233i	$0.00497325\pi$
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	−0.486409	+	0.873731i	$0.661693\pi$
$642$		0			0
$643$	−2.00000			−0.0788723			−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000			−0.0788723			−0.0394362	+	0.999222i	$0.512556\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	−0.987748	−	0.156059i	$-0.950121\pi$
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	0.358723	−	0.933444i	$-0.383212\pi$
$648$		0			0
$649$	10.0000	−	17.3205i	0.392534	−	0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	−0.986634	−	0.162951i	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$	16.0000	+	27.7128i	0.625172	+	1.08283i
$656$		0			0
$657$	6.00000			0.234082
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	0.884818	−	0.465937i	$-0.154283\pi$
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	−0.845922	+	0.533306i	$0.820949\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	+	13.8564i	−0.309761	+	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	−0.802600	−	0.596518i	$-0.796551\pi$
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	0.917899	+	0.396813i	$0.129884\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.00000	+	1.73205i	0.0382639	+	0.0662751i	0.884523	−	0.466496i	$-0.154484\pi$
$683$	1.00000	+	1.73205i	0.0382639	+	0.0662751i	−0.846259	+	0.532771i	$0.821151\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.00000	−	5.19615i	−0.114291	−	0.197958i
$690$		0			0
$691$	17.0000	−	29.4449i	0.646710	−	1.12014i	−0.337193	−	0.941435i	$-0.609478\pi$
$691$	17.0000	−	29.4449i	0.646710	−	1.12014i	0.983904	−	0.178700i	$-0.0571891\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.0000	+	27.7128i	−0.606915	+	1.05121i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	18.0000	+	31.1769i	0.678883	+	1.17586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	0.995228	+	0.0975728i	$0.0311079\pi$
$710$		0			0
$711$	−6.00000	−	10.3923i	−0.225018	−	0.389742i
$712$		0			0
$713$	−80.0000			−2.99602
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000	+	34.6410i	−0.745874	+	1.29189i	0.203911	+	0.978989i	$0.434635\pi$
$719$	−20.0000	+	34.6410i	−0.745874	+	1.29189i	−0.949785	+	0.312903i	$0.898699\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	12.0000	+	20.7846i	0.443836	+	0.768747i
$732$		0			0
$733$	13.0000	−	22.5167i	0.480166	−	0.831672i	−0.519575	−	0.854425i	$-0.673910\pi$
$733$	13.0000	−	22.5167i	0.480166	−	0.831672i	0.999741	+	0.0227529i	$0.00724310\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.0000	−	17.3205i	0.368355	−	0.638009i
$738$		0			0
$739$	−3.00000	−	5.19615i	−0.110357	−	0.191144i	0.805557	−	0.592518i	$-0.201866\pi$
$739$	−3.00000	−	5.19615i	−0.110357	−	0.191144i	−0.915914	+	0.401374i	$0.868533\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.0000			−0.513610			−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000			−0.513610			−0.256805	+	0.966463i	$0.582670\pi$
$744$		0			0
$745$	18.0000	+	31.1769i	0.659469	+	1.14223i
$746$		0			0
$747$	9.00000	−	15.5885i	0.329293	−	0.570352i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.0000			−0.436725
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	−0.655515	−	0.755182i	$-0.727548\pi$
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	0.981764	+	0.190101i	$0.0608816\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	18.0000	−	31.1769i	0.650791	−	1.12720i
$766$		0			0
$767$	−5.00000	−	8.66025i	−0.180540	−	0.312704i
$768$		0			0
$769$	54.0000			1.94729			0.973645	−	0.228069i	$-0.0732413\pi$
$769$	54.0000			1.94729			0.973645	+	0.228069i	$0.0732413\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23.0000	−	39.8372i	−0.827253	−	1.43284i	−0.900186	−	0.435507i	$-0.856569\pi$
$773$	−23.0000	−	39.8372i	−0.827253	−	1.43284i	0.0729331	−	0.997337i	$-0.476764\pi$
$774$		0			0
$775$	−5.00000	+	8.66025i	−0.179605	+	0.311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.0000	+	31.1769i	−0.644917	+	1.11703i
$780$		0			0
$781$	10.0000	+	17.3205i	0.357828	+	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.00000			0.142766
$786$		0			0
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	0.975798	+	0.218675i	$0.0701734\pi$
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	−0.298521	+	0.954403i	$0.596493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.00000	+	1.73205i	−0.0355110	+	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$	9.00000	+	15.5885i	0.317999	+	0.550791i
$802$		0			0
$803$	−2.00000	+	3.46410i	−0.0705785	+	0.122245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.00000	−	5.19615i	−0.105474	−	0.182687i	0.808458	−	0.588555i	$-0.200303\pi$
$809$	−3.00000	−	5.19615i	−0.105474	−	0.182687i	−0.913932	+	0.405868i	$0.866969\pi$
$810$		0			0
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.0000	−	17.3205i	−0.350285	−	0.606711i
$816$		0			0
$817$	−12.0000	+	20.7846i	−0.419827	+	0.727161i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	−0.882948	−	0.469471i	$-0.844445\pi$
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	0.848048	+	0.529920i	$0.177778\pi$
$822$		0			0
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	0.995768	−	0.0919029i	$-0.0292950\pi$
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	−0.577474	+	0.816409i	$0.695962\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	11.0000	+	19.0526i	0.382046	+	0.661723i	0.991355	−	0.131210i	$-0.0418863\pi$
$829$	11.0000	+	19.0526i	0.382046	+	0.661723i	−0.609309	+	0.792933i	$0.708553\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.0000			0.345238			0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000			0.345238			0.172619	+	0.984989i	$0.444777\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.00000	−	1.73205i	0.0344010	−	0.0595844i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000	+	41.5692i	0.822709	+	1.42497i
$852$		0			0
$853$	38.0000			1.30110			0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000			1.30110			0.650548	+	0.759465i	$0.274539\pi$
$854$		0			0
$855$	36.0000			1.23117
$856$		0			0
$857$	−19.0000	−	32.9090i	−0.649028	−	1.12415i	−0.983355	−	0.181692i	$-0.941843\pi$
$857$	−19.0000	−	32.9090i	−0.649028	−	1.12415i	0.334328	−	0.942457i	$-0.391491\pi$
$858$		0			0
$859$	4.00000	−	6.92820i	0.136478	−	0.236387i	−0.789683	−	0.613515i	$-0.789755\pi$
$859$	4.00000	−	6.92820i	0.136478	−	0.236387i	0.926161	+	0.377128i	$0.123088\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.0000	+	29.4449i	−0.578687	+	1.00231i	0.416944	+	0.908932i	$0.363101\pi$
$863$	−17.0000	+	29.4449i	−0.578687	+	1.00231i	−0.995630	+	0.0933825i	$0.970232\pi$
$864$		0			0
$865$	10.0000	+	17.3205i	0.340010	+	0.588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.00000			0.271381
$870$		0			0
$871$	−5.00000	−	8.66025i	−0.169419	−	0.293442i
$872$		0			0
$873$	−3.00000	+	5.19615i	−0.101535	+	0.175863i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.00000	+	15.5885i	−0.303908	+	0.526385i	−0.977018	−	0.213158i	$-0.931625\pi$
$877$	−9.00000	+	15.5885i	−0.303908	+	0.526385i	0.673109	+	0.739543i	$0.264958\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.00000	−	3.46410i	0.0671534	−	0.116313i	−0.830494	−	0.557028i	$-0.811942\pi$
$887$	2.00000	−	3.46410i	0.0671534	−	0.116313i	0.897647	+	0.440715i	$0.145275\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	9.00000	−	15.5885i	0.301511	−	0.522233i
$892$		0			0
$893$	−6.00000	−	10.3923i	−0.200782	−	0.347765i
$894$		0			0
$895$	−24.0000			−0.802232
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.0000	+	17.3205i	0.333519	+	0.577671i
$900$		0			0
$901$	18.0000	−	31.1769i	0.599667	−	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.00000	−	10.3923i	0.199447	−	0.345452i
$906$		0			0
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	−0.993163	−	0.116739i	$-0.962756\pi$
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	0.395482	−	0.918474i	$-0.370577\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	−36.0000			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$912$		0			0
$913$	6.00000	+	10.3923i	0.198571	+	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.00000	−	10.3923i	0.197922	−	0.342811i	−0.749933	−	0.661514i	$-0.769914\pi$
$919$	6.00000	−	10.3923i	0.197922	−	0.342811i	0.947854	+	0.318704i	$0.103247\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.0000			0.329154
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	12.0000	+	20.7846i	0.394132	+	0.682656i
$928$		0			0
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.0410949	+	0.999155i	$0.513085\pi$
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.844746	+	0.535167i	$0.820249\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	+	20.7846i	0.392442	+	0.679729i
$936$		0			0
$937$	−42.0000			−1.37208			−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000			−1.37208			−0.686040	+	0.727564i	$0.740653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.00000	+	8.66025i	0.162995	+	0.282316i	0.935942	−	0.352155i	$-0.114551\pi$
$941$	5.00000	+	8.66025i	0.162995	+	0.282316i	−0.772946	+	0.634472i	$0.781218\pi$
$942$		0			0
$943$	−24.0000	+	41.5692i	−0.781548	+	1.35368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	−0.291835	−	0.956469i	$-0.594266\pi$
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	0.974244	−	0.225497i	$-0.0724007\pi$
$948$		0			0
$949$	1.00000	+	1.73205i	0.0324614	+	0.0562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	4.00000	+	6.92820i	0.129437	+	0.224191i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$	−24.0000	−	41.5692i	−0.773389	−	1.33955i
$964$		0			0
$965$	−4.00000			−0.128765
$966$		0			0
$967$	50.0000			1.60789			0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000			1.60789			0.803946	+	0.594703i	$0.202730\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	0.167258	+	0.985913i	$0.446509\pi$
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	−0.937455	+	0.348107i	$0.886825\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	0.685381	−	0.728184i	$-0.259636\pi$
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	−0.973317	+	0.229465i	$0.926302\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$		0			0
$981$	42.0000			1.34096
$982$		0			0
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	0.955798	−	0.294025i	$-0.0949950\pi$
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	−0.732532	+	0.680732i	$0.761662\pi$
$984$		0			0
$985$	−6.00000	+	10.3923i	−0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000	+	27.7128i	−0.508770	+	0.881216i
$990$		0			0
$991$	−6.00000	−	10.3923i	−0.190596	−	0.330122i	0.754852	−	0.655895i	$-0.227709\pi$
$991$	−6.00000	−	10.3923i	−0.190596	−	0.330122i	−0.945448	+	0.325773i	$0.894375\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−32.0000			−1.01447
$996$		0			0
$997$	−29.0000	−	50.2295i	−0.918439	−	1.59078i	−0.801786	−	0.597611i	$-0.796117\pi$
$997$	−29.0000	−	50.2295i	−0.918439	−	1.59078i	−0.116653	−	0.993173i	$-0.537216\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2548.2.j.f.1353.1		2
7.2	even	3		2548.2.a.e.1.1		1
7.3	odd	6		2548.2.j.e.1145.1		2
7.4	even	3	inner	2548.2.j.f.1145.1		2
7.5	odd	6		52.2.a.a.1.1	✓	1
7.6	odd	2		2548.2.j.e.1353.1		2
21.5	even	6		468.2.a.b.1.1		1
28.19	even	6		208.2.a.c.1.1		1
35.12	even	12		1300.2.c.c.1249.1		2
35.19	odd	6		1300.2.a.d.1.1		1
35.33	even	12		1300.2.c.c.1249.2		2
56.5	odd	6		832.2.a.e.1.1		1
56.19	even	6		832.2.a.f.1.1		1
63.5	even	6		4212.2.i.i.2809.1		2
63.40	odd	6		4212.2.i.d.2809.1		2
63.47	even	6		4212.2.i.i.1405.1		2
63.61	odd	6		4212.2.i.d.1405.1		2
77.54	even	6		6292.2.a.g.1.1		1
84.47	odd	6		1872.2.a.f.1.1		1
91.5	even	12		676.2.d.c.337.1		2
91.12	odd	6		676.2.a.c.1.1		1
91.19	even	12		676.2.h.c.361.1		4
91.33	even	12		676.2.h.c.361.2		4
91.47	even	12		676.2.d.c.337.2		2
91.54	even	12		676.2.h.c.485.1		4
91.61	odd	6		676.2.e.c.653.1		2
91.68	odd	6		676.2.e.c.529.1		2
91.75	odd	6		676.2.e.b.529.1		2
91.82	odd	6		676.2.e.b.653.1		2
91.89	even	12		676.2.h.c.485.2		4
112.5	odd	12		3328.2.b.q.1665.2		2
112.19	even	12		3328.2.b.e.1665.1		2
112.61	odd	12		3328.2.b.q.1665.1		2
112.75	even	12		3328.2.b.e.1665.2		2
140.19	even	6		5200.2.a.q.1.1		1
168.5	even	6		7488.2.a.bn.1.1		1
168.131	odd	6		7488.2.a.bw.1.1		1
273.5	odd	12		6084.2.b.m.4393.2		2
273.47	odd	12		6084.2.b.m.4393.1		2
273.194	even	6		6084.2.a.m.1.1		1
364.47	odd	12		2704.2.f.f.337.2		2
364.103	even	6		2704.2.a.g.1.1		1
364.187	odd	12		2704.2.f.f.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	7.5	odd	6
208.2.a.c.1.1		1	28.19	even	6
468.2.a.b.1.1		1	21.5	even	6
676.2.a.c.1.1		1	91.12	odd	6
676.2.d.c.337.1		2	91.5	even	12
676.2.d.c.337.2		2	91.47	even	12
676.2.e.b.529.1		2	91.75	odd	6
676.2.e.b.653.1		2	91.82	odd	6
676.2.e.c.529.1		2	91.68	odd	6
676.2.e.c.653.1		2	91.61	odd	6
676.2.h.c.361.1		4	91.19	even	12
676.2.h.c.361.2		4	91.33	even	12
676.2.h.c.485.1		4	91.54	even	12
676.2.h.c.485.2		4	91.89	even	12
832.2.a.e.1.1		1	56.5	odd	6
832.2.a.f.1.1		1	56.19	even	6
1300.2.a.d.1.1		1	35.19	odd	6
1300.2.c.c.1249.1		2	35.12	even	12
1300.2.c.c.1249.2		2	35.33	even	12
1872.2.a.f.1.1		1	84.47	odd	6
2548.2.a.e.1.1		1	7.2	even	3
2548.2.j.e.1145.1		2	7.3	odd	6
2548.2.j.e.1353.1		2	7.6	odd	2
2548.2.j.f.1145.1		2	7.4	even	3	inner
2548.2.j.f.1353.1		2	1.1	even	1	trivial
2704.2.a.g.1.1		1	364.103	even	6
2704.2.f.f.337.1		2	364.187	odd	12
2704.2.f.f.337.2		2	364.47	odd	12
3328.2.b.e.1665.1		2	112.19	even	12
3328.2.b.e.1665.2		2	112.75	even	12
3328.2.b.q.1665.1		2	112.61	odd	12
3328.2.b.q.1665.2		2	112.5	odd	12
4212.2.i.d.1405.1		2	63.61	odd	6
4212.2.i.d.2809.1		2	63.40	odd	6
4212.2.i.i.1405.1		2	63.47	even	6
4212.2.i.i.2809.1		2	63.5	even	6
5200.2.a.q.1.1		1	140.19	even	6
6084.2.a.m.1.1		1	273.194	even	6
6084.2.b.m.4393.1		2	273.47	odd	12
6084.2.b.m.4393.2		2	273.5	odd	12
6292.2.a.g.1.1		1	77.54	even	6
7488.2.a.bn.1.1		1	168.5	even	6
7488.2.a.bw.1.1		1	168.131	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(-4.00000 + 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +2.00000 q^{29} +(5.00000 + 8.66025i) q^{31} +(3.00000 - 5.19615i) q^{37} +6.00000 q^{41} +4.00000 q^{43} +(-3.00000 - 5.19615i) q^{45} +(-1.00000 + 1.73205i) q^{47} +(-3.00000 - 5.19615i) q^{53} +4.00000 q^{55} +(-5.00000 - 8.66025i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.0000 q^{71} +(1.00000 + 1.73205i) q^{73} +(2.00000 - 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +6.00000 q^{83} +12.0000 q^{85} +(-3.00000 + 5.19615i) q^{89} +(6.00000 + 10.3923i) q^{95} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 + 3 * q^9	\( 2 q + 2 q^{5} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{17} - 6 q^{19} - 8 q^{23} + q^{25} + 4 q^{29} + 10 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 6 q^{45} - 2 q^{47} - 6 q^{53} + 8 q^{55} - 10 q^{59} - 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} + 2 q^{73} + 4 q^{79} - 9 q^{81} + 12 q^{83} + 24 q^{85} - 6 q^{89} + 12 q^{95} - 4 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^17 - 6 * q^19 - 8 * q^23 + q^25 + 4 * q^29 + 10 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 6 * q^45 - 2 * q^47 - 6 * q^53 + 8 * q^55 - 10 * q^59 - 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 + 2 * q^73 + 4 * q^79 - 9 * q^81 + 12 * q^83 + 24 * q^85 - 6 * q^89 + 12 * q^95 - 4 * q^97 + 12 * q^99

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