LMFDB - Embedded newform 3460.1.bd.a.343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3460,1,Mod(7,3460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3460, base_ring=CyclotomicField(172))

chi = DirichletCharacter(H, H._module([86, 43, 95]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3460.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3460 = 2^{2} \cdot 5 \cdot 173 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3460.bd (of order $172$, degree $84$, 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:	$1.72676494371$
Analytic rank:	$0$
Dimension:	$84$
Coefficient field:	$\Q(\zeta_{172})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{84} - x^{82} + x^{80} - x^{78} + x^{76} - x^{74} + x^{72} - x^{70} + x^{68} - x^{66} + x^{64} + \cdots + 1 $ x^84 - x^82 + x^80 - x^78 + x^76 - x^74 + x^72 - x^70 + x^68 - x^66 + x^64 - x^62 + x^60 - x^58 + x^56 - x^54 + x^52 - x^50 + x^48 - x^46 + x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{172}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{172} - \cdots)$

Embedding invariants

Embedding label			343.1
Root			$-0.994001 - 0.109371i$ of defining polynomial
Character	$\chi$	$=$	3460.343
Dual form			3460.1.bd.a.807.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.288099 + 0.957601i) q^{2} +(-0.833998 + 0.551768i) q^{4} +(0.424457 - 0.905448i) q^{5} +(-0.768647 - 0.639673i) q^{8} +(0.997332 - 0.0729953i) q^{9} +O(q^{10})$	\(q+(0.288099 + 0.957601i) q^{2} +(-0.833998 + 0.551768i) q^{4} +(0.424457 - 0.905448i) q^{5} +(-0.768647 - 0.639673i) q^{8} +(0.997332 - 0.0729953i) q^{9} +(0.989343 + 0.145601i) q^{10} +(-0.289291 + 0.538267i) q^{13} +(0.391105 - 0.920346i) q^{16} +(-0.939554 + 1.21676i) q^{17} +(0.357231 + 0.934016i) q^{18} +(0.145601 + 0.989343i) q^{20} +(-0.639673 - 0.768647i) q^{25} +(-0.598789 - 0.121952i) q^{26} +(1.65068 + 0.563135i) q^{29} +(0.994001 + 0.109371i) q^{32} +(-1.43586 - 0.549169i) q^{34} +(-0.791496 + 0.611174i) q^{36} +(1.68759 + 1.07271i) q^{37} +(-0.905448 + 0.424457i) q^{40} +(0.886285 - 1.57919i) q^{41} +(0.357231 - 0.934016i) q^{45} +(0.520940 + 0.853593i) q^{49} +(0.551768 - 0.833998i) q^{50} +(-0.0557297 - 0.608535i) q^{52} +(1.18260 + 0.309172i) q^{53} +(-0.0636980 + 1.74294i) q^{58} +(0.210076 - 1.63411i) q^{61} +(0.181637 + 0.983366i) q^{64} +(0.364581 + 0.490409i) q^{65} +(0.112215 - 1.53319i) q^{68} +(-0.813290 - 0.581859i) q^{72} +(-0.536759 - 0.779971i) q^{73} +(-0.541032 + 1.92508i) q^{74} +(-0.667319 - 0.744772i) q^{80} +(0.989343 - 0.145601i) q^{81} +(1.76757 + 0.393744i) q^{82} +(0.702916 + 1.36718i) q^{85} +(-0.0855512 - 0.777517i) q^{89} +(0.997332 + 0.0729953i) q^{90} +(1.22901 + 1.18491i) q^{97} +(-0.667319 + 0.744772i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q + 2 q^{4} + 2 q^{9}+O(q^{10}) $ 84 * q + 2 * q^4 + 2 * q^9	$ 84 q + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{26} - 2 q^{36} - 2 q^{37} + 2 q^{40} - 2 q^{49} - 2 q^{52} + 4 q^{58} - 2 q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{73} + 2 q^{74} - 2 q^{81} + 4 q^{82} + 4 q^{89} + 2 q^{90}+O(q^{100}) $ 84 * q + 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^26 - 2 * q^36 - 2 * q^37 + 2 * q^40 - 2 * q^49 - 2 * q^52 + 4 * q^58 - 2 * q^61 + 2 * q^64 - 2 * q^65 + 2 * q^73 + 2 * q^74 - 2 * q^81 + 4 * q^82 + 4 * q^89 + 2 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3460\mathbb{Z}\right)^\times$.

$n$	$521$	$1731$	$2077$
$\chi(n)$	$e\left(\frac{113}{172}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	0.288099	+	0.957601i	0.288099	+	0.957601i
$2$	0.288099	+	0.957601i	0.288099	+	0.957601i
$3$		0			0		0.999333	−	0.0365220i	$-0.0116279\pi$
$3$		0			0		−0.999333	+	0.0365220i	$0.988372\pi$
$4$	−0.833998	+	0.551768i	−0.833998	+	0.551768i
$5$	0.424457	−	0.905448i	0.424457	−	0.905448i
$5$	0.424457	−	0.905448i	0.424457	−	0.905448i
$6$		0			0
$7$		0			0		−0.872049	−	0.489418i	$-0.837209\pi$
$7$		0			0		0.872049	+	0.489418i	$0.162791\pi$
$8$	−0.768647	−	0.639673i	−0.768647	−	0.639673i
$9$	0.997332	−	0.0729953i	0.997332	−	0.0729953i
$10$	0.989343	+	0.145601i	0.989343	+	0.145601i
$11$		0			0		−0.940385	−	0.340112i	$-0.889535\pi$
$11$		0			0		0.940385	+	0.340112i	$0.110465\pi$
$12$		0			0
$13$	−0.289291	+	0.538267i	−0.289291	+	0.538267i	−0.983366	−	0.181637i	$-0.941860\pi$
$13$	−0.289291	+	0.538267i	−0.289291	+	0.538267i	0.694074	+	0.719903i	$0.255814\pi$
$14$		0			0
$15$		0			0
$16$	0.391105	−	0.920346i	0.391105	−	0.920346i
$17$	−0.939554	+	1.21676i	−0.939554	+	1.21676i	0.0365220	+	0.999333i	$0.488372\pi$
$17$	−0.939554	+	1.21676i	−0.939554	+	1.21676i	−0.976076	+	0.217430i	$0.930233\pi$
$18$	0.357231	+	0.934016i	0.357231	+	0.934016i
$19$		0			0		−0.843936	−	0.536444i	$-0.819767\pi$
$19$		0			0		0.843936	+	0.536444i	$0.180233\pi$
$20$	0.145601	+	0.989343i	0.145601	+	0.989343i
$21$		0			0
$22$		0			0
$23$		0			0		−0.340112	−	0.940385i	$-0.610465\pi$
$23$		0			0		0.340112	+	0.940385i	$0.389535\pi$
$24$		0			0
$25$	−0.639673	−	0.768647i	−0.639673	−	0.768647i
$26$	−0.598789	−	0.121952i	−0.598789	−	0.121952i
$27$		0			0
$28$		0			0
$29$	1.65068	+	0.563135i	1.65068	+	0.563135i	0.983366	−	0.181637i	$-0.0581395\pi$
$29$	1.65068	+	0.563135i	1.65068	+	0.563135i	0.667319	+	0.744772i	$0.267442\pi$
$30$		0			0
$31$		0			0		−0.999333	−	0.0365220i	$-0.988372\pi$
$31$		0			0		0.999333	+	0.0365220i	$0.0116279\pi$
$32$	0.994001	+	0.109371i	0.994001	+	0.109371i
$33$		0			0
$34$	−1.43586	−	0.549169i	−1.43586	−	0.549169i
$35$		0			0
$36$	−0.791496	+	0.611174i	−0.791496	+	0.611174i
$37$	1.68759	+	1.07271i	1.68759	+	1.07271i	0.853593	+	0.520940i	$0.174419\pi$
$37$	1.68759	+	1.07271i	1.68759	+	1.07271i	0.833998	+	0.551768i	$0.186047\pi$
$38$		0			0
$39$		0			0
$40$	−0.905448	+	0.424457i	−0.905448	+	0.424457i
$41$	0.886285	−	1.57919i	0.886285	−	1.57919i	0.0729953	−	0.997332i	$-0.476744\pi$
$41$	0.886285	−	1.57919i	0.886285	−	1.57919i	0.813290	−	0.581859i	$-0.197674\pi$
$42$		0			0
$43$		0			0		−0.199567	−	0.979884i	$-0.563953\pi$
$43$		0			0		0.199567	+	0.979884i	$0.436047\pi$
$44$		0			0
$45$	0.357231	−	0.934016i	0.357231	−	0.934016i
$46$		0			0
$47$		0			0		0.802527	−	0.596616i	$-0.203488\pi$
$47$		0			0		−0.802527	+	0.596616i	$0.796512\pi$
$48$		0			0
$49$	0.520940	+	0.853593i	0.520940	+	0.853593i
$50$	0.551768	−	0.833998i	0.551768	−	0.833998i
$51$		0			0
$52$	−0.0557297	−	0.608535i	−0.0557297	−	0.608535i
$53$	1.18260	+	0.309172i	1.18260	+	0.309172i	0.791496	−	0.611174i	$-0.209302\pi$
$53$	1.18260	+	0.309172i	1.18260	+	0.309172i	0.391105	+	0.920346i	$0.372093\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.0636980	+	1.74294i	−0.0636980	+	1.74294i
$59$		0			0		−0.971942	−	0.235221i	$-0.924419\pi$
$59$		0			0		0.971942	+	0.235221i	$0.0755814\pi$
$60$		0			0
$61$	0.210076	−	1.63411i	0.210076	−	1.63411i	−0.457242	−	0.889342i	$-0.651163\pi$
$61$	0.210076	−	1.63411i	0.210076	−	1.63411i	0.667319	−	0.744772i	$-0.267442\pi$
$62$		0			0
$63$		0			0
$64$	0.181637	+	0.983366i	0.181637	+	0.983366i
$65$	0.364581	+	0.490409i	0.364581	+	0.490409i
$66$		0			0
$67$		0			0		0.680810	−	0.732460i	$-0.261628\pi$
$67$		0			0		−0.680810	+	0.732460i	$0.738372\pi$
$68$	0.112215	−	1.53319i	0.112215	−	1.53319i
$69$		0			0
$70$		0			0
$71$		0			0		0.897545	−	0.440923i	$-0.145349\pi$
$71$		0			0		−0.897545	+	0.440923i	$0.854651\pi$
$72$	−0.813290	−	0.581859i	−0.813290	−	0.581859i
$73$	−0.536759	−	0.779971i	−0.536759	−	0.779971i	0.457242	−	0.889342i	$-0.348837\pi$
$73$	−0.536759	−	0.779971i	−0.536759	−	0.779971i	−0.994001	+	0.109371i	$0.965116\pi$
$74$	−0.541032	+	1.92508i	−0.541032	+	1.92508i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.596616	−	0.802527i	$-0.296512\pi$
$79$		0			0		−0.596616	+	0.802527i	$0.703488\pi$
$80$	−0.667319	−	0.744772i	−0.667319	−	0.744772i
$81$	0.989343	−	0.145601i	0.989343	−	0.145601i
$82$	1.76757	+	0.393744i	1.76757	+	0.393744i
$83$		0			0		0.0182641	−	0.999833i	$-0.494186\pi$
$83$		0			0		−0.0182641	+	0.999833i	$0.505814\pi$
$84$		0			0
$85$	0.702916	+	1.36718i	0.702916	+	1.36718i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.0855512	−	0.777517i	−0.0855512	−	0.777517i	−0.957601	−	0.288099i	$-0.906977\pi$
$89$	−0.0855512	−	0.777517i	−0.0855512	−	0.777517i	0.872049	−	0.489418i	$-0.162791\pi$
$90$	0.997332	+	0.0729953i	0.997332	+	0.0729953i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.22901	+	1.18491i	1.22901	+	1.18491i	0.976076	+	0.217430i	$0.0697674\pi$
$97$	1.22901	+	1.18491i	1.22901	+	1.18491i	0.252933	+	0.967484i	$0.418605\pi$
$98$	−0.667319	+	0.744772i	−0.667319	+	0.744772i
$99$		0			0
$100$	0.957601	+	0.288099i	0.957601	+	0.288099i
$101$	−1.93335	+	0.320709i	−1.93335	+	0.320709i	−0.999333	−	0.0365220i	$-0.988372\pi$
$101$	−1.93335	+	0.320709i	−1.93335	+	0.320709i	−0.934016	+	0.357231i	$0.883721\pi$
$102$		0			0
$103$		0			0		−0.181637	−	0.983366i	$-0.558140\pi$
$103$		0			0		0.181637	+	0.983366i	$0.441860\pi$
$104$	0.566678	−	0.228685i	0.566678	−	0.228685i
$105$		0			0
$106$	0.0446426	+	1.22153i	0.0446426	+	1.22153i
$107$		0			0		0.719903	−	0.694074i	$-0.244186\pi$
$107$		0			0		−0.719903	+	0.694074i	$0.755814\pi$
$108$		0			0
$109$	−1.86139	+	0.343817i	−1.86139	+	0.343817i	−0.989343	−	0.145601i	$-0.953488\pi$
$109$	−1.86139	+	0.343817i	−1.86139	+	0.343817i	−0.872049	+	0.489418i	$0.837209\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.307483	−	1.27053i	−0.307483	−	1.27053i	−0.889342	−	0.457242i	$-0.848837\pi$
$113$	−0.307483	−	1.27053i	−0.307483	−	1.27053i	0.581859	−	0.813290i	$-0.302326\pi$
$114$		0			0
$115$		0			0
$116$	−1.68739	+	0.441141i	−1.68739	+	0.441141i
$117$	−0.249229	+	0.557948i	−0.249229	+	0.557948i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.768647	+	0.639673i	0.768647	+	0.639673i
$122$	1.62535	−	0.269617i	1.62535	−	0.269617i
$123$		0			0
$124$		0			0
$125$	−0.967484	+	0.252933i	−0.967484	+	0.252933i
$126$		0			0
$127$		0			0		−0.989343	−	0.145601i	$-0.953488\pi$
$127$		0			0		0.989343	+	0.145601i	$0.0465116\pi$
$128$	−0.889342	+	0.457242i	−0.889342	+	0.457242i
$129$		0			0
$130$	−0.364581	+	0.490409i	−0.364581	+	0.490409i
$131$		0			0		−0.566908	−	0.823781i	$-0.691860\pi$
$131$		0			0		0.566908	+	0.823781i	$0.308140\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.50052	−	0.334254i	1.50052	−	0.334254i
$137$	−1.06413	−	0.136801i	−1.06413	−	0.136801i	−0.424457	−	0.905448i	$-0.639535\pi$
$137$	−1.06413	−	0.136801i	−1.06413	−	0.136801i	−0.639673	+	0.768647i	$0.720930\pi$
$138$		0			0
$139$		0			0		0.357231	−	0.934016i	$-0.383721\pi$
$139$		0			0		−0.357231	+	0.934016i	$0.616279\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.322880	−	0.946440i	0.322880	−	0.946440i
$145$	1.21053	−	1.25558i	1.21053	−	1.25558i
$146$	0.592261	−	0.738709i	0.592261	−	0.738709i
$147$		0			0
$148$	−1.99933	+	0.0365220i	−1.99933	+	0.0365220i
$149$	−0.674452	+	0.873444i	−0.674452	+	0.873444i	−0.997332	−	0.0729953i	$-0.976744\pi$
$149$	−0.674452	+	0.873444i	−0.674452	+	0.873444i	0.322880	+	0.946440i	$0.395349\pi$
$150$		0			0
$151$		0			0		−0.667319	−	0.744772i	$-0.732558\pi$
$151$		0			0		0.667319	+	0.744772i	$0.267442\pi$
$152$		0			0
$153$	−0.848229	+	1.28210i	−0.848229	+	1.28210i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.80263	+	0.727460i	−1.80263	+	0.727460i	−0.813290	+	0.581859i	$0.802326\pi$
$157$	−1.80263	+	0.727460i	−1.80263	+	0.727460i	−0.989343	+	0.145601i	$0.953488\pi$
$158$		0			0
$159$		0			0
$160$	0.520940	−	0.853593i	0.520940	−	0.853593i
$161$		0			0
$162$	0.424457	+	0.905448i	0.424457	+	0.905448i
$163$		0			0		−0.995833	−	0.0911985i	$-0.970930\pi$
$163$		0			0		0.995833	+	0.0911985i	$0.0290698\pi$
$164$	0.132187	+	1.80607i	0.132187	+	1.80607i
$165$		0			0
$166$		0			0
$167$		0			0		0.440923	−	0.897545i	$-0.354651\pi$
$167$		0			0		−0.440923	+	0.897545i	$0.645349\pi$
$168$		0			0
$169$	0.345726	+	0.522566i	0.345726	+	0.522566i
$170$	−1.10670	+	1.06700i	−1.10670	+	1.06700i
$171$		0			0
$172$		0			0
$173$	−0.920346	−	0.391105i	−0.920346	−	0.391105i
$173$	−0.920346	−	0.391105i	−0.920346	−	0.391105i
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.719903	−	0.305926i	0.719903	−	0.305926i
$179$		0			0		0.946440	−	0.322880i	$-0.104651\pi$
$179$		0			0		−0.946440	+	0.322880i	$0.895349\pi$
$180$	0.217430	+	0.976076i	0.217430	+	0.976076i
$181$	−1.59837	+	0.146378i	−1.59837	+	0.146378i	−0.853593	−	0.520940i	$-0.825581\pi$
$181$	−1.59837	+	0.146378i	−1.59837	+	0.146378i	−0.744772	+	0.667319i	$0.767442\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.68759	−	1.07271i	1.68759	−	1.07271i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.913050	−	0.407849i	$-0.866279\pi$
$191$		0			0		0.913050	+	0.407849i	$0.133721\pi$
$192$		0			0
$193$	−0.121756	−	0.0805529i	−0.121756	−	0.0805529i	0.489418	−	0.872049i	$-0.337209\pi$
$193$	−0.121756	−	0.0805529i	−0.121756	−	0.0805529i	−0.611174	+	0.791496i	$0.709302\pi$
$194$	−0.780598	+	1.51827i	−0.780598	+	1.51827i
$195$		0			0
$196$	−0.905448	−	0.424457i	−0.905448	−	0.424457i
$197$	−0.251577	−	1.95693i	−0.251577	−	1.95693i	−0.288099	−	0.957601i	$-0.593023\pi$
$197$	−0.251577	−	1.95693i	−0.251577	−	1.95693i	0.0365220	−	0.999333i	$-0.488372\pi$
$198$		0			0
$199$		0			0		0.199567	−	0.979884i	$-0.436047\pi$
$199$		0			0		−0.199567	+	0.979884i	$0.563953\pi$
$200$			1.00000i			1.00000i
$201$		0			0
$202$	−0.864107	−	1.75898i	−0.864107	−	1.75898i
$203$		0			0
$204$		0			0
$205$	−1.05369	−	1.47278i	−1.05369	−	1.47278i
$206$		0			0
$207$		0			0
$208$	0.382248	+	0.476767i	0.382248	+	0.476767i
$209$		0			0
$210$		0			0
$211$		0			0		−0.489418	−	0.872049i	$-0.662791\pi$
$211$		0			0		0.489418	+	0.872049i	$0.337209\pi$
$212$	−1.15688	+	0.394672i	−1.15688	+	0.394672i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.865505	−	1.68342i	−0.865505	−	1.68342i
$219$		0			0
$220$		0			0
$221$	−0.383138	−	0.857730i	−0.383138	−	0.857730i
$222$		0			0
$223$		0			0		−0.889342	−	0.457242i	$-0.848837\pi$
$223$		0			0		0.889342	+	0.457242i	$0.151163\pi$
$224$		0			0
$225$	−0.694074	−	0.719903i	−0.694074	−	0.719903i
$226$	1.12808	−	0.660485i	1.12808	−	0.660485i
$227$		0			0		0.374230	−	0.927336i	$-0.377907\pi$
$227$		0			0		−0.374230	+	0.927336i	$0.622093\pi$
$228$		0			0
$229$	0.406167	−	0.155346i	0.406167	−	0.155346i	−0.145601	−	0.989343i	$-0.546512\pi$
$229$	0.406167	−	0.155346i	0.406167	−	0.155346i	0.551768	+	0.833998i	$0.313953\pi$
$230$		0			0
$231$		0			0
$232$	−0.908571	−	1.48875i	−0.908571	−	1.48875i
$233$	0.792811	−	0.191869i	0.792811	−	0.191869i	0.181637	−	0.983366i	$-0.441860\pi$
$233$	0.792811	−	0.191869i	0.792811	−	0.191869i	0.611174	+	0.791496i	$0.290698\pi$
$234$	−0.606094	−	0.0779174i	−0.606094	−	0.0779174i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.999833	−	0.0182641i	$-0.994186\pi$
$239$		0			0		0.999833	+	0.0182641i	$0.00581395\pi$
$240$		0			0
$241$	−0.213920	−	0.247707i	−0.213920	−	0.247707i	0.639673	−	0.768647i	$-0.279070\pi$
$241$	−0.213920	−	0.247707i	−0.213920	−	0.247707i	−0.853593	+	0.520940i	$0.825581\pi$
$242$	−0.391105	+	0.920346i	−0.391105	+	0.920346i
$243$		0			0
$244$	0.726448	+	1.47876i	0.726448	+	1.47876i
$245$	0.994001	−	0.109371i	0.994001	−	0.109371i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.520940	−	0.853593i	−0.520940	−	0.853593i
$251$		0			0		0.813290	−	0.581859i	$-0.197674\pi$
$251$		0			0		−0.813290	+	0.581859i	$0.802326\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.694074	−	0.719903i	−0.694074	−	0.719903i
$257$	0.788415	+	0.632112i	0.788415	+	0.632112i	0.934016	−	0.357231i	$-0.116279\pi$
$257$	0.788415	+	0.632112i	0.788415	+	0.632112i	−0.145601	+	0.989343i	$0.546512\pi$
$258$		0			0
$259$		0			0
$260$	−0.574652	−	0.207836i	−0.574652	−	0.207836i
$261$	1.68739	+	0.441141i	1.68739	+	0.441141i
$262$		0			0
$263$		0			0		−0.999833	−	0.0182641i	$-0.994186\pi$
$263$		0			0		0.999833	+	0.0182641i	$0.00581395\pi$
$264$		0			0
$265$	0.781903	−	0.939554i	0.781903	−	0.939554i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.0369259	+	0.504517i	−0.0369259	+	0.504517i	0.946440	+	0.322880i	$0.104651\pi$
$269$	−0.0369259	+	0.504517i	−0.0369259	+	0.504517i	−0.983366	+	0.181637i	$0.941860\pi$
$270$		0			0
$271$		0			0		−0.0547678	−	0.998499i	$-0.517442\pi$
$271$		0			0		0.0547678	+	0.998499i	$0.482558\pi$
$272$	0.752379	+	1.34060i	0.752379	+	1.34060i
$273$		0			0
$274$	−0.175574	−	1.05842i	−0.175574	−	1.05842i
$275$		0			0
$276$		0			0
$277$	−1.83946	−	0.781688i	−1.83946	−	0.781688i	−0.934016	−	0.357231i	$-0.883721\pi$
$277$	−1.83946	−	0.781688i	−1.83946	−	0.781688i	−0.905448	−	0.424457i	$-0.860465\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.163709	−	1.78761i	0.163709	−	1.78761i	−0.357231	−	0.934016i	$-0.616279\pi$
$281$	0.163709	−	1.78761i	0.163709	−	1.78761i	0.520940	−	0.853593i	$-0.325581\pi$
$282$		0			0
$283$		0			0		0.424457	−	0.905448i	$-0.360465\pi$
$283$		0			0		−0.424457	+	0.905448i	$0.639535\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.999333	+	0.0365220i	0.999333	+	0.0365220i
$289$	−0.344817	−	1.31895i	−0.344817	−	1.31895i
$290$	1.55110	+	0.797476i	1.55110	+	0.797476i
$291$		0			0
$292$	0.878018	+	0.354328i	0.878018	+	0.354328i
$293$	−0.500476	+	1.91435i	−0.500476	+	1.91435i	−0.109371	+	0.994001i	$0.534884\pi$
$293$	−0.500476	+	1.91435i	−0.500476	+	1.91435i	−0.391105	+	0.920346i	$0.627907\pi$
$294$		0			0
$295$		0			0
$296$	−0.610980	−	1.90404i	−0.610980	−	1.90404i
$297$		0			0
$298$	−1.03072	−	0.394217i	−1.03072	−	0.394217i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.39044	−	0.883824i	−1.39044	−	0.883824i
$306$	−1.47211	−	0.442893i	−1.47211	−	0.442893i
$307$		0			0		−0.322880	−	0.946440i	$-0.604651\pi$
$307$		0			0		0.322880	+	0.946440i	$0.395349\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.109371	−	0.994001i	$-0.465116\pi$
$311$		0			0		−0.109371	+	0.994001i	$0.534884\pi$
$312$		0			0
$313$	−0.606094	+	1.88881i	−0.606094	+	1.88881i	−0.181637	+	0.983366i	$0.558140\pi$
$313$	−0.606094	+	1.88881i	−0.606094	+	1.88881i	−0.424457	+	0.905448i	$0.639535\pi$
$314$	−1.21595	−	1.51662i	−1.21595	−	1.51662i
$315$		0			0
$316$		0			0
$317$	−0.791496	+	0.388826i	−0.791496	+	0.388826i	−0.791496	−	0.611174i	$-0.790698\pi$
$317$	−0.791496	+	0.388826i	−0.791496	+	0.388826i			1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	0.967484	+	0.252933i	0.967484	+	0.252933i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.744772	+	0.667319i	−0.744772	+	0.667319i
$325$	0.598789	−	0.121952i	0.598789	−	0.121952i
$326$		0			0
$327$		0			0
$328$	−1.69141	+	0.646908i	−1.69141	+	0.646908i
$329$		0			0
$330$		0			0
$331$		0			0		0.905448	−	0.424457i	$-0.139535\pi$
$331$		0			0		−0.905448	+	0.424457i	$0.860465\pi$
$332$		0			0
$333$	1.76139	+	0.946660i	1.76139	+	0.946660i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.21222	−	1.40368i	1.21222	−	1.40368i	0.322880	−	0.946440i	$-0.395349\pi$
$337$	1.21222	−	1.40368i	1.21222	−	1.40368i	0.889342	−	0.457242i	$-0.151163\pi$
$338$	−0.400806	+	0.481619i	−0.400806	+	0.481619i
$339$		0			0
$340$	−1.34060	−	0.752379i	−1.34060	−	0.752379i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.109371	−	0.994001i	0.109371	−	0.994001i
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	−1.29036	−	1.19937i	−1.29036	−	1.19937i	−0.967484	−	0.252933i	$-0.918605\pi$
$349$	−1.29036	−	1.19937i	−1.29036	−	1.19937i	−0.322880	−	0.946440i	$-0.604651\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.937334	−	1.67015i	0.937334	−	1.67015i	0.217430	−	0.976076i	$-0.430233\pi$
$353$	0.937334	−	1.67015i	0.937334	−	1.67015i	0.719903	−	0.694074i	$-0.244186\pi$
$354$		0			0
$355$		0			0
$356$	0.500358	+	0.601243i	0.500358	+	0.601243i
$357$		0			0
$358$		0			0
$359$		0			0		0.957601	−	0.288099i	$-0.0930233\pi$
$359$		0			0		−0.957601	+	0.288099i	$0.906977\pi$
$360$	−0.872049	+	0.489418i	−0.872049	+	0.489418i
$361$	0.424457	+	0.905448i	0.424457	+	0.905448i
$362$	−0.600660	−	1.48842i	−0.600660	−	1.48842i
$363$		0			0
$364$		0			0
$365$	−0.934054	+	0.154943i	−0.934054	+	0.154943i
$366$		0			0
$367$		0			0		0.952179	−	0.305541i	$-0.0988372\pi$
$367$		0			0		−0.952179	+	0.305541i	$0.901163\pi$
$368$		0			0
$369$	0.768647	−	1.63967i	0.768647	−	1.63967i
$370$	1.51342	+	1.30699i	1.51342	+	1.30699i
$371$		0			0
$372$		0			0
$373$	0.182431	+	1.65799i	0.182431	+	1.65799i	0.639673	+	0.768647i	$0.279070\pi$
$373$	0.182431	+	1.65799i	0.182431	+	1.65799i	−0.457242	+	0.889342i	$0.651163\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.780646	+	0.725598i	−0.780646	+	0.725598i
$378$		0			0
$379$		0			0		−0.457242	−	0.889342i	$-0.651163\pi$
$379$		0			0		0.457242	+	0.889342i	$0.348837\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.536444	−	0.843936i	$-0.319767\pi$
$383$		0			0		−0.536444	+	0.843936i	$0.680233\pi$
$384$		0			0
$385$		0			0
$386$	0.0420598	−	0.139801i	0.0420598	−	0.139801i
$387$		0			0
$388$	−1.67879	−	0.310088i	−1.67879	−	0.310088i
$389$	0.160676	−	0.242862i	0.160676	−	0.242862i	−0.744772	−	0.667319i	$-0.767442\pi$
$389$	0.160676	−	0.242862i	0.160676	−	0.242862i	0.905448	+	0.424457i	$0.139535\pi$
$390$		0			0
$391$		0			0
$392$	0.145601	−	0.989343i	0.145601	−	0.989343i
$393$		0			0
$394$	1.80148	−	0.804701i	1.80148	−	0.804701i
$395$		0			0
$396$		0			0
$397$	0.0769861	+	0.464100i	0.0769861	+	0.464100i	0.997332	+	0.0729953i	$0.0232558\pi$
$397$	0.0769861	+	0.464100i	0.0769861	+	0.464100i	−0.920346	+	0.391105i	$0.872093\pi$
$398$		0			0
$399$		0			0
$400$	−0.957601	+	0.288099i	−0.957601	+	0.288099i
$401$	−1.24657	+	1.39126i	−1.24657	+	1.39126i	−0.357231	+	0.934016i	$0.616279\pi$
$401$	−1.24657	+	1.39126i	−1.24657	+	1.39126i	−0.889342	+	0.457242i	$0.848837\pi$
$402$		0			0
$403$		0			0
$404$	1.43545	−	1.33423i	1.43545	−	1.33423i
$405$	0.288099	−	0.957601i	0.288099	−	0.957601i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.405833	+	1.12210i	0.405833	+	1.12210i	0.957601	+	0.288099i	$0.0930233\pi$
$409$	0.405833	+	1.12210i	0.405833	+	1.12210i	−0.551768	+	0.833998i	$0.686047\pi$
$410$	1.10677	−	1.43332i	1.10677	−	1.43332i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.346427	+	0.503397i	−0.346427	+	0.503397i
$417$		0			0
$418$		0			0
$419$		0			0		0.181637	−	0.983366i	$-0.441860\pi$
$419$		0			0		−0.181637	+	0.983366i	$0.558140\pi$
$420$		0			0
$421$	0.0364733	−	0.00200056i	0.0364733	−	0.00200056i	−0.0365220	−	0.999333i	$-0.511628\pi$
$421$	0.0364733	−	0.00200056i	0.0364733	−	0.00200056i	0.0729953	+	0.997332i	$0.476744\pi$
$422$		0			0
$423$		0			0
$424$	−0.711234	−	0.994123i	−0.711234	−	0.994123i
$425$	1.53627	−	0.0561451i	1.53627	−	0.0561451i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.252933	−	0.967484i	$-0.418605\pi$
$431$		0			0		−0.252933	+	0.967484i	$0.581395\pi$
$432$		0			0
$433$	0.0210439	−	0.0699470i	0.0210439	−	0.0699470i	−0.946440	−	0.322880i	$-0.895349\pi$
$433$	0.0210439	−	0.0699470i	0.0210439	−	0.0699470i	0.967484	+	0.252933i	$0.0813953\pi$
$434$		0			0
$435$		0			0
$436$	1.36269	−	1.31380i	1.36269	−	1.31380i
$437$		0			0
$438$		0			0
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$	0.581859	+	0.813290i	0.581859	+	0.813290i
$442$	0.710981	−	0.614004i	0.710981	−	0.614004i
$443$		0			0		0.694074	−	0.719903i	$-0.255814\pi$
$443$		0			0		−0.694074	+	0.719903i	$0.744186\pi$
$444$		0			0
$445$	−0.740314	−	0.252560i	−0.740314	−	0.252560i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.385625	−	0.560356i	0.385625	−	0.560356i	−0.581859	−	0.813290i	$-0.697674\pi$
$449$	0.385625	−	0.560356i	0.385625	−	0.560356i	0.967484	+	0.252933i	$0.0813953\pi$
$450$	0.489418	−	0.872049i	0.489418	−	0.872049i
$451$		0			0
$452$	0.957479	+	0.889962i	0.957479	+	0.889962i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00800	+	0.263526i	−1.00800	+	0.263526i	−0.719903	−	0.694074i	$-0.755814\pi$
$457$	−1.00800	+	0.263526i	−1.00800	+	0.263526i	−0.288099	+	0.957601i	$0.593023\pi$
$458$	0.265775	+	0.344190i	0.265775	+	0.344190i
$459$		0			0
$460$		0			0
$461$	1.26417	−	1.17503i	1.26417	−	1.17503i	0.288099	−	0.957601i	$-0.406977\pi$
$461$	1.26417	−	1.17503i	1.26417	−	1.17503i	0.976076	−	0.217430i	$-0.0697674\pi$
$462$		0			0
$463$		0			0		−0.913050	−	0.407849i	$-0.866279\pi$
$463$		0			0		0.913050	+	0.407849i	$0.133721\pi$
$464$	1.16387	−	1.29896i	1.16387	−	1.29896i
$465$		0			0
$466$	0.412142	+	0.703919i	0.412142	+	0.703919i
$467$		0			0		0.995833	−	0.0911985i	$-0.0290698\pi$
$467$		0			0		−0.995833	+	0.0911985i	$0.970930\pi$
$468$	−0.100001	−	0.602843i	−0.100001	−	0.602843i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.20201	+	0.222023i	1.20201	+	0.222023i
$478$		0			0
$479$		0			0		0.288099	−	0.957601i	$-0.406977\pi$
$479$		0			0		−0.288099	+	0.957601i	$0.593023\pi$
$480$		0			0
$481$	−1.06561	+	0.598049i	−1.06561	+	0.598049i
$482$	0.175574	−	0.276214i	0.175574	−	0.276214i
$483$		0			0
$484$	−0.994001	−	0.109371i	−0.994001	−	0.109371i
$485$	1.59454	−	0.609860i	1.59454	−	0.609860i
$486$		0			0
$487$		0			0		0.109371	−	0.994001i	$-0.465116\pi$
$487$		0			0		−0.109371	+	0.994001i	$0.534884\pi$
$488$	−1.20677	+	1.12168i	−1.20677	+	1.12168i
$489$		0			0
$490$	0.391105	+	0.920346i	0.391105	+	0.920346i
$491$		0			0		−0.756836	−	0.653605i	$-0.773256\pi$
$491$		0			0		0.756836	+	0.653605i	$0.226744\pi$
$492$		0			0
$493$	−2.23611	+	1.47940i	−2.23611	+	1.47940i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.979884	−	0.199567i	$-0.0639535\pi$
$499$		0			0		−0.979884	+	0.199567i	$0.936047\pi$
$500$	0.667319	−	0.744772i	0.667319	−	0.744772i
$501$		0			0
$502$		0			0
$503$		0			0		−0.374230	−	0.927336i	$-0.622093\pi$
$503$		0			0		0.374230	+	0.927336i	$0.377907\pi$
$504$		0			0
$505$	−0.530238	+	1.88667i	−0.530238	+	1.88667i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.01260	−	1.21676i	−1.01260	−	1.21676i	−0.976076	−	0.217430i	$-0.930233\pi$
$509$	−1.01260	−	1.21676i	−1.01260	−	1.21676i	−0.0365220	−	0.999333i	$-0.511628\pi$
$510$		0			0
$511$		0			0
$512$	0.489418	−	0.872049i	0.489418	−	0.872049i
$513$		0			0
$514$	−0.378170	+	0.937098i	−0.378170	+	0.937098i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.0334676	−	0.610164i	0.0334676	−	0.610164i
$521$	0.911504	−	1.69598i	0.911504	−	1.69598i	0.217430	−	0.976076i	$-0.430233\pi$
$521$	0.911504	−	1.69598i	0.911504	−	1.69598i	0.694074	−	0.719903i	$-0.255814\pi$
$522$	0.0636980	+	1.74294i	0.0636980	+	1.74294i
$523$		0			0		0.199567	−	0.979884i	$-0.436047\pi$
$523$		0			0		−0.199567	+	0.979884i	$0.563953\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.768647	+	0.639673i	−0.768647	+	0.639673i
$530$	1.12498	+	0.478066i	1.12498	+	0.478066i
$531$		0			0
$532$		0			0
$533$	0.593631	+	0.933904i	0.593631	+	0.933904i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.493764	+	0.109991i	−0.493764	+	0.109991i
$539$		0			0
$540$		0			0
$541$	1.44111	−	1.29124i	1.44111	−	1.29124i	0.551768	−	0.833998i	$-0.313953\pi$
$541$	1.44111	−	1.29124i	1.44111	−	1.29124i	0.889342	−	0.457242i	$-0.151163\pi$
$542$		0			0
$543$		0			0
$544$	−1.06700	+	1.10670i	−1.06700	+	1.10670i
$545$	−0.478772	+	1.83133i	−0.478772	+	1.83133i
$546$		0			0
$547$		0			0		−0.0729953	−	0.997332i	$-0.523256\pi$
$547$		0			0		0.0729953	+	0.997332i	$0.476744\pi$
$548$	0.962964	−	0.473061i	0.962964	−	0.473061i
$549$	0.0902334	−	1.64509i	0.0902334	−	1.64509i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.218596	−	1.98668i	0.218596	−	1.98668i
$555$		0			0
$556$		0			0
$557$	−1.09153	−	0.306767i	−1.09153	−	0.306767i	−0.322880	−	0.946440i	$-0.604651\pi$
$557$	−1.09153	−	0.306767i	−1.09153	−	0.306767i	−0.768647	+	0.639673i	$0.779070\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.75898	−	0.358240i	1.75898	−	0.358240i
$563$		0			0		0.520940	−	0.853593i	$-0.325581\pi$
$563$		0			0		−0.520940	+	0.853593i	$0.674419\pi$
$564$		0			0
$565$	−1.28091	−	0.260876i	−1.28091	−	0.260876i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.557948	−	1.73877i	−0.557948	−	1.73877i	−0.667319	−	0.744772i	$-0.732558\pi$
$569$	−0.557948	−	1.73877i	−0.557948	−	1.73877i	0.109371	−	0.994001i	$-0.465116\pi$
$570$		0			0
$571$		0			0		0.639673	−	0.768647i	$-0.279070\pi$
$571$		0			0		−0.639673	+	0.768647i	$0.720930\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.252933	+	0.967484i	0.252933	+	0.967484i
$577$	−1.38722	−	0.0506980i	−1.38722	−	0.0506980i	−0.667319	−	0.744772i	$-0.732558\pi$
$577$	−1.38722	−	0.0506980i	−1.38722	−	0.0506980i	−0.719903	+	0.694074i	$0.755814\pi$
$578$	1.16368	−	0.710184i	1.16368	−	0.710184i
$579$		0			0
$580$	−0.316793	+	1.71509i	−0.316793	+	1.71509i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.0863484	+	0.942872i	−0.0863484	+	0.942872i
$585$	0.399406	+	0.462488i	0.399406	+	0.462488i
$586$	−1.97737	+	0.0722656i	−1.97737	+	0.0722656i
$587$		0			0		−0.997332	−	0.0729953i	$-0.976744\pi$
$587$		0			0		0.997332	+	0.0729953i	$0.0232558\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.64729	−	1.13363i	1.64729	−	1.13363i
$593$	−0.900868	−	1.60517i	−0.900868	−	1.60517i	−0.791496	−	0.611174i	$-0.790698\pi$
$593$	−0.900868	−	1.60517i	−0.900868	−	1.60517i	−0.109371	−	0.994001i	$-0.534884\pi$
$594$		0			0
$595$		0			0
$596$	0.0805529	−	1.10059i	0.0805529	−	1.10059i
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$	0.671121	+	1.05581i	0.671121	+	1.05581i	0.994001	+	0.109371i	$0.0348837\pi$
$601$	0.671121	+	1.05581i	0.671121	+	1.05581i	−0.322880	+	0.946440i	$0.604651\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.905448	−	0.424457i	0.905448	−	0.424457i
$606$		0			0
$607$		0			0		0.862965	−	0.505263i	$-0.168605\pi$
$607$		0			0		−0.862965	+	0.505263i	$0.831395\pi$
$608$		0			0
$609$		0			0
$610$	0.445767	−	1.58611i	0.445767	−	1.58611i
$611$		0			0
$612$		−	1.53729i		−	1.53729i
$613$	−0.968405	+	0.142520i	−0.968405	+	0.142520i	−0.611174	−	0.791496i	$-0.709302\pi$
$613$	−0.968405	+	0.142520i	−0.968405	+	0.142520i	−0.357231	+	0.934016i	$0.616279\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.45391	+	1.30271i	1.45391	+	1.30271i	0.872049	+	0.489418i	$0.162791\pi$
$617$	1.45391	+	1.30271i	1.45391	+	1.30271i	0.581859	+	0.813290i	$0.302326\pi$
$618$		0			0
$619$		0			0		−0.983366	−	0.181637i	$-0.941860\pi$
$619$		0			0		0.983366	+	0.181637i	$0.0581395\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.181637	+	0.983366i	−0.181637	+	0.983366i
$626$	−1.98334	−	0.0362300i	−1.98334	−	0.0362300i
$627$		0			0
$628$	1.10200	−	1.60133i	1.10200	−	1.60133i
$629$	−2.89081	+	1.04553i	−2.89081	+	1.04553i
$630$		0			0
$631$		0			0		−0.991838	−	0.127507i	$-0.959302\pi$
$631$		0			0		0.991838	+	0.127507i	$0.0406977\pi$
$632$		0			0
$633$		0			0
$634$	−0.600370	−	0.645917i	−0.600370	−	0.645917i
$635$		0			0
$636$		0			0
$637$	−0.610164	+	0.0334676i	−0.610164	+	0.0334676i
$638$		0			0
$639$		0			0
$640$	0.0365220	+	0.999333i	0.0365220	+	0.999333i
$641$	−0.946440	−	0.677120i	−0.946440	−	0.677120i		−	1.00000i	$-0.5\pi$
$641$	−0.946440	−	0.677120i	−0.946440	−	0.677120i	−0.946440	+	0.322880i	$0.895349\pi$
$642$		0			0
$643$		0			0		0.235221	−	0.971942i	$-0.424419\pi$
$643$		0			0		−0.235221	+	0.971942i	$0.575581\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.457242	−	0.889342i	$-0.651163\pi$
$647$		0			0		0.457242	+	0.889342i	$0.348837\pi$
$648$	−0.853593	−	0.520940i	−0.853593	−	0.520940i
$649$		0			0
$650$	0.289291	+	0.538267i	0.289291	+	0.538267i
$651$		0			0
$652$		0			0
$653$	1.88783	−	0.644038i	1.88783	−	0.644038i	0.920346	−	0.391105i	$-0.127907\pi$
$653$	1.88783	−	0.644038i	1.88783	−	0.644038i	0.967484	−	0.252933i	$-0.0813953\pi$
$654$		0			0
$655$		0			0
$656$	−1.10677	−	1.43332i	−1.10677	−	1.43332i
$657$	−0.592261	−	0.738709i	−0.592261	−	0.738709i
$658$		0			0
$659$		0			0		0.889342	−	0.457242i	$-0.151163\pi$
$659$		0			0		−0.889342	+	0.457242i	$0.848837\pi$
$660$		0			0
$661$	−0.0672258	+	1.83946i	−0.0672258	+	1.83946i	0.357231	+	0.934016i	$0.383721\pi$
$661$	−0.0672258	+	1.83946i	−0.0672258	+	1.83946i	−0.424457	+	0.905448i	$0.639535\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.399067	+	1.95944i	−0.399067	+	1.95944i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.0475612	+	0.213509i	0.0475612	+	0.213509i	0.994001	−	0.109371i	$-0.0348837\pi$
$673$	0.0475612	+	0.213509i	0.0475612	+	0.213509i	−0.946440	+	0.322880i	$0.895349\pi$
$674$	1.69341	+	0.756425i	1.69341	+	0.756425i
$675$		0			0
$676$	−0.576670	−	0.245058i	−0.576670	−	0.245058i
$677$	0.621079	−	1.71724i	0.621079	−	1.71724i	−0.0729953	−	0.997332i	$-0.523256\pi$
$677$	0.621079	−	1.71724i	0.621079	−	1.71724i	0.694074	−	0.719903i	$-0.255814\pi$
$678$		0			0
$679$		0			0
$680$	0.334254	−	1.50052i	0.334254	−	1.50052i
$681$		0			0
$682$		0			0
$683$		0			0		−0.756836	−	0.653605i	$-0.773256\pi$
$683$		0			0		0.756836	+	0.653605i	$0.226744\pi$
$684$		0			0
$685$	−0.575543	+	0.905448i	−0.575543	+	0.905448i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.508534	+	0.547114i	−0.508534	+	0.547114i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	0.983366	−	0.181637i	0.983366	−	0.181637i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.08879	+	2.56213i	1.08879	+	2.56213i
$698$	0.776768	−	1.58119i	0.776768	−	1.58119i
$699$		0			0
$700$		0			0
$701$	0.145601	+	1.98934i	0.145601	+	1.98934i	0.145601	+	0.989343i	$0.453488\pi$
$701$	0.145601	+	1.98934i	0.145601	+	1.98934i			1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.86938	+	0.416423i	1.86938	+	0.416423i
$707$		0			0
$708$		0			0
$709$	−0.173617	−	1.35051i	−0.173617	−	1.35051i	−0.813290	−	0.581859i	$-0.802326\pi$
$709$	−0.173617	−	1.35051i	−0.173617	−	1.35051i	0.639673	−	0.768647i	$-0.279070\pi$
$710$		0			0
$711$		0			0
$712$	−0.431598	+	0.652361i	−0.431598	+	0.652361i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.625528	−	0.780202i	$-0.284884\pi$
$719$		0			0		−0.625528	+	0.780202i	$0.715116\pi$
$720$	−0.719903	−	0.694074i	−0.719903	−	0.694074i
$721$		0			0
$722$	−0.744772	+	0.667319i	−0.744772	+	0.667319i
$723$		0			0
$724$	1.25227	−	1.00401i	1.25227	−	1.00401i
$725$	−0.623046	−	1.62902i	−0.623046	−	1.62902i
$726$		0			0
$727$		0			0		0.780202	−	0.625528i	$-0.215116\pi$
$727$		0			0		−0.780202	+	0.625528i	$0.784884\pi$
$728$		0			0
$729$	0.976076	−	0.217430i	0.976076	−	0.217430i
$730$	−0.417474	−	0.849812i	−0.417474	−	0.849812i
$731$		0			0
$732$		0			0
$733$	1.45722	−	0.409541i	1.45722	−	0.409541i	0.551768	−	0.833998i	$-0.313953\pi$
$733$	1.45722	−	0.409541i	1.45722	−	0.409541i	0.905448	+	0.424457i	$0.139535\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.79160	+	0.263669i	1.79160	+	0.263669i
$739$		0			0		0.145601	−	0.989343i	$-0.453488\pi$
$739$		0			0		−0.145601	+	0.989343i	$0.546512\pi$
$740$	−0.815561	+	1.82579i	−0.815561	+	1.82579i
$741$		0			0
$742$		0			0
$743$		0			0		0.986519	−	0.163646i	$-0.0523256\pi$
$743$		0			0		−0.986519	+	0.163646i	$0.947674\pi$
$744$		0			0
$745$	0.504583	+	0.981421i	0.504583	+	0.981421i
$746$	−1.53513	+	0.652361i	−1.53513	+	0.652361i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.235221	−	0.971942i	$-0.424419\pi$
$751$		0			0		−0.235221	+	0.971942i	$0.575581\pi$
$752$		0			0
$753$		0			0
$754$	−0.919737	−	0.538503i	−0.919737	−	0.538503i
$755$		0			0
$756$		0			0
$757$	−1.52807	+	1.27167i	−1.52807	+	1.27167i	−0.694074	+	0.719903i	$0.744186\pi$
$757$	−1.52807	+	1.27167i	−1.52807	+	1.27167i	−0.833998	+	0.551768i	$0.813953\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.236484	−	0.0954343i	0.236484	−	0.0954343i	−0.252933	−	0.967484i	$-0.581395\pi$
$761$	0.236484	−	0.0954343i	0.236484	−	0.0954343i	0.489418	+	0.872049i	$0.337209\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.800839	+	1.31222i	0.800839	+	1.31222i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.846649	+	0.0619667i	0.846649	+	0.0619667i	0.489418	−	0.872049i	$-0.337209\pi$
$769$	0.846649	+	0.0619667i	0.846649	+	0.0619667i	0.357231	+	0.934016i	$0.383721\pi$
$770$		0			0
$771$		0			0
$772$	0.145991			0.145991
$773$	1.50937	+	1.12210i	1.50937	+	1.12210i	0.957601	+	0.288099i	$0.0930233\pi$
$773$	1.50937	+	1.12210i	1.50937	+	1.12210i	0.551768	+	0.833998i	$0.313953\pi$
$774$		0			0
$775$		0			0
$776$	−0.186717	−	1.69694i	−0.186717	−	1.69694i
$777$		0			0
$778$	0.278856	+	0.0838951i	0.278856	+	0.0838951i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.989343	−	0.145601i	0.989343	−	0.145601i
$785$	−0.106462	+	1.94097i	−0.106462	+	1.94097i
$786$		0			0
$787$		0			0		−0.986519	−	0.163646i	$-0.947674\pi$
$787$		0			0		0.986519	+	0.163646i	$0.0523256\pi$
$788$	1.28959	+	1.49327i	1.28959	+	1.49327i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.818816	+	0.585813i	0.818816	+	0.585813i
$794$	−0.422243	+	0.207429i	−0.422243	+	0.207429i
$795$		0			0
$796$		0			0
$797$	0.0849459	−	1.16061i	0.0849459	−	1.16061i	−0.768647	−	0.639673i	$-0.779070\pi$
$797$	0.0849459	−	1.16061i	0.0849459	−	1.16061i	0.853593	−	0.520940i	$-0.174419\pi$
$798$		0			0
$799$		0			0
$800$	−0.551768	−	0.833998i	−0.551768	−	0.833998i
$801$	−0.142078	−	0.769198i	−0.142078	−	0.769198i
$802$	−1.69141	−	0.792899i	−1.69141	−	0.792899i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.69121	+	0.990199i	1.69121	+	0.990199i
$809$	−0.514344	−	1.34480i	−0.514344	−	1.34480i	−0.905448	−	0.424457i	$-0.860465\pi$
$809$	−0.514344	−	1.34480i	−0.514344	−	1.34480i	0.391105	−	0.920346i	$-0.372093\pi$
$810$	1.00000			1.00000
$811$		0			0		−0.391105	−	0.920346i	$-0.627907\pi$
$811$		0			0		0.391105	+	0.920346i	$0.372093\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.957601	+	0.711901i	−0.957601	+	0.711901i
$819$		0			0
$820$	1.69141	+	0.646908i	1.69141	+	0.646908i
$821$	−0.525940	+	0.127283i	−0.525940	+	0.127283i	−0.489418	−	0.872049i	$-0.662791\pi$
$821$	−0.525940	+	0.127283i	−0.525940	+	0.127283i	−0.0365220	+	0.999333i	$0.511628\pi$
$822$		0			0
$823$		0			0		0.181637	−	0.983366i	$-0.441860\pi$
$823$		0			0		−0.181637	+	0.983366i	$0.558140\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.270561	−	0.962703i	$-0.412791\pi$
$827$		0			0		−0.270561	+	0.962703i	$0.587209\pi$
$828$		0			0
$829$	−0.565494	+	0.436660i	−0.565494	+	0.436660i	−0.853593	−	0.520940i	$-0.825581\pi$
$829$	−0.565494	+	0.436660i	−0.565494	+	0.436660i	0.288099	+	0.957601i	$0.406977\pi$
$830$		0			0
$831$		0			0
$832$	−0.581859	−	0.186710i	−0.581859	−	0.186710i
$833$	−1.52807	−	0.168136i	−1.52807	−	0.168136i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.979884	−	0.199567i	$-0.936047\pi$
$839$		0			0		0.979884	+	0.199567i	$0.0639535\pi$
$840$		0			0
$841$	1.61614	+	1.24794i	1.61614	+	1.24794i
$842$	0.0124237	+	0.0343505i	0.0124237	+	0.0343505i
$843$		0			0
$844$		0			0
$845$	0.619902	−	0.0912307i	0.619902	−	0.0912307i
$846$		0			0
$847$		0			0
$848$	0.747067	−	0.967484i	0.747067	−	0.967484i
$849$		0			0
$850$	0.496362	+	1.45496i	0.496362	+	1.45496i
$851$		0			0
$852$		0			0
$853$	0.525191	−	0.375742i	0.525191	−	0.375742i	−0.288099	−	0.957601i	$-0.593023\pi$
$853$	0.525191	−	0.375742i	0.525191	−	0.375742i	0.813290	+	0.581859i	$0.197674\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.10670	−	0.921006i	−1.10670	−	0.921006i	−0.109371	−	0.994001i	$-0.534884\pi$
$857$	−1.10670	−	0.921006i	−1.10670	−	0.921006i	−0.997332	+	0.0729953i	$0.976744\pi$
$858$		0			0
$859$		0			0		−0.322880	−	0.946440i	$-0.604651\pi$
$859$		0			0		0.322880	+	0.946440i	$0.395349\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.288099	−	0.957601i	$-0.593023\pi$
$863$		0			0		0.288099	+	0.957601i	$0.406977\pi$
$864$		0			0
$865$	−0.744772	+	0.667319i	−0.744772	+	0.667319i
$866$	0.0730440			0.0730440
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.65068	+	0.926409i	1.65068	+	0.926409i
$873$	1.31222	+	1.09204i	1.31222	+	1.09204i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.17098	−	0.837765i	1.17098	−	0.837765i	0.181637	−	0.983366i	$-0.441860\pi$
$877$	1.17098	−	0.837765i	1.17098	−	0.837765i	0.989343	+	0.145601i	$0.0465116\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.500358	+	1.17744i	−0.500358	+	1.17744i	0.457242	+	0.889342i	$0.348837\pi$
$881$	−0.500358	+	1.17744i	−0.500358	+	1.17744i	−0.957601	+	0.288099i	$0.906977\pi$
$882$	−0.611174	+	0.791496i	−0.611174	+	0.791496i
$883$		0			0		−0.357231	−	0.934016i	$-0.616279\pi$
$883$		0			0		0.357231	+	0.934016i	$0.383721\pi$
$884$	0.792804	+	0.503941i	0.792804	+	0.503941i
$885$		0			0
$886$		0			0
$887$		0			0		0.0547678	−	0.998499i	$-0.482558\pi$
$887$		0			0		−0.0547678	+	0.998499i	$0.517442\pi$
$888$		0			0
$889$		0			0
$890$	0.0285679	−	0.781688i	0.0285679	−	0.781688i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.647696	+	0.207836i	0.647696	+	0.207836i
$899$		0			0
$900$	0.976076	+	0.217430i	0.976076	+	0.217430i
$901$	−1.48731	+	1.14846i	−1.48731	+	1.14846i
$902$		0			0
$903$		0			0
$904$	−0.576379	+	1.17328i	−0.576379	+	1.17328i
$905$	−0.545899	+	1.50937i	−0.545899	+	1.50937i
$906$		0			0
$907$		0			0		0.181637	−	0.983366i	$-0.441860\pi$
$907$		0			0		−0.181637	+	0.983366i	$0.558140\pi$
$908$		0			0
$909$	−1.90478	+	0.460979i	−1.90478	+	0.460979i
$910$		0			0
$911$		0			0		0.802527	−	0.596616i	$-0.203488\pi$
$911$		0			0		−0.802527	+	0.596616i	$0.796512\pi$
$912$		0			0
$913$		0			0
$914$	−0.542758	−	0.889342i	−0.542758	−	0.889342i
$915$		0			0
$916$	−0.253027	+	0.353667i	−0.253027	+	0.353667i
$917$		0			0
$918$		0			0
$919$		0			0		−0.391105	−	0.920346i	$-0.627907\pi$
$919$		0			0		0.391105	+	0.920346i	$0.372093\pi$
$920$		0			0
$921$		0			0
$922$	1.48942	+	0.872049i	1.48942	+	0.872049i
$923$		0			0
$924$		0			0
$925$	−0.254972	−	1.98334i	−0.254972	−	1.98334i
$926$		0			0
$927$		0			0
$928$	1.57919	+	0.740294i	1.57919	+	0.740294i
$929$	−0.354583	−	1.91968i	−0.354583	−	1.91968i	−0.391105	−	0.920346i	$-0.627907\pi$
$929$	−0.354583	−	1.91968i	−0.354583	−	1.91968i	0.0365220	−	0.999333i	$-0.488372\pi$
$930$		0			0
$931$		0			0
$932$	−0.555335	+	0.597466i	−0.555335	+	0.597466i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.548473	−	0.269440i	0.548473	−	0.269440i
$937$	−1.51925	−	1.08693i	−1.51925	−	1.08693i	−0.967484	−	0.252933i	$-0.918605\pi$
$937$	−1.51925	−	1.08693i	−1.51925	−	1.08693i	−0.551768	−	0.833998i	$-0.686047\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.541032	−	0.00988310i	0.541032	−	0.00988310i	0.252933	−	0.967484i	$-0.418605\pi$
$941$	0.541032	−	0.00988310i	0.541032	−	0.00988310i	0.288099	+	0.957601i	$0.406977\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.976076	−	0.217430i	$-0.930233\pi$
$947$		0			0		0.976076	+	0.217430i	$0.0697674\pi$
$948$		0			0
$949$	0.575112	−	0.0632803i	0.575112	−	0.0632803i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.602896	−	1.02972i	−0.602896	−	1.02972i	−0.994001	−	0.109371i	$-0.965116\pi$
$953$	−0.602896	−	1.02972i	−0.602896	−	1.02972i	0.391105	−	0.920346i	$-0.372093\pi$
$954$	0.133690	+	1.21501i	0.133690	+	1.21501i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.997332	+	0.0729953i	0.997332	+	0.0729953i
$962$	−0.879693	−	0.848130i	−0.879693	−	0.848130i
$963$		0			0
$964$	0.315085	+	0.0885527i	0.315085	+	0.0885527i
$965$	−0.124617	+	0.0760524i	−0.124617	+	0.0760524i
$966$		0			0
$967$		0			0		−0.946440	−	0.322880i	$-0.895349\pi$
$967$		0			0		0.946440	+	0.322880i	$0.104651\pi$
$968$	−0.181637	−	0.983366i	−0.181637	−	0.983366i
$969$		0			0
$970$	1.04339	+	1.35123i	1.04339	+	1.35123i
$971$		0			0		−0.0365220	−	0.999333i	$-0.511628\pi$
$971$		0			0		0.0365220	+	0.999333i	$0.488372\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.42179	−	0.832453i	−1.42179	−	0.832453i
$977$	0.230486	−	0.177975i	0.230486	−	0.177975i	−0.489418	−	0.872049i	$-0.662791\pi$
$977$	0.230486	−	0.177975i	0.230486	−	0.177975i	0.719903	+	0.694074i	$0.244186\pi$
$978$		0			0
$979$		0			0
$980$	−0.768647	+	0.639673i	−0.768647	+	0.639673i
$981$	−1.83133	+	0.478772i	−1.83133	+	0.478772i
$982$		0			0
$983$		0			0		−0.0547678	−	0.998499i	$-0.517442\pi$
$983$		0			0		0.0547678	+	0.998499i	$0.482558\pi$
$984$		0			0
$985$	−1.87869	−	0.602843i	−1.87869	−	0.602843i
$986$	−2.06089	−	1.71509i	−2.06089	−	1.71509i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.145601	−	0.989343i	$-0.453488\pi$
$991$		0			0		−0.145601	+	0.989343i	$0.546512\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.175594	−	0.0493496i	0.175594	−	0.0493496i	−0.181637	−	0.983366i	$-0.558140\pi$
$997$	0.175594	−	0.0493496i	0.175594	−	0.0493496i	0.357231	+	0.934016i	$0.383721\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3460.1.bd.a.343.1	✓	84
4.3	odd	2	CM	3460.1.bd.a.343.1	✓	84
5.2	odd	4		3460.1.bm.a.1727.1	yes	84
20.7	even	4		3460.1.bm.a.1727.1	yes	84
173.115	odd	172		3460.1.bm.a.2883.1	yes	84
692.115	even	172		3460.1.bm.a.2883.1	yes	84
865.807	even	172	inner	3460.1.bd.a.807.1	yes	84
3460.807	odd	172	inner	3460.1.bd.a.807.1	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3460.1.bd.a.343.1	✓	84	1.1	even	1	trivial
3460.1.bd.a.343.1	✓	84	4.3	odd	2	CM
3460.1.bd.a.807.1	yes	84	865.807	even	172	inner
3460.1.bd.a.807.1	yes	84	3460.807	odd	172	inner
3460.1.bm.a.1727.1	yes	84	5.2	odd	4
3460.1.bm.a.1727.1	yes	84	20.7	even	4
3460.1.bm.a.2883.1	yes	84	173.115	odd	172
3460.1.bm.a.2883.1	yes	84	692.115	even	172

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:	\( N \)	\(=\)	\( 3460 = 2^{2} \cdot 5 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3460.bd (of order \(172\), degree \(84\), minimal)

\(f(q)\)	\(=\)	\(q+(0.288099 + 0.957601i) q^{2} +(-0.833998 + 0.551768i) q^{4} +(0.424457 - 0.905448i) q^{5} +(-0.768647 - 0.639673i) q^{8} +(0.997332 - 0.0729953i) q^{9} +O(q^{10})\)	\(q+(0.288099 + 0.957601i) q^{2} +(-0.833998 + 0.551768i) q^{4} +(0.424457 - 0.905448i) q^{5} +(-0.768647 - 0.639673i) q^{8} +(0.997332 - 0.0729953i) q^{9} +(0.989343 + 0.145601i) q^{10} +(-0.289291 + 0.538267i) q^{13} +(0.391105 - 0.920346i) q^{16} +(-0.939554 + 1.21676i) q^{17} +(0.357231 + 0.934016i) q^{18} +(0.145601 + 0.989343i) q^{20} +(-0.639673 - 0.768647i) q^{25} +(-0.598789 - 0.121952i) q^{26} +(1.65068 + 0.563135i) q^{29} +(0.994001 + 0.109371i) q^{32} +(-1.43586 - 0.549169i) q^{34} +(-0.791496 + 0.611174i) q^{36} +(1.68759 + 1.07271i) q^{37} +(-0.905448 + 0.424457i) q^{40} +(0.886285 - 1.57919i) q^{41} +(0.357231 - 0.934016i) q^{45} +(0.520940 + 0.853593i) q^{49} +(0.551768 - 0.833998i) q^{50} +(-0.0557297 - 0.608535i) q^{52} +(1.18260 + 0.309172i) q^{53} +(-0.0636980 + 1.74294i) q^{58} +(0.210076 - 1.63411i) q^{61} +(0.181637 + 0.983366i) q^{64} +(0.364581 + 0.490409i) q^{65} +(0.112215 - 1.53319i) q^{68} +(-0.813290 - 0.581859i) q^{72} +(-0.536759 - 0.779971i) q^{73} +(-0.541032 + 1.92508i) q^{74} +(-0.667319 - 0.744772i) q^{80} +(0.989343 - 0.145601i) q^{81} +(1.76757 + 0.393744i) q^{82} +(0.702916 + 1.36718i) q^{85} +(-0.0855512 - 0.777517i) q^{89} +(0.997332 + 0.0729953i) q^{90} +(1.22901 + 1.18491i) q^{97} +(-0.667319 + 0.744772i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 84 * q + 2 * q^4 + 2 * q^9	\( 84 q + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{26} - 2 q^{36} - 2 q^{37} + 2 q^{40} - 2 q^{49} - 2 q^{52} + 4 q^{58} - 2 q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{73} + 2 q^{74} - 2 q^{81} + 4 q^{82} + 4 q^{89} + 2 q^{90}+O(q^{100}) \) 84 * q + 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^26 - 2 * q^36 - 2 * q^37 + 2 * q^40 - 2 * q^49 - 2 * q^52 + 4 * q^58 - 2 * q^61 + 2 * q^64 - 2 * q^65 + 2 * q^73 + 2 * q^74 - 2 * q^81 + 4 * q^82 + 4 * q^89 + 2 * q^90

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