LMFDB - Embedded newform 3460.1.bd.a.123.1

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			123.1
Root			$0.145601 - 0.989343i$ of defining polynomial
Character	$\chi$	$=$	3460.123
Dual form			3460.1.bd.a.647.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.611174 - 0.791496i) q^{2} +(-0.252933 - 0.967484i) q^{4} +(0.551768 - 0.833998i) q^{5} +(-0.920346 - 0.391105i) q^{8} +(0.581859 + 0.813290i) q^{9} +O(q^{10})$	\(q+(0.611174 - 0.791496i) q^{2} +(-0.252933 - 0.967484i) q^{4} +(0.551768 - 0.833998i) q^{5} +(-0.920346 - 0.391105i) q^{8} +(0.581859 + 0.813290i) q^{9} +(-0.322880 - 0.946440i) q^{10} +(-1.24084 + 0.159519i) q^{13} +(-0.872049 + 0.489418i) q^{16} +(1.41484 - 1.17744i) q^{17} +(0.999333 + 0.0365220i) q^{18} +(-0.946440 - 0.322880i) q^{20} +(-0.391105 - 0.920346i) q^{25} +(-0.632112 + 1.07962i) q^{26} +(0.792899 - 1.69141i) q^{29} +(-0.145601 + 0.989343i) q^{32} +(-0.0672258 - 1.83946i) q^{34} +(0.639673 - 0.768647i) q^{36} +(-0.414385 - 0.222712i) q^{37} +(-0.833998 + 0.551768i) q^{40} +(-0.595860 - 1.55793i) q^{41} +(0.999333 - 0.0365220i) q^{45} +(0.744772 - 0.667319i) q^{49} +(-0.967484 - 0.252933i) q^{50} +(0.468183 + 1.16015i) q^{52} +(-1.51172 + 0.279229i) q^{53} +(-0.854143 - 1.66132i) q^{58} +(0.182367 + 1.99133i) q^{61} +(0.694074 + 0.719903i) q^{64} +(-0.551619 + 1.12288i) q^{65} +(-1.49702 - 1.07102i) q^{68} +(-0.217430 - 0.976076i) q^{72} +(0.0362300 + 1.98334i) q^{73} +(-0.429537 + 0.191869i) q^{74} +(-0.0729953 + 0.997332i) q^{80} +(-0.322880 + 0.946440i) q^{81} +(-1.59727 - 0.480548i) q^{82} +(-0.201319 - 1.82965i) q^{85} +(1.72551 - 0.253943i) q^{89} +(0.581859 - 0.813290i) q^{90} +(-1.13924 + 0.695267i) q^{97} +(-0.0729953 - 0.997332i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3460\mathbb{Z}\right)^\times$.

$n$	$521$	$1731$	$2077$
$\chi(n)$	$e\left(\frac{165}{172}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	0.611174	−	0.791496i	0.611174	−	0.791496i
$2$	0.611174	−	0.791496i	0.611174	−	0.791496i
$3$		0			0		−0.889342	−	0.457242i	$-0.848837\pi$
$3$		0			0		0.889342	+	0.457242i	$0.151163\pi$
$4$	−0.252933	−	0.967484i	−0.252933	−	0.967484i
$5$	0.551768	−	0.833998i	0.551768	−	0.833998i
$5$	0.551768	−	0.833998i	0.551768	−	0.833998i
$6$		0			0
$7$		0			0		0.934016	−	0.357231i	$-0.116279\pi$
$7$		0			0		−0.934016	+	0.357231i	$0.883721\pi$
$8$	−0.920346	−	0.391105i	−0.920346	−	0.391105i
$9$	0.581859	+	0.813290i	0.581859	+	0.813290i
$10$	−0.322880	−	0.946440i	−0.322880	−	0.946440i
$11$		0			0		−0.979884	−	0.199567i	$-0.936047\pi$
$11$		0			0		0.979884	+	0.199567i	$0.0639535\pi$
$12$		0			0
$13$	−1.24084	+	0.159519i	−1.24084	+	0.159519i	−0.719903	−	0.694074i	$-0.755814\pi$
$13$	−1.24084	+	0.159519i	−1.24084	+	0.159519i	−0.520940	+	0.853593i	$0.674419\pi$
$14$		0			0
$15$		0			0
$16$	−0.872049	+	0.489418i	−0.872049	+	0.489418i
$17$	1.41484	−	1.17744i	1.41484	−	1.17744i	0.457242	−	0.889342i	$-0.348837\pi$
$17$	1.41484	−	1.17744i	1.41484	−	1.17744i	0.957601	−	0.288099i	$-0.0930233\pi$
$18$	0.999333	+	0.0365220i	0.999333	+	0.0365220i
$19$		0			0		−0.880843	−	0.473409i	$-0.843023\pi$
$19$		0			0		0.880843	+	0.473409i	$0.156977\pi$
$20$	−0.946440	−	0.322880i	−0.946440	−	0.322880i
$21$		0			0
$22$		0			0
$23$		0			0		−0.199567	−	0.979884i	$-0.563953\pi$
$23$		0			0		0.199567	+	0.979884i	$0.436047\pi$
$24$		0			0
$25$	−0.391105	−	0.920346i	−0.391105	−	0.920346i
$26$	−0.632112	+	1.07962i	−0.632112	+	1.07962i
$27$		0			0
$28$		0			0
$29$	0.792899	−	1.69141i	0.792899	−	1.69141i	0.0729953	−	0.997332i	$-0.476744\pi$
$29$	0.792899	−	1.69141i	0.792899	−	1.69141i	0.719903	−	0.694074i	$-0.244186\pi$
$30$		0			0
$31$		0			0		0.889342	−	0.457242i	$-0.151163\pi$
$31$		0			0		−0.889342	+	0.457242i	$0.848837\pi$
$32$	−0.145601	+	0.989343i	−0.145601	+	0.989343i
$33$		0			0
$34$	−0.0672258	−	1.83946i	−0.0672258	−	1.83946i
$35$		0			0
$36$	0.639673	−	0.768647i	0.639673	−	0.768647i
$37$	−0.414385	−	0.222712i	−0.414385	−	0.222712i	0.252933	−	0.967484i	$-0.418605\pi$
$37$	−0.414385	−	0.222712i	−0.414385	−	0.222712i	−0.667319	+	0.744772i	$0.732558\pi$
$38$		0			0
$39$		0			0
$40$	−0.833998	+	0.551768i	−0.833998	+	0.551768i
$41$	−0.595860	−	1.55793i	−0.595860	−	1.55793i	−0.813290	−	0.581859i	$-0.802326\pi$
$41$	−0.595860	−	1.55793i	−0.595860	−	1.55793i	0.217430	−	0.976076i	$-0.430233\pi$
$42$		0			0
$43$		0			0		0.862965	−	0.505263i	$-0.168605\pi$
$43$		0			0		−0.862965	+	0.505263i	$0.831395\pi$
$44$		0			0
$45$	0.999333	−	0.0365220i	0.999333	−	0.0365220i
$46$		0			0
$47$		0			0		−0.897545	−	0.440923i	$-0.854651\pi$
$47$		0			0		0.897545	+	0.440923i	$0.145349\pi$
$48$		0			0
$49$	0.744772	−	0.667319i	0.744772	−	0.667319i
$50$	−0.967484	−	0.252933i	−0.967484	−	0.252933i
$51$		0			0
$52$	0.468183	+	1.16015i	0.468183	+	1.16015i
$53$	−1.51172	+	0.279229i	−1.51172	+	0.279229i	−0.872049	−	0.489418i	$-0.837209\pi$
$53$	−1.51172	+	0.279229i	−1.51172	+	0.279229i	−0.639673	+	0.768647i	$0.720930\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.854143	−	1.66132i	−0.854143	−	1.66132i
$59$		0			0		0.0547678	−	0.998499i	$-0.482558\pi$
$59$		0			0		−0.0547678	+	0.998499i	$0.517442\pi$
$60$		0			0
$61$	0.182367	+	1.99133i	0.182367	+	1.99133i	0.109371	+	0.994001i	$0.465116\pi$
$61$	0.182367	+	1.99133i	0.182367	+	1.99133i	0.0729953	+	0.997332i	$0.476744\pi$
$62$		0			0
$63$		0			0
$64$	0.694074	+	0.719903i	0.694074	+	0.719903i
$65$	−0.551619	+	1.12288i	−0.551619	+	1.12288i
$66$		0			0
$67$		0			0		0.952179	−	0.305541i	$-0.0988372\pi$
$67$		0			0		−0.952179	+	0.305541i	$0.901163\pi$
$68$	−1.49702	−	1.07102i	−1.49702	−	1.07102i
$69$		0			0
$70$		0			0
$71$		0			0		−0.340112	−	0.940385i	$-0.610465\pi$
$71$		0			0		0.340112	+	0.940385i	$0.389535\pi$
$72$	−0.217430	−	0.976076i	−0.217430	−	0.976076i
$73$	0.0362300	+	1.98334i	0.0362300	+	1.98334i	0.145601	+	0.989343i	$0.453488\pi$
$73$	0.0362300	+	1.98334i	0.0362300	+	1.98334i	−0.109371	+	0.994001i	$0.534884\pi$
$74$	−0.429537	+	0.191869i	−0.429537	+	0.191869i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.440923	−	0.897545i	$-0.645349\pi$
$79$		0			0		0.440923	+	0.897545i	$0.354651\pi$
$80$	−0.0729953	+	0.997332i	−0.0729953	+	0.997332i
$81$	−0.322880	+	0.946440i	−0.322880	+	0.946440i
$82$	−1.59727	−	0.480548i	−1.59727	−	0.480548i
$83$		0			0		0.971942	−	0.235221i	$-0.0755814\pi$
$83$		0			0		−0.971942	+	0.235221i	$0.924419\pi$
$84$		0			0
$85$	−0.201319	−	1.82965i	−0.201319	−	1.82965i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.72551	−	0.253943i	1.72551	−	0.253943i	0.791496	−	0.611174i	$-0.209302\pi$
$89$	1.72551	−	0.253943i	1.72551	−	0.253943i	0.934016	+	0.357231i	$0.116279\pi$
$90$	0.581859	−	0.813290i	0.581859	−	0.813290i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.13924	+	0.695267i	−1.13924	+	0.695267i	−0.957601	−	0.288099i	$-0.906977\pi$
$97$	−1.13924	+	0.695267i	−1.13924	+	0.695267i	−0.181637	+	0.983366i	$0.558140\pi$
$98$	−0.0729953	−	0.997332i	−0.0729953	−	0.997332i
$99$		0			0
$100$	−0.791496	+	0.611174i	−0.791496	+	0.611174i
$101$	0.852820	+	0.542091i	0.852820	+	0.542091i	0.889342	−	0.457242i	$-0.151163\pi$
$101$	0.852820	+	0.542091i	0.852820	+	0.542091i	−0.0365220	+	0.999333i	$0.511628\pi$
$102$		0			0
$103$		0			0		−0.694074	−	0.719903i	$-0.744186\pi$
$103$		0			0		0.694074	+	0.719903i	$0.255814\pi$
$104$	1.20439	+	0.338487i	1.20439	+	0.338487i
$105$		0			0
$106$	−0.702916	+	1.36718i	−0.702916	+	1.36718i
$107$		0			0		−0.853593	−	0.520940i	$-0.825581\pi$
$107$		0			0		0.853593	+	0.520940i	$0.174419\pi$
$108$		0			0
$109$	−0.611136	+	0.589209i	−0.611136	+	0.589209i	−0.934016	−	0.357231i	$-0.883721\pi$
$109$	−0.611136	+	0.589209i	−0.611136	+	0.589209i	0.322880	+	0.946440i	$0.395349\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.97008	−	0.108059i	1.97008	−	0.108059i	0.976076	−	0.217430i	$-0.0697674\pi$
$113$	1.97008	−	0.108059i	1.97008	−	0.108059i	0.994001	+	0.109371i	$0.0348837\pi$
$114$		0			0
$115$		0			0
$116$	−1.83696	−	0.339303i	−1.83696	−	0.339303i
$117$	−0.851731	−	0.916348i	−0.851731	−	0.916348i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.920346	+	0.391105i	0.920346	+	0.391105i
$122$	1.68759	+	1.07271i	1.68759	+	1.07271i
$123$		0			0
$124$		0			0
$125$	−0.983366	−	0.181637i	−0.983366	−	0.181637i
$126$		0			0
$127$		0			0		−0.322880	−	0.946440i	$-0.604651\pi$
$127$		0			0		0.322880	+	0.946440i	$0.395349\pi$
$128$	0.994001	−	0.109371i	0.994001	−	0.109371i
$129$		0			0
$130$	0.551619	+	1.12288i	0.551619	+	1.12288i
$131$		0			0		−0.0182641	−	0.999833i	$-0.505814\pi$
$131$		0			0		0.0182641	+	0.999833i	$0.494186\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.76265	+	0.530302i	−1.76265	+	0.530302i
$137$	−0.942872	+	0.0863484i	−0.942872	+	0.0863484i	−0.551768	−	0.833998i	$-0.686047\pi$
$137$	−0.942872	+	0.0863484i	−0.942872	+	0.0863484i	−0.391105	+	0.920346i	$0.627907\pi$
$138$		0			0
$139$		0			0		0.999333	−	0.0365220i	$-0.0116279\pi$
$139$		0			0		−0.999333	+	0.0365220i	$0.988372\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.905448	−	0.424457i	−0.905448	−	0.424457i
$145$	−0.973133	−	1.59454i	−0.973133	−	1.59454i
$146$	1.59195	+	1.18349i	1.59195	+	1.18349i
$147$		0			0
$148$	−0.110658	+	0.457242i	−0.110658	+	0.457242i
$149$	−1.48731	+	1.23775i	−1.48731	+	1.23775i	−0.581859	+	0.813290i	$0.697674\pi$
$149$	−1.48731	+	1.23775i	−1.48731	+	1.23775i	−0.905448	+	0.424457i	$0.860465\pi$
$150$		0			0
$151$		0			0		0.0729953	−	0.997332i	$-0.476744\pi$
$151$		0			0		−0.0729953	+	0.997332i	$0.523256\pi$
$152$		0			0
$153$	1.78084	+	0.465573i	1.78084	+	0.465573i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.105450	+	0.0296361i	0.105450	+	0.0296361i	0.322880	−	0.946440i	$-0.395349\pi$
$157$	0.105450	+	0.0296361i	0.105450	+	0.0296361i	−0.217430	+	0.976076i	$0.569767\pi$
$158$		0			0
$159$		0			0
$160$	0.744772	+	0.667319i	0.744772	+	0.667319i
$161$		0			0
$162$	0.551768	+	0.833998i	0.551768	+	0.833998i
$163$		0			0		−0.927336	−	0.374230i	$-0.877907\pi$
$163$		0			0		0.927336	+	0.374230i	$0.122093\pi$
$164$	−1.35656	+	0.970538i	−1.35656	+	0.970538i
$165$		0			0
$166$		0			0
$167$		0			0		−0.940385	−	0.340112i	$-0.889535\pi$
$167$		0			0		0.940385	+	0.340112i	$0.110465\pi$
$168$		0			0
$169$	0.546763	−	0.142943i	0.546763	−	0.142943i
$170$	−1.57120	−	0.958891i	−1.57120	−	0.958891i
$171$		0			0
$172$		0			0
$173$	0.489418	+	0.872049i	0.489418	+	0.872049i
$173$	0.489418	+	0.872049i	0.489418	+	0.872049i
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.853593	−	1.52094i	0.853593	−	1.52094i
$179$		0			0		−0.424457	−	0.905448i	$-0.639535\pi$
$179$		0			0		0.424457	+	0.905448i	$0.360465\pi$
$180$	−0.288099	−	0.957601i	−0.288099	−	0.957601i
$181$	1.66465	−	0.671777i	1.66465	−	0.671777i	0.667319	−	0.744772i	$-0.267442\pi$
$181$	1.66465	−	0.671777i	1.66465	−	0.671777i	0.997332	+	0.0729953i	$0.0232558\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.414385	+	0.222712i	−0.414385	+	0.222712i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.732460	−	0.680810i	$-0.238372\pi$
$191$		0			0		−0.732460	+	0.680810i	$0.761628\pi$
$192$		0			0
$193$	0.411416	−	1.57369i	0.411416	−	1.57369i	−0.357231	−	0.934016i	$-0.616279\pi$
$193$	0.411416	−	1.57369i	0.411416	−	1.57369i	0.768647	−	0.639673i	$-0.220930\pi$
$194$	−0.145971	+	1.32663i	−0.145971	+	1.32663i
$195$		0			0
$196$	−0.833998	−	0.551768i	−0.833998	−	0.551768i
$197$	−0.153931	+	1.68084i	−0.153931	+	1.68084i	0.457242	+	0.889342i	$0.348837\pi$
$197$	−0.153931	+	1.68084i	−0.153931	+	1.68084i	−0.611174	+	0.791496i	$0.709302\pi$
$198$		0			0
$199$		0			0		−0.862965	−	0.505263i	$-0.831395\pi$
$199$		0			0		0.862965	+	0.505263i	$0.168605\pi$
$200$			1.00000i			1.00000i
$201$		0			0
$202$	0.950284	−	0.343693i	0.950284	−	0.343693i
$203$		0			0
$204$		0			0
$205$	−1.62809	−	0.362673i	−1.62809	−	0.362673i
$206$		0			0
$207$		0			0
$208$	1.00401	−	0.746399i	1.00401	−	0.746399i
$209$		0			0
$210$		0			0
$211$		0			0		0.357231	−	0.934016i	$-0.383721\pi$
$211$		0			0		−0.357231	+	0.934016i	$0.616279\pi$
$212$	0.652515	+	1.39194i	0.652515	+	1.39194i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.0928467	+	0.843821i	0.0928467	+	0.843821i
$219$		0			0
$220$		0			0
$221$	−1.56777	+	1.68671i	−1.56777	+	1.68671i
$222$		0			0
$223$		0			0		−0.994001	−	0.109371i	$-0.965116\pi$
$223$		0			0		0.994001	+	0.109371i	$0.0348837\pi$
$224$		0			0
$225$	0.520940	−	0.853593i	0.520940	−	0.853593i
$226$	1.11853	−	1.62535i	1.11853	−	1.62535i
$227$		0			0		−0.270561	−	0.962703i	$-0.587209\pi$
$227$		0			0		0.270561	+	0.962703i	$0.412791\pi$
$228$		0			0
$229$	−0.0210439	+	0.575814i	−0.0210439	+	0.575814i	0.946440	+	0.322880i	$0.104651\pi$
$229$	−0.0210439	+	0.575814i	−0.0210439	+	0.575814i	−0.967484	+	0.252933i	$0.918605\pi$
$230$		0			0
$231$		0			0
$232$	−1.39126	+	1.24657i	−1.39126	+	1.24657i
$233$	−0.0745729	−	1.35958i	−0.0745729	−	1.35958i	−0.768647	−	0.639673i	$-0.779070\pi$
$233$	−0.0745729	−	1.35958i	−0.0745729	−	1.35958i	0.694074	−	0.719903i	$-0.255814\pi$
$234$	−1.24584	+	0.114094i	−1.24584	+	0.114094i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.235221	−	0.971942i	$-0.575581\pi$
$239$		0			0		0.235221	+	0.971942i	$0.424419\pi$
$240$		0			0
$241$	1.05842	−	0.175574i	1.05842	−	0.175574i	0.391105	−	0.920346i	$-0.372093\pi$
$241$	1.05842	−	0.175574i	1.05842	−	0.175574i	0.667319	+	0.744772i	$0.267442\pi$
$242$	0.872049	−	0.489418i	0.872049	−	0.489418i
$243$		0			0
$244$	1.88046	−	0.680111i	1.88046	−	0.680111i
$245$	−0.145601	−	0.989343i	−0.145601	−	0.989343i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.744772	+	0.667319i	−0.744772	+	0.667319i
$251$		0			0		0.217430	−	0.976076i	$-0.430233\pi$
$251$		0			0		−0.217430	+	0.976076i	$0.569767\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.520940	−	0.853593i	0.520940	−	0.853593i
$257$	0.982962	−	1.32221i	0.982962	−	1.32221i	0.0365220	−	0.999333i	$-0.488372\pi$
$257$	0.982962	−	1.32221i	0.982962	−	1.32221i	0.946440	−	0.322880i	$-0.104651\pi$
$258$		0			0
$259$		0			0
$260$	1.22589	+	0.249669i	1.22589	+	0.249669i
$261$	1.83696	−	0.339303i	1.83696	−	0.339303i
$262$		0			0
$263$		0			0		−0.235221	−	0.971942i	$-0.575581\pi$
$263$		0			0		0.235221	+	0.971942i	$0.424419\pi$
$264$		0			0
$265$	−0.601243	+	1.41484i	−0.601243	+	1.41484i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.295447	−	0.211374i	−0.295447	−	0.211374i	0.424457	−	0.905448i	$-0.360465\pi$
$269$	−0.295447	−	0.211374i	−0.295447	−	0.211374i	−0.719903	+	0.694074i	$0.755814\pi$
$270$		0			0
$271$		0			0		−0.756836	−	0.653605i	$-0.773256\pi$
$271$		0			0		0.756836	+	0.653605i	$0.226744\pi$
$272$	−0.657552	+	1.71924i	−0.657552	+	1.71924i
$273$		0			0
$274$	−0.507914	+	0.799054i	−0.507914	+	0.799054i
$275$		0			0
$276$		0			0
$277$	−0.870520	−	1.55110i	−0.870520	−	1.55110i	−0.833998	−	0.551768i	$-0.813953\pi$
$277$	−0.870520	−	1.55110i	−0.870520	−	1.55110i	−0.0365220	−	0.999333i	$-0.511628\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.254561	+	0.630797i	−0.254561	+	0.630797i	−0.999333	−	0.0365220i	$-0.988372\pi$
$281$	−0.254561	+	0.630797i	−0.254561	+	0.630797i	0.744772	+	0.667319i	$0.232558\pi$
$282$		0			0
$283$		0			0		0.551768	−	0.833998i	$-0.313953\pi$
$283$		0			0		−0.551768	+	0.833998i	$0.686047\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.889342	+	0.457242i	−0.889342	+	0.457242i
$289$	0.433776	−	2.34842i	0.433776	−	2.34842i
$290$	−1.85683	−	0.204309i	−1.85683	−	0.204309i
$291$		0			0
$292$	1.90969	−	0.536706i	1.90969	−	0.536706i
$293$	−0.117294	−	0.635019i	−0.117294	−	0.635019i	−0.989343	−	0.145601i	$-0.953488\pi$
$293$	−0.117294	−	0.635019i	−0.117294	−	0.635019i	0.872049	−	0.489418i	$-0.162791\pi$
$294$		0			0
$295$		0			0
$296$	0.294275	+	0.367040i	0.294275	+	0.367040i
$297$		0			0
$298$	0.0706689	+	1.93368i	0.0706689	+	1.93368i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.76139	+	0.946660i	1.76139	+	0.946660i
$306$	1.45690	−	1.12498i	1.45690	−	1.12498i
$307$		0			0		0.905448	−	0.424457i	$-0.139535\pi$
$307$		0			0		−0.905448	+	0.424457i	$0.860465\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.989343	−	0.145601i	$-0.953488\pi$
$311$		0			0		0.989343	+	0.145601i	$0.0465116\pi$
$312$		0			0
$313$	−1.24584	+	1.55390i	−1.24584	+	1.55390i	−0.551768	+	0.833998i	$0.686047\pi$
$313$	−1.24584	+	1.55390i	−1.24584	+	1.55390i	−0.694074	+	0.719903i	$0.744186\pi$
$314$	0.0879053	−	0.0653507i	0.0879053	−	0.0653507i
$315$		0			0
$316$		0			0
$317$	0.639673	+	1.76865i	0.639673	+	1.76865i	0.639673	+	0.768647i	$0.279070\pi$
$317$	0.639673	+	1.76865i	0.639673	+	1.76865i			1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	0.983366	−	0.181637i	0.983366	−	0.181637i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.997332	+	0.0729953i	0.997332	+	0.0729953i
$325$	0.632112	+	1.07962i	0.632112	+	1.07962i
$326$		0			0
$327$		0			0
$328$	−0.0609186	+	1.66688i	−0.0609186	+	1.66688i
$329$		0			0
$330$		0			0
$331$		0			0		0.833998	−	0.551768i	$-0.186047\pi$
$331$		0			0		−0.833998	+	0.551768i	$0.813953\pi$
$332$		0			0
$333$	−0.0599849	−	0.466602i	−0.0599849	−	0.466602i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.89945	−	0.315085i	−1.89945	−	0.315085i	−0.905448	−	0.424457i	$-0.860465\pi$
$337$	−1.89945	−	0.315085i	−1.89945	−	0.315085i	−0.994001	+	0.109371i	$0.965116\pi$
$338$	0.221029	−	0.520124i	0.221029	−	0.520124i
$339$		0			0
$340$	−1.71924	+	0.657552i	−1.71924	+	0.657552i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.989343	+	0.145601i	0.989343	+	0.145601i
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	−0.0779174	−	0.242820i	−0.0779174	−	0.242820i	0.905448	−	0.424457i	$-0.139535\pi$
$349$	−0.0779174	−	0.242820i	−0.0779174	−	0.242820i	−0.983366	+	0.181637i	$0.941860\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.565494	+	1.47854i	0.565494	+	1.47854i	0.853593	+	0.520940i	$0.174419\pi$
$353$	0.565494	+	1.47854i	0.565494	+	1.47854i	−0.288099	+	0.957601i	$0.593023\pi$
$354$		0			0
$355$		0			0
$356$	−0.682125	−	1.60517i	−0.682125	−	1.60517i
$357$		0			0
$358$		0			0
$359$		0			0		−0.791496	−	0.611174i	$-0.790698\pi$
$359$		0			0		0.791496	+	0.611174i	$0.209302\pi$
$360$	−0.934016	−	0.357231i	−0.934016	−	0.357231i
$361$	0.551768	+	0.833998i	0.551768	+	0.833998i
$362$	0.485682	−	1.72814i	0.485682	−	1.72814i
$363$		0			0
$364$		0			0
$365$	1.67410	+	1.06413i	1.67410	+	1.06413i
$366$		0			0
$367$		0			0		0.780202	−	0.625528i	$-0.215116\pi$
$367$		0			0		−0.780202	+	0.625528i	$0.784884\pi$
$368$		0			0
$369$	0.920346	−	1.39110i	0.920346	−	1.39110i
$370$	−0.0769861	+	0.464100i	−0.0769861	+	0.464100i
$371$		0			0
$372$		0			0
$373$	0.500476	−	0.0736548i	0.500476	−	0.0736548i	0.109371	−	0.994001i	$-0.465116\pi$
$373$	0.500476	−	0.0736548i	0.500476	−	0.0736548i	0.391105	+	0.920346i	$0.372093\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.714052	+	2.22525i	−0.714052	+	2.22525i
$378$		0			0
$379$		0			0		−0.109371	−	0.994001i	$-0.534884\pi$
$379$		0			0		0.109371	+	0.994001i	$0.465116\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.473409	−	0.880843i	$-0.343023\pi$
$383$		0			0		−0.473409	+	0.880843i	$0.656977\pi$
$384$		0			0
$385$		0			0
$386$	−0.994123	−	1.28743i	−0.994123	−	1.28743i
$387$		0			0
$388$	0.960810	+	0.926338i	0.960810	+	0.926338i
$389$	1.83133	+	0.478772i	1.83133	+	0.478772i	0.997332	−	0.0729953i	$-0.0232558\pi$
$389$	1.83133	+	0.478772i	1.83133	+	0.478772i	0.833998	+	0.551768i	$0.186047\pi$
$390$		0			0
$391$		0			0
$392$	−0.946440	+	0.322880i	−0.946440	+	0.322880i
$393$		0			0
$394$	1.23630	+	1.14912i	1.23630	+	1.14912i
$395$		0			0
$396$		0			0
$397$	1.07128	−	1.68534i	1.07128	−	1.68534i	0.489418	−	0.872049i	$-0.337209\pi$
$397$	1.07128	−	1.68534i	1.07128	−	1.68534i	0.581859	−	0.813290i	$-0.302326\pi$
$398$		0			0
$399$		0			0
$400$	0.791496	+	0.611174i	0.791496	+	0.611174i
$401$	−0.00533187	−	0.0728492i	−0.00533187	−	0.0728492i	0.994001	−	0.109371i	$-0.0348837\pi$
$401$	−0.00533187	−	0.0728492i	−0.00533187	−	0.0728492i	−0.999333	+	0.0365220i	$0.988372\pi$
$402$		0			0
$403$		0			0
$404$	0.308757	−	0.962202i	0.308757	−	0.962202i
$405$	0.611174	+	0.791496i	0.611174	+	0.791496i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.175987	+	0.864107i	0.175987	+	0.864107i	0.967484	+	0.252933i	$0.0813953\pi$
$409$	0.175987	+	0.864107i	0.175987	+	0.864107i	−0.791496	+	0.611174i	$0.790698\pi$
$410$	−1.28210	+	1.06697i	−1.28210	+	1.06697i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.0228493	−	1.25085i	0.0228493	−	1.25085i
$417$		0			0
$418$		0			0
$419$		0			0		0.694074	−	0.719903i	$-0.255814\pi$
$419$		0			0		−0.694074	+	0.719903i	$0.744186\pi$
$420$		0			0
$421$	−1.27053	+	1.47120i	−1.27053	+	1.47120i	−0.457242	+	0.889342i	$0.651163\pi$
$421$	−1.27053	+	1.47120i	−1.27053	+	1.47120i	−0.813290	+	0.581859i	$0.802326\pi$
$422$		0			0
$423$		0			0
$424$	1.50052	+	0.334254i	1.50052	+	0.334254i
$425$	−1.63701	−	0.841642i	−1.63701	−	0.841642i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.181637	−	0.983366i	$-0.558140\pi$
$431$		0			0		0.181637	+	0.983366i	$0.441860\pi$
$432$		0			0
$433$	0.558909	+	0.723811i	0.558909	+	0.723811i	0.983366	−	0.181637i	$-0.0581395\pi$
$433$	0.558909	+	0.723811i	0.558909	+	0.723811i	−0.424457	+	0.905448i	$0.639535\pi$
$434$		0			0
$435$		0			0
$436$	0.724627	+	0.442233i	0.724627	+	0.442233i
$437$		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
$441$	0.976076	+	0.217430i	0.976076	+	0.217430i
$442$	0.376846	+	2.27176i	0.376846	+	2.27176i
$443$		0			0		−0.520940	−	0.853593i	$-0.674419\pi$
$443$		0			0		0.520940	+	0.853593i	$0.325581\pi$
$444$		0			0
$445$	0.740294	−	1.57919i	0.740294	−	1.57919i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.00728980	−	0.399067i	0.00728980	−	0.399067i	−0.976076	−	0.217430i	$-0.930233\pi$
$449$	0.00728980	−	0.399067i	0.00728980	−	0.399067i	0.983366	−	0.181637i	$-0.0581395\pi$
$450$	−0.357231	−	0.934016i	−0.357231	−	0.934016i
$451$		0			0
$452$	−0.602843	−	1.87869i	−0.602843	−	1.87869i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.46477	−	0.270556i	−1.46477	−	0.270556i	−0.611174	−	0.791496i	$-0.709302\pi$
$457$	−1.46477	−	0.270556i	−1.46477	−	0.270556i	−0.853593	+	0.520940i	$0.825581\pi$
$458$	0.442893	+	0.368578i	0.442893	+	0.368578i
$459$		0			0
$460$		0			0
$461$	−0.346427	+	1.07960i	−0.346427	+	1.07960i	0.611174	+	0.791496i	$0.290698\pi$
$461$	−0.346427	+	1.07960i	−0.346427	+	1.07960i	−0.957601	+	0.288099i	$0.906977\pi$
$462$		0			0
$463$		0			0		0.732460	−	0.680810i	$-0.238372\pi$
$463$		0			0		−0.732460	+	0.680810i	$0.761628\pi$
$464$	0.136358	+	1.86305i	0.136358	+	1.86305i
$465$		0			0
$466$	−1.12168	−	0.771913i	−1.12168	−	0.771913i
$467$		0			0		0.927336	−	0.374230i	$-0.122093\pi$
$467$		0			0		−0.927336	+	0.374230i	$0.877907\pi$
$468$	−0.671121	+	1.05581i	−0.671121	+	1.05581i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.10670	−	1.06700i	−1.10670	−	1.06700i
$478$		0			0
$479$		0			0		−0.611174	−	0.791496i	$-0.709302\pi$
$479$		0			0		0.611174	+	0.791496i	$0.290698\pi$
$480$		0			0
$481$	0.549714	+	0.210248i	0.549714	+	0.210248i
$482$	0.507914	−	0.945045i	0.507914	−	0.945045i
$483$		0			0
$484$	0.145601	−	0.989343i	0.145601	−	0.989343i
$485$	−0.0487437	+	1.33375i	−0.0487437	+	1.33375i
$486$		0			0
$487$		0			0		−0.989343	−	0.145601i	$-0.953488\pi$
$487$		0			0		0.989343	+	0.145601i	$0.0465116\pi$
$488$	0.610980	−	1.90404i	0.610980	−	1.90404i
$489$		0			0
$490$	−0.872049	−	0.489418i	−0.872049	−	0.489418i
$491$		0			0		0.163646	−	0.986519i	$-0.447674\pi$
$491$		0			0		−0.163646	+	0.986519i	$0.552326\pi$
$492$		0			0
$493$	−0.869705	−	3.32667i	−0.869705	−	3.32667i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.505263	−	0.862965i	$-0.668605\pi$
$499$		0			0		0.505263	+	0.862965i	$0.331395\pi$
$500$	0.0729953	+	0.997332i	0.0729953	+	0.997332i
$501$		0			0
$502$		0			0
$503$		0			0		0.270561	−	0.962703i	$-0.412791\pi$
$503$		0			0		−0.270561	+	0.962703i	$0.587209\pi$
$504$		0			0
$505$	0.922661	−	0.412142i	0.922661	−	0.412142i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.500358	+	1.17744i	0.500358	+	1.17744i	0.957601	+	0.288099i	$0.0930233\pi$
$509$	0.500358	+	1.17744i	0.500358	+	1.17744i	−0.457242	+	0.889342i	$0.651163\pi$
$510$		0			0
$511$		0			0
$512$	−0.357231	−	0.934016i	−0.357231	−	0.934016i
$513$		0			0
$514$	−0.445767	−	1.58611i	−0.445767	−	1.58611i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.946844	−	0.817696i	0.946844	−	0.817696i
$521$	−0.809039	+	0.104008i	−0.809039	+	0.104008i	−0.520940	−	0.853593i	$-0.674419\pi$
$521$	−0.809039	+	0.104008i	−0.809039	+	0.104008i	−0.288099	+	0.957601i	$0.593023\pi$
$522$	0.854143	−	1.66132i	0.854143	−	1.66132i
$523$		0			0		−0.862965	−	0.505263i	$-0.831395\pi$
$523$		0			0		0.862965	+	0.505263i	$0.168605\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.920346	+	0.391105i	−0.920346	+	0.391105i
$530$	0.752379	+	1.34060i	0.752379	+	1.34060i
$531$		0			0
$532$		0			0
$533$	0.987889	+	1.83810i	0.987889	+	1.83810i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.347871	+	0.104659i	−0.347871	+	0.104659i
$539$		0			0
$540$		0			0
$541$	−1.96148	−	0.143562i	−1.96148	−	0.143562i	−0.967484	−	0.252933i	$-0.918605\pi$
$541$	−1.96148	−	0.143562i	−1.96148	−	0.143562i	−0.994001	+	0.109371i	$0.965116\pi$
$542$		0			0
$543$		0			0
$544$	0.958891	+	1.57120i	0.958891	+	1.57120i
$545$	0.154194	+	0.834792i	0.154194	+	0.834792i
$546$		0			0
$547$		0			0		0.813290	−	0.581859i	$-0.197674\pi$
$547$		0			0		−0.813290	+	0.581859i	$0.802326\pi$
$548$	0.322025	+	0.890373i	0.322025	+	0.890373i
$549$	−1.51342	+	1.30699i	−1.51342	+	1.30699i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.75973	−	0.258979i	−1.75973	−	0.258979i
$555$		0			0
$556$		0			0
$557$	−0.0148979	−	0.0333520i	−0.0148979	−	0.0333520i	0.905448	−	0.424457i	$-0.139535\pi$
$557$	−0.0148979	−	0.0333520i	−0.0148979	−	0.0333520i	−0.920346	+	0.391105i	$0.872093\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.343693	+	0.587010i	0.343693	+	0.587010i
$563$		0			0		−0.744772	−	0.667319i	$-0.767442\pi$
$563$		0			0		0.744772	+	0.667319i	$0.232558\pi$
$564$		0			0
$565$	0.996904	−	1.70266i	0.996904	−	1.70266i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.916348	+	1.14293i	0.916348	+	1.14293i	0.989343	+	0.145601i	$0.0465116\pi$
$569$	0.916348	+	1.14293i	0.916348	+	1.14293i	−0.0729953	+	0.997332i	$0.523256\pi$
$570$		0			0
$571$		0			0		0.391105	−	0.920346i	$-0.372093\pi$
$571$		0			0		−0.391105	+	0.920346i	$0.627907\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.181637	+	0.983366i	−0.181637	+	0.983366i
$577$	−0.926588	+	0.476392i	−0.926588	+	0.476392i	−0.853593	−	0.520940i	$-0.825581\pi$
$577$	−0.926588	+	0.476392i	−0.926588	+	0.476392i	−0.0729953	+	0.997332i	$0.523256\pi$
$578$	−1.59366	−	1.77863i	−1.59366	−	1.77863i
$579$		0			0
$580$	−1.29655	+	1.34480i	−1.29655	+	1.34480i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.742351	−	1.83953i	0.742351	−	1.83953i
$585$	−1.23419	+	0.204731i	−1.23419	+	0.204731i
$586$	−0.574302	−	0.295269i	−0.574302	−	0.295269i
$587$		0			0		0.581859	−	0.813290i	$-0.302326\pi$
$587$		0			0		−0.581859	+	0.813290i	$0.697674\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.470364	−	0.00859218i	0.470364	−	0.00859218i
$593$	−0.349670	+	0.914248i	−0.349670	+	0.914248i	0.639673	+	0.768647i	$0.279070\pi$
$593$	−0.349670	+	0.914248i	−0.349670	+	0.914248i	−0.989343	+	0.145601i	$0.953488\pi$
$594$		0			0
$595$		0			0
$596$	1.57369	+	1.12588i	1.57369	+	1.12588i
$597$		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
$601$	0.759847	+	1.41380i	0.759847	+	1.41380i	0.905448	+	0.424457i	$0.139535\pi$
$601$	0.759847	+	1.41380i	0.759847	+	1.41380i	−0.145601	+	0.989343i	$0.546512\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.833998	−	0.551768i	0.833998	−	0.551768i
$606$		0			0
$607$		0			0		0.566908	−	0.823781i	$-0.308140\pi$
$607$		0			0		−0.566908	+	0.823781i	$0.691860\pi$
$608$		0			0
$609$		0			0
$610$	1.82579	−	0.815561i	1.82579	−	0.815561i
$611$		0			0
$612$		−	1.84069i		−	1.84069i
$613$	−0.230686	+	0.676195i	−0.230686	+	0.676195i	0.768647	+	0.639673i	$0.220930\pi$
$613$	−0.230686	+	0.676195i	−0.230686	+	0.676195i	−0.999333	+	0.0365220i	$0.988372\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.91009	−	0.139801i	1.91009	−	0.139801i	0.934016	−	0.357231i	$-0.116279\pi$
$617$	1.91009	−	0.139801i	1.91009	−	0.139801i	0.976076	+	0.217430i	$0.0697674\pi$
$618$		0			0
$619$		0			0		−0.719903	−	0.694074i	$-0.755814\pi$
$619$		0			0		0.719903	+	0.694074i	$0.244186\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.694074	+	0.719903i	−0.694074	+	0.719903i
$626$	0.468482	+	1.93578i	0.468482	+	1.93578i
$627$		0			0
$628$	0.00200056	−	0.109517i	0.00200056	−	0.109517i
$629$	−0.848520	+	0.172813i	−0.848520	+	0.172813i
$630$		0			0
$631$		0			0		0.995833	−	0.0911985i	$-0.0290698\pi$
$631$		0			0		−0.995833	+	0.0911985i	$0.970930\pi$
$632$		0			0
$633$		0			0
$634$	1.79083	+	0.574652i	1.79083	+	0.574652i
$635$		0			0
$636$		0			0
$637$	−0.817696	+	0.946844i	−0.817696	+	0.946844i
$638$		0			0
$639$		0			0
$640$	0.457242	−	0.889342i	0.457242	−	0.889342i
$641$	−0.424457	−	1.90545i	−0.424457	−	1.90545i	−0.424457	−	0.905448i	$-0.639535\pi$
$641$	−0.424457	−	1.90545i	−0.424457	−	1.90545i		−	1.00000i	$-0.5\pi$
$642$		0			0
$643$		0			0		−0.998499	−	0.0547678i	$-0.982558\pi$
$643$		0			0		0.998499	+	0.0547678i	$0.0174419\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.109371	−	0.994001i	$-0.534884\pi$
$647$		0			0		0.109371	+	0.994001i	$0.465116\pi$
$648$	0.667319	−	0.744772i	0.667319	−	0.744772i
$649$		0			0
$650$	1.24084	+	0.159519i	1.24084	+	0.159519i
$651$		0			0
$652$		0			0
$653$	0.493948	+	1.05369i	0.493948	+	1.05369i	0.983366	+	0.181637i	$0.0581395\pi$
$653$	0.493948	+	1.05369i	0.493948	+	1.05369i	−0.489418	+	0.872049i	$0.662791\pi$
$654$		0			0
$655$		0			0
$656$	1.28210	+	1.06697i	1.28210	+	1.06697i
$657$	−1.59195	+	1.18349i	−1.59195	+	1.18349i
$658$		0			0
$659$		0			0		0.994001	−	0.109371i	$-0.0348837\pi$
$659$		0			0		−0.994001	+	0.109371i	$0.965116\pi$
$660$		0			0
$661$	0.447565	+	0.870520i	0.447565	+	0.870520i	0.999333	+	0.0365220i	$0.0116279\pi$
$661$	0.447565	+	0.870520i	0.447565	+	0.870520i	−0.551768	+	0.833998i	$0.686047\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.405975	−	0.237697i	−0.405975	−	0.237697i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.570058	−	1.89479i	−0.570058	−	1.89479i	−0.424457	−	0.905448i	$-0.639535\pi$
$673$	−0.570058	−	1.89479i	−0.570058	−	1.89479i	−0.145601	−	0.989343i	$-0.546512\pi$
$674$	−1.41028	+	1.31084i	−1.41028	+	1.31084i
$675$		0			0
$676$	−0.276589	−	0.492830i	−0.276589	−	0.492830i
$677$	0.292349	−	1.43545i	0.292349	−	1.43545i	−0.520940	−	0.853593i	$-0.674419\pi$
$677$	0.292349	−	1.43545i	0.292349	−	1.43545i	0.813290	−	0.581859i	$-0.197674\pi$
$678$		0			0
$679$		0			0
$680$	−0.530302	+	1.76265i	−0.530302	+	1.76265i
$681$		0			0
$682$		0			0
$683$		0			0		0.163646	−	0.986519i	$-0.447674\pi$
$683$		0			0		−0.163646	+	0.986519i	$0.552326\pi$
$684$		0			0
$685$	−0.448232	+	0.833998i	−0.448232	+	0.833998i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.83127	−	0.587628i	1.83127	−	0.587628i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	0.719903	−	0.694074i	0.719903	−	0.694074i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.67742	−	1.50264i	−2.67742	−	1.50264i
$698$	−0.239812	−	0.0867337i	−0.239812	−	0.0867337i
$699$		0			0
$700$		0			0
$701$	−0.946440	+	0.677120i	−0.946440	+	0.677120i	−0.946440	−	0.322880i	$-0.895349\pi$
$701$	−0.946440	+	0.677120i	−0.946440	+	0.677120i			1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.51588	+	0.456059i	1.51588	+	0.456059i
$707$		0			0
$708$		0			0
$709$	0.173675	−	1.89642i	0.173675	−	1.89642i	−0.217430	−	0.976076i	$-0.569767\pi$
$709$	0.173675	−	1.89642i	0.173675	−	1.89642i	0.391105	−	0.920346i	$-0.372093\pi$
$710$		0			0
$711$		0			0
$712$	−1.68739	−	0.441141i	−1.68739	−	0.441141i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.802527	−	0.596616i	$-0.796512\pi$
$719$		0			0		0.802527	+	0.596616i	$0.203488\pi$
$720$	−0.853593	+	0.520940i	−0.853593	+	0.520940i
$721$		0			0
$722$	0.997332	+	0.0729953i	0.997332	+	0.0729953i
$723$		0			0
$724$	−1.07098	−	1.44061i	−1.07098	−	1.44061i
$725$	−1.86679	−	0.0682243i	−1.86679	−	0.0682243i
$726$		0			0
$727$		0			0		−0.596616	−	0.802527i	$-0.703488\pi$
$727$		0			0		0.596616	+	0.802527i	$0.296512\pi$
$728$		0			0
$729$	−0.957601	+	0.288099i	−0.957601	+	0.288099i
$730$	1.86542	−	0.674672i	1.86542	−	0.674672i
$731$		0			0
$732$		0			0
$733$	−0.133486	+	0.298834i	−0.133486	+	0.298834i	−0.967484	−	0.252933i	$-0.918605\pi$
$733$	−0.133486	+	0.298834i	−0.133486	+	0.298834i	0.833998	+	0.551768i	$0.186047\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.538563	−	1.57866i	−0.538563	−	1.57866i
$739$		0			0		0.946440	−	0.322880i	$-0.104651\pi$
$739$		0			0		−0.946440	+	0.322880i	$0.895349\pi$
$740$	0.320282	+	0.344580i	0.320282	+	0.344580i
$741$		0			0
$742$		0			0
$743$		0			0		−0.843936	−	0.536444i	$-0.819767\pi$
$743$		0			0		0.843936	+	0.536444i	$0.180233\pi$
$744$		0			0
$745$	0.211630	+	1.92336i	0.211630	+	1.92336i
$746$	0.247580	−	0.441141i	0.247580	−	0.441141i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.998499	−	0.0547678i	$-0.982558\pi$
$751$		0			0		0.998499	+	0.0547678i	$0.0174419\pi$
$752$		0			0
$753$		0			0
$754$	1.32487	+	1.92519i	1.32487	+	1.92519i
$755$		0			0
$756$		0			0
$757$	0.268007	−	0.113891i	0.268007	−	0.113891i	−0.252933	−	0.967484i	$-0.581395\pi$
$757$	0.268007	−	0.113891i	0.268007	−	0.113891i	0.520940	+	0.853593i	$0.325581\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.175594	−	0.0493496i	−0.175594	−	0.0493496i	0.181637	−	0.983366i	$-0.441860\pi$
$761$	−0.175594	−	0.0493496i	−0.175594	−	0.0493496i	−0.357231	+	0.934016i	$0.616279\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.37090	−	1.22833i	1.37090	−	1.22833i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.642102	−	0.897494i	0.642102	−	0.897494i	−0.357231	−	0.934016i	$-0.616279\pi$
$769$	0.642102	−	0.897494i	0.642102	−	0.897494i	0.999333	+	0.0365220i	$0.0116279\pi$
$770$		0			0
$771$		0			0
$772$	−1.62658			−1.62658
$773$	−1.75898	+	0.864107i	−1.75898	+	0.864107i	−0.791496	+	0.611174i	$0.790698\pi$
$773$	−1.75898	+	0.864107i	−1.75898	+	0.864107i	−0.967484	+	0.252933i	$0.918605\pi$
$774$		0			0
$775$		0			0
$776$	1.32041	−	0.194325i	1.32041	−	0.194325i
$777$		0			0
$778$	1.49821	−	1.15688i	1.49821	−	1.15688i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.322880	+	0.946440i	−0.322880	+	0.946440i
$785$	0.0829005	−	0.0715930i	0.0829005	−	0.0715930i
$786$		0			0
$787$		0			0		0.843936	−	0.536444i	$-0.180233\pi$
$787$		0			0		−0.843936	+	0.536444i	$0.819767\pi$
$788$	1.66512	−	0.276214i	1.66512	−	0.276214i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.543944	−	2.44184i	−0.543944	−	2.44184i
$794$	−0.679204	−	1.87795i	−0.679204	−	1.87795i
$795$		0			0
$796$		0			0
$797$	−1.58766	−	1.13588i	−1.58766	−	1.13588i	−0.920346	−	0.391105i	$-0.872093\pi$
$797$	−1.58766	−	1.13588i	−1.58766	−	1.13588i	−0.667319	−	0.744772i	$-0.732558\pi$
$798$		0			0
$799$		0			0
$800$	0.967484	−	0.252933i	0.967484	−	0.252933i
$801$	1.21053	+	1.25558i	1.21053	+	1.25558i
$802$	−0.0609186	−	0.0403033i	−0.0609186	−	0.0403033i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.572876	−	0.832453i	−0.572876	−	0.832453i
$809$	−1.70605	−	0.0623499i	−1.70605	−	0.0623499i	−0.833998	−	0.551768i	$-0.813953\pi$
$809$	−1.70605	−	0.0623499i	−1.70605	−	0.0623499i	−0.872049	+	0.489418i	$0.837209\pi$
$810$	1.00000			1.00000
$811$		0			0		−0.872049	−	0.489418i	$-0.837209\pi$
$811$		0			0		0.872049	+	0.489418i	$0.162791\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.791496	+	0.388826i	0.791496	+	0.388826i
$819$		0			0
$820$	0.0609186	+	1.66688i	0.0609186	+	1.66688i
$821$	−0.100011	−	1.82336i	−0.100011	−	1.82336i	−0.457242	−	0.889342i	$-0.651163\pi$
$821$	−0.100011	−	1.82336i	−0.100011	−	1.82336i	0.357231	−	0.934016i	$-0.383721\pi$
$822$		0			0
$823$		0			0		0.694074	−	0.719903i	$-0.255814\pi$
$823$		0			0		−0.694074	+	0.719903i	$0.744186\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.913050	−	0.407849i	$-0.133721\pi$
$827$		0			0		−0.913050	+	0.407849i	$0.866279\pi$
$828$		0			0
$829$	1.27849	−	1.53627i	1.27849	−	1.53627i	0.611174	−	0.791496i	$-0.290698\pi$
$829$	1.27849	−	1.53627i	1.27849	−	1.53627i	0.667319	−	0.744772i	$-0.267442\pi$
$830$		0			0
$831$		0			0
$832$	−0.976076	−	0.782570i	−0.976076	−	0.782570i
$833$	0.268007	−	1.82108i	0.268007	−	1.82108i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.505263	−	0.862965i	$-0.331395\pi$
$839$		0			0		−0.505263	+	0.862965i	$0.668605\pi$
$840$		0			0
$841$	−1.59249	−	1.91358i	−1.59249	−	1.91358i
$842$	0.387935	+	1.90478i	0.387935	+	1.90478i
$843$		0			0
$844$		0			0
$845$	0.182473	−	0.534871i	0.182473	−	0.534871i
$846$		0			0
$847$		0			0
$848$	1.18164	−	0.983366i	1.18164	−	0.983366i
$849$		0			0
$850$	−1.66665	+	0.781294i	−1.66665	+	0.781294i
$851$		0			0
$852$		0			0
$853$	−0.393744	+	1.76757i	−0.393744	+	1.76757i	0.217430	+	0.976076i	$0.430233\pi$
$853$	−0.393744	+	1.76757i	−0.393744	+	1.76757i	−0.611174	+	0.791496i	$0.709302\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.57120	−	0.667689i	−1.57120	−	0.667689i	−0.581859	−	0.813290i	$-0.697674\pi$
$857$	−1.57120	−	0.667689i	−1.57120	−	0.667689i	−0.989343	+	0.145601i	$0.953488\pi$
$858$		0			0
$859$		0			0		0.905448	−	0.424457i	$-0.139535\pi$
$859$		0			0		−0.905448	+	0.424457i	$0.860465\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.611174	−	0.791496i	$-0.290698\pi$
$863$		0			0		−0.611174	+	0.791496i	$0.709302\pi$
$864$		0			0
$865$	0.997332	+	0.0729953i	0.997332	+	0.0729953i
$866$	0.914485			0.914485
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.792899	−	0.303258i	0.792899	−	0.303258i
$873$	−1.22833	−	0.521983i	−1.22833	−	0.521983i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.371194	−	1.66634i	0.371194	−	1.66634i	−0.322880	−	0.946440i	$-0.604651\pi$
$877$	0.371194	−	1.66634i	0.371194	−	1.66634i	0.694074	−	0.719903i	$-0.255814\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.682125	−	0.382827i	0.682125	−	0.382827i	−0.109371	−	0.994001i	$-0.534884\pi$
$881$	0.682125	−	0.382827i	0.682125	−	0.382827i	0.791496	+	0.611174i	$0.209302\pi$
$882$	0.768647	−	0.639673i	0.768647	−	0.639673i
$883$		0			0		−0.999333	−	0.0365220i	$-0.988372\pi$
$883$		0			0		0.999333	+	0.0365220i	$0.0116279\pi$
$884$	2.02841	+	1.09017i	2.02841	+	1.09017i
$885$		0			0
$886$		0			0
$887$		0			0		0.756836	−	0.653605i	$-0.226744\pi$
$887$		0			0		−0.756836	+	0.653605i	$0.773256\pi$
$888$		0			0
$889$		0			0
$890$	−0.797476	−	1.55110i	−0.797476	−	1.55110i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.311405	−	0.249669i	−0.311405	−	0.249669i
$899$		0			0
$900$	−0.957601	−	0.288099i	−0.957601	−	0.288099i
$901$	−1.81007	+	2.17503i	−1.81007	+	2.17503i
$902$		0			0
$903$		0			0
$904$	−1.85542	−	0.671055i	−1.85542	−	0.671055i
$905$	0.358240	−	1.75898i	0.358240	−	1.75898i
$906$		0			0
$907$		0			0		0.694074	−	0.719903i	$-0.255814\pi$
$907$		0			0		−0.694074	+	0.719903i	$0.744186\pi$
$908$		0			0
$909$	0.0553443	+	1.00901i	0.0553443	+	1.00901i
$910$		0			0
$911$		0			0		−0.897545	−	0.440923i	$-0.854651\pi$
$911$		0			0		0.897545	+	0.440923i	$0.145349\pi$
$912$		0			0
$913$		0			0
$914$	−1.10937	+	0.994001i	−1.10937	+	0.994001i
$915$		0			0
$916$	0.562413	−	0.125283i	0.562413	−	0.125283i
$917$		0			0
$918$		0			0
$919$		0			0		−0.872049	−	0.489418i	$-0.837209\pi$
$919$		0			0		0.872049	+	0.489418i	$0.162791\pi$
$920$		0			0
$921$		0			0
$922$	0.642769	+	0.934016i	0.642769	+	0.934016i
$923$		0			0
$924$		0			0
$925$	−0.0429036	+	0.468482i	−0.0429036	+	0.468482i
$926$		0			0
$927$		0			0
$928$	1.55793	+	1.03072i	1.55793	+	1.03072i
$929$	1.32929	+	1.37876i	1.32929	+	1.37876i	0.872049	+	0.489418i	$0.162791\pi$
$929$	1.32929	+	1.37876i	1.32929	+	1.37876i	0.457242	+	0.889342i	$0.348837\pi$
$930$		0			0
$931$		0			0
$932$	−1.29651	+	0.416030i	−1.29651	+	0.416030i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.425499	+	1.17647i	0.425499	+	1.17647i
$937$	−0.0158820	−	0.0712965i	−0.0158820	−	0.0712965i	0.967484	−	0.252933i	$-0.0813953\pi$
$937$	−0.0158820	−	0.0712965i	−0.0158820	−	0.0712965i	−0.983366	+	0.181637i	$0.941860\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.429537	−	1.77486i	0.429537	−	1.77486i	−0.181637	−	0.983366i	$-0.558140\pi$
$941$	0.429537	−	1.77486i	0.429537	−	1.77486i	0.611174	−	0.791496i	$-0.290698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.957601	−	0.288099i	$-0.906977\pi$
$947$		0			0		0.957601	+	0.288099i	$0.0930233\pi$
$948$		0			0
$949$	−0.361337	−	2.45524i	−0.361337	−	2.45524i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.726448	−	0.499926i	−0.726448	−	0.499926i	0.145601	−	0.989343i	$-0.453488\pi$
$953$	−0.726448	−	0.499926i	−0.726448	−	0.499926i	−0.872049	+	0.489418i	$0.837209\pi$
$954$	−1.52091	+	0.223832i	−1.52091	+	0.223832i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.581859	−	0.813290i	0.581859	−	0.813290i
$962$	0.502381	−	0.306599i	0.502381	−	0.306599i
$963$		0			0
$964$	−0.437576	−	0.979599i	−0.437576	−	0.979599i
$965$	−1.08545	−	1.21143i	−1.08545	−	1.21143i
$966$		0			0
$967$		0			0		0.424457	−	0.905448i	$-0.360465\pi$
$967$		0			0		−0.424457	+	0.905448i	$0.639535\pi$
$968$	−0.694074	−	0.719903i	−0.694074	−	0.719903i
$969$		0			0
$970$	1.02587	+	0.853732i	1.02587	+	0.853732i
$971$		0			0		0.457242	−	0.889342i	$-0.348837\pi$
$971$		0			0		−0.457242	+	0.889342i	$0.651163\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.13363	−	1.64729i	−1.13363	−	1.64729i
$977$	1.21082	−	1.45496i	1.21082	−	1.45496i	0.357231	−	0.934016i	$-0.383721\pi$
$977$	1.21082	−	1.45496i	1.21082	−	1.45496i	0.853593	−	0.520940i	$-0.174419\pi$
$978$		0			0
$979$		0			0
$980$	−0.920346	+	0.391105i	−0.920346	+	0.391105i
$981$	−0.834792	−	0.154194i	−0.834792	−	0.154194i
$982$		0			0
$983$		0			0		−0.756836	−	0.653605i	$-0.773256\pi$
$983$		0			0		0.756836	+	0.653605i	$0.226744\pi$
$984$		0			0
$985$	1.31688	+	1.05581i	1.31688	+	1.05581i
$986$	−3.16459	−	1.34480i	−3.16459	−	1.34480i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.946440	−	0.322880i	$-0.104651\pi$
$991$		0			0		−0.946440	+	0.322880i	$0.895349\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.305259	−	0.683381i	0.305259	−	0.683381i	−0.694074	−	0.719903i	$-0.744186\pi$
$997$	0.305259	−	0.683381i	0.305259	−	0.683381i	0.999333	+	0.0365220i	$0.0116279\pi$
$998$		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.123.1	✓	84
4.3	odd	2	CM	3460.1.bd.a.123.1	✓	84
5.2	odd	4		3460.1.bm.a.1507.1	yes	84
20.7	even	4		3460.1.bm.a.1507.1	yes	84
173.128	odd	172		3460.1.bm.a.2723.1	yes	84
692.647	even	172		3460.1.bm.a.2723.1	yes	84
865.647	even	172	inner	3460.1.bd.a.647.1	yes	84
3460.647	odd	172	inner	3460.1.bd.a.647.1	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3460.1.bd.a.123.1	✓	84	1.1	even	1	trivial
3460.1.bd.a.123.1	✓	84	4.3	odd	2	CM
3460.1.bd.a.647.1	yes	84	865.647	even	172	inner
3460.1.bd.a.647.1	yes	84	3460.647	odd	172	inner
3460.1.bm.a.1507.1	yes	84	5.2	odd	4
3460.1.bm.a.1507.1	yes	84	20.7	even	4
3460.1.bm.a.2723.1	yes	84	173.128	odd	172
3460.1.bm.a.2723.1	yes	84	692.647	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)

\(f(q)\)	\(=\)	\(q+(0.611174 - 0.791496i) q^{2} +(-0.252933 - 0.967484i) q^{4} +(0.551768 - 0.833998i) q^{5} +(-0.920346 - 0.391105i) q^{8} +(0.581859 + 0.813290i) q^{9} +O(q^{10})\)	\(q+(0.611174 - 0.791496i) q^{2} +(-0.252933 - 0.967484i) q^{4} +(0.551768 - 0.833998i) q^{5} +(-0.920346 - 0.391105i) q^{8} +(0.581859 + 0.813290i) q^{9} +(-0.322880 - 0.946440i) q^{10} +(-1.24084 + 0.159519i) q^{13} +(-0.872049 + 0.489418i) q^{16} +(1.41484 - 1.17744i) q^{17} +(0.999333 + 0.0365220i) q^{18} +(-0.946440 - 0.322880i) q^{20} +(-0.391105 - 0.920346i) q^{25} +(-0.632112 + 1.07962i) q^{26} +(0.792899 - 1.69141i) q^{29} +(-0.145601 + 0.989343i) q^{32} +(-0.0672258 - 1.83946i) q^{34} +(0.639673 - 0.768647i) q^{36} +(-0.414385 - 0.222712i) q^{37} +(-0.833998 + 0.551768i) q^{40} +(-0.595860 - 1.55793i) q^{41} +(0.999333 - 0.0365220i) q^{45} +(0.744772 - 0.667319i) q^{49} +(-0.967484 - 0.252933i) q^{50} +(0.468183 + 1.16015i) q^{52} +(-1.51172 + 0.279229i) q^{53} +(-0.854143 - 1.66132i) q^{58} +(0.182367 + 1.99133i) q^{61} +(0.694074 + 0.719903i) q^{64} +(-0.551619 + 1.12288i) q^{65} +(-1.49702 - 1.07102i) q^{68} +(-0.217430 - 0.976076i) q^{72} +(0.0362300 + 1.98334i) q^{73} +(-0.429537 + 0.191869i) q^{74} +(-0.0729953 + 0.997332i) q^{80} +(-0.322880 + 0.946440i) q^{81} +(-1.59727 - 0.480548i) q^{82} +(-0.201319 - 1.82965i) q^{85} +(1.72551 - 0.253943i) q^{89} +(0.581859 - 0.813290i) q^{90} +(-1.13924 + 0.695267i) q^{97} +(-0.0729953 - 0.997332i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 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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