Embedded newform 58.3.f.b.27.1

• Label: 58.3.f.b.27.1
• Level: $58$
• Weight: $3$
• Character: 58.27
• Analytic conductor: $1.580$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	58.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.58038553329\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(3\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.1
Character	\(\chi\)	\(=\)	58.27
Dual form			58.3.f.b.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.158342 - 1.40532i) q^{2} +(-2.85077 + 4.53698i) q^{3} +(-1.94986 - 0.445042i) q^{4} +(-1.01211 + 0.807134i) q^{5} +(5.92452 + 4.72465i) q^{6} +(3.01755 + 13.2207i) q^{7} +(-0.934170 + 2.66971i) q^{8} +(-8.55233 - 17.7591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{15}{28}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.158342	−	1.40532i	0.0791708	−	0.702661i
							\(3\)	−2.85077	+	4.53698i	−0.950258	+	1.51233i	−0.0941318	+	0.995560i	\(0.530008\pi\)
−0.856126	+	0.516767i	\(0.827135\pi\)
\(5\)	−1.01211	+	0.807134i	−0.202423	+	0.161427i	−0.719455	−	0.694539i	\(-0.755608\pi\)
							0.517032	+	0.855966i	\(0.327037\pi\)
\(7\)	3.01755	+	13.2207i	0.431078	+	1.88868i	0.457839	+	0.889035i	\(0.348624\pi\)
							−0.0267610	+	0.999642i	\(0.508519\pi\)
\(11\)	−2.00245	−	5.72268i	−0.182041	−	0.520244i	0.816312	−	0.577611i	\(-0.196015\pi\)
							−0.998353	+	0.0573674i	\(0.981729\pi\)
\(13\)	0.481401	−	0.999640i	0.0370309	−	0.0768954i	−0.881633	−	0.471936i	\(-0.843555\pi\)
							0.918664	+	0.395041i	\(0.129270\pi\)
\(17\)	8.04861	+	8.04861i	0.473447	+	0.473447i	0.903028	−	0.429581i	\(-0.141339\pi\)
							−0.429581	+	0.903028i	\(0.641339\pi\)
\(19\)	19.2386	−	12.0884i	1.01256	−	0.636231i	0.0798266	−	0.996809i	\(-0.474563\pi\)
							0.932729	+	0.360578i	\(0.117420\pi\)
\(23\)	−2.83664	+	3.55703i	−0.123332	+	0.154654i	−0.839664	−	0.543106i	\(-0.817248\pi\)
							0.716332	+	0.697760i	\(0.245820\pi\)
\(29\)	28.8772	−	2.66562i	0.995767	−	0.0919179i
							\(31\)	−4.84733	+	43.0213i	−0.156366	+	1.38778i	0.633139	+	0.774038i	\(0.281766\pi\)
−0.789505	+	0.613744i	\(0.789663\pi\)
\(37\)	−0.918286	+	2.62431i	−0.0248185	+	0.0709273i	−0.955626	−	0.294582i	\(-0.904820\pi\)
							0.930808	+	0.365510i	\(0.119105\pi\)
\(41\)	43.6242	−	43.6242i	1.06400	−	1.06400i	0.0661983	−	0.997806i	\(-0.478913\pi\)
							0.997806	−	0.0661983i	\(-0.0210870\pi\)
\(43\)	−12.6232	+	1.42229i	−0.293563	+	0.0330766i	−0.257518	−	0.966274i	\(-0.582905\pi\)
							−0.0360452	+	0.999350i	\(0.511476\pi\)
\(47\)	46.9636	−	16.4333i	0.999226	−	0.349644i	0.219396	−	0.975636i	\(-0.429591\pi\)
							0.779830	+	0.625992i	\(0.215306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.27.1	✓	36
29.14	odd	28	inner	58.3.f.b.43.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.27.1	✓	36	1.1	even	1	trivial
58.3.f.b.43.1	yes	36	29.14	odd	28	inner