Embedded newform 58.3.f.b.11.2

• Label: 58.3.f.b.11.2
• Level: $58$
• Weight: $3$
• Character: 58.11
• Analytic conductor: $1.580$
• Analytic rank: $0$
• Dimension: $36$
• 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			11.2
Character	\(\chi\)	\(=\)	58.11
Dual form			58.3.f.b.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33485 - 0.467085i) q^{2} +(-0.123918 - 1.09981i) q^{3} +(1.56366 - 1.24698i) q^{4} +(2.16499 - 4.49565i) q^{5} +(-0.679116 - 1.41020i) q^{6} +(-6.05241 + 7.58949i) q^{7} +(1.50481 - 2.39490i) q^{8} +(7.58013 - 1.73012i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33485	−	0.467085i	0.667426	−	0.233543i
							\(3\)	−0.123918	−	1.09981i	−0.0413061	−	0.366602i	−0.997313	−	0.0732578i	\(-0.976660\pi\)
0.956007	−	0.293344i	\(-0.0947682\pi\)
\(5\)	2.16499	−	4.49565i	0.432998	−	0.899130i	−0.564293	−	0.825575i	\(-0.690851\pi\)
							0.997291	−	0.0735553i	\(-0.0234345\pi\)
\(7\)	−6.05241	+	7.58949i	−0.864631	+	1.08421i	0.131051	+	0.991376i	\(0.458165\pi\)
							−0.995681	+	0.0928369i	\(0.970406\pi\)
\(11\)	7.51630	+	11.9621i	0.683300	+	1.08747i	0.990723	+	0.135893i	\(0.0433904\pi\)
							−0.307423	+	0.951573i	\(0.599467\pi\)
\(13\)	−20.2431	−	4.62037i	−1.55717	−	0.355413i	−0.644657	−	0.764472i	\(-0.723000\pi\)
							−0.912508	+	0.409059i	\(0.865857\pi\)
\(17\)	−16.8975	+	16.8975i	−0.993972	+	0.993972i	−0.999982	−	0.00600965i	\(-0.998087\pi\)
							0.00600965	+	0.999982i	\(0.498087\pi\)
\(19\)	11.9477	+	1.34618i	0.628826	+	0.0708517i	0.420625	−	0.907234i	\(-0.361811\pi\)
							0.208201	+	0.978086i	\(0.433239\pi\)
\(23\)	−1.98039	+	0.953706i	−0.0861039	+	0.0414655i	−0.476440	−	0.879207i	\(-0.658073\pi\)
							0.390336	+	0.920672i	\(0.372359\pi\)
\(29\)	−8.77955	−	27.6391i	−0.302743	−	0.953072i
							\(31\)	9.99612	−	3.49779i	0.322456	−	0.112832i	−0.164197	−	0.986428i	\(-0.552503\pi\)
0.486653	+	0.873595i	\(0.338218\pi\)
\(37\)	26.7742	−	42.6108i	0.723626	−	1.15164i	−0.258969	−	0.965886i	\(-0.583383\pi\)
							0.982595	−	0.185759i	\(-0.0594744\pi\)
\(41\)	−19.5241	−	19.5241i	−0.476198	−	0.476198i	0.427715	−	0.903914i	\(-0.359319\pi\)
							−0.903914	+	0.427715i	\(0.859319\pi\)
\(43\)	11.5941	−	33.1341i	0.269631	−	0.770561i	−0.726854	−	0.686792i	\(-0.759018\pi\)
							0.996485	−	0.0837695i	\(-0.0266959\pi\)
\(47\)	−33.8653	+	21.2790i	−0.720538	+	0.452744i	−0.841740	−	0.539884i	\(-0.818468\pi\)
							0.121201	+	0.992628i	\(0.461325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.11.2	✓	36
29.8	odd	28	inner	58.3.f.b.37.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.11.2	✓	36	1.1	even	1	trivial
58.3.f.b.37.2	yes	36	29.8	odd	28	inner