Embedded newform 58.3.f.b.21.1

• Label: 58.3.f.b.21.1
• Level: $58$
• Weight: $3$
• Character: 58.21
• 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			21.1
Character	\(\chi\)	\(=\)	58.21
Dual form			58.3.f.b.47.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.467085 - 1.33485i) q^{2} +(-3.82213 - 0.430651i) q^{3} +(-1.56366 - 1.24698i) q^{4} +(-1.57560 - 3.27177i) q^{5} +(-2.36012 + 4.90083i) q^{6} +(-3.99451 - 5.00896i) q^{7} +(-2.39490 + 1.50481i) q^{8} +(5.64887 + 1.28932i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{17}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.467085	−	1.33485i	0.233543	−	0.667426i
							\(3\)	−3.82213	−	0.430651i	−1.27404	−	0.143550i	−0.551089	−	0.834446i	\(-0.685788\pi\)
−0.722954	+	0.690896i	\(0.757216\pi\)
\(5\)	−1.57560	−	3.27177i	−0.315120	−	0.654354i	0.681904	−	0.731442i	\(-0.261152\pi\)
							−0.997024	+	0.0770876i	\(0.975438\pi\)
\(7\)	−3.99451	−	5.00896i	−0.570644	−	0.715565i	0.409841	−	0.912157i	\(-0.365584\pi\)
							−0.980485	+	0.196591i	\(0.937013\pi\)
\(11\)	2.60954	+	1.63968i	0.237231	+	0.149062i	0.645401	−	0.763844i	\(-0.276690\pi\)
							−0.408171	+	0.912906i	\(0.633833\pi\)
\(13\)	17.9599	−	4.09923i	1.38153	−	0.315325i	0.533737	−	0.845651i	\(-0.320787\pi\)
							0.847794	+	0.530325i	\(0.177930\pi\)
\(17\)	6.94525	−	6.94525i	0.408544	−	0.408544i	−0.472686	−	0.881231i	\(-0.656716\pi\)
							0.881231	+	0.472686i	\(0.156716\pi\)
\(19\)	−1.44182	−	12.7965i	−0.0758855	−	0.673502i	−0.972856	−	0.231412i	\(-0.925666\pi\)
							0.896970	−	0.442090i	\(-0.145763\pi\)
\(23\)	−36.2729	−	17.4681i	−1.57708	−	0.759483i	−0.578657	−	0.815571i	\(-0.696423\pi\)
							−0.998426	+	0.0560875i	\(0.982137\pi\)
\(29\)	−15.5071	+	24.5057i	−0.534728	+	0.845024i
							\(31\)	−14.6250	+	41.7960i	−0.471776	+	1.34826i	0.424768	+	0.905302i	\(0.360356\pi\)
−0.896544	+	0.442955i	\(0.853930\pi\)
\(37\)	35.8776	−	22.5434i	0.969665	−	0.609281i	0.0486530	−	0.998816i	\(-0.484507\pi\)
							0.921012	+	0.389535i	\(0.127364\pi\)
\(41\)	−19.2352	−	19.2352i	−0.469152	−	0.469152i	0.432488	−	0.901640i	\(-0.357636\pi\)
							−0.901640	+	0.432488i	\(0.857636\pi\)
\(43\)	8.72044	−	3.05141i	0.202801	−	0.0709631i	−0.226967	−	0.973902i	\(-0.572881\pi\)
							0.429768	+	0.902939i	\(0.358595\pi\)
\(47\)	45.0367	−	71.6755i	0.958228	−	1.52501i	0.110973	−	0.993823i	\(-0.464603\pi\)
							0.847255	−	0.531187i	\(-0.178254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.21.1	✓	36
29.18	odd	28	inner	58.3.f.b.47.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.21.1	✓	36	1.1	even	1	trivial
58.3.f.b.47.1	yes	36	29.18	odd	28	inner