Embedded newform 58.3.f.b.19.2

• Label: 58.3.f.b.19.2
• Level: $58$
• Weight: $3$
• Character: 58.19
• 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			19.2
Character	\(\chi\)	\(=\)	58.19
Dual form			58.3.f.b.55.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.752407 - 1.19745i) q^{2} +(-0.674793 + 1.92845i) q^{3} +(-0.867767 + 1.80194i) q^{4} +(6.79118 + 1.55004i) q^{5} +(2.81694 - 0.642947i) q^{6} +(3.48966 - 1.68053i) q^{7} +(2.81064 - 0.316683i) q^{8} +(3.77292 + 3.00880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.752407	−	1.19745i	−0.376203	−	0.598724i
							\(3\)	−0.674793	+	1.92845i	−0.224931	+	0.642816i	0.775010	+	0.631949i	\(0.217745\pi\)
−0.999941	+	0.0108668i	\(0.996541\pi\)
\(5\)	6.79118	+	1.55004i	1.35824	+	0.310009i	0.838772	−	0.544483i	\(-0.183274\pi\)
							0.519465	+	0.854492i	\(0.326131\pi\)
\(7\)	3.48966	−	1.68053i	0.498523	−	0.240076i	−0.167687	−	0.985840i	\(-0.553630\pi\)
							0.666210	+	0.745764i	\(0.267915\pi\)
\(11\)	−9.38585	−	1.05753i	−0.853259	−	0.0961392i	−0.325509	−	0.945539i	\(-0.605536\pi\)
							−0.527750	+	0.849400i	\(0.676964\pi\)
\(13\)	4.94770	−	3.94566i	0.380592	−	0.303512i	−0.414443	−	0.910075i	\(-0.636024\pi\)
							0.795035	+	0.606563i	\(0.207452\pi\)
\(17\)	−12.8097	+	12.8097i	−0.753515	+	0.753515i	−0.975133	−	0.221619i	\(-0.928866\pi\)
							0.221619	+	0.975133i	\(0.428866\pi\)
\(19\)	−2.29335	+	0.802480i	−0.120703	+	0.0422358i	−0.389955	−	0.920834i	\(-0.627509\pi\)
							0.269252	+	0.963070i	\(0.413224\pi\)
\(23\)	−8.68673	−	38.0591i	−0.377684	−	1.65474i	−0.704537	−	0.709667i	\(-0.748845\pi\)
							0.326853	−	0.945075i	\(-0.394012\pi\)
\(29\)	−14.9552	−	24.8463i	−0.515698	−	0.856770i
							\(31\)	−7.47568	−	11.8975i	−0.241151	−	0.383789i	0.704117	−	0.710084i	\(-0.251343\pi\)
−0.945268	+	0.326294i	\(0.894200\pi\)
\(37\)	71.3292	−	8.03687i	1.92782	−	0.217213i	0.936692	−	0.350155i	\(-0.113871\pi\)
							0.991124	+	0.132942i	\(0.0424424\pi\)
\(41\)	−44.9202	−	44.9202i	−1.09561	−	1.09561i	−0.994917	−	0.100697i	\(-0.967893\pi\)
							−0.100697	−	0.994917i	\(-0.532107\pi\)
\(43\)	22.6494	+	14.2316i	0.526730	+	0.330967i	0.769007	−	0.639241i	\(-0.220751\pi\)
							−0.242276	+	0.970207i	\(0.577894\pi\)
\(47\)	−6.48129	+	57.5231i	−0.137900	+	1.22390i	0.713983	+	0.700163i	\(0.246889\pi\)
							−0.851883	+	0.523732i	\(0.824539\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.19.2	✓	36
29.26	odd	28	inner	58.3.f.b.55.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.19.2	✓	36	1.1	even	1	trivial
58.3.f.b.55.2	yes	36	29.26	odd	28	inner