Embedded newform 58.3.f.b.31.2

• Label: 58.3.f.b.31.2
• Level: $58$
• Weight: $3$
• Character: 58.31
• 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			31.2
Character	\(\chi\)	\(=\)	58.31
Dual form			58.3.f.b.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40532 - 0.158342i) q^{2} +(-1.32900 - 0.835065i) q^{3} +(1.94986 + 0.445042i) q^{4} +(2.32581 - 1.85477i) q^{5} +(1.73544 + 1.38397i) q^{6} +(-2.63944 - 11.5641i) q^{7} +(-2.66971 - 0.934170i) q^{8} +(-2.83605 - 5.88912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40532	−	0.158342i	−0.702661	−	0.0791708i
							\(3\)	−1.32900	−	0.835065i	−0.442999	−	0.278355i	0.292006	−	0.956417i	\(-0.405677\pi\)
−0.735005	+	0.678062i	\(0.762820\pi\)
\(5\)	2.32581	−	1.85477i	0.465162	−	0.370955i	−0.362682	−	0.931913i	\(-0.618139\pi\)
							0.827844	+	0.560959i	\(0.189567\pi\)
\(7\)	−2.63944	−	11.5641i	−0.377063	−	1.65202i	−0.706403	−	0.707810i	\(-0.749683\pi\)
							0.329340	−	0.944211i	\(-0.393174\pi\)
\(11\)	18.1316	−	6.34453i	1.64833	−	0.576776i	0.663178	−	0.748462i	\(-0.269207\pi\)
							0.985152	+	0.171686i	\(0.0549215\pi\)
\(13\)	−10.7691	+	22.3623i	−0.828394	+	1.72018i	−0.145883	+	0.989302i	\(0.546602\pi\)
							−0.682511	+	0.730875i	\(0.739112\pi\)
\(17\)	3.75395	−	3.75395i	0.220820	−	0.220820i	−0.588023	−	0.808844i	\(-0.700094\pi\)
							0.808844	+	0.588023i	\(0.200094\pi\)
\(19\)	−0.452402	−	0.719994i	−0.0238106	−	0.0378944i	0.834607	−	0.550846i	\(-0.185695\pi\)
							−0.858417	+	0.512952i	\(0.828552\pi\)
\(23\)	−0.288432	+	0.361682i	−0.0125405	+	0.0157253i	−0.788062	−	0.615596i	\(-0.788915\pi\)
							0.775522	+	0.631321i	\(0.217487\pi\)
\(29\)	20.4193	−	20.5925i	0.704115	−	0.710086i
							\(31\)	11.7286	+	1.32149i	0.378341	+	0.0426288i	0.299088	−	0.954226i	\(-0.403318\pi\)
0.0792537	+	0.996854i	\(0.474746\pi\)
\(37\)	14.2179	+	4.97507i	0.384268	+	0.134461i	0.515498	−	0.856891i	\(-0.327607\pi\)
							−0.131230	+	0.991352i	\(0.541893\pi\)
\(41\)	21.5061	+	21.5061i	0.524539	+	0.524539i	0.918939	−	0.394400i	\(-0.129048\pi\)
							−0.394400	+	0.918939i	\(0.629048\pi\)
\(43\)	−7.83188	−	69.5099i	−0.182137	−	1.61651i	−0.669159	−	0.743119i	\(-0.733346\pi\)
							0.487022	−	0.873390i	\(-0.338083\pi\)
\(47\)	14.3747	+	41.0805i	0.305844	+	0.874053i	0.989537	+	0.144282i	\(0.0460873\pi\)
							−0.683692	+	0.729770i	\(0.739627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.31.2	yes	36
29.15	odd	28	inner	58.3.f.b.15.2	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.15.2	✓	36	29.15	odd	28	inner
58.3.f.b.31.2	yes	36	1.1	even	1	trivial