Embedded newform 74.5.g.a.23.6

• Label: 74.5.g.a.23.6
• Level: $74$
• Weight: $5$
• Character: 74.23
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.6
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.732051 + 2.73205i) q^{2} +(7.29967 - 4.21447i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(-12.4312 + 46.3938i) q^{5} +(16.8579 + 16.8579i) q^{6} +(-18.9531 - 32.8277i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(-4.97651 + 8.61956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.732051	+	2.73205i	0.183013	+	0.683013i
							\(3\)	7.29967	−	4.21447i	0.811075	−	0.468274i	−0.0362541	−	0.999343i	\(-0.511543\pi\)
0.847329	+	0.531068i	\(0.178209\pi\)
\(5\)	−12.4312	+	46.3938i	−0.497247	+	1.85575i	0.0198176	+	0.999804i	\(0.493691\pi\)
							−0.517064	+	0.855947i	\(0.672975\pi\)
\(7\)	−18.9531	−	32.8277i	−0.386798	−	0.669953i	0.605219	−	0.796059i	\(-0.293086\pi\)
							−0.992017	+	0.126106i	\(0.959752\pi\)
\(11\)			114.993i			0.950359i	0.879889	+	0.475180i	\(0.157617\pi\)
							−0.879889	+	0.475180i	\(0.842383\pi\)
\(13\)	−5.81441	+	21.6997i	−0.0344048	+	0.128400i	−0.980992	−	0.194047i	\(-0.937839\pi\)
							0.946588	+	0.322447i	\(0.104505\pi\)
\(17\)	253.519	−	67.9302i	0.877229	−	0.235053i	0.208018	−	0.978125i	\(-0.433299\pi\)
							0.669211	+	0.743072i	\(0.266632\pi\)
\(19\)	−96.4642	+	360.009i	−0.267214	+	0.997256i	0.693668	+	0.720295i	\(0.255994\pi\)
							−0.960881	+	0.276960i	\(0.910673\pi\)
\(23\)	104.820	+	104.820i	0.198147	+	0.198147i	0.799205	−	0.601058i	\(-0.205254\pi\)
							−0.601058	+	0.799205i	\(0.705254\pi\)
\(29\)	638.560	−	638.560i	0.759287	−	0.759287i	−0.216906	−	0.976193i	\(-0.569596\pi\)
							0.976193	+	0.216906i	\(0.0695964\pi\)
\(31\)	212.313	−	212.313i	0.220929	−	0.220929i	−0.587960	−	0.808890i	\(-0.700069\pi\)
							0.808890	+	0.587960i	\(0.200069\pi\)
\(37\)	1203.34	−	652.785i	0.878994	−	0.476834i
							\(41\)	1652.14	−	953.864i	0.982832	−	0.567438i	0.0797081	−	0.996818i	\(-0.474601\pi\)
0.903124	+	0.429380i	\(0.141268\pi\)
\(43\)	1202.02	+	1202.02i	0.650090	+	0.650090i	0.953015	−	0.302925i	\(-0.0979630\pi\)
							−0.302925	+	0.953015i	\(0.597963\pi\)
\(47\)	352.755			0.159690			0.0798450	−	0.996807i	\(-0.474557\pi\)
							0.0798450	+	0.996807i	\(0.474557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.a.23.6	✓	28
37.29	odd	12	inner	74.5.g.a.29.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.a.23.6	✓	28	1.1	even	1	trivial
74.5.g.a.29.6	yes	28	37.29	odd	12	inner