Embedded newform 74.5.g.b.23.1

• Label: 74.5.g.b.23.1
• 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			23.1
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.b.29.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732051 - 2.73205i) q^{2} +(-13.9920 + 8.07829i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(-9.30753 + 34.7362i) q^{5} +(32.3132 + 32.3132i) q^{6} +(-30.9246 - 53.5629i) q^{7} +(16.0000 + 16.0000i) q^{8} +(90.0175 - 155.915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \)

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
							\(3\)	−13.9920	+	8.07829i	−1.55467	+	0.897588i	−0.556915	+	0.830569i	\(0.688015\pi\)
−0.997752	+	0.0670184i	\(0.978651\pi\)
\(5\)	−9.30753	+	34.7362i	−0.372301	+	1.38945i	0.484947	+	0.874544i	\(0.338839\pi\)
							−0.857248	+	0.514903i	\(0.827828\pi\)
\(7\)	−30.9246	−	53.5629i	−0.631114	−	1.09312i	−0.987324	−	0.158715i	\(-0.949265\pi\)
							0.356211	−	0.934406i	\(-0.384069\pi\)
\(11\)			185.693i			1.53465i	0.641258	+	0.767325i	\(0.278413\pi\)
							−0.641258	+	0.767325i	\(0.721587\pi\)
\(13\)	72.0010	−	268.711i	0.426042	−	1.59001i	−0.335596	−	0.942006i	\(-0.608938\pi\)
							0.761638	−	0.648003i	\(-0.224396\pi\)
\(17\)	−102.654	+	27.5059i	−0.355203	+	0.0951763i	−0.432008	−	0.901870i	\(-0.642195\pi\)
							0.0768052	+	0.997046i	\(0.475528\pi\)
\(19\)	2.22976	−	8.32158i	0.00617662	−	0.0230515i	−0.962769	−	0.270326i	\(-0.912868\pi\)
							0.968945	+	0.247275i	\(0.0795351\pi\)
\(23\)	46.1045	+	46.1045i	0.0871541	+	0.0871541i	0.749340	−	0.662186i	\(-0.230371\pi\)
							−0.662186	+	0.749340i	\(0.730371\pi\)
\(29\)	734.634	−	734.634i	0.873524	−	0.873524i	−0.119330	−	0.992855i	\(-0.538075\pi\)
							0.992855	+	0.119330i	\(0.0380748\pi\)
\(31\)	78.9357	−	78.9357i	0.0821392	−	0.0821392i	−0.664844	−	0.746983i	\(-0.731502\pi\)
							0.746983	+	0.664844i	\(0.231502\pi\)
\(37\)	1350.62	+	223.547i	0.986578	+	0.163292i
							\(41\)	−266.034	+	153.595i	−0.158259	+	0.0913709i	−0.577038	−	0.816717i	\(-0.695792\pi\)
0.418779	+	0.908088i	\(0.362458\pi\)
\(43\)	−2039.33	−	2039.33i	−1.10293	−	1.10293i	−0.994055	−	0.108880i	\(-0.965274\pi\)
							−0.108880	−	0.994055i	\(-0.534726\pi\)
\(47\)	−821.208			−0.371756			−0.185878	−	0.982573i	\(-0.559513\pi\)
							−0.185878	+	0.982573i	\(0.559513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.b.23.1	✓	28
37.29	odd	12	inner	74.5.g.b.29.1	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.b.23.1	✓	28	1.1	even	1	trivial
74.5.g.b.29.1	yes	28	37.29	odd	12	inner