Embedded newform 74.5.g.a.45.2

• Label: 74.5.g.a.45.2
• Level: $74$
• Weight: $5$
• Character: 74.45
• 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			45.2
Character	\(\chi\)	\(=\)	74.45
Dual form			74.5.g.a.51.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.73205 - 0.732051i) q^{2} +(-9.68326 - 5.59063i) q^{3} +(6.92820 + 4.00000i) q^{4} +(-24.8057 + 6.64666i) q^{5} +(22.3625 + 22.3625i) q^{6} +(-19.9156 + 34.4949i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(22.0103 + 38.1230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.73205	−	0.732051i	−0.683013	−	0.183013i
							\(3\)	−9.68326	−	5.59063i	−1.07592	−	0.621181i	−0.146125	−	0.989266i	\(-0.546680\pi\)
−0.929792	+	0.368085i	\(0.880014\pi\)
\(5\)	−24.8057	+	6.64666i	−0.992227	+	0.265866i	−0.718186	−	0.695852i	\(-0.755027\pi\)
							−0.274041	+	0.961718i	\(0.588360\pi\)
\(7\)	−19.9156	+	34.4949i	−0.406442	+	0.703978i	−0.994488	−	0.104850i	\(-0.966564\pi\)
							0.588046	+	0.808827i	\(0.299897\pi\)
\(11\)		−	64.8670i		−	0.536091i	−0.963406	−	0.268045i	\(-0.913622\pi\)
							0.963406	−	0.268045i	\(-0.0863777\pi\)
\(13\)	111.821	−	29.9623i	0.661662	−	0.177292i	0.0876661	−	0.996150i	\(-0.472059\pi\)
							0.573996	+	0.818858i	\(0.305392\pi\)
\(17\)	−15.2932	+	57.0751i	−0.0529178	+	0.197492i	−0.987324	−	0.158717i	\(-0.949264\pi\)
							0.934406	+	0.356209i	\(0.115931\pi\)
\(19\)	628.263	−	168.342i	1.74034	−	0.466323i	0.757817	−	0.652467i	\(-0.226266\pi\)
							0.982522	+	0.186144i	\(0.0595992\pi\)
\(23\)	152.604	+	152.604i	0.288477	+	0.288477i	0.836478	−	0.548001i	\(-0.184611\pi\)
							−0.548001	+	0.836478i	\(0.684611\pi\)
\(29\)	312.534	−	312.534i	0.371622	−	0.371622i	−0.496446	−	0.868068i	\(-0.665362\pi\)
							0.868068	+	0.496446i	\(0.165362\pi\)
\(31\)	−68.3566	+	68.3566i	−0.0711307	+	0.0711307i	−0.741777	−	0.670646i	\(-0.766017\pi\)
							0.670646	+	0.741777i	\(0.266017\pi\)
\(37\)	−17.1371	+	1368.89i	−0.0125180	+	0.999922i
							\(41\)	−1704.80	−	984.269i	−1.01416	−	0.585526i	−0.101754	−	0.994810i	\(-0.532445\pi\)
−0.912407	+	0.409283i	\(0.865779\pi\)
\(43\)	908.495	+	908.495i	0.491344	+	0.491344i	0.908730	−	0.417385i	\(-0.137054\pi\)
							−0.417385	+	0.908730i	\(0.637054\pi\)
\(47\)	2232.24			1.01052			0.505261	−	0.862967i	\(-0.331396\pi\)
							0.505261	+	0.862967i	\(0.331396\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.a.45.2	✓	28
37.14	odd	12	inner	74.5.g.a.51.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.a.45.2	✓	28	1.1	even	1	trivial
74.5.g.a.51.2	yes	28	37.14	odd	12	inner