Embedded newform 74.5.i.b.13.4

• Label: 74.5.i.b.13.4
• Level: $74$
• Weight: $5$
• Character: 74.13
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	74.13
Dual form			74.5.i.b.57.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.62232 + 2.31691i) q^{2} +(0.340097 - 0.0599682i) q^{3} +(-2.73616 + 7.51754i) q^{4} +(27.1996 + 2.37966i) q^{5} +(0.690686 + 0.690686i) q^{6} +(70.2106 - 58.9137i) q^{7} +(-21.8564 + 5.85641i) q^{8} +(-76.0030 + 27.6628i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 114 q^{5} - 576 q^{8} - 210 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{11}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.62232	+	2.31691i	0.405580	+	0.579228i
							\(3\)	0.340097	−	0.0599682i	0.0377885	−	0.00666313i	−0.154722	−	0.987958i	\(-0.549448\pi\)
0.192510	+	0.981295i	\(0.438337\pi\)
\(5\)	27.1996	+	2.37966i	1.08798	+	0.0951863i	0.617006	−	0.786958i	\(-0.288345\pi\)
							0.470978	+	0.882145i	\(0.343901\pi\)
\(7\)	70.2106	−	58.9137i	1.43287	−	1.20232i	0.488876	−	0.872353i	\(-0.337407\pi\)
							0.943993	−	0.329967i	\(-0.107037\pi\)
\(11\)	17.0590	+	9.84901i	0.140983	+	0.0813968i	0.568833	−	0.822453i	\(-0.307395\pi\)
							−0.427849	+	0.903850i	\(0.640729\pi\)
\(13\)	137.922	+	295.775i	0.816106	+	1.75015i	0.643937	+	0.765078i	\(0.277300\pi\)
							0.172169	+	0.985067i	\(0.444923\pi\)
\(17\)	90.9533	+	42.4122i	0.314717	+	0.146755i	0.573558	−	0.819165i	\(-0.305563\pi\)
							−0.258841	+	0.965920i	\(0.583341\pi\)
\(19\)	105.232	−	150.286i	0.291500	−	0.416306i	−0.646514	−	0.762902i	\(-0.723774\pi\)
							0.938014	+	0.346597i	\(0.112663\pi\)
\(23\)	14.1373	−	52.7611i	0.0267246	−	0.0997374i	−0.951275	−	0.308343i	\(-0.900226\pi\)
							0.978000	+	0.208605i	\(0.0668924\pi\)
\(29\)	−330.378	−	1232.99i	−0.392840	−	1.46610i	−0.825429	−	0.564506i	\(-0.809067\pi\)
							0.432589	−	0.901591i	\(-0.357600\pi\)
\(31\)	−596.737	+	596.737i	−0.620954	+	0.620954i	−0.945775	−	0.324821i	\(-0.894696\pi\)
							0.324821	+	0.945775i	\(0.394696\pi\)
\(37\)	−1060.92	−	865.220i	−0.774961	−	0.632009i
							\(41\)	2.92422	−	8.03423i	0.00173957	−	0.00477943i	−0.938820	−	0.344409i	\(-0.888079\pi\)
0.940559	+	0.339629i	\(0.110302\pi\)
\(43\)	−947.209	−	947.209i	−0.512282	−	0.512282i	0.402943	−	0.915225i	\(-0.367987\pi\)
							−0.915225	+	0.402943i	\(0.867987\pi\)
\(47\)	1593.00	+	2759.15i	0.721140	+	1.24905i	0.960543	+	0.278131i	\(0.0897149\pi\)
							−0.239403	+	0.970920i	\(0.576952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.i.b.13.4	✓	72
37.20	odd	36	inner	74.5.i.b.57.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.i.b.13.4	✓	72	1.1	even	1	trivial
74.5.i.b.57.4	yes	72	37.20	odd	36	inner