Embedded newform 74.5.i.b.13.1

• Label: 74.5.i.b.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	74.13
Dual form			74.5.i.b.57.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.62232 + 2.31691i) q^{2} +(-16.5144 + 2.91194i) q^{3} +(-2.73616 + 7.51754i) q^{4} +(-18.8912 - 1.65276i) q^{5} +(-33.5383 - 33.5383i) q^{6} +(24.5278 - 20.5813i) q^{7} +(-21.8564 + 5.85641i) q^{8} +(188.131 - 68.4741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 114 q^{5} - 576 q^{8} - 210 q^{9}+O(q^{10}) \)

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
							\(3\)	−16.5144	+	2.91194i	−1.83493	+	0.323548i	−0.980576	−	0.196137i	\(-0.937160\pi\)
−0.854357	+	0.519686i	\(0.826049\pi\)
\(5\)	−18.8912	−	1.65276i	−0.755647	−	0.0661105i	−0.297179	−	0.954822i	\(-0.596046\pi\)
							−0.458467	+	0.888711i	\(0.651601\pi\)
\(7\)	24.5278	−	20.5813i	0.500568	−	0.420026i	−0.357228	−	0.934017i	\(-0.616278\pi\)
							0.857796	+	0.513991i	\(0.171834\pi\)
\(11\)	53.4651	+	30.8681i	0.441861	+	0.255108i	0.704387	−	0.709817i	\(-0.251222\pi\)
							−0.262526	+	0.964925i	\(0.584556\pi\)
\(13\)	32.3620	+	69.4006i	0.191491	+	0.410654i	0.978488	−	0.206302i	\(-0.0661431\pi\)
							−0.786997	+	0.616957i	\(0.788365\pi\)
\(17\)	−422.713	−	197.114i	−1.46267	−	0.682056i	−0.481725	−	0.876322i	\(-0.659990\pi\)
							−0.980949	+	0.194266i	\(0.937767\pi\)
\(19\)	366.067	−	522.798i	1.01404	−	1.44819i	0.124011	−	0.992281i	\(-0.460424\pi\)
							0.890025	−	0.455912i	\(-0.150687\pi\)
\(23\)	174.488	−	651.199i	0.329846	−	1.23100i	−0.579504	−	0.814969i	\(-0.696754\pi\)
							0.909350	−	0.416032i	\(-0.136579\pi\)
\(29\)	218.959	+	817.165i	0.260355	+	0.971659i	0.965032	+	0.262131i	\(0.0844252\pi\)
							−0.704677	+	0.709528i	\(0.748908\pi\)
\(31\)	314.434	−	314.434i	0.327195	−	0.327195i	−0.524324	−	0.851519i	\(-0.675682\pi\)
							0.851519	+	0.524324i	\(0.175682\pi\)
\(37\)	765.524	−	1134.96i	0.559185	−	0.829043i
							\(41\)	315.805	−	867.667i	0.187867	−	0.516161i	−0.809624	−	0.586949i	\(-0.800329\pi\)
0.997491	+	0.0707876i	\(0.0225513\pi\)
\(43\)	−151.807	−	151.807i	−0.0821020	−	0.0821020i	0.664863	−	0.746965i	\(-0.268490\pi\)
							−0.746965	+	0.664863i	\(0.768490\pi\)
\(47\)	−843.056	−	1460.22i	−0.381646	−	0.661030i	0.609652	−	0.792669i	\(-0.291309\pi\)
							−0.991298	+	0.131639i	\(0.957976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.i.b.13.1	✓	72
37.20	odd	36	inner	74.5.i.b.57.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.i.b.13.1	✓	72	1.1	even	1	trivial
74.5.i.b.57.1	yes	72	37.20	odd	36	inner