Embedded newform 65.3.p.a.11.6

• Label: 65.3.p.a.11.6
• Level: $65$
• Weight: $3$
• Character: 65.11
• Analytic conductor: $1.771$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	65.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.77112171834\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.107580 - 0.401493i) q^{2} +(-2.08519 - 3.61165i) q^{3} +(3.31448 + 1.91361i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(-1.67438 + 0.448649i) q^{6} +(-2.68917 - 10.0361i) q^{7} +(2.30053 - 2.30053i) q^{8} +(-4.19603 + 7.26773i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.107580	−	0.401493i	0.0537899	−	0.200747i	−0.933802	−	0.357791i	\(-0.883530\pi\)
							0.987592	+	0.157045i	\(0.0501966\pi\)
\(3\)	−2.08519	−	3.61165i	−0.695063	−	1.20388i	−0.970159	−	0.242468i	\(-0.922043\pi\)
							0.275096	−	0.961417i	\(-0.411290\pi\)
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	−2.68917	−	10.0361i	−0.384168	−	1.43373i	−0.839475	−	0.543399i	\(-0.817137\pi\)
0.455307	−	0.890335i	\(-0.349530\pi\)
\(11\)	−4.73818	−	1.26959i	−0.430743	−	0.115417i	0.0369317	−	0.999318i	\(-0.488242\pi\)
							−0.467675	+	0.883900i	\(0.654908\pi\)
\(13\)	7.53326	+	10.5948i	0.579481	+	0.814986i
							\(17\)	19.1458	+	11.0538i	1.12622	+	0.650226i	0.942983	−	0.332842i	\(-0.108007\pi\)
0.183242	+	0.983068i	\(0.441341\pi\)
\(19\)	33.5167	−	8.98078i	1.76404	−	0.472672i	0.776508	−	0.630107i	\(-0.216989\pi\)
							0.987529	+	0.157435i	\(0.0503223\pi\)
\(23\)	−27.6598	+	15.9694i	−1.20260	+	0.694321i	−0.961132	−	0.276088i	\(-0.910962\pi\)
							−0.241467	+	0.970409i	\(0.577628\pi\)
\(29\)	−13.5241	−	23.4245i	−0.466350	−	0.807742i	0.532911	−	0.846171i	\(-0.321098\pi\)
							−0.999261	+	0.0384292i	\(0.987765\pi\)
\(31\)	9.08360	+	9.08360i	0.293019	+	0.293019i	0.838272	−	0.545253i	\(-0.183566\pi\)
							−0.545253	+	0.838272i	\(0.683566\pi\)
\(37\)	14.3796	+	3.85300i	0.388638	+	0.104135i	0.447846	−	0.894110i	\(-0.352191\pi\)
							−0.0592089	+	0.998246i	\(0.518858\pi\)
\(41\)	−3.69614	+	13.7942i	−0.0901497	+	0.336443i	−0.996240	−	0.0866421i	\(-0.972386\pi\)
							0.906090	+	0.423085i	\(0.139053\pi\)
\(43\)	−5.13914	−	2.96708i	−0.119515	−	0.0690019i	0.439051	−	0.898462i	\(-0.355315\pi\)
							−0.558566	+	0.829460i	\(0.688648\pi\)
\(47\)	33.5097	−	33.5097i	0.712973	−	0.712973i	−0.254183	−	0.967156i	\(-0.581807\pi\)
							0.967156	+	0.254183i	\(0.0818067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.6	yes	40
5.2	odd	4		325.3.w.f.24.6		40
5.3	odd	4		325.3.w.e.24.5		40
5.4	even	2		325.3.t.d.76.5		40
13.6	odd	12	inner	65.3.p.a.6.6	✓	40
65.19	odd	12		325.3.t.d.201.5		40
65.32	even	12		325.3.w.e.149.5		40
65.58	even	12		325.3.w.f.149.6		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.6	✓	40	13.6	odd	12	inner
65.3.p.a.11.6	yes	40	1.1	even	1	trivial
325.3.t.d.76.5		40	5.4	even	2
325.3.t.d.201.5		40	65.19	odd	12
325.3.w.e.24.5		40	5.3	odd	4
325.3.w.e.149.5		40	65.32	even	12
325.3.w.f.24.6		40	5.2	odd	4
325.3.w.f.149.6		40	65.58	even	12