Embedded newform 65.3.p.a.11.10

• Label: 65.3.p.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.886220 - 3.30742i) q^{2} +(-0.0164891 - 0.0285599i) q^{3} +(-6.68953 - 3.86220i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(-0.109072 + 0.0292259i) q^{6} +(0.185607 + 0.692695i) q^{7} +(-9.01754 + 9.01754i) q^{8} +(4.49946 - 7.79329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.886220	−	3.30742i	0.443110	−	1.65371i	−0.277768	−	0.960648i	\(-0.589595\pi\)
							0.720878	−	0.693062i	\(-0.243739\pi\)
\(3\)	−0.0164891	−	0.0285599i	−0.00549635	−	0.00951996i	0.863264	−	0.504752i	\(-0.168416\pi\)
							−0.868760	+	0.495232i	\(0.835083\pi\)
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	0.185607	+	0.692695i	0.0265153	+	0.0989564i	0.977915	−	0.209001i	\(-0.0670212\pi\)
−0.951400	+	0.307957i	\(0.900355\pi\)
\(11\)	15.5985	+	4.17961i	1.41805	+	0.379965i	0.884791	−	0.465988i	\(-0.154301\pi\)
							0.533257	+	0.845953i	\(0.320968\pi\)
\(13\)	−2.16135	+	12.8191i	−0.166257	+	0.986082i
							\(17\)	−1.49033	−	0.860444i	−0.0876666	−	0.0506143i	0.455526	−	0.890223i	\(-0.349451\pi\)
−0.543192	+	0.839608i	\(0.682785\pi\)
\(19\)	6.58643	−	1.76483i	0.346654	−	0.0928857i	−0.0812910	−	0.996690i	\(-0.525904\pi\)
							0.427945	+	0.903805i	\(0.359238\pi\)
\(23\)	1.16760	−	0.674114i	0.0507652	−	0.0293093i	−0.474403	−	0.880308i	\(-0.657336\pi\)
							0.525168	+	0.850999i	\(0.324003\pi\)
\(29\)	23.7681	+	41.1675i	0.819588	+	1.41957i	0.905986	+	0.423308i	\(0.139131\pi\)
							−0.0863974	+	0.996261i	\(0.527535\pi\)
\(31\)	−21.6495	−	21.6495i	−0.698371	−	0.698371i	0.265688	−	0.964059i	\(-0.414401\pi\)
							−0.964059	+	0.265688i	\(0.914401\pi\)
\(37\)	−42.2011	−	11.3078i	−1.14057	−	0.305615i	−0.361390	−	0.932415i	\(-0.617698\pi\)
							−0.779180	+	0.626800i	\(0.784364\pi\)
\(41\)	−14.6912	+	54.8283i	−0.358322	+	1.33728i	0.517930	+	0.855423i	\(0.326703\pi\)
							−0.876252	+	0.481853i	\(0.839964\pi\)
\(43\)	22.7934	+	13.1597i	0.530078	+	0.306041i	0.741048	−	0.671452i	\(-0.234329\pi\)
							−0.210970	+	0.977492i	\(0.567662\pi\)
\(47\)	−57.0372	+	57.0372i	−1.21356	+	1.21356i	−0.243710	+	0.969848i	\(0.578364\pi\)
							−0.969848	+	0.243710i	\(0.921636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.10	yes	40
5.2	odd	4		325.3.w.f.24.10		40
5.3	odd	4		325.3.w.e.24.1		40
5.4	even	2		325.3.t.d.76.1		40
13.6	odd	12	inner	65.3.p.a.6.10	✓	40
65.19	odd	12		325.3.t.d.201.1		40
65.32	even	12		325.3.w.e.149.1		40
65.58	even	12		325.3.w.f.149.10		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.10	✓	40	13.6	odd	12	inner
65.3.p.a.11.10	yes	40	1.1	even	1	trivial
325.3.t.d.76.1		40	5.4	even	2
325.3.t.d.201.1		40	65.19	odd	12
325.3.w.e.24.1		40	5.3	odd	4
325.3.w.e.149.1		40	65.32	even	12
325.3.w.f.24.10		40	5.2	odd	4
325.3.w.f.149.10		40	65.58	even	12