Embedded newform 65.3.p.a.11.7

• Label: 65.3.p.a.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.312366 - 1.16577i) q^{2} +(1.19180 + 2.06426i) q^{3} +(2.20266 + 1.27171i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(2.77872 - 0.744556i) q^{6} +(2.00836 + 7.49531i) q^{7} +(5.58415 - 5.58415i) q^{8} +(1.65922 - 2.87385i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.312366	−	1.16577i	0.156183	−	0.582883i	−0.842818	−	0.538198i	\(-0.819105\pi\)
							0.999001	−	0.0446844i	\(-0.0142282\pi\)
\(3\)	1.19180	+	2.06426i	0.397267	+	0.688087i	0.993388	−	0.114808i	\(-0.0366253\pi\)
							−0.596120	+	0.802895i	\(0.703292\pi\)
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	2.00836	+	7.49531i	0.286909	+	1.07076i	0.947433	+	0.319954i	\(0.103667\pi\)
−0.660524	+	0.750805i	\(0.729666\pi\)
\(11\)	−16.8511	−	4.51524i	−1.53192	−	0.410477i	−0.608275	−	0.793726i	\(-0.708138\pi\)
							−0.923645	+	0.383250i	\(0.874805\pi\)
\(13\)	9.78343	−	8.56064i	0.752571	−	0.658511i
							\(17\)	−6.48447	−	3.74381i	−0.381439	−	0.220224i	0.297005	−	0.954876i	\(-0.404012\pi\)
−0.678444	+	0.734652i	\(0.737346\pi\)
\(19\)	−22.8831	+	6.13151i	−1.20437	+	0.322711i	−0.804552	−	0.593882i	\(-0.797595\pi\)
							−0.399822	+	0.916593i	\(0.630928\pi\)
\(23\)	−20.4308	+	11.7957i	−0.888295	+	0.512857i	−0.873384	−	0.487031i	\(-0.838080\pi\)
							−0.0149106	+	0.999889i	\(0.504746\pi\)
\(29\)	9.01400	+	15.6127i	0.310828	+	0.538369i	0.978542	−	0.206048i	\(-0.0660604\pi\)
							−0.667714	+	0.744418i	\(0.732727\pi\)
\(31\)	−39.1605	−	39.1605i	−1.26324	−	1.26324i	−0.949512	−	0.313730i	\(-0.898421\pi\)
							−0.313730	−	0.949512i	\(-0.601579\pi\)
\(37\)	32.5768	+	8.72892i	0.880453	+	0.235917i	0.670603	−	0.741817i	\(-0.266036\pi\)
							0.209851	+	0.977733i	\(0.432702\pi\)
\(41\)	12.9857	−	48.4634i	0.316725	−	1.18203i	−0.605648	−	0.795733i	\(-0.707086\pi\)
							0.922373	−	0.386301i	\(-0.126247\pi\)
\(43\)	39.3952	+	22.7448i	0.916167	+	0.528949i	0.882410	−	0.470481i	\(-0.155920\pi\)
							0.0337567	+	0.999430i	\(0.489253\pi\)
\(47\)	−2.92652	+	2.92652i	−0.0622664	+	0.0622664i	−0.737554	−	0.675288i	\(-0.764019\pi\)
							0.675288	+	0.737554i	\(0.264019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.7	yes	40
5.2	odd	4		325.3.w.f.24.7		40
5.3	odd	4		325.3.w.e.24.4		40
5.4	even	2		325.3.t.d.76.4		40
13.6	odd	12	inner	65.3.p.a.6.7	✓	40
65.19	odd	12		325.3.t.d.201.4		40
65.32	even	12		325.3.w.e.149.4		40
65.58	even	12		325.3.w.f.149.7		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.7	✓	40	13.6	odd	12	inner
65.3.p.a.11.7	yes	40	1.1	even	1	trivial
325.3.t.d.76.4		40	5.4	even	2
325.3.t.d.201.4		40	65.19	odd	12
325.3.w.e.24.4		40	5.3	odd	4
325.3.w.e.149.4		40	65.32	even	12
325.3.w.f.24.7		40	5.2	odd	4
325.3.w.f.149.7		40	65.58	even	12