Embedded newform 65.3.p.a.11.9

• Label: 65.3.p.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.729421 - 2.72224i) q^{2} +(2.39711 + 4.15192i) q^{3} +(-3.41442 - 1.97132i) q^{4} +(1.58114 + 1.58114i) q^{5} +(13.0510 - 3.49701i) q^{6} +(-1.27402 - 4.75472i) q^{7} +(0.114321 - 0.114321i) q^{8} +(-6.99232 + 12.1111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.729421	−	2.72224i	0.364711	−	1.36112i	−0.503102	−	0.864227i	\(-0.667808\pi\)
							0.867813	−	0.496892i	\(-0.165525\pi\)
\(3\)	2.39711	+	4.15192i	0.799038	+	1.38397i	0.920243	+	0.391347i	\(0.127991\pi\)
							−0.121205	+	0.992628i	\(0.538676\pi\)
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−1.27402	−	4.75472i	−0.182003	−	0.679245i	−0.995252	−	0.0973293i	\(-0.968970\pi\)
0.813249	−	0.581916i	\(-0.197697\pi\)
\(11\)	−10.3054	−	2.76133i	−0.936858	−	0.251030i	−0.242082	−	0.970256i	\(-0.577830\pi\)
							−0.694777	+	0.719226i	\(0.744497\pi\)
\(13\)	−12.9147	+	1.48643i	−0.993442	+	0.114340i
							\(17\)	5.66061	+	3.26816i	0.332977	+	0.192244i	0.657162	−	0.753749i	\(-0.271757\pi\)
−0.324185	+	0.945994i	\(0.605090\pi\)
\(19\)	22.9152	−	6.14012i	1.20607	−	0.323164i	0.400848	−	0.916144i	\(-0.368715\pi\)
							0.805217	+	0.592980i	\(0.202049\pi\)
\(23\)	−29.1620	+	16.8367i	−1.26791	+	0.732030i	−0.974593	−	0.223984i	\(-0.928094\pi\)
							−0.293320	+	0.956014i	\(0.594760\pi\)
\(29\)	−9.74539	−	16.8795i	−0.336048	−	0.582052i	0.647638	−	0.761949i	\(-0.275757\pi\)
							−0.983686	+	0.179896i	\(0.942424\pi\)
\(31\)	37.3590	+	37.3590i	1.20513	+	1.20513i	0.972586	+	0.232542i	\(0.0747044\pi\)
							0.232542	+	0.972586i	\(0.425296\pi\)
\(37\)	−8.92172	−	2.39057i	−0.241127	−	0.0646099i	0.136231	−	0.990677i	\(-0.456501\pi\)
							−0.377359	+	0.926067i	\(0.623168\pi\)
\(41\)	19.1574	−	71.4963i	0.467253	−	1.74381i	−0.182057	−	0.983288i	\(-0.558276\pi\)
							0.649310	−	0.760524i	\(-0.275058\pi\)
\(43\)	25.3415	+	14.6309i	0.589337	+	0.340254i	0.764835	−	0.644226i	\(-0.222820\pi\)
							−0.175499	+	0.984480i	\(0.556154\pi\)
\(47\)	−56.5604	+	56.5604i	−1.20341	+	1.20341i	−0.230290	+	0.973122i	\(0.573968\pi\)
							−0.973122	+	0.230290i	\(0.926032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.9	yes	40
5.2	odd	4		325.3.w.f.24.9		40
5.3	odd	4		325.3.w.e.24.2		40
5.4	even	2		325.3.t.d.76.2		40
13.6	odd	12	inner	65.3.p.a.6.9	✓	40
65.19	odd	12		325.3.t.d.201.2		40
65.32	even	12		325.3.w.e.149.2		40
65.58	even	12		325.3.w.f.149.9		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.9	✓	40	13.6	odd	12	inner
65.3.p.a.11.9	yes	40	1.1	even	1	trivial
325.3.t.d.76.2		40	5.4	even	2
325.3.t.d.201.2		40	65.19	odd	12
325.3.w.e.24.2		40	5.3	odd	4
325.3.w.e.149.2		40	65.32	even	12
325.3.w.f.24.9		40	5.2	odd	4
325.3.w.f.149.9		40	65.58	even	12