Embedded newform 65.3.p.a.11.5

• Label: 65.3.p.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0928988 - 0.346703i) q^{2} +(-0.295143 - 0.511202i) q^{3} +(3.35253 + 1.93558i) q^{4} +(1.58114 + 1.58114i) q^{5} +(-0.204654 + 0.0548368i) q^{6} +(-0.0877254 - 0.327396i) q^{7} +(1.99774 - 1.99774i) q^{8} +(4.32578 - 7.49247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0928988	−	0.346703i	0.0464494	−	0.173352i	−0.938804	−	0.344451i	\(-0.888065\pi\)
							0.985254	+	0.171099i	\(0.0547318\pi\)
\(3\)	−0.295143	−	0.511202i	−0.0983809	−	0.170401i	0.812634	−	0.582775i	\(-0.198033\pi\)
							−0.911015	+	0.412374i	\(0.864700\pi\)
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−0.0877254	−	0.327396i	−0.0125322	−	0.0467708i	0.959377	−	0.282129i	\(-0.0910405\pi\)
−0.971909	+	0.235358i	\(0.924374\pi\)
\(11\)	6.41839	+	1.71980i	0.583490	+	0.156346i	0.538476	−	0.842641i	\(-0.319000\pi\)
							0.0450138	+	0.998986i	\(0.485667\pi\)
\(13\)	−9.96434	+	8.34936i	−0.766488	+	0.642259i
							\(17\)	−25.1734	−	14.5339i	−1.48079	−	0.854933i	−0.481024	−	0.876707i	\(-0.659735\pi\)
−0.999763	+	0.0217742i	\(0.993069\pi\)
\(19\)	−27.1027	+	7.26215i	−1.42646	+	0.382218i	−0.887770	−	0.460287i	\(-0.847746\pi\)
							−0.538688	+	0.842505i	\(0.681080\pi\)
\(23\)	10.5817	−	6.10936i	0.460075	−	0.265624i	−0.252001	−	0.967727i	\(-0.581089\pi\)
							0.712076	+	0.702103i	\(0.247755\pi\)
\(29\)	−19.0696	−	33.0295i	−0.657572	−	1.13895i	−0.981242	−	0.192778i	\(-0.938250\pi\)
							0.323670	−	0.946170i	\(-0.395083\pi\)
\(31\)	27.5966	+	27.5966i	0.890211	+	0.890211i	0.994543	−	0.104331i	\(-0.0332702\pi\)
							−0.104331	+	0.994543i	\(0.533270\pi\)
\(37\)	−26.6159	−	7.13170i	−0.719348	−	0.192749i	−0.119467	−	0.992838i	\(-0.538119\pi\)
							−0.599881	+	0.800090i	\(0.704785\pi\)
\(41\)	−8.23262	+	30.7246i	−0.200796	+	0.749380i	0.789894	+	0.613243i	\(0.210135\pi\)
							−0.990690	+	0.136137i	\(0.956531\pi\)
\(43\)	−0.391444	−	0.226000i	−0.00910334	−	0.00525582i	0.495441	−	0.868641i	\(-0.335006\pi\)
							−0.504545	+	0.863386i	\(0.668340\pi\)
\(47\)	45.8281	−	45.8281i	0.975066	−	0.975066i	−0.0246302	−	0.999697i	\(-0.507841\pi\)
							0.999697	+	0.0246302i	\(0.00784083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.5	yes	40
5.2	odd	4		325.3.w.f.24.5		40
5.3	odd	4		325.3.w.e.24.6		40
5.4	even	2		325.3.t.d.76.6		40
13.6	odd	12	inner	65.3.p.a.6.5	✓	40
65.19	odd	12		325.3.t.d.201.6		40
65.32	even	12		325.3.w.e.149.6		40
65.58	even	12		325.3.w.f.149.5		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.5	✓	40	13.6	odd	12	inner
65.3.p.a.11.5	yes	40	1.1	even	1	trivial
325.3.t.d.76.6		40	5.4	even	2
325.3.t.d.201.6		40	65.19	odd	12
325.3.w.e.24.6		40	5.3	odd	4
325.3.w.e.149.6		40	65.32	even	12
325.3.w.f.24.5		40	5.2	odd	4
325.3.w.f.149.5		40	65.58	even	12