Embedded newform 65.3.p.a.11.3

• Label: 65.3.p.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	65.11
Dual form			65.3.p.a.6.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.590063 + 2.20214i) q^{2} +(1.05817 + 1.83281i) q^{3} +(-1.03716 - 0.598805i) q^{4} +(1.58114 + 1.58114i) q^{5} +(-4.66050 + 1.24878i) q^{6} +(-0.314408 - 1.17338i) q^{7} +(-4.51768 + 4.51768i) q^{8} +(2.26053 - 3.91536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.590063	+	2.20214i	−0.295031	+	1.10107i	0.646161	+	0.763201i	\(0.276373\pi\)
							−0.941193	+	0.337871i	\(0.890293\pi\)
\(3\)	1.05817	+	1.83281i	0.352725	+	0.610937i	0.986726	−	0.162395i	\(-0.0519218\pi\)
							−0.634001	+	0.773332i	\(0.718589\pi\)
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−0.314408	−	1.17338i	−0.0449154	−	0.167626i	0.939825	−	0.341656i	\(-0.110988\pi\)
−0.984740	+	0.174030i	\(0.944321\pi\)
\(11\)	−13.3250	−	3.57041i	−1.21136	−	0.324583i	−0.404065	−	0.914730i	\(-0.632403\pi\)
							−0.807296	+	0.590147i	\(0.799070\pi\)
\(13\)	12.5229	+	3.48944i	0.963302	+	0.268419i
							\(17\)	8.20286	+	4.73592i	0.482521	+	0.278584i	0.721467	−	0.692449i	\(-0.243468\pi\)
−0.238945	+	0.971033i	\(0.576802\pi\)
\(19\)	23.4493	−	6.28321i	1.23417	−	0.330695i	0.417969	−	0.908461i	\(-0.362742\pi\)
							0.816203	+	0.577766i	\(0.196075\pi\)
\(23\)	10.0605	−	5.80846i	0.437415	−	0.252542i	−0.265085	−	0.964225i	\(-0.585400\pi\)
							0.702501	+	0.711683i	\(0.252067\pi\)
\(29\)	−2.30788	−	3.99736i	−0.0795819	−	0.137840i	0.823488	−	0.567334i	\(-0.192025\pi\)
							−0.903070	+	0.429494i	\(0.858692\pi\)
\(31\)	−21.8447	−	21.8447i	−0.704669	−	0.704669i	0.260740	−	0.965409i	\(-0.416033\pi\)
							−0.965409	+	0.260740i	\(0.916033\pi\)
\(37\)	−19.1950	−	5.14329i	−0.518785	−	0.139008i	−0.0100804	−	0.999949i	\(-0.503209\pi\)
							−0.508704	+	0.860941i	\(0.669875\pi\)
\(41\)	−14.6840	+	54.8015i	−0.358147	+	1.33662i	0.518331	+	0.855180i	\(0.326553\pi\)
							−0.876478	+	0.481442i	\(0.840113\pi\)
\(43\)	−61.4835	−	35.4975i	−1.42985	−	0.825523i	−0.432740	−	0.901519i	\(-0.642453\pi\)
							−0.997108	+	0.0759958i	\(0.975786\pi\)
\(47\)	−22.8093	+	22.8093i	−0.485303	+	0.485303i	−0.906820	−	0.421517i	\(-0.861498\pi\)
							0.421517	+	0.906820i	\(0.361498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.3.p.a.11.3	yes	40
5.2	odd	4		325.3.w.f.24.3		40
5.3	odd	4		325.3.w.e.24.8		40
5.4	even	2		325.3.t.d.76.8		40
13.6	odd	12	inner	65.3.p.a.6.3	✓	40
65.19	odd	12		325.3.t.d.201.8		40
65.32	even	12		325.3.w.e.149.8		40
65.58	even	12		325.3.w.f.149.3		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.3.p.a.6.3	✓	40	13.6	odd	12	inner
65.3.p.a.11.3	yes	40	1.1	even	1	trivial
325.3.t.d.76.8		40	5.4	even	2
325.3.t.d.201.8		40	65.19	odd	12
325.3.w.e.24.8		40	5.3	odd	4
325.3.w.e.149.8		40	65.32	even	12
325.3.w.f.24.3		40	5.2	odd	4
325.3.w.f.149.3		40	65.58	even	12