Embedded newform 154.3.g.a.65.8

• Label: 154.3.g.a.65.8
• Level: $154$
• Weight: $3$
• Character: 154.65
• Analytic conductor: $4.196$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 154 = 2 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	154.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19619607115\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.8
Character	\(\chi\)	\(=\)	154.65
Dual form			154.3.g.a.109.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(2.16323 + 3.74682i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.09885 + 3.63531i) q^{5} -6.11853i q^{6} +(-4.12992 + 5.65188i) q^{7} -2.82843i q^{8} +(-4.85911 + 8.41622i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} + 4 q^{5} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	−	0.707107i	−0.612372	−	0.353553i
							\(3\)	2.16323	+	3.74682i	0.721076	+	1.24894i	0.960569	+	0.278042i	\(0.0896854\pi\)
−0.239493	+	0.970898i	\(0.576981\pi\)
\(5\)	−2.09885	+	3.63531i	−0.419770	+	0.727063i	−0.995916	−	0.0902838i	\(-0.971223\pi\)
							0.576146	+	0.817347i	\(0.304556\pi\)
\(7\)	−4.12992	+	5.65188i	−0.589988	+	0.807412i
							\(11\)	−7.78313	−	7.77322i	−0.707557	−	0.706656i
\(13\)		−	7.57538i		−	0.582722i								−0.956613	−	0.291361i	\(-0.905892\pi\)
							0.956613	−	0.291361i	\(-0.0941080\pi\)
\(17\)	−7.46509	+	4.30997i	−0.439123	+	0.253528i	−0.703226	−	0.710967i	\(-0.748258\pi\)
							0.264103	+	0.964495i	\(0.414924\pi\)
\(19\)	8.87506	+	5.12402i	0.467108	+	0.269685i	0.715028	−	0.699095i	\(-0.246414\pi\)
							−0.247920	+	0.968780i	\(0.579747\pi\)
\(23\)	2.81061	−	4.86812i	0.122201	−	0.211658i	−0.798435	−	0.602082i	\(-0.794338\pi\)
							0.920635	+	0.390424i	\(0.127672\pi\)
\(29\)			54.2152i			1.86949i	0.355321	+	0.934744i	\(0.384371\pi\)
							−0.355321	+	0.934744i	\(0.615629\pi\)
\(31\)	25.8359	+	44.7492i	0.833417	+	1.44352i	0.895313	+	0.445438i	\(0.146952\pi\)
							−0.0618953	+	0.998083i	\(0.519714\pi\)
\(37\)	−17.7195	+	30.6911i	−0.478906	+	0.829489i	−0.999707	−	0.0241885i	\(-0.992300\pi\)
							0.520802	+	0.853678i	\(0.325633\pi\)
\(41\)		−	13.0646i		−	0.318650i	−0.987226	−	0.159325i	\(-0.949068\pi\)
							0.987226	−	0.159325i	\(-0.0509317\pi\)
\(43\)		−	2.86553i		−	0.0666402i	−0.999445	−	0.0333201i	\(-0.989392\pi\)
							0.999445	−	0.0333201i	\(-0.0106081\pi\)
\(47\)	42.6190	−	73.8183i	0.906787	−	1.57060i	0.0882872	−	0.996095i	\(-0.471861\pi\)
							0.818500	−	0.574506i	\(-0.194806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	154.3.g.a.65.8	✓	32
7.2	even	3		1078.3.d.d.197.9		16
7.4	even	3	inner	154.3.g.a.109.16	yes	32
7.5	odd	6		1078.3.d.e.197.16		16
11.10	odd	2	inner	154.3.g.a.65.16	yes	32
77.32	odd	6	inner	154.3.g.a.109.8	yes	32
77.54	even	6		1078.3.d.e.197.8		16
77.65	odd	6		1078.3.d.d.197.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.3.g.a.65.8	✓	32	1.1	even	1	trivial
154.3.g.a.65.16	yes	32	11.10	odd	2	inner
154.3.g.a.109.8	yes	32	77.32	odd	6	inner
154.3.g.a.109.16	yes	32	7.4	even	3	inner
1078.3.d.d.197.1		16	77.65	odd	6
1078.3.d.d.197.9		16	7.2	even	3
1078.3.d.e.197.8		16	77.54	even	6
1078.3.d.e.197.16		16	7.5	odd	6