Embedded newform 147.2.k.a.104.3

• Label: 147.2.k.a.104.3
• Level: $147$
• Weight: $2$
• Character: 147.104
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.k (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			104.3
Character	\(\chi\)	\(=\)	147.104
Dual form			147.2.k.a.41.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77201 - 1.41313i) q^{2} +(-1.67647 - 0.435258i) q^{3} +(0.698044 + 3.05833i) q^{4} +(3.64925 + 1.75739i) q^{5} +(2.35565 + 3.14036i) q^{6} +(0.439442 - 2.60900i) q^{7} +(1.11810 - 2.32176i) q^{8} +(2.62110 + 1.45939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77201	−	1.41313i	−1.25300	−	0.999237i	−0.999491	−	0.0319043i	\(-0.989843\pi\)
							−0.253512	−	0.967332i	\(-0.581586\pi\)
\(3\)	−1.67647	−	0.435258i	−0.967910	−	0.251296i
							\(5\)	3.64925	+	1.75739i	1.63199	+	0.785927i	0.999939	+	0.0110027i	\(0.00350232\pi\)
0.632054	+	0.774924i	\(0.282212\pi\)
\(7\)	0.439442	−	2.60900i	0.166094	−	0.986110i
							\(11\)	−0.206062	−	0.164329i	−0.0621300	−	0.0495470i	0.591932	−	0.805988i	\(-0.298365\pi\)
−0.654062	+	0.756441i	\(0.726937\pi\)
\(13\)	−0.541512	−	0.431842i	−0.150188	−	0.119771i	0.545513	−	0.838102i	\(-0.316335\pi\)
							−0.695701	+	0.718331i	\(0.744906\pi\)
\(17\)	0.756197	−	3.31312i	0.183405	−	0.803549i	−0.796589	−	0.604521i	\(-0.793365\pi\)
							0.979994	−	0.199028i	\(-0.0637783\pi\)
\(19\)		−	2.65532i		−	0.609172i	−0.952485	−	0.304586i	\(-0.901482\pi\)
							0.952485	−	0.304586i	\(-0.0985181\pi\)
\(23\)	7.19179	−	1.64148i	1.49959	−	0.342272i	0.607571	−	0.794265i	\(-0.292144\pi\)
							0.892022	+	0.451993i	\(0.149287\pi\)
\(29\)	−0.664450	−	0.151656i	−0.123385	−	0.0281619i	0.160382	−	0.987055i	\(-0.448727\pi\)
							−0.283768	+	0.958893i	\(0.591584\pi\)
\(31\)			4.03886i			0.725401i	0.931906	+	0.362701i	\(0.118145\pi\)
							−0.931906	+	0.362701i	\(0.881855\pi\)
\(37\)	1.08700	−	4.76246i	0.178702	−	0.782943i	−0.803529	−	0.595265i	\(-0.797047\pi\)
							0.982231	−	0.187677i	\(-0.0600960\pi\)
\(41\)	4.22692	+	2.03558i	0.660133	+	0.317904i	0.733786	−	0.679380i	\(-0.237751\pi\)
							−0.0736528	+	0.997284i	\(0.523466\pi\)
\(43\)	−1.86843	+	0.899788i	−0.284933	+	0.137216i	−0.570891	−	0.821026i	\(-0.693402\pi\)
							0.285959	+	0.958242i	\(0.407688\pi\)
\(47\)	−4.97816	+	6.24242i	−0.726139	+	0.910550i	−0.998668	−	0.0515978i	\(-0.983569\pi\)
							0.272529	+	0.962148i	\(0.412140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.104.3	yes	96
3.2	odd	2	inner	147.2.k.a.104.14	yes	96
49.41	odd	14	inner	147.2.k.a.41.14	yes	96
147.41	even	14	inner	147.2.k.a.41.3	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.41.3	✓	96	147.41	even	14	inner
147.2.k.a.41.14	yes	96	49.41	odd	14	inner
147.2.k.a.104.3	yes	96	1.1	even	1	trivial
147.2.k.a.104.14	yes	96	3.2	odd	2	inner