Embedded newform 147.4.o.a.101.2

• Label: 147.4.o.a.101.2
• Level: $147$
• Weight: $4$
• Character: 147.101
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.2
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.97448 - 1.95234i) q^{2} +(3.58231 + 3.76391i) q^{3} +(15.0694 + 13.9824i) q^{4} +(2.11564 + 1.44242i) q^{5} +(-10.4717 - 25.7174i) q^{6} +(-9.97572 - 15.6040i) q^{7} +(-29.1151 - 60.4581i) q^{8} +(-1.33405 + 26.9670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.97448	−	1.95234i	−1.75874	−	0.690256i	−0.999208	−	0.0397903i	\(-0.987331\pi\)
							−0.759536	−	0.650466i	\(-0.774574\pi\)
\(3\)	3.58231	+	3.76391i	0.689417	+	0.724365i
							\(5\)	2.11564	+	1.44242i	0.189229	+	0.129014i	0.654229	−	0.756297i	\(-0.272993\pi\)
−0.465000	+	0.885311i	\(0.653946\pi\)
\(7\)	−9.97572	−	15.6040i	−0.538638	−	0.842537i
							\(11\)	−8.12131	−	53.8814i	−0.222606	−	1.47690i	−0.768936	−	0.639325i	\(-0.779214\pi\)
0.546330	−	0.837570i	\(-0.316024\pi\)
\(13\)	15.0967	−	12.0392i	0.322082	−	0.256852i	−0.449079	−	0.893492i	\(-0.648248\pi\)
							0.771161	+	0.636640i	\(0.219676\pi\)
\(17\)	110.570	+	34.1063i	1.57748	+	0.486588i	0.955139	−	0.296157i	\(-0.0957052\pi\)
							0.622342	+	0.782746i	\(0.286181\pi\)
\(19\)	−11.5307	+	6.65727i	−0.139228	+	0.0803833i	−0.567996	−	0.823031i	\(-0.692281\pi\)
							0.428768	+	0.903415i	\(0.358948\pi\)
\(23\)	−20.5613	−	66.6580i	−0.186405	−	0.604311i	−0.999746	−	0.0225238i	\(-0.992830\pi\)
							0.813341	−	0.581787i	\(-0.197646\pi\)
\(29\)	215.079	−	49.0903i	1.37721	−	0.314339i	0.531083	−	0.847320i	\(-0.321785\pi\)
							0.846127	+	0.532981i	\(0.178928\pi\)
\(31\)	96.7364	+	55.8508i	0.560464	+	0.323584i	0.753332	−	0.657641i	\(-0.228446\pi\)
							−0.192868	+	0.981225i	\(0.561779\pi\)
\(37\)	192.863	−	178.951i	0.856932	−	0.795117i	−0.123424	−	0.992354i	\(-0.539387\pi\)
							0.980356	+	0.197237i	\(0.0631970\pi\)
\(41\)	−54.3502	+	26.1737i	−0.207026	+	0.0996986i	−0.534524	−	0.845153i	\(-0.679509\pi\)
							0.327497	+	0.944852i	\(0.393795\pi\)
\(43\)	136.054	+	65.5200i	0.482512	+	0.232365i	0.659295	−	0.751884i	\(-0.270855\pi\)
							−0.176783	+	0.984250i	\(0.556569\pi\)
\(47\)	−14.6391	+	37.2998i	−0.0454325	+	0.115760i	−0.951789	−	0.306754i	\(-0.900757\pi\)
							0.906356	+	0.422514i	\(0.138852\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.2	✓	648
3.2	odd	2	inner	147.4.o.a.101.53	yes	648
49.33	odd	42	inner	147.4.o.a.131.53	yes	648
147.131	even	42	inner	147.4.o.a.131.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.2	✓	648	1.1	even	1	trivial
147.4.o.a.101.53	yes	648	3.2	odd	2	inner
147.4.o.a.131.2	yes	648	147.131	even	42	inner
147.4.o.a.131.53	yes	648	49.33	odd	42	inner