Embedded newform 147.4.o.a.101.16

• Label: 147.4.o.a.101.16
• 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			101.16
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.47431 - 0.971095i) q^{2} +(-4.43170 - 2.71293i) q^{3} +(-0.685237 - 0.635807i) q^{4} +(1.93254 + 1.31759i) q^{5} +(8.33088 + 11.0162i) q^{6} +(18.3735 - 2.32679i) q^{7} +(10.3043 + 21.3972i) q^{8} +(12.2800 + 24.0458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.47431	−	0.971095i	−0.874800	−	0.343334i	−0.114903	−	0.993377i	\(-0.536656\pi\)
							−0.759897	+	0.650043i	\(0.774751\pi\)
\(3\)	−4.43170	−	2.71293i	−0.852882	−	0.522104i
							\(5\)	1.93254	+	1.31759i	0.172852	+	0.117848i	0.646656	−	0.762782i	\(-0.276167\pi\)
−0.473804	+	0.880630i	\(0.657119\pi\)
\(7\)	18.3735	−	2.32679i	0.992077	−	0.125635i
							\(11\)	1.51266	+	10.0358i	0.0414621	+	0.275083i	0.999949	−	0.0100520i	\(-0.00319969\pi\)
−0.958487	+	0.285135i	\(0.907962\pi\)
\(13\)	20.5332	−	16.3747i	0.438069	−	0.349348i	−0.379488	−	0.925197i	\(-0.623900\pi\)
							0.817556	+	0.575849i	\(0.195328\pi\)
\(17\)	−50.7750	−	15.6620i	−0.724397	−	0.223447i	−0.0894374	−	0.995992i	\(-0.528507\pi\)
							−0.634959	+	0.772546i	\(0.718983\pi\)
\(19\)	92.1390	−	53.1965i	1.11253	−	0.642321i	0.173049	−	0.984913i	\(-0.444638\pi\)
							0.939484	+	0.342592i	\(0.111305\pi\)
\(23\)	26.5625	+	86.1135i	0.240811	+	0.780692i	0.992615	+	0.121305i	\(0.0387078\pi\)
							−0.751804	+	0.659387i	\(0.770816\pi\)
\(29\)	12.1045	−	2.76276i	0.0775083	−	0.0176908i	−0.183591	−	0.983003i	\(-0.558772\pi\)
							0.261099	+	0.965312i	\(0.415915\pi\)
\(31\)	−36.5717	−	21.1147i	−0.211886	−	0.122332i	0.390302	−	0.920687i	\(-0.372371\pi\)
							−0.602188	+	0.798355i	\(0.705704\pi\)
\(37\)	50.7939	−	47.1298i	0.225688	−	0.209408i	−0.559221	−	0.829018i	\(-0.688900\pi\)
							0.784910	+	0.619610i	\(0.212709\pi\)
\(41\)	285.763	−	137.616i	1.08850	−	0.524196i	0.198479	−	0.980105i	\(-0.436400\pi\)
							0.890026	+	0.455909i	\(0.150686\pi\)
\(43\)	−218.883	−	105.408i	−0.776262	−	0.373828i	0.00342768	−	0.999994i	\(-0.498909\pi\)
							−0.779690	+	0.626166i	\(0.784623\pi\)
\(47\)	−9.58887	+	24.4320i	−0.0297591	+	0.0758251i	−0.944981	−	0.327125i	\(-0.893920\pi\)
							0.915222	+	0.402950i	\(0.132015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.16	✓	648
3.2	odd	2	inner	147.4.o.a.101.39	yes	648
49.33	odd	42	inner	147.4.o.a.131.39	yes	648
147.131	even	42	inner	147.4.o.a.131.16	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.16	✓	648	1.1	even	1	trivial
147.4.o.a.101.39	yes	648	3.2	odd	2	inner
147.4.o.a.131.16	yes	648	147.131	even	42	inner
147.4.o.a.131.39	yes	648	49.33	odd	42	inner