Embedded newform 147.4.o.a.101.39

• Label: 147.4.o.a.101.39
• 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			101.39
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.47431 + 0.971095i) q^{2} +(-4.78655 + 2.02212i) q^{3} +(-0.685237 - 0.635807i) q^{4} +(-1.93254 - 1.31759i) q^{5} +(-13.8071 + 0.355166i) q^{6} +(18.3735 - 2.32679i) q^{7} +(-10.3043 - 21.3972i) q^{8} +(18.8220 - 19.3580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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	0.759897	−	0.650043i	\(-0.225249\pi\)
							0.114903	+	0.993377i	\(0.463344\pi\)
\(3\)	−4.78655	+	2.02212i	−0.921171	+	0.389158i
							\(5\)	−1.93254	−	1.31759i	−0.172852	−	0.117848i	0.473804	−	0.880630i	\(-0.342881\pi\)
−0.646656	+	0.762782i	\(0.723833\pi\)
\(7\)	18.3735	−	2.32679i	0.992077	−	0.125635i
							\(11\)	−1.51266	−	10.0358i	−0.0414621	−	0.275083i	0.958487	−	0.285135i	\(-0.0920384\pi\)
−0.999949	+	0.0100520i	\(0.996800\pi\)
\(13\)	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	0.634959	−	0.772546i	\(-0.281017\pi\)
							0.0894374	+	0.995992i	\(0.471493\pi\)
\(19\)	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	\(-0.961292\pi\)
							0.751804	−	0.659387i	\(-0.229184\pi\)
\(29\)	−12.1045	+	2.76276i	−0.0775083	+	0.0176908i	−0.261099	−	0.965312i	\(-0.584085\pi\)
							0.183591	+	0.983003i	\(0.441228\pi\)
\(31\)	−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	−0.890026	−	0.455909i	\(-0.849314\pi\)
							−0.198479	+	0.980105i	\(0.563600\pi\)
\(43\)	−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	−0.915222	−	0.402950i	\(-0.867985\pi\)
							0.944981	+	0.327125i	\(0.106080\pi\)

(See \(a_n\) instead)

Twists

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

    

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