Embedded newform 147.4.o.a.131.42

• Label: 147.4.o.a.131.42
• Level: $147$
• Weight: $4$
• Character: 147.131
• 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			131.42
Character	\(\chi\)	\(=\)	147.131
Dual form			147.4.o.a.101.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.31144 - 1.29964i) q^{2} +(-2.61396 - 4.49079i) q^{3} +(3.41215 - 3.16601i) q^{4} +(12.8821 - 8.78287i) q^{5} +(-14.4924 - 11.4738i) q^{6} +(-5.96320 - 17.5340i) q^{7} +(-5.16336 + 10.7218i) q^{8} +(-13.3344 + 23.4775i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.31144	−	1.29964i	1.17077	−	0.459494i	0.301325	−	0.953521i	\(-0.402571\pi\)
							0.869446	+	0.494028i	\(0.164476\pi\)
\(3\)	−2.61396	−	4.49079i	−0.503058	−	0.864253i
							\(5\)	12.8821	−	8.78287i	1.15221	−	0.785563i	0.172308	−	0.985043i	\(-0.444878\pi\)
0.979902	+	0.199480i	\(0.0639252\pi\)
\(7\)	−5.96320	−	17.5340i	−0.321982	−	0.946746i
							\(11\)	3.82129	−	25.3526i	0.104742	−	0.694918i	−0.873403	−	0.486997i	\(-0.838092\pi\)
0.978146	−	0.207921i	\(-0.0666698\pi\)
\(13\)	−7.06195	−	5.63172i	−0.150664	−	0.120151i	0.545257	−	0.838269i	\(-0.316432\pi\)
							−0.695921	+	0.718118i	\(0.745004\pi\)
\(17\)	31.1727	−	9.61551i	0.444735	−	0.137183i	−0.0642957	−	0.997931i	\(-0.520480\pi\)
							0.509031	+	0.860748i	\(0.330004\pi\)
\(19\)	−15.4664	−	8.92954i	−0.186749	−	0.107820i	0.403711	−	0.914887i	\(-0.367720\pi\)
							−0.590460	+	0.807067i	\(0.701054\pi\)
\(23\)	24.5067	−	79.4488i	0.222174	−	0.720270i	−0.774024	−	0.633156i	\(-0.781759\pi\)
							0.996198	−	0.0871145i	\(-0.0277646\pi\)
\(29\)	−160.052	−	36.5308i	−1.02486	−	0.233918i	−0.323125	−	0.946356i	\(-0.604734\pi\)
							−0.701734	+	0.712439i	\(0.747591\pi\)
\(31\)	52.5092	−	30.3162i	0.304224	−	0.175644i	−0.340115	−	0.940384i	\(-0.610466\pi\)
							0.644339	+	0.764740i	\(0.277133\pi\)
\(37\)	223.035	+	206.946i	0.990992	+	0.919506i	0.996756	−	0.0804776i	\(-0.0256446\pi\)
							−0.00576490	+	0.999983i	\(0.501835\pi\)
\(41\)	440.806	+	212.281i	1.67908	+	0.808603i	0.997006	+	0.0773207i	\(0.0246365\pi\)
							0.682075	+	0.731282i	\(0.261078\pi\)
\(43\)	470.532	−	226.596i	1.66873	−	0.803619i	0.670647	−	0.741777i	\(-0.266017\pi\)
							0.998086	−	0.0618428i	\(-0.0196977\pi\)
\(47\)	136.422	+	347.596i	0.423386	+	1.07877i	0.970369	+	0.241626i	\(0.0776807\pi\)
							−0.546984	+	0.837143i	\(0.684224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.131.42	yes	648
3.2	odd	2	inner	147.4.o.a.131.13	yes	648
49.3	odd	42	inner	147.4.o.a.101.13	✓	648
147.101	even	42	inner	147.4.o.a.101.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.13	✓	648	49.3	odd	42	inner
147.4.o.a.101.42	yes	648	147.101	even	42	inner
147.4.o.a.131.13	yes	648	3.2	odd	2	inner
147.4.o.a.131.42	yes	648	1.1	even	1	trivial