Embedded newform 147.4.o.a.131.11

• Label: 147.4.o.a.131.11
• 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			131.11
Character	\(\chi\)	\(=\)	147.131
Dual form			147.4.o.a.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.61324 + 1.41809i) q^{2} +(5.16133 + 0.600543i) q^{3} +(5.18012 - 4.80645i) q^{4} +(-14.1175 + 9.62516i) q^{5} +(-19.5008 + 5.14935i) q^{6} +(-17.7302 + 5.35166i) q^{7} +(1.57212 - 3.26455i) q^{8} +(26.2787 + 6.19920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.61324	+	1.41809i	−1.27747	+	0.501372i	−0.904501	−	0.426470i	\(-0.859757\pi\)
							−0.372973	+	0.927842i	\(0.621662\pi\)
\(3\)	5.16133	+	0.600543i	0.993299	+	0.115574i
							\(5\)	−14.1175	+	9.62516i	−1.26271	+	0.860900i	−0.994730	−	0.102525i	\(-0.967308\pi\)
−0.267978	+	0.963425i	\(0.586356\pi\)
\(7\)	−17.7302	+	5.35166i	−0.957340	+	0.288962i
							\(11\)	4.02106	−	26.6780i	0.110218	−	0.731246i	−0.863644	−	0.504103i	\(-0.831823\pi\)
0.973861	−	0.227144i	\(-0.0729387\pi\)
\(13\)	−20.2605	−	16.1572i	−0.432251	−	0.344708i	0.383069	−	0.923720i	\(-0.374867\pi\)
							−0.815319	+	0.579012i	\(0.803439\pi\)
\(17\)	25.6621	−	7.91570i	0.366116	−	0.112932i	−0.106238	−	0.994341i	\(-0.533881\pi\)
							0.472354	+	0.881409i	\(0.343404\pi\)
\(19\)	−39.0366	−	22.5378i	−0.471348	−	0.272133i	0.245456	−	0.969408i	\(-0.421062\pi\)
							−0.716804	+	0.697275i	\(0.754396\pi\)
\(23\)	35.6104	−	115.446i	0.322838	−	1.04662i	−0.638123	−	0.769934i	\(-0.720289\pi\)
							0.960961	−	0.276682i	\(-0.0892349\pi\)
\(29\)	−172.654	−	39.4071i	−1.10555	−	0.252335i	−0.369480	−	0.929239i	\(-0.620464\pi\)
							−0.736071	+	0.676904i	\(0.763321\pi\)
\(31\)	219.808	−	126.906i	1.27351	−	0.735259i	0.297860	−	0.954609i	\(-0.403727\pi\)
							0.975646	+	0.219350i	\(0.0703937\pi\)
\(37\)	−256.546	−	238.040i	−1.13989	−	1.05766i	−0.997697	−	0.0678241i	\(-0.978394\pi\)
							−0.142192	−	0.989839i	\(-0.545415\pi\)
\(41\)	232.198	+	111.821i	0.884468	+	0.425937i	0.820255	−	0.571999i	\(-0.193832\pi\)
							0.0642137	+	0.997936i	\(0.479546\pi\)
\(43\)	−279.154	+	134.433i	−0.990014	+	0.476765i	−0.857538	−	0.514421i	\(-0.828007\pi\)
							−0.132476	+	0.991186i	\(0.542293\pi\)
\(47\)	−149.141	−	380.004i	−0.462860	−	1.17935i	−0.951715	−	0.306983i	\(-0.900681\pi\)
							0.488855	−	0.872365i	\(-0.337415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.131.11	yes	648
3.2	odd	2	inner	147.4.o.a.131.44	yes	648
49.3	odd	42	inner	147.4.o.a.101.44	yes	648
147.101	even	42	inner	147.4.o.a.101.11	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.11	✓	648	147.101	even	42	inner
147.4.o.a.101.44	yes	648	49.3	odd	42	inner
147.4.o.a.131.11	yes	648	1.1	even	1	trivial
147.4.o.a.131.44	yes	648	3.2	odd	2	inner