Embedded newform 147.4.o.a.101.11

• Label: 147.4.o.a.101.11
• 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			101.11
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.11

$q$-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 \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.11	✓	648	1.1	even	1	trivial
147.4.o.a.101.44	yes	648	3.2	odd	2	inner
147.4.o.a.131.11	yes	648	147.131	even	42	inner
147.4.o.a.131.44	yes	648	49.33	odd	42	inner