Embedded newform 147.3.j.a.13.3

• Label: 147.3.j.a.13.3
• Level: $147$
• Weight: $3$
• Character: 147.13
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $108$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	147.13
Dual form			147.3.j.a.34.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.706945 - 3.09733i) q^{2} +(-1.35417 + 1.07992i) q^{3} +(-5.48980 + 2.64375i) q^{4} +(6.01934 - 4.80027i) q^{5} +(4.30218 + 3.43087i) q^{6} +(0.865390 - 6.94630i) q^{7} +(4.14628 + 5.19927i) q^{8} +(0.667563 - 2.92478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 32 q^{4} - 2 q^{7} + 12 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.706945	−	3.09733i	−0.353473	−	1.54866i	−0.769099	−	0.639129i	\(-0.779295\pi\)
							0.415627	−	0.909535i	\(-0.363562\pi\)
\(3\)	−1.35417	+	1.07992i	−0.451391	+	0.359972i
							\(5\)	6.01934	−	4.80027i	1.20387	−	0.960053i	0.204048	−	0.978961i	\(-0.434590\pi\)
0.999821	+	0.0189075i	\(0.00601880\pi\)
\(7\)	0.865390	−	6.94630i	0.123627	−	0.992329i
							\(11\)	0.494754	+	2.16766i	0.0449776	+	0.197060i	0.992425	−	0.122852i	\(-0.0392040\pi\)
−0.947447	+	0.319911i	\(0.896347\pi\)
\(13\)	−13.9628	+	3.18692i	−1.07406	+	0.245148i	−0.722750	−	0.691110i	\(-0.757122\pi\)
							−0.351314	+	0.936258i	\(0.614265\pi\)
\(17\)	−6.25315	+	12.9848i	−0.367832	+	0.763811i	−0.999939	−	0.0110701i	\(-0.996476\pi\)
							0.632107	+	0.774881i	\(0.282190\pi\)
\(19\)		−	29.9360i		−	1.57558i	−0.615945	−	0.787789i	\(-0.711226\pi\)
							0.615945	−	0.787789i	\(-0.288774\pi\)
\(23\)	21.2514	−	10.2341i	0.923975	−	0.444963i	0.0894864	−	0.995988i	\(-0.471477\pi\)
							0.834489	+	0.551025i	\(0.185763\pi\)
\(29\)	−27.9679	−	13.4686i	−0.964411	−	0.464436i	−0.115695	−	0.993285i	\(-0.536910\pi\)
							−0.848716	+	0.528849i	\(0.822624\pi\)
\(31\)			19.0945i			0.615952i	0.951394	+	0.307976i	\(0.0996517\pi\)
							−0.951394	+	0.307976i	\(0.900348\pi\)
\(37\)	9.51163	+	4.58056i	0.257071	+	0.123799i	0.557981	−	0.829854i	\(-0.311576\pi\)
							−0.300910	+	0.953652i	\(0.597290\pi\)
\(41\)	−23.7300	+	18.9241i	−0.578781	+	0.461563i	−0.868597	−	0.495520i	\(-0.834978\pi\)
							0.289815	+	0.957083i	\(0.406406\pi\)
\(43\)	33.8196	−	42.4085i	0.786503	−	0.986244i	−0.213454	−	0.976953i	\(-0.568471\pi\)
							0.999957	−	0.00929049i	\(-0.00295730\pi\)
\(47\)	87.8533	−	20.0519i	1.86922	−	0.426637i	0.871279	−	0.490787i	\(-0.163291\pi\)
							0.997940	+	0.0641503i	\(0.0204337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.j.a.13.3	✓	108
3.2	odd	2		441.3.v.c.307.16		108
49.34	odd	14	inner	147.3.j.a.34.3	yes	108
147.83	even	14		441.3.v.c.181.16		108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.j.a.13.3	✓	108	1.1	even	1	trivial
147.3.j.a.34.3	yes	108	49.34	odd	14	inner
441.3.v.c.181.16		108	147.83	even	14
441.3.v.c.307.16		108	3.2	odd	2