Embedded newform 147.3.j.a.13.1

• Label: 147.3.j.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	147.13
Dual form			147.3.j.a.34.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.828971 - 3.63196i) q^{2} +(1.35417 - 1.07992i) q^{3} +(-8.90007 + 4.28605i) q^{4} +(-5.61004 + 4.47386i) q^{5} +(-5.04478 - 4.02308i) q^{6} +(-3.05350 - 6.29890i) q^{7} +(13.6538 + 17.1213i) q^{8} +(0.667563 - 2.92478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 32 q^{4} - 2 q^{7} + 12 q^{8} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.828971	−	3.63196i	−0.414486	−	1.81598i	−0.562261	−	0.826960i	\(-0.690069\pi\)
							0.147776	−	0.989021i	\(-0.452789\pi\)
\(3\)	1.35417	−	1.07992i	0.451391	−	0.359972i
							\(5\)	−5.61004	+	4.47386i	−1.12201	+	0.894772i	−0.995268	−	0.0971639i	\(-0.969023\pi\)
−0.126740	+	0.991936i	\(0.540451\pi\)
\(7\)	−3.05350	−	6.29890i	−0.436214	−	0.899843i
							\(11\)	2.53997	+	11.1283i	0.230906	+	1.01167i	0.948890	+	0.315607i	\(0.102208\pi\)
−0.717984	+	0.696060i	\(0.754935\pi\)
\(13\)	−7.83503	+	1.78830i	−0.602695	+	0.137561i	−0.512972	−	0.858406i	\(-0.671455\pi\)
							−0.0897232	+	0.995967i	\(0.528598\pi\)
\(17\)	−3.00924	+	6.24875i	−0.177014	+	0.367573i	−0.970532	−	0.240974i	\(-0.922533\pi\)
							0.793518	+	0.608547i	\(0.208247\pi\)
\(19\)		−	17.1421i		−	0.902217i	−0.892469	−	0.451108i	\(-0.851029\pi\)
							0.892469	−	0.451108i	\(-0.148971\pi\)
\(23\)	−37.9077	+	18.2554i	−1.64816	+	0.793712i	−0.648692	+	0.761051i	\(0.724684\pi\)
							−0.999466	+	0.0326608i	\(0.989602\pi\)
\(29\)	−32.5458	−	15.6733i	−1.12227	−	0.540457i	−0.221677	−	0.975120i	\(-0.571153\pi\)
							−0.900593	+	0.434663i	\(0.856867\pi\)
\(31\)			5.25973i			0.169669i	0.996395	+	0.0848344i	\(0.0270361\pi\)
							−0.996395	+	0.0848344i	\(0.972964\pi\)
\(37\)	−52.5363	−	25.3001i	−1.41990	−	0.683787i	−0.442810	−	0.896616i	\(-0.646018\pi\)
							−0.977090	+	0.212828i	\(0.931733\pi\)
\(41\)	46.9147	−	37.4132i	1.14426	−	0.912518i	0.147199	−	0.989107i	\(-0.452974\pi\)
							0.997063	+	0.0765888i	\(0.0244029\pi\)
\(43\)	−14.4335	+	18.0990i	−0.335662	+	0.420907i	−0.920805	−	0.390023i	\(-0.872467\pi\)
							0.585143	+	0.810930i	\(0.301038\pi\)
\(47\)	15.0781	−	3.44147i	0.320810	−	0.0732229i	−0.0590833	−	0.998253i	\(-0.518818\pi\)
							0.379894	+	0.925030i	\(0.375961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.j.a.13.1	✓	108
3.2	odd	2		441.3.v.c.307.18		108
49.34	odd	14	inner	147.3.j.a.34.1	yes	108
147.83	even	14		441.3.v.c.181.18		108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.j.a.13.1	✓	108	1.1	even	1	trivial
147.3.j.a.34.1	yes	108	49.34	odd	14	inner
441.3.v.c.181.18		108	147.83	even	14
441.3.v.c.307.18		108	3.2	odd	2