Embedded newform 147.2.k.a.125.5

• Label: 147.2.k.a.125.5
• Level: $147$
• Weight: $2$
• Character: 147.125
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			125.5
Character	\(\chi\)	\(=\)	147.125
Dual form			147.2.k.a.20.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.649912 - 1.34956i) q^{2} +(-1.71516 + 0.241291i) q^{3} +(-0.151939 + 0.190525i) q^{4} +(-0.852820 + 3.73645i) q^{5} +(1.44034 + 2.15789i) q^{6} +(1.09133 + 2.41019i) q^{7} +(-2.56481 - 0.585401i) q^{8} +(2.88356 - 0.827705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.649912	−	1.34956i	−0.459557	−	0.954281i	−0.994031	−	0.109096i	\(-0.965204\pi\)
							0.534474	−	0.845185i	\(-0.320510\pi\)
\(3\)	−1.71516	+	0.241291i	−0.990249	+	0.139309i
							\(5\)	−0.852820	+	3.73645i	−0.381393	+	1.67099i	0.311730	+	0.950171i	\(0.399092\pi\)
−0.693122	+	0.720820i	\(0.743765\pi\)
\(7\)	1.09133	+	2.41019i	0.412484	+	0.910965i
							\(11\)	1.21923	+	2.53175i	0.367611	+	0.763352i	0.999936	−	0.0113513i	\(-0.00361332\pi\)
−0.632324	+	0.774704i	\(0.717899\pi\)
\(13\)	−1.31157	−	2.72351i	−0.363764	−	0.755365i	0.636104	−	0.771604i	\(-0.280545\pi\)
							−0.999868	+	0.0162389i	\(0.994831\pi\)
\(17\)	0.778223	+	0.975861i	0.188747	+	0.236681i	0.867197	−	0.497966i	\(-0.165919\pi\)
							−0.678450	+	0.734647i	\(0.737348\pi\)
\(19\)			8.48718i			1.94709i	0.228490	+	0.973546i	\(0.426621\pi\)
							−0.228490	+	0.973546i	\(0.573379\pi\)
\(23\)	−1.22262	−	0.975008i	−0.254934	−	0.203303i	0.487680	−	0.873022i	\(-0.337843\pi\)
							−0.742615	+	0.669719i	\(0.766415\pi\)
\(29\)	2.10937	−	1.68217i	0.391701	−	0.312371i	−0.407760	−	0.913089i	\(-0.633690\pi\)
							0.799461	+	0.600718i	\(0.205119\pi\)
\(31\)			0.536032i			0.0962742i	0.998841	+	0.0481371i	\(0.0153284\pi\)
							−0.998841	+	0.0481371i	\(0.984672\pi\)
\(37\)	0.425539	+	0.533609i	0.0699581	+	0.0877247i	0.815579	−	0.578645i	\(-0.196418\pi\)
							−0.745621	+	0.666370i	\(0.767847\pi\)
\(41\)	1.98871	−	8.71310i	0.310584	−	1.36076i	−0.542969	−	0.839753i	\(-0.682700\pi\)
							0.853553	−	0.521006i	\(-0.174443\pi\)
\(43\)	0.131403	+	0.575716i	0.0200388	+	0.0877959i	0.983959	−	0.178397i	\(-0.0570912\pi\)
							−0.963920	+	0.266193i	\(0.914234\pi\)
\(47\)	2.61236	−	1.25804i	0.381051	−	0.183505i	−0.233546	−	0.972346i	\(-0.575033\pi\)
							0.614597	+	0.788841i	\(0.289319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.125.5	yes	96
3.2	odd	2	inner	147.2.k.a.125.12	yes	96
49.20	odd	14	inner	147.2.k.a.20.12	yes	96
147.20	even	14	inner	147.2.k.a.20.5	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.20.5	✓	96	147.20	even	14	inner
147.2.k.a.20.12	yes	96	49.20	odd	14	inner
147.2.k.a.125.5	yes	96	1.1	even	1	trivial
147.2.k.a.125.12	yes	96	3.2	odd	2	inner