Embedded newform 147.2.k.a.83.10

• Label: 147.2.k.a.83.10
• Level: $147$
• Weight: $2$
• Character: 147.83
• 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			83.10
Character	\(\chi\)	\(=\)	147.83
Dual form			147.2.k.a.62.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500364 + 0.114205i) q^{2} +(-1.71421 + 0.247991i) q^{3} +(-1.56462 - 0.753479i) q^{4} +(1.88314 - 2.36138i) q^{5} +(-0.886049 - 0.0716847i) q^{6} +(-1.31378 - 2.29652i) q^{7} +(-1.49935 - 1.19569i) q^{8} +(2.87700 - 0.850216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500364	+	0.114205i	0.353811	+	0.0807551i	0.395732	−	0.918366i	\(-0.370491\pi\)
							−0.0419214	+	0.999121i	\(0.513348\pi\)
\(3\)	−1.71421	+	0.247991i	−0.989697	+	0.143178i
							\(5\)	1.88314	−	2.36138i	0.842165	−	1.05604i	−0.155506	−	0.987835i	\(-0.549701\pi\)
0.997671	−	0.0682066i	\(-0.0217277\pi\)
\(7\)	−1.31378	−	2.29652i	−0.496561	−	0.868002i
							\(11\)	1.61022	+	0.367523i	0.485501	+	0.110812i	0.458263	−	0.888817i	\(-0.348472\pi\)
0.0272378	+	0.999629i	\(0.491329\pi\)
\(13\)	−2.41719	−	0.551708i	−0.670408	−	0.153016i	−0.126249	−	0.991999i	\(-0.540294\pi\)
							−0.544159	+	0.838982i	\(0.683151\pi\)
\(17\)	1.01473	−	0.488668i	0.246108	−	0.118519i	−0.306764	−	0.951786i	\(-0.599246\pi\)
							0.552872	+	0.833266i	\(0.313532\pi\)
\(19\)		−	5.26007i		−	1.20674i	−0.797461	−	0.603371i	\(-0.793824\pi\)
							0.797461	−	0.603371i	\(-0.206176\pi\)
\(23\)	−2.57810	+	5.35348i	−0.537571	+	1.11628i	0.438481	+	0.898740i	\(0.355517\pi\)
							−0.976052	+	0.217537i	\(0.930198\pi\)
\(29\)	2.83323	+	5.88325i	0.526117	+	1.09249i	0.979550	+	0.201202i	\(0.0644849\pi\)
							−0.453433	+	0.891290i	\(0.649801\pi\)
\(31\)			1.14077i			0.204889i	0.994739	+	0.102444i	\(0.0326664\pi\)
							−0.994739	+	0.102444i	\(0.967334\pi\)
\(37\)	10.9387	−	5.26779i	1.79831	−	0.866020i	0.871634	−	0.490157i	\(-0.163060\pi\)
							0.926675	−	0.375863i	\(-0.122654\pi\)
\(41\)	7.40491	−	9.28546i	1.15645	−	1.45014i	0.285770	−	0.958298i	\(-0.407751\pi\)
							0.870682	−	0.491847i	\(-0.163678\pi\)
\(43\)	0.729648	+	0.914950i	0.111270	+	0.139529i	0.834348	−	0.551238i	\(-0.185845\pi\)
							−0.723078	+	0.690767i	\(0.757273\pi\)
\(47\)	0.915444	−	4.01082i	0.133531	−	0.585038i	−0.863243	−	0.504788i	\(-0.831571\pi\)
							0.996775	−	0.0802507i	\(-0.0255721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.83.10	yes	96
3.2	odd	2	inner	147.2.k.a.83.7	yes	96
49.13	odd	14	inner	147.2.k.a.62.7	✓	96
147.62	even	14	inner	147.2.k.a.62.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.62.7	✓	96	49.13	odd	14	inner
147.2.k.a.62.10	yes	96	147.62	even	14	inner
147.2.k.a.83.7	yes	96	3.2	odd	2	inner
147.2.k.a.83.10	yes	96	1.1	even	1	trivial