Embedded newform 147.2.k.a.104.11

• Label: 147.2.k.a.104.11
• Level: $147$
• Weight: $2$
• Character: 147.104
• 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			104.11
Character	\(\chi\)	\(=\)	147.104
Dual form			147.2.k.a.41.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.708830 + 0.565273i) q^{2} +(0.562979 - 1.63800i) q^{3} +(-0.262135 - 1.14849i) q^{4} +(-3.11976 - 1.50240i) q^{5} +(1.32498 - 0.842829i) q^{6} +(1.97338 + 1.76232i) q^{7} +(1.25014 - 2.59595i) q^{8} +(-2.36611 - 1.84432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.708830	+	0.565273i	0.501219	+	0.399709i	0.841200	−	0.540723i	\(-0.181849\pi\)
							−0.339982	+	0.940432i	\(0.610421\pi\)
\(3\)	0.562979	−	1.63800i	0.325036	−	0.945702i
							\(5\)	−3.11976	−	1.50240i	−1.39520	−	0.671891i	−0.423016	−	0.906122i	\(-0.639029\pi\)
−0.972181	+	0.234231i	\(0.924743\pi\)
\(7\)	1.97338	+	1.76232i	0.745868	+	0.666093i
							\(11\)	4.39120	+	3.50187i	1.32400	+	1.05585i	0.993711	+	0.111971i	\(0.0357164\pi\)
0.330285	+	0.943881i	\(0.392855\pi\)
\(13\)	0.177428	+	0.141494i	0.0492095	+	0.0392433i	0.647786	−	0.761823i	\(-0.275695\pi\)
							−0.598576	+	0.801066i	\(0.704267\pi\)
\(17\)	0.323366	−	1.41676i	0.0784277	−	0.343614i	−0.920456	−	0.390846i	\(-0.872183\pi\)
							0.998884	+	0.0472313i	\(0.0150398\pi\)
\(19\)			5.02827i			1.15356i	0.816898	+	0.576782i	\(0.195692\pi\)
							−0.816898	+	0.576782i	\(0.804308\pi\)
\(23\)	3.57128	−	0.815122i	0.744664	−	0.169965i	0.166678	−	0.986011i	\(-0.446696\pi\)
							0.577986	+	0.816047i	\(0.303839\pi\)
\(29\)	0.768658	+	0.175441i	0.142736	+	0.0325786i	0.293292	−	0.956023i	\(-0.405249\pi\)
							−0.150556	+	0.988602i	\(0.548106\pi\)
\(31\)		−	4.63475i		−	0.832426i	−0.909267	−	0.416213i	\(-0.863357\pi\)
							0.909267	−	0.416213i	\(-0.136643\pi\)
\(37\)	−0.757363	+	3.31822i	−0.124510	+	0.545513i	0.873741	+	0.486391i	\(0.161687\pi\)
							−0.998251	+	0.0591213i	\(0.981170\pi\)
\(41\)	−5.72497	−	2.75700i	−0.894090	−	0.430571i	−0.0703391	−	0.997523i	\(-0.522408\pi\)
							−0.823751	+	0.566952i	\(0.808122\pi\)
\(43\)	−1.08373	+	0.521899i	−0.165268	+	0.0795888i	−0.514689	−	0.857377i	\(-0.672093\pi\)
							0.349421	+	0.936966i	\(0.386378\pi\)
\(47\)	−3.10226	+	3.89012i	−0.452512	+	0.567432i	−0.954793	−	0.297273i	\(-0.903923\pi\)
							0.502281	+	0.864704i	\(0.332494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.104.11	yes	96
3.2	odd	2	inner	147.2.k.a.104.6	yes	96
49.41	odd	14	inner	147.2.k.a.41.6	✓	96
147.41	even	14	inner	147.2.k.a.41.11	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.41.6	✓	96	49.41	odd	14	inner
147.2.k.a.41.11	yes	96	147.41	even	14	inner
147.2.k.a.104.6	yes	96	3.2	odd	2	inner
147.2.k.a.104.11	yes	96	1.1	even	1	trivial