Embedded newform 147.2.k.a.104.4

• Label: 147.2.k.a.104.4
• 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			104.4
Character	\(\chi\)	\(=\)	147.104
Dual form			147.2.k.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17656 - 0.938273i) q^{2} +(-1.31872 + 1.12293i) q^{3} +(0.0588885 + 0.258007i) q^{4} +(-0.160135 - 0.0771168i) q^{5} +(2.60517 - 0.0838773i) q^{6} +(0.931491 + 2.47635i) q^{7} +(-1.13308 + 2.35287i) q^{8} +(0.478039 - 2.96167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17656	−	0.938273i	−0.831951	−	0.663459i	0.111940	−	0.993715i	\(-0.464294\pi\)
							−0.943891	+	0.330256i	\(0.892865\pi\)
\(3\)	−1.31872	+	1.12293i	−0.761363	+	0.648326i
							\(5\)	−0.160135	−	0.0771168i	−0.0716144	−	0.0344877i	0.397734	−	0.917501i	\(-0.369797\pi\)
−0.469348	+	0.883013i	\(0.655511\pi\)
\(7\)	0.931491	+	2.47635i	0.352071	+	0.935973i
							\(11\)	3.30885	+	2.63872i	0.997655	+	0.795604i	0.978925	−	0.204221i	\(-0.0654661\pi\)
0.0187305	+	0.999825i	\(0.494038\pi\)
\(13\)	2.71138	+	2.16226i	0.752002	+	0.599702i	0.922654	−	0.385630i	\(-0.126016\pi\)
							−0.170651	+	0.985331i	\(0.554587\pi\)
\(17\)	−1.12571	+	4.93208i	−0.273026	+	1.19620i	0.633395	+	0.773829i	\(0.281661\pi\)
							−0.906421	+	0.422376i	\(0.861196\pi\)
\(19\)		−	0.363498i		−	0.0833922i	−0.999130	−	0.0416961i	\(-0.986724\pi\)
							0.999130	−	0.0416961i	\(-0.0132761\pi\)
\(23\)	1.58516	−	0.361803i	0.330529	−	0.0754411i	−0.0540360	−	0.998539i	\(-0.517209\pi\)
							0.384565	+	0.923098i	\(0.374351\pi\)
\(29\)	−6.82709	−	1.55824i	−1.26776	−	0.289358i	−0.464813	−	0.885409i	\(-0.653878\pi\)
							−0.802946	+	0.596051i	\(0.796735\pi\)
\(31\)			7.18306i			1.29011i	0.764134	+	0.645057i	\(0.223167\pi\)
							−0.764134	+	0.645057i	\(0.776833\pi\)
\(37\)	1.12601	−	4.93339i	0.185116	−	0.811044i	−0.794029	−	0.607880i	\(-0.792020\pi\)
							0.979145	−	0.203165i	\(-0.0651226\pi\)
\(41\)	1.68678	+	0.812311i	0.263431	+	0.126862i	0.560938	−	0.827858i	\(-0.310441\pi\)
							−0.297507	+	0.954720i	\(0.596155\pi\)
\(43\)	−5.20198	+	2.50514i	−0.793294	+	0.382030i	−0.786222	−	0.617944i	\(-0.787966\pi\)
							−0.00707276	+	0.999975i	\(0.502251\pi\)
\(47\)	0.502413	−	0.630007i	0.0732845	−	0.0918959i	−0.743838	−	0.668360i	\(-0.766997\pi\)
							0.817122	+	0.576464i	\(0.195568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.104.4	yes	96
3.2	odd	2	inner	147.2.k.a.104.13	yes	96
49.41	odd	14	inner	147.2.k.a.41.13	yes	96
147.41	even	14	inner	147.2.k.a.41.4	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.41.4	✓	96	147.41	even	14	inner
147.2.k.a.41.13	yes	96	49.41	odd	14	inner
147.2.k.a.104.4	yes	96	1.1	even	1	trivial
147.2.k.a.104.13	yes	96	3.2	odd	2	inner