Embedded newform 147.2.o.b.26.5

• Label: 147.2.o.b.26.5
• Level: $147$
• Weight: $2$
• Character: 147.26
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			26.5
Character	\(\chi\)	\(=\)	147.26
Dual form			147.2.o.b.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34521 - 0.100810i) q^{2} +(0.896512 + 1.48198i) q^{3} +(-0.178230 - 0.0268638i) q^{4} +(-0.507807 - 0.471176i) q^{5} +(-1.05660 - 2.08395i) q^{6} +(1.99487 + 1.73796i) q^{7} +(2.86737 + 0.654459i) q^{8} +(-1.39253 + 2.65723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34521	−	0.100810i	−0.951208	−	0.0712832i	−0.409921	−	0.912121i	\(-0.634444\pi\)
							−0.541287	+	0.840838i	\(0.682063\pi\)
\(3\)	0.896512	+	1.48198i	0.517602	+	0.855622i
							\(5\)	−0.507807	−	0.471176i	−0.227098	−	0.210716i	0.558414	−	0.829562i	\(-0.311410\pi\)
−0.785512	+	0.618846i	\(0.787600\pi\)
\(7\)	1.99487	+	1.73796i	0.753990	+	0.656886i
							\(11\)	−2.98468	+	4.37772i	−0.899914	+	1.31993i	0.0471511	+	0.998888i	\(0.484986\pi\)
−0.947065	+	0.321043i	\(0.895967\pi\)
\(13\)	−0.223326	−	0.463742i	−0.0619396	−	0.128619i	0.867701	−	0.497086i	\(-0.165597\pi\)
							−0.929641	+	0.368468i	\(0.879883\pi\)
\(17\)	−1.27577	+	3.25061i	−0.309419	+	0.788388i	0.688580	+	0.725161i	\(0.258235\pi\)
							−0.997999	+	0.0632272i	\(0.979861\pi\)
\(19\)	2.46140	+	1.42109i	0.564685	+	0.326021i	0.755024	−	0.655697i	\(-0.227625\pi\)
							−0.190339	+	0.981718i	\(0.560959\pi\)
\(23\)	7.23403	−	2.83915i	1.50840	−	0.592003i	0.540057	−	0.841629i	\(-0.318403\pi\)
							0.968342	+	0.249626i	\(0.0803075\pi\)
\(29\)	1.66990	−	1.33170i	0.310093	−	0.247291i	−0.456060	−	0.889949i	\(-0.650740\pi\)
							0.766153	+	0.642658i	\(0.222168\pi\)
\(31\)	3.27064	−	1.88831i	0.587425	−	0.339150i	−0.176654	−	0.984273i	\(-0.556527\pi\)
							0.764079	+	0.645123i	\(0.223194\pi\)
\(37\)	−2.78669	+	0.420027i	−0.458130	+	0.0690520i	−0.374053	−	0.927407i	\(-0.622032\pi\)
							−0.0840768	+	0.996459i	\(0.526794\pi\)
\(41\)	2.45406	−	10.7519i	0.383259	−	1.67917i	−0.303933	−	0.952693i	\(-0.598300\pi\)
							0.687193	−	0.726475i	\(-0.258843\pi\)
\(43\)	−1.18057	−	5.17244i	−0.180036	−	0.788789i	−0.981611	−	0.190894i	\(-0.938861\pi\)
							0.801575	−	0.597895i	\(-0.203996\pi\)
\(47\)	−0.418868	+	5.58941i	−0.0610982	+	0.815299i	0.879526	+	0.475850i	\(0.157860\pi\)
							−0.940625	+	0.339449i	\(0.889760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.26.5	yes	192
3.2	odd	2	inner	147.2.o.b.26.12	yes	192
49.17	odd	42	inner	147.2.o.b.17.12	yes	192
147.17	even	42	inner	147.2.o.b.17.5	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.17.5	✓	192	147.17	even	42	inner
147.2.o.b.17.12	yes	192	49.17	odd	42	inner
147.2.o.b.26.5	yes	192	1.1	even	1	trivial
147.2.o.b.26.12	yes	192	3.2	odd	2	inner