Embedded newform 136.3.j.b.115.17

• Label: 136.3.j.b.115.17
• Level: $136$
• Weight: $3$
• Character: 136.115
• Analytic conductor: $3.706$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	136.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70573159530\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			115.17
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.462601 - 1.94576i) q^{2} +(-1.38331 - 1.38331i) q^{3} +(-3.57200 - 1.80023i) q^{4} +(-5.77712 + 5.77712i) q^{5} +(-3.33151 + 2.05167i) q^{6} +(3.98488 + 3.98488i) q^{7} +(-5.15523 + 6.11748i) q^{8} -5.17291i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 12 q^{4} + 26 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\)	\(69\)	\(103\)	\(105\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.462601	−	1.94576i	0.231301	−	0.972882i
							\(3\)	−1.38331	−	1.38331i	−0.461103	−	0.461103i	0.437914	−	0.899017i	\(-0.355717\pi\)
−0.899017	+	0.437914i	\(0.855717\pi\)
\(5\)	−5.77712	+	5.77712i	−1.15542	+	1.15542i	−0.169977	+	0.985448i	\(0.554369\pi\)
							−0.985448	+	0.169977i	\(0.945631\pi\)
\(7\)	3.98488	+	3.98488i	0.569268	+	0.569268i	0.931923	−	0.362655i	\(-0.118130\pi\)
							−0.362655	+	0.931923i	\(0.618130\pi\)
\(11\)	−10.1617	+	10.1617i	−0.923792	+	0.923792i	−0.997295	−	0.0735034i	\(-0.976582\pi\)
							0.0735034	+	0.997295i	\(0.476582\pi\)
\(13\)		−	2.42739i		−	0.186723i	−0.995632	−	0.0933613i	\(-0.970239\pi\)
							0.995632	−	0.0933613i	\(-0.0297612\pi\)
\(17\)	−16.7853	−	2.69331i	−0.987370	−	0.158430i
							\(19\)		−	1.82948i		−	0.0962885i	−0.998840	−	0.0481443i	\(-0.984669\pi\)
0.998840	−	0.0481443i	\(-0.0153307\pi\)
\(23\)	−9.59140	−	9.59140i	−0.417018	−	0.417018i	0.467157	−	0.884174i	\(-0.345278\pi\)
							−0.884174	+	0.467157i	\(0.845278\pi\)
\(29\)	−16.9451	+	16.9451i	−0.584314	+	0.584314i	−0.936086	−	0.351772i	\(-0.885579\pi\)
							0.351772	+	0.936086i	\(0.385579\pi\)
\(31\)	−40.2011	+	40.2011i	−1.29681	+	1.29681i	−0.366320	+	0.930489i	\(0.619382\pi\)
							−0.930489	+	0.366320i	\(0.880618\pi\)
\(37\)	19.1560	−	19.1560i	0.517729	−	0.517729i	−0.399155	−	0.916884i	\(-0.630696\pi\)
							0.916884	+	0.399155i	\(0.130696\pi\)
\(41\)	35.0793	−	35.0793i	0.855594	−	0.855594i	−0.135222	−	0.990815i	\(-0.543175\pi\)
							0.990815	+	0.135222i	\(0.0431747\pi\)
\(43\)		−	25.5319i		−	0.593764i	−0.954914	−	0.296882i	\(-0.904053\pi\)
							0.954914	−	0.296882i	\(-0.0959469\pi\)
\(47\)			45.1888i			0.961464i	0.876868	+	0.480732i	\(0.159629\pi\)
							−0.876868	+	0.480732i	\(0.840371\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.j.b.115.17	yes	64
4.3	odd	2		544.3.n.b.47.20		64
8.3	odd	2	inner	136.3.j.b.115.16	✓	64
8.5	even	2		544.3.n.b.47.19		64
17.4	even	4	inner	136.3.j.b.123.16	yes	64
68.55	odd	4		544.3.n.b.463.19		64
136.21	even	4		544.3.n.b.463.20		64
136.123	odd	4	inner	136.3.j.b.123.17	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.j.b.115.16	✓	64	8.3	odd	2	inner
136.3.j.b.115.17	yes	64	1.1	even	1	trivial
136.3.j.b.123.16	yes	64	17.4	even	4	inner
136.3.j.b.123.17	yes	64	136.123	odd	4	inner
544.3.n.b.47.19		64	8.5	even	2
544.3.n.b.47.20		64	4.3	odd	2
544.3.n.b.463.19		64	68.55	odd	4
544.3.n.b.463.20		64	136.21	even	4