Embedded newform 136.3.j.b.115.16

• Label: 136.3.j.b.115.16
• 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			115.16
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.16

$q$-expansion

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

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	\(-0.445631\pi\)
							0.985448	−	0.169977i	\(-0.0543692\pi\)
\(7\)	−3.98488	−	3.98488i	−0.569268	−	0.569268i	0.362655	−	0.931923i	\(-0.381870\pi\)
							−0.931923	+	0.362655i	\(0.881870\pi\)
\(11\)	−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	\(0.0297612\pi\)
							−0.995632	+	0.0933613i	\(0.970239\pi\)
\(17\)	−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	0.884174	−	0.467157i	\(-0.154722\pi\)
							−0.467157	+	0.884174i	\(0.654722\pi\)
\(29\)	16.9451	−	16.9451i	0.584314	−	0.584314i	−0.351772	−	0.936086i	\(-0.614421\pi\)
							0.936086	+	0.351772i	\(0.114421\pi\)
\(31\)	40.2011	−	40.2011i	1.29681	−	1.29681i	0.366320	−	0.930489i	\(-0.380618\pi\)
							0.930489	−	0.366320i	\(-0.119382\pi\)
\(37\)	−19.1560	+	19.1560i	−0.517729	+	0.517729i	−0.916884	−	0.399155i	\(-0.869304\pi\)
							0.399155	+	0.916884i	\(0.369304\pi\)
\(41\)	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	\(-0.840371\pi\)
							0.876868	−	0.480732i	\(-0.159629\pi\)

(See \(a_n\) instead)

Twists

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

    

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