Embedded newform 136.3.j.b.115.15

• Label: 136.3.j.b.115.15
• 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			115.15
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.603225 - 1.90686i) q^{2} +(0.735236 + 0.735236i) q^{3} +(-3.27224 + 2.30053i) q^{4} +(0.444064 - 0.444064i) q^{5} +(0.958480 - 1.84551i) q^{6} +(8.56417 + 8.56417i) q^{7} +(6.36070 + 4.85196i) q^{8} -7.91886i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 12 q^{4} + 26 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.603225	−	1.90686i	−0.301613	−	0.953431i
							\(3\)	0.735236	+	0.735236i	0.245079	+	0.245079i	0.818947	−	0.573869i	\(-0.194558\pi\)
−0.573869	+	0.818947i	\(0.694558\pi\)
\(5\)	0.444064	−	0.444064i	0.0888127	−	0.0888127i	−0.661305	−	0.750117i	\(-0.729997\pi\)
							0.750117	+	0.661305i	\(0.229997\pi\)
\(7\)	8.56417	+	8.56417i	1.22345	+	1.22345i	0.966396	+	0.257056i	\(0.0827525\pi\)
							0.257056	+	0.966396i	\(0.417247\pi\)
\(11\)	9.50644	−	9.50644i	0.864222	−	0.864222i	−0.127603	−	0.991825i	\(-0.540728\pi\)
							0.991825	+	0.127603i	\(0.0407284\pi\)
\(13\)			5.01388i			0.385683i	0.981230	+	0.192842i	\(0.0617704\pi\)
							−0.981230	+	0.192842i	\(0.938230\pi\)
\(17\)	10.3801	+	13.4630i	0.610595	+	0.791943i
							\(19\)		−	6.72316i		−	0.353851i	−0.984224	−	0.176925i	\(-0.943385\pi\)
0.984224	−	0.176925i	\(-0.0566151\pi\)
\(23\)	−10.9205	−	10.9205i	−0.474804	−	0.474804i	0.428661	−	0.903465i	\(-0.358985\pi\)
							−0.903465	+	0.428661i	\(0.858985\pi\)
\(29\)	21.0627	−	21.0627i	0.726301	−	0.726301i	−0.243580	−	0.969881i	\(-0.578322\pi\)
							0.969881	+	0.243580i	\(0.0783218\pi\)
\(31\)	5.93996	−	5.93996i	0.191612	−	0.191612i	−0.604781	−	0.796392i	\(-0.706739\pi\)
							0.796392	+	0.604781i	\(0.206739\pi\)
\(37\)	−40.0989	+	40.0989i	−1.08375	+	1.08375i	−0.0875980	+	0.996156i	\(0.527919\pi\)
							−0.996156	+	0.0875980i	\(0.972081\pi\)
\(41\)	−3.97160	+	3.97160i	−0.0968683	+	0.0968683i	−0.753880	−	0.657012i	\(-0.771820\pi\)
							0.657012	+	0.753880i	\(0.271820\pi\)
\(43\)		−	27.8793i		−	0.648357i	−0.945996	−	0.324178i	\(-0.894912\pi\)
							0.945996	−	0.324178i	\(-0.105088\pi\)
\(47\)			91.5150i			1.94713i	0.228417	+	0.973563i	\(0.426645\pi\)
							−0.228417	+	0.973563i	\(0.573355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.j.b.115.15	✓	64
4.3	odd	2		544.3.n.b.47.15		64
8.3	odd	2	inner	136.3.j.b.115.18	yes	64
8.5	even	2		544.3.n.b.47.16		64
17.4	even	4	inner	136.3.j.b.123.18	yes	64
68.55	odd	4		544.3.n.b.463.16		64
136.21	even	4		544.3.n.b.463.15		64
136.123	odd	4	inner	136.3.j.b.123.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.j.b.115.15	✓	64	1.1	even	1	trivial
136.3.j.b.115.18	yes	64	8.3	odd	2	inner
136.3.j.b.123.15	yes	64	136.123	odd	4	inner
136.3.j.b.123.18	yes	64	17.4	even	4	inner
544.3.n.b.47.15		64	4.3	odd	2
544.3.n.b.47.16		64	8.5	even	2
544.3.n.b.463.15		64	136.21	even	4
544.3.n.b.463.16		64	68.55	odd	4