Embedded newform 136.3.j.b.115.10

• Label: 136.3.j.b.115.10
• 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			115.10
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17987 - 1.61490i) q^{2} +(-3.41961 - 3.41961i) q^{3} +(-1.21583 + 3.81074i) q^{4} +(-2.66341 + 2.66341i) q^{5} +(-1.48766 + 9.55703i) q^{6} +(-1.35907 - 1.35907i) q^{7} +(7.58850 - 2.53271i) q^{8} +14.3875i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 12 q^{4} + 26 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17987	−	1.61490i	−0.589933	−	0.807452i
							\(3\)	−3.41961	−	3.41961i	−1.13987	−	1.13987i	−0.988473	−	0.151398i	\(-0.951623\pi\)
−0.151398	−	0.988473i	\(-0.548377\pi\)
\(5\)	−2.66341	+	2.66341i	−0.532683	+	0.532683i	−0.921370	−	0.388687i	\(-0.872929\pi\)
							0.388687	+	0.921370i	\(0.372929\pi\)
\(7\)	−1.35907	−	1.35907i	−0.194153	−	0.194153i	0.603335	−	0.797488i	\(-0.293838\pi\)
							−0.797488	+	0.603335i	\(0.793838\pi\)
\(11\)	−1.28155	+	1.28155i	−0.116504	+	0.116504i	−0.762955	−	0.646451i	\(-0.776252\pi\)
							0.646451	+	0.762955i	\(0.276252\pi\)
\(13\)			1.81011i			0.139239i	0.997574	+	0.0696196i	\(0.0221786\pi\)
							−0.997574	+	0.0696196i	\(0.977821\pi\)
\(17\)	12.5746	+	11.4402i	0.739685	+	0.672953i
							\(19\)		−	22.4367i		−	1.18088i	−0.807082	−	0.590439i	\(-0.798955\pi\)
0.807082	−	0.590439i	\(-0.201045\pi\)
\(23\)	27.5269	+	27.5269i	1.19682	+	1.19682i	0.975113	+	0.221707i	\(0.0711630\pi\)
							0.221707	+	0.975113i	\(0.428837\pi\)
\(29\)	−30.5896	+	30.5896i	−1.05481	+	1.05481i	−0.0564056	+	0.998408i	\(0.517964\pi\)
							−0.998408	+	0.0564056i	\(0.982036\pi\)
\(31\)	−23.2276	+	23.2276i	−0.749279	+	0.749279i	−0.974344	−	0.225065i	\(-0.927741\pi\)
							0.225065	+	0.974344i	\(0.427741\pi\)
\(37\)	−26.0233	+	26.0233i	−0.703332	+	0.703332i	−0.965124	−	0.261792i	\(-0.915686\pi\)
							0.261792	+	0.965124i	\(0.415686\pi\)
\(41\)	−9.47879	+	9.47879i	−0.231190	+	0.231190i	−0.813189	−	0.581999i	\(-0.802271\pi\)
							0.581999	+	0.813189i	\(0.302271\pi\)
\(43\)			16.0287i			0.372761i	0.982478	+	0.186381i	\(0.0596758\pi\)
							−0.982478	+	0.186381i	\(0.940324\pi\)
\(47\)		−	25.0657i		−	0.533312i	−0.963792	−	0.266656i	\(-0.914081\pi\)
							0.963792	−	0.266656i	\(-0.0859188\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.j.b.115.10	✓	64
4.3	odd	2		544.3.n.b.47.29		64
8.3	odd	2	inner	136.3.j.b.115.23	yes	64
8.5	even	2		544.3.n.b.47.30		64
17.4	even	4	inner	136.3.j.b.123.23	yes	64
68.55	odd	4		544.3.n.b.463.30		64
136.21	even	4		544.3.n.b.463.29		64
136.123	odd	4	inner	136.3.j.b.123.10	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.j.b.115.10	✓	64	1.1	even	1	trivial
136.3.j.b.115.23	yes	64	8.3	odd	2	inner
136.3.j.b.123.10	yes	64	136.123	odd	4	inner
136.3.j.b.123.23	yes	64	17.4	even	4	inner
544.3.n.b.47.29		64	4.3	odd	2
544.3.n.b.47.30		64	8.5	even	2
544.3.n.b.463.29		64	136.21	even	4
544.3.n.b.463.30		64	68.55	odd	4