Embedded newform 136.3.j.b.115.13

• Label: 136.3.j.b.115.13
• 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			115.13
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.775308 + 1.84361i) q^{2} +(-2.20075 - 2.20075i) q^{3} +(-2.79779 - 2.85873i) q^{4} +(-0.154410 + 0.154410i) q^{5} +(5.76358 - 2.35106i) q^{6} +(6.51226 + 6.51226i) q^{7} +(7.43954 - 2.94164i) q^{8} +0.686594i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 12 q^{4} + 26 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.775308	+	1.84361i	−0.387654	+	0.921805i
							\(3\)	−2.20075	−	2.20075i	−0.733583	−	0.733583i	0.237745	−	0.971328i	\(-0.423592\pi\)
−0.971328	+	0.237745i	\(0.923592\pi\)
\(5\)	−0.154410	+	0.154410i	−0.0308820	+	0.0308820i	−0.722379	−	0.691497i	\(-0.756951\pi\)
							0.691497	+	0.722379i	\(0.256951\pi\)
\(7\)	6.51226	+	6.51226i	0.930323	+	0.930323i	0.997726	−	0.0674029i	\(-0.0214713\pi\)
							−0.0674029	+	0.997726i	\(0.521471\pi\)
\(11\)	−13.9217	+	13.9217i	−1.26561	+	1.26561i	−0.317276	+	0.948333i	\(0.602768\pi\)
							−0.948333	+	0.317276i	\(0.897232\pi\)
\(13\)			19.1727i			1.47483i	0.675443	+	0.737413i	\(0.263953\pi\)
							−0.675443	+	0.737413i	\(0.736047\pi\)
\(17\)	13.5698	−	10.2401i	0.798225	−	0.602359i
							\(19\)			7.59530i			0.399752i	0.979821	+	0.199876i	\(0.0640540\pi\)
−0.979821	+	0.199876i	\(0.935946\pi\)
\(23\)	14.6882	+	14.6882i	0.638617	+	0.638617i	0.950214	−	0.311597i	\(-0.100864\pi\)
							−0.311597	+	0.950214i	\(0.600864\pi\)
\(29\)	39.4616	−	39.4616i	1.36074	−	1.36074i	0.487773	−	0.872971i	\(-0.337810\pi\)
							0.872971	−	0.487773i	\(-0.162190\pi\)
\(31\)	−24.6384	+	24.6384i	−0.794787	+	0.794787i	−0.982268	−	0.187481i	\(-0.939968\pi\)
							0.187481	+	0.982268i	\(0.439968\pi\)
\(37\)	−6.28065	+	6.28065i	−0.169747	+	0.169747i	−0.786868	−	0.617121i	\(-0.788299\pi\)
							0.617121	+	0.786868i	\(0.288299\pi\)
\(41\)	−2.88295	+	2.88295i	−0.0703159	+	0.0703159i	−0.741390	−	0.671074i	\(-0.765833\pi\)
							0.671074	+	0.741390i	\(0.265833\pi\)
\(43\)		−	17.2266i		−	0.400618i	−0.979733	−	0.200309i	\(-0.935805\pi\)
							0.979733	−	0.200309i	\(-0.0641945\pi\)
\(47\)		−	27.4245i		−	0.583500i	−0.956495	−	0.291750i	\(-0.905763\pi\)
							0.956495	−	0.291750i	\(-0.0942375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.j.b.115.13	✓	64
4.3	odd	2		544.3.n.b.47.26		64
8.3	odd	2	inner	136.3.j.b.115.20	yes	64
8.5	even	2		544.3.n.b.47.25		64
17.4	even	4	inner	136.3.j.b.123.20	yes	64
68.55	odd	4		544.3.n.b.463.25		64
136.21	even	4		544.3.n.b.463.26		64
136.123	odd	4	inner	136.3.j.b.123.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.j.b.115.13	✓	64	1.1	even	1	trivial
136.3.j.b.115.20	yes	64	8.3	odd	2	inner
136.3.j.b.123.13	yes	64	136.123	odd	4	inner
136.3.j.b.123.20	yes	64	17.4	even	4	inner
544.3.n.b.47.25		64	8.5	even	2
544.3.n.b.47.26		64	4.3	odd	2
544.3.n.b.463.25		64	68.55	odd	4
544.3.n.b.463.26		64	136.21	even	4