Embedded newform 136.3.j.b.115.14

• Label: 136.3.j.b.115.14
• 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			115.14
Character	\(\chi\)	\(=\)	136.115
Dual form			136.3.j.b.123.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.678905 - 1.88125i) q^{2} +(3.22659 + 3.22659i) q^{3} +(-3.07817 + 2.55438i) q^{4} +(-4.60664 + 4.60664i) q^{5} +(3.87946 - 8.26055i) q^{6} +(-3.51075 - 3.51075i) q^{7} +(6.89520 + 4.05662i) q^{8} +11.8217i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 12 q^{4} + 26 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.678905	−	1.88125i	−0.339453	−	0.940623i
							\(3\)	3.22659	+	3.22659i	1.07553	+	1.07553i	0.996904	+	0.0786248i	\(0.0250529\pi\)
0.0786248	+	0.996904i	\(0.474947\pi\)
\(5\)	−4.60664	+	4.60664i	−0.921328	+	0.921328i	−0.997123	−	0.0757953i	\(-0.975850\pi\)
							0.0757953	+	0.997123i	\(0.475850\pi\)
\(7\)	−3.51075	−	3.51075i	−0.501536	−	0.501536i	0.410379	−	0.911915i	\(-0.365396\pi\)
							−0.911915	+	0.410379i	\(0.865396\pi\)
\(11\)	−7.92177	+	7.92177i	−0.720161	+	0.720161i	−0.968638	−	0.248477i	\(-0.920070\pi\)
							0.248477	+	0.968638i	\(0.420070\pi\)
\(13\)			14.5417i			1.11859i	0.828969	+	0.559295i	\(0.188928\pi\)
							−0.828969	+	0.559295i	\(0.811072\pi\)
\(17\)	8.98632	−	14.4307i	0.528607	−	0.848867i
							\(19\)			15.3710i			0.809002i	0.914538	+	0.404501i	\(0.132555\pi\)
−0.914538	+	0.404501i	\(0.867445\pi\)
\(23\)	17.2188	+	17.2188i	0.748644	+	0.748644i	0.974225	−	0.225580i	\(-0.0724279\pi\)
							−0.225580	+	0.974225i	\(0.572428\pi\)
\(29\)	−0.205362	+	0.205362i	−0.00708145	+	0.00708145i	−0.710639	−	0.703557i	\(-0.751594\pi\)
							0.703557	+	0.710639i	\(0.251594\pi\)
\(31\)	38.4634	−	38.4634i	1.24076	−	1.24076i	0.281067	−	0.959688i	\(-0.409312\pi\)
							0.959688	−	0.281067i	\(-0.0906884\pi\)
\(37\)	36.5781	−	36.5781i	0.988598	−	0.988598i	−0.0113378	−	0.999936i	\(-0.503609\pi\)
							0.999936	+	0.0113378i	\(0.00360900\pi\)
\(41\)	−42.1320	+	42.1320i	−1.02761	+	1.02761i	−0.0280011	+	0.999608i	\(0.508914\pi\)
							−0.999608	+	0.0280011i	\(0.991086\pi\)
\(43\)		−	6.08831i		−	0.141589i	−0.997491	−	0.0707943i	\(-0.977447\pi\)
							0.997491	−	0.0707943i	\(-0.0225534\pi\)
\(47\)			47.8791i			1.01870i	0.860558	+	0.509352i	\(0.170115\pi\)
							−0.860558	+	0.509352i	\(0.829885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.j.b.115.14	✓	64
4.3	odd	2		544.3.n.b.47.4		64
8.3	odd	2	inner	136.3.j.b.115.19	yes	64
8.5	even	2		544.3.n.b.47.3		64
17.4	even	4	inner	136.3.j.b.123.19	yes	64
68.55	odd	4		544.3.n.b.463.3		64
136.21	even	4		544.3.n.b.463.4		64
136.123	odd	4	inner	136.3.j.b.123.14	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.j.b.115.14	✓	64	1.1	even	1	trivial
136.3.j.b.115.19	yes	64	8.3	odd	2	inner
136.3.j.b.123.14	yes	64	136.123	odd	4	inner
136.3.j.b.123.19	yes	64	17.4	even	4	inner
544.3.n.b.47.3		64	8.5	even	2
544.3.n.b.47.4		64	4.3	odd	2
544.3.n.b.463.3		64	68.55	odd	4
544.3.n.b.463.4		64	136.21	even	4