Embedded newform 136.3.q.a.5.1

• Label: 136.3.q.a.5.1
• Level: $136$
• Weight: $3$
• Character: 136.5
• Analytic conductor: $3.706$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	136.q (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70573159530\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(34\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99615 - 0.123996i) q^{2} +(-2.49131 + 1.66464i) q^{3} +(3.96925 + 0.495030i) q^{4} +(-5.28331 - 1.05092i) q^{5} +(5.17944 - 3.01396i) q^{6} +(0.430227 + 2.16290i) q^{7} +(-7.86185 - 1.48033i) q^{8} +(-0.00856088 + 0.0206678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\)	\(69\)	\(103\)	\(105\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99615	−	0.123996i	−0.998076	−	0.0619980i
							\(3\)	−2.49131	+	1.66464i	−0.830436	+	0.554879i	−0.896555	−	0.442932i	\(-0.853938\pi\)
0.0661194	+	0.997812i	\(0.478938\pi\)
\(5\)	−5.28331	−	1.05092i	−1.05666	−	0.210183i	−0.363966	−	0.931412i	\(-0.618578\pi\)
							−0.692697	+	0.721229i	\(0.743578\pi\)
\(7\)	0.430227	+	2.16290i	0.0614610	+	0.308986i	0.999272	−	0.0381620i	\(-0.0121503\pi\)
							−0.937811	+	0.347148i	\(0.887150\pi\)
\(11\)	7.82631	−	11.7129i	0.711483	−	1.06481i	−0.282919	−	0.959144i	\(-0.591303\pi\)
							0.994402	−	0.105666i	\(-0.0336974\pi\)
\(13\)	1.09364	−	1.09364i	0.0841258	−	0.0841258i	−0.663792	−	0.747917i	\(-0.731054\pi\)
							0.747917	+	0.663792i	\(0.231054\pi\)
\(17\)	3.88499	−	16.5501i	0.228529	−	0.973537i
							\(19\)	23.1056	−	9.57066i	1.21609	−	0.503719i	0.319922	−	0.947444i	\(-0.396343\pi\)
0.896163	+	0.443725i	\(0.146343\pi\)
\(23\)	10.3598	−	15.5046i	0.450428	−	0.674113i	−0.534875	−	0.844931i	\(-0.679641\pi\)
							0.985302	+	0.170819i	\(0.0546413\pi\)
\(29\)	1.01077	−	5.08147i	0.0348541	−	0.175223i	−0.959437	−	0.281922i	\(-0.909028\pi\)
							0.994291	+	0.106699i	\(0.0340281\pi\)
\(31\)	7.79835	−	5.21069i	0.251560	−	0.168087i	−0.423397	−	0.905944i	\(-0.639162\pi\)
							0.674956	+	0.737858i	\(0.264162\pi\)
\(37\)	−43.5483	+	29.0980i	−1.17698	+	0.786433i	−0.980968	−	0.194168i	\(-0.937799\pi\)
							−0.196012	+	0.980601i	\(0.562799\pi\)
\(41\)	7.56567	+	38.0352i	0.184529	+	0.927688i	0.956434	+	0.291950i	\(0.0943041\pi\)
							−0.771905	+	0.635738i	\(0.780696\pi\)
\(43\)	21.7637	+	9.01480i	0.506131	+	0.209647i	0.621113	−	0.783721i	\(-0.286681\pi\)
							−0.114981	+	0.993368i	\(0.536681\pi\)
\(47\)	−21.5201	−	21.5201i	−0.457873	−	0.457873i	0.440083	−	0.897957i	\(-0.354949\pi\)
							−0.897957	+	0.440083i	\(0.854949\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.1	✓	272
8.5	even	2	inner	136.3.q.a.5.25	yes	272
17.7	odd	16	inner	136.3.q.a.109.25	yes	272
136.109	odd	16	inner	136.3.q.a.109.1	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.1	✓	272	1.1	even	1	trivial
136.3.q.a.5.25	yes	272	8.5	even	2	inner
136.3.q.a.109.1	yes	272	136.109	odd	16	inner
136.3.q.a.109.25	yes	272	17.7	odd	16	inner