Embedded newform 136.3.q.a.5.15

• Label: 136.3.q.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.301940 + 1.97708i) q^{2} +(2.58349 - 1.72623i) q^{3} +(-3.81766 - 1.19392i) q^{4} +(-9.15038 - 1.82012i) q^{5} +(2.63284 + 5.62898i) q^{6} +(-1.42493 - 7.16362i) q^{7} +(3.51317 - 7.18732i) q^{8} +(0.250398 - 0.604514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.301940	+	1.97708i	−0.150970	+	0.988538i
							\(3\)	2.58349	−	1.72623i	0.861164	−	0.575412i	−0.0446901	−	0.999001i	\(-0.514230\pi\)
0.905854	+	0.423589i	\(0.139230\pi\)
\(5\)	−9.15038	−	1.82012i	−1.83008	−	0.364025i	−0.844812	−	0.535063i	\(-0.820288\pi\)
							−0.985264	+	0.171038i	\(0.945288\pi\)
\(7\)	−1.42493	−	7.16362i	−0.203562	−	1.02337i	−0.938510	−	0.345251i	\(-0.887794\pi\)
							0.734949	−	0.678123i	\(-0.237206\pi\)
\(11\)	9.54225	−	14.2810i	0.867477	−	1.29827i	−0.0858422	−	0.996309i	\(-0.527358\pi\)
							0.953320	−	0.301963i	\(-0.0976419\pi\)
\(13\)	−7.08179	+	7.08179i	−0.544753	+	0.544753i	−0.924918	−	0.380166i	\(-0.875867\pi\)
							0.380166	+	0.924918i	\(0.375867\pi\)
\(17\)	−15.0044	−	7.99178i	−0.882611	−	0.470105i
							\(19\)	−17.6734	+	7.32057i	−0.930181	+	0.385293i	−0.795747	−	0.605629i	\(-0.792921\pi\)
−0.134434	+	0.990923i	\(0.542921\pi\)
\(23\)	11.9878	−	17.9410i	0.521208	−	0.780043i	−0.473715	−	0.880678i	\(-0.657087\pi\)
							0.994923	+	0.100635i	\(0.0320874\pi\)
\(29\)	3.06709	−	15.4193i	0.105762	−	0.531700i	−0.891187	−	0.453637i	\(-0.850126\pi\)
							0.996948	−	0.0780632i	\(-0.0248736\pi\)
\(31\)	12.5684	−	8.39793i	0.405432	−	0.270901i	−0.336089	−	0.941830i	\(-0.609104\pi\)
							0.741522	+	0.670929i	\(0.234104\pi\)
\(37\)	−29.4483	+	19.6767i	−0.795899	+	0.531803i	−0.885756	−	0.464152i	\(-0.846359\pi\)
							0.0898563	+	0.995955i	\(0.471359\pi\)
\(41\)	−9.07734	−	45.6349i	−0.221398	−	1.11305i	−0.918302	−	0.395880i	\(-0.870439\pi\)
							0.696904	−	0.717165i	\(-0.254561\pi\)
\(43\)	−17.0731	−	7.07192i	−0.397050	−	0.164463i	0.175219	−	0.984529i	\(-0.443937\pi\)
							−0.572269	+	0.820066i	\(0.693937\pi\)
\(47\)	15.9079	+	15.9079i	0.338467	+	0.338467i	0.855790	−	0.517323i	\(-0.173072\pi\)
							−0.517323	+	0.855790i	\(0.673072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.15	✓	272
8.5	even	2	inner	136.3.q.a.5.28	yes	272
17.7	odd	16	inner	136.3.q.a.109.28	yes	272
136.109	odd	16	inner	136.3.q.a.109.15	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.15	✓	272	1.1	even	1	trivial
136.3.q.a.5.28	yes	272	8.5	even	2	inner
136.3.q.a.109.15	yes	272	136.109	odd	16	inner
136.3.q.a.109.28	yes	272	17.7	odd	16	inner