Embedded newform 136.3.q.a.5.19

• Label: 136.3.q.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.108110 + 1.99708i) q^{2} +(-3.69899 + 2.47159i) q^{3} +(-3.97662 + 0.431807i) q^{4} +(-1.02401 - 0.203689i) q^{5} +(-5.33584 - 7.11996i) q^{6} +(0.149919 + 0.753693i) q^{7} +(-1.29226 - 7.89494i) q^{8} +(4.12963 - 9.96982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.108110	+	1.99708i	0.0540549	+	0.998538i
							\(3\)	−3.69899	+	2.47159i	−1.23300	+	0.823862i	−0.989287	−	0.145986i	\(-0.953364\pi\)
−0.243710	+	0.969848i	\(0.578364\pi\)
\(5\)	−1.02401	−	0.203689i	−0.204803	−	0.0407378i	0.0916228	−	0.995794i	\(-0.470795\pi\)
							−0.296425	+	0.955056i	\(0.595795\pi\)
\(7\)	0.149919	+	0.753693i	0.0214170	+	0.107670i	0.990015	−	0.140961i	\(-0.0450192\pi\)
							−0.968598	+	0.248631i	\(0.920019\pi\)
\(11\)	−2.30941	+	3.45627i	−0.209946	+	0.314206i	−0.921466	−	0.388460i	\(-0.873007\pi\)
							0.711520	+	0.702666i	\(0.248007\pi\)
\(13\)	4.67837	−	4.67837i	0.359875	−	0.359875i	−0.503892	−	0.863767i	\(-0.668099\pi\)
							0.863767	+	0.503892i	\(0.168099\pi\)
\(17\)	−8.07486	−	14.9598i	−0.474992	−	0.879990i
							\(19\)	−14.3345	+	5.93756i	−0.754450	+	0.312503i	−0.726556	−	0.687108i	\(-0.758880\pi\)
−0.0278940	+	0.999611i	\(0.508880\pi\)
\(23\)	−3.98061	+	5.95740i	−0.173070	+	0.259017i	−0.907857	−	0.419281i	\(-0.862282\pi\)
							0.734787	+	0.678298i	\(0.237282\pi\)
\(29\)	3.27357	−	16.4573i	0.112882	−	0.567495i	−0.882403	−	0.470495i	\(-0.844075\pi\)
							0.995284	−	0.0969997i	\(-0.0309246\pi\)
\(31\)	−46.7745	+	31.2537i	−1.50885	+	1.00818i	−0.520869	+	0.853636i	\(0.674392\pi\)
							−0.987985	+	0.154548i	\(0.950608\pi\)
\(37\)	−29.6361	+	19.8022i	−0.800975	+	0.535194i	−0.887365	−	0.461067i	\(-0.847467\pi\)
							0.0863906	+	0.996261i	\(0.472467\pi\)
\(41\)	−1.09896	−	5.52484i	−0.0268039	−	0.134752i	0.965066	−	0.262006i	\(-0.0843839\pi\)
							−0.991870	+	0.127253i	\(0.959384\pi\)
\(43\)	10.6620	+	4.41635i	0.247954	+	0.102706i	0.503199	−	0.864170i	\(-0.332156\pi\)
							−0.255245	+	0.966876i	\(0.582156\pi\)
\(47\)	−22.3489	−	22.3489i	−0.475509	−	0.475509i	0.428183	−	0.903692i	\(-0.359154\pi\)
							−0.903692	+	0.428183i	\(0.859154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.19	✓	272
8.5	even	2	inner	136.3.q.a.5.26	yes	272
17.7	odd	16	inner	136.3.q.a.109.26	yes	272
136.109	odd	16	inner	136.3.q.a.109.19	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.19	✓	272	1.1	even	1	trivial
136.3.q.a.5.26	yes	272	8.5	even	2	inner
136.3.q.a.109.19	yes	272	136.109	odd	16	inner
136.3.q.a.109.26	yes	272	17.7	odd	16	inner