Embedded newform 136.3.q.a.5.20

• Label: 136.3.q.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.506868 + 1.93471i) q^{2} +(-0.155374 + 0.103818i) q^{3} +(-3.48617 + 1.96128i) q^{4} +(6.64253 + 1.32128i) q^{5} +(-0.279611 - 0.247982i) q^{6} +(0.946911 + 4.76044i) q^{7} +(-5.56153 - 5.75060i) q^{8} +(-3.43079 + 8.28265i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.506868	+	1.93471i	0.253434	+	0.967353i
							\(3\)	−0.155374	+	0.103818i	−0.0517914	+	0.0346059i	−0.581196	−	0.813763i	\(-0.697415\pi\)
0.529405	+	0.848369i	\(0.322415\pi\)
\(5\)	6.64253	+	1.32128i	1.32851	+	0.264256i	0.807789	−	0.589472i	\(-0.200664\pi\)
							0.520718	+	0.853729i	\(0.325664\pi\)
\(7\)	0.946911	+	4.76044i	0.135273	+	0.680063i	0.987592	+	0.157041i	\(0.0501955\pi\)
							−0.852319	+	0.523022i	\(0.824805\pi\)
\(11\)	7.90483	−	11.8304i	0.718621	−	1.07549i	−0.274858	−	0.961485i	\(-0.588631\pi\)
							0.993480	−	0.114008i	\(-0.0363690\pi\)
\(13\)	−11.3789	+	11.3789i	−0.875298	+	0.875298i	−0.993044	−	0.117745i	\(-0.962433\pi\)
							0.117745	+	0.993044i	\(0.462433\pi\)
\(17\)	4.35954	+	16.4315i	0.256443	+	0.966559i
							\(19\)	−13.3977	+	5.54953i	−0.705144	+	0.292080i	−0.706294	−	0.707919i	\(-0.749634\pi\)
0.00114941	+	0.999999i	\(0.499634\pi\)
\(23\)	16.7616	−	25.0854i	0.728763	−	1.09067i	−0.263273	−	0.964721i	\(-0.584802\pi\)
							0.992037	−	0.125950i	\(-0.0401979\pi\)
\(29\)	7.53036	−	37.8577i	0.259667	−	1.30544i	−0.602217	−	0.798332i	\(-0.705716\pi\)
							0.861885	−	0.507104i	\(-0.169284\pi\)
\(31\)	17.8790	−	11.9464i	0.576741	−	0.385366i	−0.232716	−	0.972545i	\(-0.574761\pi\)
							0.809458	+	0.587178i	\(0.199761\pi\)
\(37\)	53.6487	−	35.8469i	1.44996	−	0.968835i	0.452956	−	0.891533i	\(-0.350369\pi\)
							0.997008	−	0.0773022i	\(-0.0246306\pi\)
\(41\)	10.7940	+	54.2652i	0.263269	+	1.32354i	0.855512	+	0.517782i	\(0.173242\pi\)
							−0.592244	+	0.805759i	\(0.701758\pi\)
\(43\)	11.7208	+	4.85493i	0.272578	+	0.112905i	0.514785	−	0.857319i	\(-0.327872\pi\)
							−0.242208	+	0.970224i	\(0.577872\pi\)
\(47\)	−8.36611	−	8.36611i	−0.178002	−	0.178002i	0.612482	−	0.790484i	\(-0.290171\pi\)
							−0.790484	+	0.612482i	\(0.790171\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.20	✓	272
8.5	even	2	inner	136.3.q.a.5.23	yes	272
17.7	odd	16	inner	136.3.q.a.109.23	yes	272
136.109	odd	16	inner	136.3.q.a.109.20	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.20	✓	272	1.1	even	1	trivial
136.3.q.a.5.23	yes	272	8.5	even	2	inner
136.3.q.a.109.20	yes	272	136.109	odd	16	inner
136.3.q.a.109.23	yes	272	17.7	odd	16	inner