Embedded newform 136.3.q.a.5.18

• Label: 136.3.q.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0311599 - 1.99976i) q^{2} +(-1.32480 + 0.885206i) q^{3} +(-3.99806 + 0.124624i) q^{4} +(-0.988766 - 0.196678i) q^{5} +(1.81148 + 2.62170i) q^{6} +(-1.85512 - 9.32633i) q^{7} +(0.373797 + 7.99126i) q^{8} +(-2.47263 + 5.96947i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0311599	−	1.99976i	−0.0155799	−	0.999879i
							\(3\)	−1.32480	+	0.885206i	−0.441602	+	0.295069i	−0.756424	−	0.654082i	\(-0.773055\pi\)
0.314822	+	0.949151i	\(0.398055\pi\)
\(5\)	−0.988766	−	0.196678i	−0.197753	−	0.0393356i	0.0952198	−	0.995456i	\(-0.469645\pi\)
							−0.292973	+	0.956121i	\(0.594645\pi\)
\(7\)	−1.85512	−	9.32633i	−0.265018	−	1.33233i	−0.852343	−	0.522983i	\(-0.824819\pi\)
							0.587326	−	0.809351i	\(-0.300181\pi\)
\(11\)	−6.78959	+	10.1613i	−0.617235	+	0.923758i	−1.00000		8.79020e-5i	\(-0.999972\pi\)
							0.382765	+	0.923846i	\(0.374972\pi\)
\(13\)	−14.9740	+	14.9740i	−1.15185	+	1.15185i	−0.165668	+	0.986182i	\(0.552978\pi\)
							−0.986182	+	0.165668i	\(0.947022\pi\)
\(17\)	−14.6701	−	8.58995i	−0.862949	−	0.505291i
							\(19\)	6.84156	−	2.83387i	0.360082	−	0.149151i	−0.195305	−	0.980742i	\(-0.562570\pi\)
0.555388	+	0.831591i	\(0.312570\pi\)
\(23\)	4.86797	−	7.28543i	0.211651	−	0.316758i	−0.710421	−	0.703777i	\(-0.751495\pi\)
							0.922072	+	0.387019i	\(0.126495\pi\)
\(29\)	5.68793	−	28.5951i	0.196135	−	0.986039i	−0.749795	−	0.661670i	\(-0.769848\pi\)
							0.945931	−	0.324369i	\(-0.105152\pi\)
\(31\)	29.6566	−	19.8159i	0.956665	−	0.639223i	0.0239027	−	0.999714i	\(-0.492391\pi\)
							0.932763	+	0.360491i	\(0.117391\pi\)
\(37\)	−42.6288	+	28.4837i	−1.15213	+	0.769829i	−0.976689	−	0.214661i	\(-0.931135\pi\)
							−0.175442	+	0.984490i	\(0.556135\pi\)
\(41\)	13.7808	+	69.2808i	0.336117	+	1.68977i	0.666160	+	0.745809i	\(0.267937\pi\)
							−0.330043	+	0.943966i	\(0.607063\pi\)
\(43\)	−63.5359	−	26.3174i	−1.47758	−	0.612033i	−0.509006	−	0.860763i	\(-0.669987\pi\)
							−0.968573	+	0.248730i	\(0.919987\pi\)
\(47\)	−9.92836	−	9.92836i	−0.211242	−	0.211242i	0.593553	−	0.804795i	\(-0.297725\pi\)
							−0.804795	+	0.593553i	\(0.797725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.18	yes	272
8.5	even	2	inner	136.3.q.a.5.9	✓	272
17.7	odd	16	inner	136.3.q.a.109.9	yes	272
136.109	odd	16	inner	136.3.q.a.109.18	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.9	✓	272	8.5	even	2	inner
136.3.q.a.5.18	yes	272	1.1	even	1	trivial
136.3.q.a.109.9	yes	272	17.7	odd	16	inner
136.3.q.a.109.18	yes	272	136.109	odd	16	inner