Embedded newform 136.3.q.a.5.3

• Label: 136.3.q.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93868 + 0.491460i) q^{2} +(1.98785 - 1.32824i) q^{3} +(3.51693 - 1.90556i) q^{4} +(8.85748 + 1.76186i) q^{5} +(-3.20102 + 3.55198i) q^{6} +(-1.81827 - 9.14104i) q^{7} +(-5.88169 + 5.42271i) q^{8} +(-1.25682 + 3.03423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93868	+	0.491460i	−0.969338	+	0.245730i
							\(3\)	1.98785	−	1.32824i	0.662617	−	0.442747i	−0.178252	−	0.983985i	\(-0.557044\pi\)
0.840869	+	0.541238i	\(0.182044\pi\)
\(5\)	8.85748	+	1.76186i	1.77150	+	0.352372i	0.969525	−	0.244991i	\(-0.0787851\pi\)
							0.801970	+	0.597364i	\(0.203785\pi\)
\(7\)	−1.81827	−	9.14104i	−0.259752	−	1.30586i	−0.861737	−	0.507355i	\(-0.830623\pi\)
							0.601985	−	0.798508i	\(-0.294377\pi\)
\(11\)	4.05592	−	6.07011i	0.368720	−	0.551828i	−0.599995	−	0.800004i	\(-0.704831\pi\)
							0.968715	+	0.248175i	\(0.0798309\pi\)
\(13\)	−10.8181	+	10.8181i	−0.832161	+	0.832161i	−0.987812	−	0.155651i	\(-0.950253\pi\)
							0.155651	+	0.987812i	\(0.450253\pi\)
\(17\)	8.91188	−	14.4768i	0.524228	−	0.851578i
							\(19\)	10.7136	−	4.43770i	0.563871	−	0.233563i	−0.0824937	−	0.996592i	\(-0.526288\pi\)
0.646365	+	0.763029i	\(0.276288\pi\)
\(23\)	−9.74282	+	14.5812i	−0.423601	+	0.633963i	−0.980477	−	0.196632i	\(-0.937000\pi\)
							0.556877	+	0.830595i	\(0.312000\pi\)
\(29\)	3.51239	−	17.6580i	0.121117	−	0.608895i	−0.871778	−	0.489901i	\(-0.837033\pi\)
							0.992895	−	0.118994i	\(-0.0379671\pi\)
\(31\)	−1.68909	+	1.12862i	−0.0544869	+	0.0364070i	−0.582515	−	0.812820i	\(-0.697931\pi\)
							0.528028	+	0.849227i	\(0.322931\pi\)
\(37\)	−53.0926	+	35.4754i	−1.43494	+	0.958794i	−0.436689	+	0.899612i	\(0.643849\pi\)
							−0.998247	+	0.0591814i	\(0.981151\pi\)
\(41\)	−6.16975	−	31.0174i	−0.150482	−	0.756522i	−0.980149	−	0.198264i	\(-0.936470\pi\)
							0.829667	−	0.558259i	\(-0.188530\pi\)
\(43\)	23.8153	+	9.86462i	0.553844	+	0.229410i	0.642010	−	0.766696i	\(-0.278101\pi\)
							−0.0881662	+	0.996106i	\(0.528101\pi\)
\(47\)	−26.7248	−	26.7248i	−0.568613	−	0.568613i	0.363127	−	0.931740i	\(-0.381709\pi\)
							−0.931740	+	0.363127i	\(0.881709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.3	✓	272
8.5	even	2	inner	136.3.q.a.5.29	yes	272
17.7	odd	16	inner	136.3.q.a.109.29	yes	272
136.109	odd	16	inner	136.3.q.a.109.3	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.3	✓	272	1.1	even	1	trivial
136.3.q.a.5.29	yes	272	8.5	even	2	inner
136.3.q.a.109.3	yes	272	136.109	odd	16	inner
136.3.q.a.109.29	yes	272	17.7	odd	16	inner