Embedded newform 136.3.q.a.5.16

• Label: 136.3.q.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.295486 + 1.97805i) q^{2} +(3.57309 - 2.38746i) q^{3} +(-3.82538 - 1.16897i) q^{4} +(5.68249 + 1.13032i) q^{5} +(3.66673 + 7.77323i) q^{6} +(-1.07342 - 5.39645i) q^{7} +(3.44263 - 7.22138i) q^{8} +(3.62286 - 8.74635i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.295486	+	1.97805i	−0.147743	+	0.989026i
							\(3\)	3.57309	−	2.38746i	1.19103	−	0.795822i	0.207798	−	0.978172i	\(-0.433370\pi\)
0.983234	+	0.182350i	\(0.0583705\pi\)
\(5\)	5.68249	+	1.13032i	1.13650	+	0.226064i	0.727259	−	0.686363i	\(-0.240794\pi\)
							0.409239	+	0.912427i	\(0.365794\pi\)
\(7\)	−1.07342	−	5.39645i	−0.153346	−	0.770922i	−0.978538	−	0.206069i	\(-0.933933\pi\)
							0.825192	−	0.564853i	\(-0.191067\pi\)
\(11\)	−7.04483	+	10.5433i	−0.640439	+	0.958484i	0.359243	+	0.933244i	\(0.383035\pi\)
							−0.999681	+	0.0252402i	\(0.991965\pi\)
\(13\)	17.7432	−	17.7432i	1.36486	−	1.36486i	0.497254	−	0.867605i	\(-0.334342\pi\)
							0.867605	−	0.497254i	\(-0.165658\pi\)
\(17\)	−3.74527	+	16.5823i	−0.220310	+	0.975430i
							\(19\)	−4.52423	+	1.87400i	−0.238117	+	0.0986314i	−0.498551	−	0.866860i	\(-0.666134\pi\)
0.260434	+	0.965492i	\(0.416134\pi\)
\(23\)	−22.7001	+	33.9730i	−0.986959	+	1.47709i	−0.111519	+	0.993762i	\(0.535571\pi\)
							−0.875440	+	0.483326i	\(0.839429\pi\)
\(29\)	2.74917	−	13.8210i	0.0947990	−	0.476587i	−0.903997	−	0.427538i	\(-0.859381\pi\)
							0.998796	−	0.0490487i	\(-0.0156189\pi\)
\(31\)	9.53688	−	6.37234i	0.307641	−	0.205559i	−0.392160	−	0.919897i	\(-0.628272\pi\)
							0.699801	+	0.714338i	\(0.253272\pi\)
\(37\)	−15.9204	+	10.6377i	−0.430281	+	0.287505i	−0.751791	−	0.659402i	\(-0.770810\pi\)
							0.321510	+	0.946906i	\(0.395810\pi\)
\(41\)	1.96041	+	9.85567i	0.0478150	+	0.240382i	0.997300	−	0.0734288i	\(-0.0233942\pi\)
							−0.949485	+	0.313811i	\(0.898394\pi\)
\(43\)	−69.4988	−	28.7873i	−1.61625	−	0.669473i	−0.622659	−	0.782494i	\(-0.713947\pi\)
							−0.993593	+	0.113020i	\(0.963947\pi\)
\(47\)	−25.5203	−	25.5203i	−0.542985	−	0.542985i	0.381418	−	0.924403i	\(-0.375436\pi\)
							−0.924403	+	0.381418i	\(0.875436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.16	✓	272
8.5	even	2	inner	136.3.q.a.5.27	yes	272
17.7	odd	16	inner	136.3.q.a.109.27	yes	272
136.109	odd	16	inner	136.3.q.a.109.16	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.16	✓	272	1.1	even	1	trivial
136.3.q.a.5.27	yes	272	8.5	even	2	inner
136.3.q.a.109.16	yes	272	136.109	odd	16	inner
136.3.q.a.109.27	yes	272	17.7	odd	16	inner