Embedded newform 136.3.q.a.5.11

• Label: 136.3.q.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	136.5
Dual form			136.3.q.a.109.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18540 + 1.61084i) q^{2} +(-2.34076 + 1.56405i) q^{3} +(-1.18963 - 3.81900i) q^{4} +(0.703425 + 0.139920i) q^{5} +(0.255314 - 5.62462i) q^{6} +(-1.92692 - 9.68727i) q^{7} +(7.56201 + 2.61075i) q^{8} +(-0.411234 + 0.992807i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18540	+	1.61084i	−0.592702	+	0.805422i
							\(3\)	−2.34076	+	1.56405i	−0.780253	+	0.521349i	−0.880746	−	0.473589i	\(-0.842958\pi\)
0.100493	+	0.994938i	\(0.467958\pi\)
\(5\)	0.703425	+	0.139920i	0.140685	+	0.0279840i	0.264930	−	0.964268i	\(-0.414651\pi\)
							−0.124245	+	0.992252i	\(0.539651\pi\)
\(7\)	−1.92692	−	9.68727i	−0.275274	−	1.38390i	−0.832726	−	0.553686i	\(-0.813221\pi\)
							0.557452	−	0.830209i	\(-0.311779\pi\)
\(11\)	4.90202	−	7.33639i	0.445638	−	0.666945i	−0.538849	−	0.842403i	\(-0.681141\pi\)
							0.984487	+	0.175458i	\(0.0561406\pi\)
\(13\)	5.85176	−	5.85176i	0.450136	−	0.450136i	−0.445264	−	0.895399i	\(-0.646890\pi\)
							0.895399	+	0.445264i	\(0.146890\pi\)
\(17\)	11.3488	+	12.6572i	0.667576	+	0.744541i
							\(19\)	15.6222	−	6.47092i	0.822220	−	0.340575i	0.0684020	−	0.997658i	\(-0.478210\pi\)
0.753818	+	0.657083i	\(0.228210\pi\)
\(23\)	17.1858	−	25.7204i	0.747209	−	1.11828i	−0.241786	−	0.970330i	\(-0.577733\pi\)
							0.988995	−	0.147948i	\(-0.0472669\pi\)
\(29\)	−0.272819	+	1.37155i	−0.00940754	+	0.0472949i	−0.985204	−	0.171386i	\(-0.945175\pi\)
							0.975796	+	0.218681i	\(0.0701754\pi\)
\(31\)	2.05019	−	1.36989i	0.0661352	−	0.0441901i	−0.522063	−	0.852907i	\(-0.674837\pi\)
							0.588198	+	0.808717i	\(0.299837\pi\)
\(37\)	26.0875	−	17.4311i	0.705067	−	0.471111i	−0.150629	−	0.988590i	\(-0.548130\pi\)
							0.855695	+	0.517480i	\(0.173130\pi\)
\(41\)	−8.51838	−	42.8248i	−0.207765	−	1.04451i	−0.934058	−	0.357121i	\(-0.883758\pi\)
							0.726293	−	0.687386i	\(-0.241242\pi\)
\(43\)	−42.8505	−	17.7493i	−0.996524	−	0.412774i	−0.176003	−	0.984390i	\(-0.556317\pi\)
							−0.820521	+	0.571616i	\(0.806317\pi\)
\(47\)	−32.7874	−	32.7874i	−0.697604	−	0.697604i	0.266289	−	0.963893i	\(-0.414202\pi\)
							−0.963893	+	0.266289i	\(0.914202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.3.q.a.5.11	✓	272
8.5	even	2	inner	136.3.q.a.5.33	yes	272
17.7	odd	16	inner	136.3.q.a.109.33	yes	272
136.109	odd	16	inner	136.3.q.a.109.11	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.3.q.a.5.11	✓	272	1.1	even	1	trivial
136.3.q.a.5.33	yes	272	8.5	even	2	inner
136.3.q.a.109.11	yes	272	136.109	odd	16	inner
136.3.q.a.109.33	yes	272	17.7	odd	16	inner