Embedded newform 187.2.m.a.10.2

• Label: 187.2.m.a.10.2
• Level: $187$
• Weight: $2$
• Character: 187.10
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.m (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	187.10
Dual form			187.2.m.a.131.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25162 - 0.932652i) q^{2} +(-2.66909 - 1.78343i) q^{3} +(2.78574 + 2.78574i) q^{4} +(2.54233 - 0.505701i) q^{5} +(4.34645 + 6.50493i) q^{6} +(-0.607350 + 3.05336i) q^{7} +(-1.80901 - 4.36732i) q^{8} +(2.79536 + 6.74860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25162	−	0.932652i	−1.59214	−	0.659484i	−0.601860	−	0.798601i	\(-0.705574\pi\)
							−0.990276	+	0.139117i	\(0.955574\pi\)
\(3\)	−2.66909	−	1.78343i	−1.54100	−	1.02966i	−0.979323	−	0.202303i	\(-0.935157\pi\)
							−0.561674	−	0.827358i	\(-0.689843\pi\)
\(5\)	2.54233	−	0.505701i	1.13697	−	0.226156i	0.409506	−	0.912307i	\(-0.365701\pi\)
							0.727459	+	0.686151i	\(0.240701\pi\)
\(7\)	−0.607350	+	3.05336i	−0.229557	+	1.15406i	0.678302	+	0.734783i	\(0.262716\pi\)
							−0.907859	+	0.419276i	\(0.862284\pi\)
\(11\)	3.29541	−	0.374521i	0.993604	−	0.112922i
							\(13\)	−0.635496	−	0.635496i	−0.176255	−	0.176255i	0.613466	−	0.789721i	\(-0.289775\pi\)
−0.789721	+	0.613466i	\(0.789775\pi\)
\(17\)	4.11200	+	0.302469i	0.997306	+	0.0733596i
							\(19\)	−0.134678	−	0.0557855i	−0.0308973	−	0.0127981i	0.367181	−	0.930149i	\(-0.380323\pi\)
−0.398079	+	0.917351i	\(0.630323\pi\)
\(23\)	−2.79819	+	1.86969i	−0.583462	+	0.389857i	−0.811982	−	0.583682i	\(-0.801611\pi\)
							0.228520	+	0.973539i	\(0.426611\pi\)
\(29\)	4.51218	−	0.897529i	0.837892	−	0.166667i	0.242549	−	0.970139i	\(-0.422016\pi\)
							0.595342	+	0.803472i	\(0.297016\pi\)
\(31\)	−1.47824	+	2.21234i	−0.265499	+	0.397348i	−0.940135	−	0.340804i	\(-0.889301\pi\)
							0.674635	+	0.738151i	\(0.264301\pi\)
\(37\)	6.78648	+	4.53458i	1.11569	+	0.745480i	0.969820	−	0.243823i	\(-0.0784018\pi\)
							0.145870	+	0.989304i	\(0.453402\pi\)
\(41\)	0.970630	−	4.87969i	0.151587	−	0.762079i	−0.827948	−	0.560805i	\(-0.810492\pi\)
							0.979535	−	0.201274i	\(-0.0645082\pi\)
\(43\)	5.85136	−	2.42371i	0.892324	−	0.369613i	0.111060	−	0.993814i	\(-0.464575\pi\)
							0.781264	+	0.624201i	\(0.214575\pi\)
\(47\)	6.83807	−	6.83807i	0.997435	−	0.997435i	−0.00256132	−	0.999997i	\(-0.500815\pi\)
							0.999997	+	0.00256132i	\(0.000815296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.m.a.10.2	✓	128
11.10	odd	2	inner	187.2.m.a.10.15	yes	128
17.12	odd	16	inner	187.2.m.a.131.15	yes	128
187.131	even	16	inner	187.2.m.a.131.2	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.m.a.10.2	✓	128	1.1	even	1	trivial
187.2.m.a.10.15	yes	128	11.10	odd	2	inner
187.2.m.a.131.2	yes	128	187.131	even	16	inner
187.2.m.a.131.15	yes	128	17.12	odd	16	inner