Embedded newform 187.2.j.a.16.7

• Label: 187.2.j.a.16.7
• Level: $187$
• Weight: $2$
• Character: 187.16
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			16.7
Character	\(\chi\)	\(=\)	187.16
Dual form			187.2.j.a.152.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.150351 - 0.462733i) q^{2} +(-1.01595 + 1.39834i) q^{3} +(1.42652 - 1.03643i) q^{4} +(0.411135 + 0.133586i) q^{5} +(0.799806 + 0.259873i) q^{6} +(1.36610 + 1.88027i) q^{7} +(-1.48131 - 1.07624i) q^{8} +(0.00385940 + 0.0118780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.150351	−	0.462733i	−0.106314	−	0.327201i	0.883722	−	0.468011i	\(-0.155029\pi\)
							−0.990037	+	0.140810i	\(0.955029\pi\)
\(3\)	−1.01595	+	1.39834i	−0.586560	+	0.807331i	−0.994395	−	0.105724i	\(-0.966284\pi\)
							0.407835	+	0.913056i	\(0.366284\pi\)
\(5\)	0.411135	+	0.133586i	0.183865	+	0.0597414i	0.399502	−	0.916732i	\(-0.369183\pi\)
							−0.215637	+	0.976474i	\(0.569183\pi\)
\(7\)	1.36610	+	1.88027i	0.516336	+	0.710676i	0.984972	−	0.172716i	\(-0.0552542\pi\)
							−0.468636	+	0.883392i	\(0.655254\pi\)
\(11\)	0.896081	−	3.19328i	0.270178	−	0.962810i
							\(13\)	1.55373	+	4.78190i	0.430928	+	1.32626i	0.897202	+	0.441620i	\(0.145596\pi\)
−0.466274	+	0.884640i	\(0.654404\pi\)
\(17\)	3.49633	−	2.18532i	0.847986	−	0.530019i
							\(19\)	3.47951	+	2.52801i	0.798255	+	0.579966i	0.910402	−	0.413726i	\(-0.135773\pi\)
−0.112147	+	0.993692i	\(0.535773\pi\)
\(23\)			2.37337i			0.494882i	0.968903	+	0.247441i	\(0.0795896\pi\)
							−0.968903	+	0.247441i	\(0.920410\pi\)
\(29\)	−2.24111	−	3.08463i	−0.416164	−	0.572801i	0.548544	−	0.836122i	\(-0.315182\pi\)
							−0.964708	+	0.263320i	\(0.915182\pi\)
\(31\)	−8.80885	+	2.86217i	−1.58212	+	0.514061i	−0.962601	−	0.270924i	\(-0.912671\pi\)
							−0.619515	+	0.784985i	\(0.712671\pi\)
\(37\)	−2.44435	−	3.36436i	−0.401849	−	0.553097i	0.559358	−	0.828926i	\(-0.311048\pi\)
							−0.961207	+	0.275829i	\(0.911048\pi\)
\(41\)	6.87388	−	9.46108i	1.07352	−	1.47757i	0.207059	−	0.978329i	\(-0.433611\pi\)
							0.866461	−	0.499245i	\(-0.166389\pi\)
\(43\)	−6.49757			−0.990870			−0.495435	−	0.868645i	\(-0.664991\pi\)
							−0.495435	+	0.868645i	\(0.664991\pi\)
\(47\)	−4.92926	−	3.58132i	−0.719007	−	0.522389i	0.167060	−	0.985947i	\(-0.446573\pi\)
							−0.886067	+	0.463558i	\(0.846573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.16.7	✓	64
11.9	even	5	inner	187.2.j.a.152.8	yes	64
17.16	even	2	inner	187.2.j.a.16.8	yes	64
187.152	even	10	inner	187.2.j.a.152.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.16.7	✓	64	1.1	even	1	trivial
187.2.j.a.16.8	yes	64	17.16	even	2	inner
187.2.j.a.152.7	yes	64	187.152	even	10	inner
187.2.j.a.152.8	yes	64	11.9	even	5	inner