Embedded newform 187.2.m.a.131.4

• Label: 187.2.m.a.131.4
• Level: $187$
• Weight: $2$
• Character: 187.131
• 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			131.4
Character	\(\chi\)	\(=\)	187.131
Dual form			187.2.m.a.10.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50318 + 0.622639i) q^{2} +(1.02275 - 0.683380i) q^{3} +(0.457668 - 0.457668i) q^{4} +(0.908813 + 0.180774i) q^{5} +(-1.11188 + 1.66405i) q^{6} +(0.274197 + 1.37848i) q^{7} +(0.842281 - 2.03345i) q^{8} +(-0.569040 + 1.37378i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50318	+	0.622639i	−1.06291	+	0.440272i	−0.844484	−	0.535581i	\(-0.820093\pi\)
							−0.218427	+	0.975853i	\(0.570093\pi\)
\(3\)	1.02275	−	0.683380i	0.590485	−	0.394550i	−0.224123	−	0.974561i	\(-0.571952\pi\)
							0.814609	+	0.580011i	\(0.196952\pi\)
\(5\)	0.908813	+	0.180774i	0.406434	+	0.0808447i	0.394073	−	0.919079i	\(-0.371066\pi\)
							0.0123601	+	0.999924i	\(0.496066\pi\)
\(7\)	0.274197	+	1.37848i	0.103637	+	0.521017i	0.997374	+	0.0724256i	\(0.0230740\pi\)
							−0.893737	+	0.448591i	\(0.851926\pi\)
\(11\)	3.30396	−	0.289591i	0.996181	−	0.0873149i
							\(13\)	1.11553	−	1.11553i	0.309391	−	0.309391i	−0.535282	−	0.844673i	\(-0.679795\pi\)
0.844673	+	0.535282i	\(0.179795\pi\)
\(17\)	3.74754	+	1.71928i	0.908913	+	0.416986i
							\(19\)	−1.09531	+	0.453691i	−0.251281	+	0.104084i	−0.504769	−	0.863255i	\(-0.668422\pi\)
0.253488	+	0.967339i	\(0.418422\pi\)
\(23\)	5.77248	+	3.85705i	1.20364	+	0.804250i	0.985168	−	0.171592i	\(-0.0548912\pi\)
							0.218477	+	0.975842i	\(0.429891\pi\)
\(29\)	0.380882	+	0.0757622i	0.0707281	+	0.0140687i	0.230327	−	0.973113i	\(-0.426020\pi\)
							−0.159599	+	0.987182i	\(0.551020\pi\)
\(31\)	−3.10456	−	4.64630i	−0.557595	−	0.834500i	0.440399	−	0.897802i	\(-0.354837\pi\)
							−0.997994	+	0.0633020i	\(0.979837\pi\)
\(37\)	−0.388302	+	0.259455i	−0.0638365	+	0.0426542i	−0.587079	−	0.809530i	\(-0.699722\pi\)
							0.523242	+	0.852184i	\(0.324722\pi\)
\(41\)	−1.30051	−	6.53809i	−0.203105	−	1.02108i	−0.938983	−	0.343964i	\(-0.888230\pi\)
							0.735878	−	0.677114i	\(-0.236770\pi\)
\(43\)	−9.81295	−	4.06466i	−1.49646	−	0.619855i	−0.523750	−	0.851872i	\(-0.675467\pi\)
							−0.972712	+	0.232018i	\(0.925467\pi\)
\(47\)	−2.38298	−	2.38298i	−0.347593	−	0.347593i	0.511619	−	0.859212i	\(-0.329046\pi\)
							−0.859212	+	0.511619i	\(0.829046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.m.a.131.4	yes	128
11.10	odd	2	inner	187.2.m.a.131.13	yes	128
17.10	odd	16	inner	187.2.m.a.10.13	yes	128
187.10	even	16	inner	187.2.m.a.10.4	✓	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.m.a.10.4	✓	128	187.10	even	16	inner
187.2.m.a.10.13	yes	128	17.10	odd	16	inner
187.2.m.a.131.4	yes	128	1.1	even	1	trivial
187.2.m.a.131.13	yes	128	11.10	odd	2	inner