Embedded newform 187.2.r.a.9.14

• Label: 187.2.r.a.9.14
• Level: $187$
• Weight: $2$
• Character: 187.9
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.r (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			9.14
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.864466 + 1.69661i) q^{2} +(-1.81207 + 0.435039i) q^{3} +(-0.955613 + 1.31529i) q^{4} +(-3.43817 - 0.270590i) q^{5} +(-2.30456 - 2.69830i) q^{6} +(-0.552442 + 2.30109i) q^{7} +(0.703788 + 0.111469i) q^{8} +(0.421312 - 0.214669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.864466	+	1.69661i	0.611270	+	1.19968i	0.964484	+	0.264142i	\(0.0850887\pi\)
							−0.353214	+	0.935542i	\(0.614911\pi\)
\(3\)	−1.81207	+	0.435039i	−1.04620	+	0.251170i	−0.719886	−	0.694092i	\(-0.755806\pi\)
							−0.326312	+	0.945262i	\(0.605806\pi\)
\(5\)	−3.43817	−	0.270590i	−1.53760	−	0.121012i	−0.718850	−	0.695165i	\(-0.755331\pi\)
							−0.818748	+	0.574153i	\(0.805331\pi\)
\(7\)	−0.552442	+	2.30109i	−0.208803	+	0.869729i	0.765309	+	0.643664i	\(0.222586\pi\)
							−0.974112	+	0.226066i	\(0.927414\pi\)
\(11\)	−1.86143	−	2.74501i	−0.561241	−	0.827653i
							\(13\)	−0.667538	−	0.216896i	−0.185142	−	0.0601562i	0.214979	−	0.976619i	\(-0.431032\pi\)
−0.400121	+	0.916463i	\(0.631032\pi\)
\(17\)	−1.24179	+	3.93166i	−0.301179	+	0.953567i
							\(19\)	−1.11327	+	7.02891i	−0.255402	+	1.61254i	0.442786	+	0.896627i	\(0.353990\pi\)
−0.698188	+	0.715915i	\(0.746010\pi\)
\(23\)	7.01751	−	2.90675i	1.46325	−	0.606099i	0.497943	−	0.867210i	\(-0.334089\pi\)
							0.965309	+	0.261111i	\(0.0840890\pi\)
\(29\)	−5.14065	+	3.15019i	−0.954595	+	0.584976i	−0.910259	−	0.414040i	\(-0.864117\pi\)
							−0.0443362	+	0.999017i	\(0.514117\pi\)
\(31\)	−2.00690	+	2.34977i	−0.360449	+	0.422032i	−0.910765	−	0.412925i	\(-0.864507\pi\)
							0.550316	+	0.834957i	\(0.314507\pi\)
\(37\)	2.34072	+	3.81970i	0.384811	+	0.627955i	0.985131	−	0.171805i	\(-0.0549599\pi\)
							−0.600320	+	0.799760i	\(0.704960\pi\)
\(41\)	3.80721	+	2.33306i	0.594586	+	0.364363i	0.787002	−	0.616951i	\(-0.211632\pi\)
							−0.192416	+	0.981313i	\(0.561632\pi\)
\(43\)	−5.02674	−	5.02674i	−0.766571	−	0.766571i	0.210930	−	0.977501i	\(-0.432351\pi\)
							−0.977501	+	0.210930i	\(0.932351\pi\)
\(47\)	−3.15103	−	4.33703i	−0.459626	−	0.632620i	0.514805	−	0.857307i	\(-0.327864\pi\)
							−0.974431	+	0.224687i	\(0.927864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.14	✓	256
11.5	even	5	inner	187.2.r.a.60.3	yes	256
17.2	even	8	inner	187.2.r.a.53.3	yes	256
187.104	even	40	inner	187.2.r.a.104.14	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.14	✓	256	1.1	even	1	trivial
187.2.r.a.53.3	yes	256	17.2	even	8	inner
187.2.r.a.60.3	yes	256	11.5	even	5	inner
187.2.r.a.104.14	yes	256	187.104	even	40	inner