Embedded newform 19.9.f.a.3.6

• Label: 19.9.f.a.3.6
• Level: $19$
• Weight: $9$
• Character: 19.3
• Analytic conductor: $7.740$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.74019359116\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.6
Character	\(\chi\)	\(=\)	19.3
Dual form			19.9.f.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.04200 + 2.43356i) q^{2} +(36.3630 + 99.9066i) q^{3} +(42.7015 + 242.172i) q^{4} +(-2.12030 + 12.0248i) q^{5} +(-317.382 - 115.518i) q^{6} +(-453.880 - 786.144i) q^{7} +(-1380.84 - 797.228i) q^{8} +(-3633.05 + 3048.49i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04200	+	2.43356i	−0.127625	+	0.152098i	−0.826073	−	0.563564i	\(-0.809430\pi\)
							0.698448	+	0.715661i	\(0.253874\pi\)
\(3\)	36.3630	+	99.9066i	0.448926	+	1.23342i	0.933472	+	0.358649i	\(0.116763\pi\)
							−0.484546	+	0.874766i	\(0.661015\pi\)
\(5\)	−2.12030	+	12.0248i	−0.00339248	+	0.0192397i	−0.986457	−	0.164020i	\(-0.947554\pi\)
							0.983065	+	0.183260i	\(0.0586649\pi\)
\(7\)	−453.880	−	786.144i	−0.189038	−	0.327424i	0.755892	−	0.654697i	\(-0.227204\pi\)
							−0.944930	+	0.327273i	\(0.893870\pi\)
\(11\)	−3697.06	+	6403.49i	−0.252514	+	0.437367i	−0.964217	−	0.265113i	\(-0.914591\pi\)
							0.711703	+	0.702480i	\(0.247924\pi\)
\(13\)	−5633.25	+	15477.2i	−0.197236	+	0.541901i	−0.998400	−	0.0565435i	\(-0.981992\pi\)
							0.801164	+	0.598444i	\(0.204214\pi\)
\(17\)	−24368.1	−	20447.3i	−0.291761	−	0.244816i	0.485144	−	0.874434i	\(-0.338767\pi\)
							−0.776905	+	0.629618i	\(0.783211\pi\)
\(19\)	129281.	−	16432.1i	0.992019	−	0.126089i
							\(23\)	3832.69	+	21736.3i	0.0136960	+	0.0776737i	0.990890	−	0.134676i	\(-0.0429994\pi\)
−0.977194	+	0.212350i	\(0.931888\pi\)
\(29\)	−79568.4	−	94825.9i	−0.112499	−	0.134071i	0.706856	−	0.707357i	\(-0.250113\pi\)
							−0.819355	+	0.573286i	\(0.805668\pi\)
\(31\)	−26559.5	+	15334.1i	−0.0287590	+	0.0166040i	−0.514311	−	0.857604i	\(-0.671952\pi\)
							0.485552	+	0.874208i	\(0.338619\pi\)
\(37\)			2.32989e6i			1.24316i	0.783349	+	0.621582i	\(0.213510\pi\)
							−0.783349	+	0.621582i	\(0.786490\pi\)
\(41\)	1.02677e6	+	2.82104e6i	0.363362	+	0.998329i	0.977832	+	0.209389i	\(0.0671474\pi\)
							−0.614471	+	0.788940i	\(0.710630\pi\)
\(43\)	−443257.	+	2.51383e6i	−0.129653	+	0.735297i	0.848782	+	0.528742i	\(0.177336\pi\)
							−0.978435	+	0.206554i	\(0.933775\pi\)
\(47\)	1.29647e6	−	1.08787e6i	0.265687	−	0.222938i	−0.500205	−	0.865907i	\(-0.666742\pi\)
							0.765892	+	0.642969i	\(0.222298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.3.6	✓	72
19.13	odd	18	inner	19.9.f.a.13.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.3.6	✓	72	1.1	even	1	trivial
19.9.f.a.13.6	yes	72	19.13	odd	18	inner