Embedded newform 19.9.f.a.2.6

• Label: 19.9.f.a.2.6
• Level: $19$
• Weight: $9$
• Character: 19.2
• 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			2.6
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18052 - 0.208158i) q^{2} +(63.0914 - 75.1894i) q^{3} +(-239.211 - 87.0657i) q^{4} +(-753.457 + 274.236i) q^{5} +(-90.1321 + 75.6298i) q^{6} +(994.028 + 1721.71i) q^{7} +(530.033 + 306.015i) q^{8} +(-533.617 - 3026.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18052	−	0.208158i	−0.0737826	−	0.0130099i	0.136635	−	0.990621i	\(-0.456371\pi\)
							−0.210418	+	0.977612i	\(0.567482\pi\)
\(3\)	63.0914	−	75.1894i	0.778906	−	0.928265i	−0.219977	−	0.975505i	\(-0.570598\pi\)
							0.998884	+	0.0472405i	\(0.0150427\pi\)
\(5\)	−753.457	+	274.236i	−1.20553	+	0.438778i	−0.865152	−	0.501511i	\(-0.832778\pi\)
							−0.340380	+	0.940288i	\(0.610556\pi\)
\(7\)	994.028	+	1721.71i	0.414006	+	0.717079i	0.995324	−	0.0965977i	\(-0.0307960\pi\)
							−0.581318	+	0.813677i	\(0.697463\pi\)
\(11\)	−11385.9	+	19721.0i	−0.777675	+	1.34697i	0.155604	+	0.987820i	\(0.450268\pi\)
							−0.933279	+	0.359153i	\(0.883066\pi\)
\(13\)	−33374.5	−	39774.2i	−1.16853	−	1.39260i	−0.903622	−	0.428332i	\(-0.859102\pi\)
							−0.264912	−	0.964273i	\(-0.585343\pi\)
\(17\)	12858.3	−	72923.0i	0.153953	−	0.873110i	−0.805784	−	0.592210i	\(-0.798256\pi\)
							0.959737	−	0.280901i	\(-0.0906331\pi\)
\(19\)	−122811.	−	43600.1i	−0.942375	−	0.334560i
							\(23\)	−281301.	−	102385.i	−1.00522	−	0.365869i	−0.213623	−	0.976916i	\(-0.568527\pi\)
−0.791595	+	0.611047i	\(0.790749\pi\)
\(29\)	−368404.	+	64959.6i	−0.520874	+	0.0918441i	−0.427903	−	0.903825i	\(-0.640747\pi\)
							−0.0929709	+	0.995669i	\(0.529636\pi\)
\(31\)	1.07980e6	−	623422.i	1.16922	−	0.675049i	0.215723	−	0.976455i	\(-0.430789\pi\)
							0.953496	+	0.301405i	\(0.0974558\pi\)
\(37\)			482755.i			0.257585i	0.991672	+	0.128792i	\(0.0411101\pi\)
							−0.991672	+	0.128792i	\(0.958890\pi\)
\(41\)	−280639.	+	334453.i	−0.0993146	+	0.118359i	−0.813412	−	0.581687i	\(-0.802393\pi\)
							0.714098	+	0.700046i	\(0.246837\pi\)
\(43\)	−4.11704e6	+	1.49848e6i	−1.20424	+	0.438306i	−0.864701	−	0.502287i	\(-0.832492\pi\)
							−0.339535	+	0.940593i	\(0.610270\pi\)
\(47\)	489472.	+	2.77593e6i	0.100308	+	0.568876i	0.992991	+	0.118191i	\(0.0377094\pi\)
							−0.892683	+	0.450686i	\(0.851180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.6	✓	72
19.10	odd	18	inner	19.9.f.a.10.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.6	✓	72	1.1	even	1	trivial
19.9.f.a.10.6	yes	72	19.10	odd	18	inner