Embedded newform 19.9.f.a.2.4

• Label: 19.9.f.a.2.4
• Level: $19$
• Weight: $9$
• Character: 19.2
• Analytic conductor: $7.740$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• 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			2.4
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.6035 - 2.39867i) q^{2} +(15.5877 - 18.5766i) q^{3} +(-61.2592 - 22.2965i) q^{4} +(-36.1678 + 13.1640i) q^{5} +(-256.606 + 215.318i) q^{6} +(-1384.17 - 2397.46i) q^{7} +(3842.32 + 2218.36i) q^{8} +(1037.19 + 5882.19i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.6035	−	2.39867i	−0.850220	−	0.149917i	−0.268474	−	0.963287i	\(-0.586519\pi\)
							−0.581746	+	0.813370i	\(0.697630\pi\)
\(3\)	15.5877	−	18.5766i	0.192440	−	0.229341i	−0.661193	−	0.750216i	\(-0.729950\pi\)
							0.853633	+	0.520875i	\(0.174394\pi\)
\(5\)	−36.1678	+	13.1640i	−0.0578684	+	0.0210624i	−0.370792	−	0.928716i	\(-0.620914\pi\)
							0.312924	+	0.949778i	\(0.398692\pi\)
\(7\)	−1384.17	−	2397.46i	−0.576498	−	0.998524i	−0.995877	−	0.0907131i	\(-0.971085\pi\)
							0.419379	−	0.907811i	\(-0.362248\pi\)
\(11\)	−7293.66	+	12633.0i	−0.498167	+	0.862850i	−0.999998	−	0.00211568i	\(-0.999327\pi\)
							0.501831	+	0.864966i	\(0.332660\pi\)
\(13\)	34343.3	+	40928.8i	1.20246	+	1.43303i	0.872208	+	0.489136i	\(0.162688\pi\)
							0.330248	+	0.943894i	\(0.392868\pi\)
\(17\)	−13465.8	+	76368.1i	−0.161226	+	0.914358i	0.791645	+	0.610981i	\(0.209225\pi\)
							−0.952871	+	0.303376i	\(0.901886\pi\)
\(19\)	7165.93	−	130124.i	0.0549868	−	0.998487i
							\(23\)	−129519.	−	47141.0i	−0.462831	−	0.168457i	0.100071	−	0.994980i	\(-0.468093\pi\)
−0.562902	+	0.826524i	\(0.690315\pi\)
\(29\)	1.22575e6	−	216133.i	1.73305	−	0.305583i	0.784011	−	0.620746i	\(-0.213170\pi\)
							0.949038	+	0.315163i	\(0.102059\pi\)
\(31\)	−1.16027e6	+	669881.i	−1.25635	+	0.725356i	−0.972364	−	0.233472i	\(-0.924991\pi\)
							−0.283990	+	0.958827i	\(0.591658\pi\)
\(37\)			1.43425e6i			0.765276i	0.923898	+	0.382638i	\(0.124984\pi\)
							−0.923898	+	0.382638i	\(0.875016\pi\)
\(41\)	−601355.	+	716667.i	−0.212812	+	0.253619i	−0.861881	−	0.507110i	\(-0.830714\pi\)
							0.649070	+	0.760729i	\(0.275158\pi\)
\(43\)	−1.95742e6	+	712441.i	−0.572544	+	0.208389i	−0.612035	−	0.790831i	\(-0.709649\pi\)
							0.0394904	+	0.999220i	\(0.487427\pi\)
\(47\)	481692.	+	2.73181e6i	0.0987137	+	0.559833i	0.993546	+	0.113431i	\(0.0361840\pi\)
							−0.894832	+	0.446403i	\(0.852705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.4	✓	72
19.10	odd	18	inner	19.9.f.a.10.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.4	✓	72	1.1	even	1	trivial
19.9.f.a.10.4	yes	72	19.10	odd	18	inner