Embedded newform 19.9.f.a.10.4

• Label: 19.9.f.a.10.4
• Level: $19$
• Weight: $9$
• Character: 19.10
• 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			10.4
Character	\(\chi\)	\(=\)	19.10
Dual form			19.9.f.a.2.4

$q$-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 \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.4	✓	72	19.2	odd	18	inner
19.9.f.a.10.4	yes	72	1.1	even	1	trivial