Embedded newform 19.7.f.a.3.7

• Label: 19.7.f.a.3.7
• Level: $19$
• Weight: $7$
• Character: 19.3
• Analytic conductor: $4.371$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.37102758878\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.29634 - 5.12017i) q^{2} +(10.1613 + 27.9178i) q^{3} +(3.35580 + 19.0317i) q^{4} +(-23.2992 + 132.137i) q^{5} +(186.600 + 67.9170i) q^{6} +(-124.815 - 216.186i) q^{7} +(482.323 + 278.470i) q^{8} +(-117.708 + 98.7685i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.29634	−	5.12017i	0.537042	−	0.640022i	−0.427480	−	0.904025i	\(-0.640598\pi\)
							0.964522	+	0.264003i	\(0.0850429\pi\)
\(3\)	10.1613	+	27.9178i	0.376343	+	1.03399i	0.972860	+	0.231393i	\(0.0743285\pi\)
							−0.596517	+	0.802600i	\(0.703449\pi\)
\(5\)	−23.2992	+	132.137i	−0.186394	+	1.05709i	0.737757	+	0.675066i	\(0.235885\pi\)
							−0.924151	+	0.382027i	\(0.875226\pi\)
\(7\)	−124.815	−	216.186i	−0.363891	−	0.630278i	0.624706	−	0.780860i	\(-0.285219\pi\)
							−0.988598	+	0.150581i	\(0.951885\pi\)
\(11\)	1244.31	−	2155.22i	0.934872	−	1.61925i	0.160010	−	0.987115i	\(-0.448847\pi\)
							0.774862	−	0.632130i	\(-0.217819\pi\)
\(13\)	−909.574	+	2499.03i	−0.414007	+	1.13748i	0.541034	+	0.841001i	\(0.318033\pi\)
							−0.955041	+	0.296474i	\(0.904189\pi\)
\(17\)	−1445.53	−	1212.94i	−0.294225	−	0.246884i	0.483711	−	0.875228i	\(-0.339289\pi\)
							−0.777936	+	0.628344i	\(0.783733\pi\)
\(19\)	921.105	−	6796.87i	0.134291	−	0.990942i
							\(23\)	971.553	+	5509.95i	0.0798515	+	0.452860i	0.998349	+	0.0574343i	\(0.0182920\pi\)
−0.918498	+	0.395426i	\(0.870597\pi\)
\(29\)	−9290.64	−	11072.2i	−0.380936	−	0.453982i	0.541174	−	0.840911i	\(-0.317980\pi\)
							−0.922109	+	0.386929i	\(0.873536\pi\)
\(31\)	4387.12	−	2532.91i	0.147263	−	0.0850226i	−0.424558	−	0.905401i	\(-0.639570\pi\)
							0.571821	+	0.820378i	\(0.306237\pi\)
\(37\)		−	39054.7i		−	0.771024i	−0.922703	−	0.385512i	\(-0.874025\pi\)
							0.922703	−	0.385512i	\(-0.125975\pi\)
\(41\)	−14855.7	−	40815.8i	−0.215547	−	0.592211i	0.784047	−	0.620702i	\(-0.213152\pi\)
							−0.999594	+	0.0284905i	\(0.990930\pi\)
\(43\)	15361.5	−	87119.5i	0.193210	−	1.09575i	−0.721736	−	0.692169i	\(-0.756655\pi\)
							0.914945	−	0.403578i	\(-0.132233\pi\)
\(47\)	−134106.	+	112528.i	−1.29168	+	1.08385i	−0.300159	+	0.953889i	\(0.597040\pi\)
							−0.991520	+	0.129957i	\(0.958516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.7	✓	54
19.13	odd	18	inner	19.7.f.a.13.7	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.7	✓	54	1.1	even	1	trivial
19.7.f.a.13.7	yes	54	19.13	odd	18	inner