Embedded newform 19.7.f.a.10.1

• Label: 19.7.f.a.10.1
• Level: $19$
• Weight: $7$
• Character: 19.10
• 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			10.1
Character	\(\chi\)	\(=\)	19.10
Dual form			19.7.f.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.8473 + 2.44165i) q^{2} +(19.5791 + 23.3335i) q^{3} +(125.645 - 45.7311i) q^{4} +(43.8845 + 15.9727i) q^{5} +(-328.090 - 275.300i) q^{6} +(-286.306 + 495.896i) q^{7} +(-848.852 + 490.085i) q^{8} +(-34.5199 + 195.772i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.8473	+	2.44165i	−1.73091	+	0.305206i	−0.948316	−	0.317326i	\(-0.897215\pi\)
							−0.782594	+	0.622532i	\(0.786104\pi\)
\(3\)	19.5791	+	23.3335i	0.725152	+	0.864203i	0.995121	−	0.0986665i	\(-0.0314577\pi\)
							−0.269968	+	0.962869i	\(0.587013\pi\)
\(5\)	43.8845	+	15.9727i	0.351076	+	0.127781i	0.511538	−	0.859261i	\(-0.329076\pi\)
							−0.160462	+	0.987042i	\(0.551298\pi\)
\(7\)	−286.306	+	495.896i	−0.834711	+	1.44576i	0.0595543	+	0.998225i	\(0.481032\pi\)
							−0.894265	+	0.447537i	\(0.852301\pi\)
\(11\)	−671.666	−	1163.36i	−0.504632	−	0.874049i	−0.999986	−	0.00535740i	\(-0.998295\pi\)
							0.495353	−	0.868692i	\(-0.335039\pi\)
\(13\)	−2204.40	+	2627.10i	−1.00337	+	1.19577i	−0.0227706	+	0.999741i	\(0.507249\pi\)
							−0.980598	+	0.196028i	\(0.937196\pi\)
\(17\)	108.563	+	615.689i	0.0220970	+	0.125318i	0.993861	−	0.110637i	\(-0.0352892\pi\)
							−0.971764	+	0.235955i	\(0.924178\pi\)
\(19\)	5414.75	+	4210.27i	0.789437	+	0.613832i
							\(23\)	−11079.2	+	4032.50i	−0.910594	+	0.331429i	−0.754490	−	0.656312i	\(-0.772116\pi\)
−0.156104	+	0.987741i	\(0.549893\pi\)
\(29\)	9134.21	+	1610.61i	0.374522	+	0.0660383i	0.357741	−	0.933821i	\(-0.383547\pi\)
							0.0167809	+	0.999859i	\(0.494658\pi\)
\(31\)	29610.3	+	17095.5i	0.993936	+	0.573849i	0.906448	−	0.422317i	\(-0.138783\pi\)
							0.0874874	+	0.996166i	\(0.472116\pi\)
\(37\)			19256.2i			0.380160i	0.981769	+	0.190080i	\(0.0608747\pi\)
							−0.981769	+	0.190080i	\(0.939125\pi\)
\(41\)	61720.5	+	73555.7i	0.895526	+	1.06725i	0.997372	+	0.0724461i	\(0.0230805\pi\)
							−0.101846	+	0.994800i	\(0.532475\pi\)
\(43\)	82814.7	+	30142.1i	1.04160	+	0.379112i	0.805487	−	0.592613i	\(-0.201904\pi\)
							0.236115	+	0.971725i	\(0.424126\pi\)
\(47\)	−8680.91	+	49231.9i	−0.0836125	+	0.474190i	0.914035	+	0.405636i	\(0.132950\pi\)
							−0.997647	+	0.0685546i	\(0.978161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.10.1	yes	54
19.2	odd	18	inner	19.7.f.a.2.1	✓	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.2.1	✓	54	19.2	odd	18	inner
19.7.f.a.10.1	yes	54	1.1	even	1	trivial