Embedded newform 19.7.f.a.3.6

• Label: 19.7.f.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25256 - 1.49274i) q^{2} +(1.82612 + 5.01723i) q^{3} +(10.4541 + 59.2882i) q^{4} +(-2.44738 + 13.8798i) q^{5} +(9.77677 + 3.55845i) q^{6} +(212.620 + 368.269i) q^{7} +(209.601 + 121.013i) q^{8} +(536.609 - 450.268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25256	−	1.49274i	0.156570	−	0.186593i	−0.682057	−	0.731299i	\(-0.738914\pi\)
							0.838627	+	0.544706i	\(0.183359\pi\)
\(3\)	1.82612	+	5.01723i	0.0676341	+	0.185823i	0.968905	−	0.247433i	\(-0.0795869\pi\)
							−0.901271	+	0.433256i	\(0.857365\pi\)
\(5\)	−2.44738	+	13.8798i	−0.0195791	+	0.111038i	−0.993031	−	0.117853i	\(-0.962399\pi\)
							0.973452	+	0.228891i	\(0.0735100\pi\)
\(7\)	212.620	+	368.269i	0.619885	+	1.07367i	0.989506	+	0.144489i	\(0.0461540\pi\)
							−0.369622	+	0.929182i	\(0.620513\pi\)
\(11\)	−1226.16	+	2123.77i	−0.921232	+	1.59562i	−0.123721	+	0.992317i	\(0.539483\pi\)
							−0.797511	+	0.603304i	\(0.793851\pi\)
\(13\)	1213.92	−	3335.22i	0.552536	−	1.51808i	−0.277699	−	0.960668i	\(-0.589572\pi\)
							0.830235	−	0.557413i	\(-0.188206\pi\)
\(17\)	−2361.26	−	1981.33i	−0.480614	−	0.403283i	0.370034	−	0.929018i	\(-0.379346\pi\)
							−0.850649	+	0.525735i	\(0.823790\pi\)
\(19\)	−586.158	−	6833.91i	−0.0854582	−	0.996342i
							\(23\)	−1208.04	−	6851.15i	−0.0992885	−	0.563093i	−0.993349	−	0.115146i	\(-0.963266\pi\)
0.894060	−	0.447947i	\(-0.147845\pi\)
\(29\)	19846.2	+	23651.8i	0.813735	+	0.969772i	0.999919	−	0.0127428i	\(-0.00405628\pi\)
							−0.186183	+	0.982515i	\(0.559612\pi\)
\(31\)	21298.1	−	12296.4i	0.714916	−	0.412757i	−0.0979628	−	0.995190i	\(-0.531233\pi\)
							0.812878	+	0.582433i	\(0.197899\pi\)
\(37\)		−	63695.2i		−	1.25748i	−0.777615	−	0.628740i	\(-0.783571\pi\)
							0.777615	−	0.628740i	\(-0.216429\pi\)
\(41\)	−28941.4	−	79515.8i	−0.419921	−	1.15372i	−0.951751	−	0.306873i	\(-0.900717\pi\)
							0.531829	−	0.846852i	\(-0.321505\pi\)
\(43\)	−8455.27	+	47952.2i	−0.106346	+	0.603119i	0.884328	+	0.466866i	\(0.154617\pi\)
							−0.990674	+	0.136253i	\(0.956494\pi\)
\(47\)	54068.5	−	45368.9i	0.520776	−	0.436983i	−0.344126	−	0.938923i	\(-0.611825\pi\)
							0.864902	+	0.501940i	\(0.167380\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.6	✓	54
19.13	odd	18	inner	19.7.f.a.13.6	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.6	✓	54	1.1	even	1	trivial
19.7.f.a.13.6	yes	54	19.13	odd	18	inner