Embedded newform 1008.6.a.ba.1.1

• Label: 1008.6.a.ba.1.1
• Level: $1008$
• Weight: $6$
• Character: 1008.1
• Self dual: yes
• Analytic conductor: $161.667$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(161.666890371\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1008.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+72.0000 q^{5} -49.0000 q^{7} -414.000 q^{11} -1054.00 q^{13} +1848.00 q^{17} -236.000 q^{19} +2898.00 q^{23} +2059.00 q^{25} +6522.00 q^{29} -6200.00 q^{31} -3528.00 q^{35} +9650.00 q^{37} -8484.00 q^{41} +10804.0 q^{43} +60.0000 q^{47} +2401.00 q^{49} -22506.0 q^{53} -29808.0 q^{55} -28176.0 q^{59} -35194.0 q^{61} -75888.0 q^{65} +28216.0 q^{67} -6642.00 q^{71} -52090.0 q^{73} +20286.0 q^{77} -43340.0 q^{79} +25716.0 q^{83} +133056. q^{85} -98724.0 q^{89} +51646.0 q^{91} -16992.0 q^{95} -148954. q^{97} +O(q^{100})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	72.0000	1.28798				0.643988	−	0.765036i	\(-0.277279\pi\)
			0.643988	+	0.765036i	\(0.277279\pi\)
\(7\)	−49.0000	−0.377964
			\(11\)	−414.000	−1.03162	−0.515809	−	0.856704i	\(-0.672508\pi\)
−0.515809	+	0.856704i				\(0.672508\pi\)
\(13\)	−1054.00	−1.72975	−0.864873	−	0.501991i	\(-0.832601\pi\)
			−0.864873	+	0.501991i	\(0.832601\pi\)
\(17\)	1848.00	1.55089	0.775443	−	0.631418i	\(-0.217527\pi\)
			0.775443	+	0.631418i	\(0.217527\pi\)
\(19\)	−236.000	−0.149978	−0.0749891	−	0.997184i	\(-0.523892\pi\)
			−0.0749891	+	0.997184i	\(0.523892\pi\)
\(23\)	2898.00	1.14230	0.571148	−	0.820847i	\(-0.306498\pi\)
			0.571148	+	0.820847i	\(0.306498\pi\)
\(29\)	6522.00	1.44008	0.720039	−	0.693934i	\(-0.244124\pi\)
			0.720039	+	0.693934i	\(0.244124\pi\)
\(31\)	−6200.00	−1.15874	−0.579372	−	0.815063i	\(-0.696702\pi\)
			−0.579372	+	0.815063i	\(0.696702\pi\)
\(37\)	9650.00	1.15884	0.579419	−	0.815030i	\(-0.303279\pi\)
			0.579419	+	0.815030i	\(0.303279\pi\)
\(41\)	−8484.00	−0.788208	−0.394104	−	0.919066i	\(-0.628945\pi\)
			−0.394104	+	0.919066i	\(0.628945\pi\)
\(43\)	10804.0	0.891073	0.445537	−	0.895264i	\(-0.353013\pi\)
			0.445537	+	0.895264i	\(0.353013\pi\)
\(47\)	60.0000	0.00396193	0.00198096	−	0.999998i	\(-0.499369\pi\)
			0.00198096	+	0.999998i	\(0.499369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.ba.1.1		1
3.2	odd	2		336.6.a.b.1.1		1
4.3	odd	2		126.6.a.l.1.1		1
12.11	even	2		42.6.a.c.1.1	✓	1
28.27	even	2		882.6.a.n.1.1		1
60.23	odd	4		1050.6.g.b.799.2		2
60.47	odd	4		1050.6.g.b.799.1		2
60.59	even	2		1050.6.a.g.1.1		1
84.11	even	6		294.6.e.l.79.1		2
84.23	even	6		294.6.e.l.67.1		2
84.47	odd	6		294.6.e.n.67.1		2
84.59	odd	6		294.6.e.n.79.1		2
84.83	odd	2		294.6.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.c.1.1	✓	1	12.11	even	2
126.6.a.l.1.1		1	4.3	odd	2
294.6.a.c.1.1		1	84.83	odd	2
294.6.e.l.67.1		2	84.23	even	6
294.6.e.l.79.1		2	84.11	even	6
294.6.e.n.67.1		2	84.47	odd	6
294.6.e.n.79.1		2	84.59	odd	6
336.6.a.b.1.1		1	3.2	odd	2
882.6.a.n.1.1		1	28.27	even	2
1008.6.a.ba.1.1		1	1.1	even	1	trivial
1050.6.a.g.1.1		1	60.59	even	2
1050.6.g.b.799.1		2	60.47	odd	4
1050.6.g.b.799.2		2	60.23	odd	4