Embedded newform 1008.6.a.d.1.1

• Label: 1008.6.a.d.1.1
• Level: $1008$
• Weight: $6$
• Character: 1008.1
• Self dual: yes
• Analytic conductor: $161.667$
• Analytic rank: $0$
• 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:	\(0\)
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

\(f(q)\)

\(=\)

\(q-76.0000 q^{5} +49.0000 q^{7} +650.000 q^{11} +762.000 q^{13} +556.000 q^{17} +2452.00 q^{19} -2950.00 q^{23} +2651.00 q^{25} +674.000 q^{29} +3024.00 q^{31} -3724.00 q^{35} +7730.00 q^{37} +17016.0 q^{41} -21836.0 q^{43} -23940.0 q^{47} +2401.00 q^{49} -15594.0 q^{53} -49400.0 q^{55} +5608.00 q^{59} +150.000 q^{61} -57912.0 q^{65} +43784.0 q^{67} -39178.0 q^{71} -23570.0 q^{73} +31850.0 q^{77} +17892.0 q^{79} +38972.0 q^{83} -42256.0 q^{85} -6024.00 q^{89} +37338.0 q^{91} -186352. q^{95} +108430. q^{97} +O(q^{100})\)

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
\(5\)	−76.0000	−1.35953				−0.679765	−	0.733430i	\(-0.737918\pi\)
			−0.679765	+	0.733430i	\(0.737918\pi\)
\(7\)	49.0000	0.377964
			\(11\)	650.000	1.61969	0.809845	−	0.586645i	\(-0.199551\pi\)
0.809845	+	0.586645i				\(0.199551\pi\)
\(13\)	762.000	1.25054	0.625269	−	0.780410i	\(-0.284989\pi\)
			0.625269	+	0.780410i	\(0.284989\pi\)
\(17\)	556.000	0.466608	0.233304	−	0.972404i	\(-0.425046\pi\)
			0.233304	+	0.972404i	\(0.425046\pi\)
\(19\)	2452.00	1.55825	0.779124	−	0.626870i	\(-0.215664\pi\)
			0.779124	+	0.626870i	\(0.215664\pi\)
\(23\)	−2950.00	−1.16279	−0.581397	−	0.813620i	\(-0.697493\pi\)
			−0.581397	+	0.813620i	\(0.697493\pi\)
\(29\)	674.000	0.148821	0.0744106	−	0.997228i	\(-0.476292\pi\)
			0.0744106	+	0.997228i	\(0.476292\pi\)
\(31\)	3024.00	0.565168	0.282584	−	0.959243i	\(-0.408808\pi\)
			0.282584	+	0.959243i	\(0.408808\pi\)
\(37\)	7730.00	0.928272	0.464136	−	0.885764i	\(-0.346365\pi\)
			0.464136	+	0.885764i	\(0.346365\pi\)
\(41\)	17016.0	1.58088	0.790438	−	0.612542i	\(-0.209853\pi\)
			0.790438	+	0.612542i	\(0.209853\pi\)
\(43\)	−21836.0	−1.80095	−0.900476	−	0.434907i	\(-0.856781\pi\)
			−0.900476	+	0.434907i	\(0.856781\pi\)
\(47\)	−23940.0	−1.58081	−0.790405	−	0.612585i	\(-0.790130\pi\)
			−0.790405	+	0.612585i	\(0.790130\pi\)

(See \(a_n\) instead)

Twists

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

    

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