Embedded newform 1008.6.a.x.1.1

• Label: 1008.6.a.x.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

\(f(q)\)

\(=\)

\(q+54.0000 q^{5} -49.0000 q^{7} +216.000 q^{11} +998.000 q^{13} -1302.00 q^{17} -884.000 q^{19} -2268.00 q^{23} -209.000 q^{25} +1482.00 q^{29} -8360.00 q^{31} -2646.00 q^{35} -4714.00 q^{37} +9786.00 q^{41} -19436.0 q^{43} +22200.0 q^{47} +2401.00 q^{49} -26790.0 q^{53} +11664.0 q^{55} +28092.0 q^{59} -38866.0 q^{61} +53892.0 q^{65} -23948.0 q^{67} -20628.0 q^{71} +290.000 q^{73} -10584.0 q^{77} +99544.0 q^{79} +19308.0 q^{83} -70308.0 q^{85} -36390.0 q^{89} -48902.0 q^{91} -47736.0 q^{95} -79078.0 q^{97} +O(q^{100})\)

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
\(5\)	54.0000	0.965981				0.482991	−	0.875625i	\(-0.339550\pi\)
			0.482991	+	0.875625i	\(0.339550\pi\)
\(7\)	−49.0000	−0.377964
			\(11\)	216.000	0.538235	0.269118	−	0.963107i	\(-0.413268\pi\)
0.269118	+	0.963107i				\(0.413268\pi\)
\(13\)	998.000	1.63784	0.818921	−	0.573906i	\(-0.194572\pi\)
			0.818921	+	0.573906i	\(0.194572\pi\)
\(17\)	−1302.00	−1.09267	−0.546335	−	0.837567i	\(-0.683977\pi\)
			−0.546335	+	0.837567i	\(0.683977\pi\)
\(19\)	−884.000	−0.561783	−0.280891	−	0.959740i	\(-0.590630\pi\)
			−0.280891	+	0.959740i	\(0.590630\pi\)
\(23\)	−2268.00	−0.893971	−0.446986	−	0.894541i	\(-0.647502\pi\)
			−0.446986	+	0.894541i	\(0.647502\pi\)
\(29\)	1482.00	0.327230	0.163615	−	0.986524i	\(-0.447685\pi\)
			0.163615	+	0.986524i	\(0.447685\pi\)
\(31\)	−8360.00	−1.56244	−0.781218	−	0.624259i	\(-0.785401\pi\)
			−0.781218	+	0.624259i	\(0.785401\pi\)
\(37\)	−4714.00	−0.566090	−0.283045	−	0.959107i	\(-0.591345\pi\)
			−0.283045	+	0.959107i	\(0.591345\pi\)
\(41\)	9786.00	0.909171	0.454585	−	0.890703i	\(-0.349787\pi\)
			0.454585	+	0.890703i	\(0.349787\pi\)
\(43\)	−19436.0	−1.60301	−0.801504	−	0.597989i	\(-0.795967\pi\)
			−0.801504	+	0.597989i	\(0.795967\pi\)
\(47\)	22200.0	1.46591	0.732957	−	0.680275i	\(-0.238140\pi\)
			0.732957	+	0.680275i	\(0.238140\pi\)

(See \(a_n\) instead)

Twists

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

    

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