Embedded newform 8112.2.a.j.1.1

• Label: 8112.2.a.j.1.1
• Level: $8112$
• Weight: $2$
• Character: 8112.1
• Self dual: yes
• Analytic conductor: $64.775$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

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Newspace parameters

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(64.7746461197\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	8112.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{15} +2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} -10.0000 q^{31} -4.00000 q^{35} -8.00000 q^{37} +10.0000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -12.0000 q^{47} -3.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} -6.00000 q^{57} +4.00000 q^{59} +2.00000 q^{61} -2.00000 q^{63} +2.00000 q^{67} -4.00000 q^{69} +4.00000 q^{73} +1.00000 q^{75} +1.00000 q^{81} +4.00000 q^{83} +4.00000 q^{85} +10.0000 q^{87} +6.00000 q^{89} +10.0000 q^{93} +12.0000 q^{95} -12.0000 q^{97} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350
\(5\)	2.00000	0.894427				0.447214	−	0.894427i	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
			−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	0	0
			\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
0.242536	+	0.970143i				\(0.422021\pi\)
\(19\)	6.00000	1.37649	0.688247	−	0.725476i	\(-0.258380\pi\)
			0.688247	+	0.725476i	\(0.258380\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−10.0000	−1.85695	−0.928477	−	0.371391i	\(-0.878881\pi\)
			−0.928477	+	0.371391i	\(0.878881\pi\)
\(31\)	−10.0000	−1.79605	−0.898027	−	0.439941i	\(-0.854999\pi\)
			−0.898027	+	0.439941i	\(0.854999\pi\)
\(37\)	−8.00000	−1.31519	−0.657596	−	0.753371i	\(-0.728427\pi\)
			−0.657596	+	0.753371i	\(0.728427\pi\)
\(41\)	10.0000	1.56174	0.780869	−	0.624695i	\(-0.214777\pi\)
			0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−12.0000	−1.75038	−0.875190	−	0.483779i	\(-0.839264\pi\)
			−0.875190	+	0.483779i	\(0.839264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.j.1.1		1
4.3	odd	2		1014.2.a.g.1.1		1
12.11	even	2		3042.2.a.c.1.1		1
13.5	odd	4		624.2.c.a.337.1		2
13.8	odd	4		624.2.c.a.337.2		2
13.12	even	2		8112.2.a.g.1.1		1
39.5	even	4		1872.2.c.b.1585.2		2
39.8	even	4		1872.2.c.b.1585.1		2
52.3	odd	6		1014.2.e.b.529.1		2
52.7	even	12		1014.2.i.c.361.2		4
52.11	even	12		1014.2.i.c.823.1		4
52.15	even	12		1014.2.i.c.823.2		4
52.19	even	12		1014.2.i.c.361.1		4
52.23	odd	6		1014.2.e.e.529.1		2
52.31	even	4		78.2.b.a.25.1	✓	2
52.35	odd	6		1014.2.e.b.991.1		2
52.43	odd	6		1014.2.e.e.991.1		2
52.47	even	4		78.2.b.a.25.2	yes	2
52.51	odd	2		1014.2.a.b.1.1		1
104.5	odd	4		2496.2.c.m.961.2		2
104.21	odd	4		2496.2.c.m.961.1		2
104.83	even	4		2496.2.c.f.961.2		2
104.99	even	4		2496.2.c.f.961.1		2
156.47	odd	4		234.2.b.a.181.1		2
156.83	odd	4		234.2.b.a.181.2		2
156.155	even	2		3042.2.a.n.1.1		1
260.47	odd	4		1950.2.f.d.649.1		2
260.83	odd	4		1950.2.f.d.649.2		2
260.99	even	4		1950.2.b.c.1351.1		2
260.187	odd	4		1950.2.f.g.649.1		2
260.203	odd	4		1950.2.f.g.649.2		2
260.239	even	4		1950.2.b.c.1351.2		2
364.83	odd	4		3822.2.c.d.883.1		2
364.307	odd	4		3822.2.c.d.883.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	52.31	even	4
78.2.b.a.25.2	yes	2	52.47	even	4
234.2.b.a.181.1		2	156.47	odd	4
234.2.b.a.181.2		2	156.83	odd	4
624.2.c.a.337.1		2	13.5	odd	4
624.2.c.a.337.2		2	13.8	odd	4
1014.2.a.b.1.1		1	52.51	odd	2
1014.2.a.g.1.1		1	4.3	odd	2
1014.2.e.b.529.1		2	52.3	odd	6
1014.2.e.b.991.1		2	52.35	odd	6
1014.2.e.e.529.1		2	52.23	odd	6
1014.2.e.e.991.1		2	52.43	odd	6
1014.2.i.c.361.1		4	52.19	even	12
1014.2.i.c.361.2		4	52.7	even	12
1014.2.i.c.823.1		4	52.11	even	12
1014.2.i.c.823.2		4	52.15	even	12
1872.2.c.b.1585.1		2	39.8	even	4
1872.2.c.b.1585.2		2	39.5	even	4
1950.2.b.c.1351.1		2	260.99	even	4
1950.2.b.c.1351.2		2	260.239	even	4
1950.2.f.d.649.1		2	260.47	odd	4
1950.2.f.d.649.2		2	260.83	odd	4
1950.2.f.g.649.1		2	260.187	odd	4
1950.2.f.g.649.2		2	260.203	odd	4
2496.2.c.f.961.1		2	104.99	even	4
2496.2.c.f.961.2		2	104.83	even	4
2496.2.c.m.961.1		2	104.21	odd	4
2496.2.c.m.961.2		2	104.5	odd	4
3042.2.a.c.1.1		1	12.11	even	2
3042.2.a.n.1.1		1	156.155	even	2
3822.2.c.d.883.1		2	364.83	odd	4
3822.2.c.d.883.2		2	364.307	odd	4
8112.2.a.g.1.1		1	13.12	even	2
8112.2.a.j.1.1		1	1.1	even	1	trivial