Embedded newform 56.2.a.b.1.1

• Label: 56.2.a.b.1.1
• Level: $56$
• Weight: $2$
• Character: 56.1
• Self dual: yes
• Analytic conductor: $0.447$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.447162251319\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	56.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -8.00000 q^{15} -2.00000 q^{17} -2.00000 q^{19} +2.00000 q^{21} +8.00000 q^{23} +11.0000 q^{25} -4.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} -4.00000 q^{35} -6.00000 q^{37} -2.00000 q^{41} +8.00000 q^{43} -4.00000 q^{45} -4.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} -10.0000 q^{53} -4.00000 q^{57} +6.00000 q^{59} +4.00000 q^{61} +1.00000 q^{63} -12.0000 q^{67} +16.0000 q^{69} -14.0000 q^{73} +22.0000 q^{75} -8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{83} +8.00000 q^{85} +4.00000 q^{87} +10.0000 q^{89} +8.00000 q^{93} +8.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)

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
			\(3\)	2.00000	1.15470	0.577350	−	0.816497i	\(-0.304087\pi\)
0.577350	+	0.816497i				\(0.304087\pi\)
\(5\)	−4.00000	−1.78885	−0.894427	−	0.447214i	\(-0.852416\pi\)
			−0.894427	+	0.447214i	\(0.852416\pi\)
\(7\)	1.00000	0.377964
			\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	−2.00000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.2.a.b.1.1	✓	1
3.2	odd	2		504.2.a.h.1.1		1
4.3	odd	2		112.2.a.a.1.1		1
5.2	odd	4		1400.2.g.b.449.1		2
5.3	odd	4		1400.2.g.b.449.2		2
5.4	even	2		1400.2.a.a.1.1		1
7.2	even	3		392.2.i.a.361.1		2
7.3	odd	6		392.2.i.e.177.1		2
7.4	even	3		392.2.i.a.177.1		2
7.5	odd	6		392.2.i.e.361.1		2
7.6	odd	2		392.2.a.b.1.1		1
8.3	odd	2		448.2.a.h.1.1		1
8.5	even	2		448.2.a.c.1.1		1
11.10	odd	2		6776.2.a.h.1.1		1
12.11	even	2		1008.2.a.m.1.1		1
13.12	even	2		9464.2.a.h.1.1		1
16.3	odd	4		1792.2.b.h.897.1		2
16.5	even	4		1792.2.b.a.897.1		2
16.11	odd	4		1792.2.b.h.897.2		2
16.13	even	4		1792.2.b.a.897.2		2
20.3	even	4		2800.2.g.g.449.1		2
20.7	even	4		2800.2.g.g.449.2		2
20.19	odd	2		2800.2.a.bd.1.1		1
21.2	odd	6		3528.2.s.a.361.1		2
21.5	even	6		3528.2.s.ba.361.1		2
21.11	odd	6		3528.2.s.a.3313.1		2
21.17	even	6		3528.2.s.ba.3313.1		2
21.20	even	2		3528.2.a.b.1.1		1
24.5	odd	2		4032.2.a.d.1.1		1
24.11	even	2		4032.2.a.a.1.1		1
28.3	even	6		784.2.i.b.177.1		2
28.11	odd	6		784.2.i.j.177.1		2
28.19	even	6		784.2.i.b.753.1		2
28.23	odd	6		784.2.i.j.753.1		2
28.27	even	2		784.2.a.i.1.1		1
35.34	odd	2		9800.2.a.bj.1.1		1
56.13	odd	2		3136.2.a.w.1.1		1
56.27	even	2		3136.2.a.c.1.1		1
84.83	odd	2		7056.2.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.a.b.1.1	✓	1	1.1	even	1	trivial
112.2.a.a.1.1		1	4.3	odd	2
392.2.a.b.1.1		1	7.6	odd	2
392.2.i.a.177.1		2	7.4	even	3
392.2.i.a.361.1		2	7.2	even	3
392.2.i.e.177.1		2	7.3	odd	6
392.2.i.e.361.1		2	7.5	odd	6
448.2.a.c.1.1		1	8.5	even	2
448.2.a.h.1.1		1	8.3	odd	2
504.2.a.h.1.1		1	3.2	odd	2
784.2.a.i.1.1		1	28.27	even	2
784.2.i.b.177.1		2	28.3	even	6
784.2.i.b.753.1		2	28.19	even	6
784.2.i.j.177.1		2	28.11	odd	6
784.2.i.j.753.1		2	28.23	odd	6
1008.2.a.m.1.1		1	12.11	even	2
1400.2.a.a.1.1		1	5.4	even	2
1400.2.g.b.449.1		2	5.2	odd	4
1400.2.g.b.449.2		2	5.3	odd	4
1792.2.b.a.897.1		2	16.5	even	4
1792.2.b.a.897.2		2	16.13	even	4
1792.2.b.h.897.1		2	16.3	odd	4
1792.2.b.h.897.2		2	16.11	odd	4
2800.2.a.bd.1.1		1	20.19	odd	2
2800.2.g.g.449.1		2	20.3	even	4
2800.2.g.g.449.2		2	20.7	even	4
3136.2.a.c.1.1		1	56.27	even	2
3136.2.a.w.1.1		1	56.13	odd	2
3528.2.a.b.1.1		1	21.20	even	2
3528.2.s.a.361.1		2	21.2	odd	6
3528.2.s.a.3313.1		2	21.11	odd	6
3528.2.s.ba.361.1		2	21.5	even	6
3528.2.s.ba.3313.1		2	21.17	even	6
4032.2.a.a.1.1		1	24.11	even	2
4032.2.a.d.1.1		1	24.5	odd	2
6776.2.a.h.1.1		1	11.10	odd	2
7056.2.a.c.1.1		1	84.83	odd	2
9464.2.a.h.1.1		1	13.12	even	2
9800.2.a.bj.1.1		1	35.34	odd	2