Embedded newform 2112.2.a.e.1.1

• Label: 2112.2.a.e.1.1
• Level: $2112$
• Weight: $2$
• Character: 2112.1
• Self dual: yes
• Analytic conductor: $16.864$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.8644049069\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2112.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)

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	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
0.832050	+	0.554700i				\(0.187167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	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	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.a.e.1.1		1
3.2	odd	2		6336.2.a.cj.1.1		1
4.3	odd	2		2112.2.a.r.1.1		1
8.3	odd	2		66.2.a.b.1.1	✓	1
8.5	even	2		528.2.a.j.1.1		1
12.11	even	2		6336.2.a.bw.1.1		1
24.5	odd	2		1584.2.a.f.1.1		1
24.11	even	2		198.2.a.a.1.1		1
40.3	even	4		1650.2.c.e.199.1		2
40.19	odd	2		1650.2.a.k.1.1		1
40.27	even	4		1650.2.c.e.199.2		2
56.27	even	2		3234.2.a.t.1.1		1
72.11	even	6		1782.2.e.v.595.1		2
72.43	odd	6		1782.2.e.e.595.1		2
72.59	even	6		1782.2.e.v.1189.1		2
72.67	odd	6		1782.2.e.e.1189.1		2
88.3	odd	10		726.2.e.g.493.1		4
88.19	even	10		726.2.e.o.493.1		4
88.21	odd	2		5808.2.a.bc.1.1		1
88.27	odd	10		726.2.e.g.487.1		4
88.35	even	10		726.2.e.o.565.1		4
88.43	even	2		726.2.a.c.1.1		1
88.51	even	10		726.2.e.o.511.1		4
88.59	odd	10		726.2.e.g.511.1		4
88.75	odd	10		726.2.e.g.565.1		4
88.83	even	10		726.2.e.o.487.1		4
120.59	even	2		4950.2.a.bu.1.1		1
120.83	odd	4		4950.2.c.p.199.2		2
120.107	odd	4		4950.2.c.p.199.1		2
168.83	odd	2		9702.2.a.x.1.1		1
264.131	odd	2		2178.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	8.3	odd	2
198.2.a.a.1.1		1	24.11	even	2
528.2.a.j.1.1		1	8.5	even	2
726.2.a.c.1.1		1	88.43	even	2
726.2.e.g.487.1		4	88.27	odd	10
726.2.e.g.493.1		4	88.3	odd	10
726.2.e.g.511.1		4	88.59	odd	10
726.2.e.g.565.1		4	88.75	odd	10
726.2.e.o.487.1		4	88.83	even	10
726.2.e.o.493.1		4	88.19	even	10
726.2.e.o.511.1		4	88.51	even	10
726.2.e.o.565.1		4	88.35	even	10
1584.2.a.f.1.1		1	24.5	odd	2
1650.2.a.k.1.1		1	40.19	odd	2
1650.2.c.e.199.1		2	40.3	even	4
1650.2.c.e.199.2		2	40.27	even	4
1782.2.e.e.595.1		2	72.43	odd	6
1782.2.e.e.1189.1		2	72.67	odd	6
1782.2.e.v.595.1		2	72.11	even	6
1782.2.e.v.1189.1		2	72.59	even	6
2112.2.a.e.1.1		1	1.1	even	1	trivial
2112.2.a.r.1.1		1	4.3	odd	2
2178.2.a.g.1.1		1	264.131	odd	2
3234.2.a.t.1.1		1	56.27	even	2
4950.2.a.bu.1.1		1	120.59	even	2
4950.2.c.p.199.1		2	120.107	odd	4
4950.2.c.p.199.2		2	120.83	odd	4
5808.2.a.bc.1.1		1	88.21	odd	2
6336.2.a.bw.1.1		1	12.11	even	2
6336.2.a.cj.1.1		1	3.2	odd	2
9702.2.a.x.1.1		1	168.83	odd	2