Embedded newform 4410.2.a.bd.1.1

• Label: 4410.2.a.bd.1.1
• Level: $4410$
• Weight: $2$
• Character: 4410.1
• Self dual: yes
• Analytic conductor: $35.214$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(35.2140272914\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4410.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	0	0
\(5\)	−1.00000	−0.447214
			\(7\)	0	0
\(11\)	6.00000	1.80907				0.904534	−	0.426401i	\(-0.140219\pi\)
			0.904534	+	0.426401i	\(0.140219\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	−2.00000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	3.00000	0.625543	0.312772	−	0.949828i	\(-0.398743\pi\)
			0.312772	+	0.949828i	\(0.398743\pi\)
\(29\)	3.00000	0.557086	0.278543	−	0.960424i	\(-0.410149\pi\)
			0.278543	+	0.960424i	\(0.410149\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	−4.00000	−0.657596	−0.328798	−	0.944400i	\(-0.606644\pi\)
			−0.328798	+	0.944400i	\(0.606644\pi\)
\(41\)	9.00000	1.40556	0.702782	−	0.711405i	\(-0.251941\pi\)
			0.702782	+	0.711405i	\(0.251941\pi\)
\(43\)	−7.00000	−1.06749	−0.533745	−	0.845645i	\(-0.679216\pi\)
			−0.533745	+	0.845645i	\(0.679216\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bd.1.1		1
3.2	odd	2		490.2.a.b.1.1		1
7.3	odd	6		630.2.k.b.541.1		2
7.5	odd	6		630.2.k.b.361.1		2
7.6	odd	2		4410.2.a.bm.1.1		1
12.11	even	2		3920.2.a.bc.1.1		1
15.2	even	4		2450.2.c.l.99.1		2
15.8	even	4		2450.2.c.l.99.2		2
15.14	odd	2		2450.2.a.bc.1.1		1
21.2	odd	6		490.2.e.h.361.1		2
21.5	even	6		70.2.e.c.11.1	✓	2
21.11	odd	6		490.2.e.h.471.1		2
21.17	even	6		70.2.e.c.51.1	yes	2
21.20	even	2		490.2.a.c.1.1		1
84.47	odd	6		560.2.q.g.81.1		2
84.59	odd	6		560.2.q.g.401.1		2
84.83	odd	2		3920.2.a.p.1.1		1
105.17	odd	12		350.2.j.b.149.2		4
105.38	odd	12		350.2.j.b.149.1		4
105.47	odd	12		350.2.j.b.249.1		4
105.59	even	6		350.2.e.e.51.1		2
105.62	odd	4		2450.2.c.g.99.1		2
105.68	odd	12		350.2.j.b.249.2		4
105.83	odd	4		2450.2.c.g.99.2		2
105.89	even	6		350.2.e.e.151.1		2
105.104	even	2		2450.2.a.w.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.c.11.1	✓	2	21.5	even	6
70.2.e.c.51.1	yes	2	21.17	even	6
350.2.e.e.51.1		2	105.59	even	6
350.2.e.e.151.1		2	105.89	even	6
350.2.j.b.149.1		4	105.38	odd	12
350.2.j.b.149.2		4	105.17	odd	12
350.2.j.b.249.1		4	105.47	odd	12
350.2.j.b.249.2		4	105.68	odd	12
490.2.a.b.1.1		1	3.2	odd	2
490.2.a.c.1.1		1	21.20	even	2
490.2.e.h.361.1		2	21.2	odd	6
490.2.e.h.471.1		2	21.11	odd	6
560.2.q.g.81.1		2	84.47	odd	6
560.2.q.g.401.1		2	84.59	odd	6
630.2.k.b.361.1		2	7.5	odd	6
630.2.k.b.541.1		2	7.3	odd	6
2450.2.a.w.1.1		1	105.104	even	2
2450.2.a.bc.1.1		1	15.14	odd	2
2450.2.c.g.99.1		2	105.62	odd	4
2450.2.c.g.99.2		2	105.83	odd	4
2450.2.c.l.99.1		2	15.2	even	4
2450.2.c.l.99.2		2	15.8	even	4
3920.2.a.p.1.1		1	84.83	odd	2
3920.2.a.bc.1.1		1	12.11	even	2
4410.2.a.bd.1.1		1	1.1	even	1	trivial
4410.2.a.bm.1.1		1	7.6	odd	2