Embedded newform 441.2.a.f.1.1

• Label: 441.2.a.f.1.1
• Level: $441$
• Weight: $2$
• Character: 441.1
• Self dual: yes
• Analytic conductor: $3.521$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	441.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} -3.00000 q^{8} +O(q^{10})\)

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	0.353553	−	0.935414i	\(-0.384973\pi\)
			0.353553	+	0.935414i	\(0.384973\pi\)
\(3\)	0	0
			\(5\)	−2.00000	−0.894427	−0.447214	−	0.894427i	\(-0.647584\pi\)
−0.447214	+	0.894427i				\(0.647584\pi\)
\(7\)	0	0
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	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	441.2.a.f.1.1		1
3.2	odd	2		147.2.a.a.1.1		1
4.3	odd	2		7056.2.a.p.1.1		1
7.2	even	3		441.2.e.b.361.1		2
7.3	odd	6		441.2.e.a.226.1		2
7.4	even	3		441.2.e.b.226.1		2
7.5	odd	6		441.2.e.a.361.1		2
7.6	odd	2		63.2.a.a.1.1		1
12.11	even	2		2352.2.a.v.1.1		1
15.14	odd	2		3675.2.a.n.1.1		1
21.2	odd	6		147.2.e.c.67.1		2
21.5	even	6		147.2.e.b.67.1		2
21.11	odd	6		147.2.e.c.79.1		2
21.17	even	6		147.2.e.b.79.1		2
21.20	even	2		21.2.a.a.1.1	✓	1
24.5	odd	2		9408.2.a.bv.1.1		1
24.11	even	2		9408.2.a.m.1.1		1
28.27	even	2		1008.2.a.l.1.1		1
35.13	even	4		1575.2.d.a.1324.1		2
35.27	even	4		1575.2.d.a.1324.2		2
35.34	odd	2		1575.2.a.c.1.1		1
56.13	odd	2		4032.2.a.h.1.1		1
56.27	even	2		4032.2.a.k.1.1		1
63.13	odd	6		567.2.f.b.379.1		2
63.20	even	6		567.2.f.g.190.1		2
63.34	odd	6		567.2.f.b.190.1		2
63.41	even	6		567.2.f.g.379.1		2
77.76	even	2		7623.2.a.g.1.1		1
84.11	even	6		2352.2.q.e.961.1		2
84.23	even	6		2352.2.q.e.1537.1		2
84.47	odd	6		2352.2.q.x.1537.1		2
84.59	odd	6		2352.2.q.x.961.1		2
84.83	odd	2		336.2.a.a.1.1		1
105.62	odd	4		525.2.d.a.274.1		2
105.83	odd	4		525.2.d.a.274.2		2
105.104	even	2		525.2.a.d.1.1		1
168.83	odd	2		1344.2.a.s.1.1		1
168.125	even	2		1344.2.a.g.1.1		1
231.230	odd	2		2541.2.a.j.1.1		1
273.272	even	2		3549.2.a.c.1.1		1
336.83	odd	4		5376.2.c.l.2689.1		2
336.125	even	4		5376.2.c.r.2689.2		2
336.251	odd	4		5376.2.c.l.2689.2		2
336.293	even	4		5376.2.c.r.2689.1		2
357.356	even	2		6069.2.a.b.1.1		1
399.398	odd	2		7581.2.a.d.1.1		1
420.419	odd	2		8400.2.a.bn.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	21.20	even	2
63.2.a.a.1.1		1	7.6	odd	2
147.2.a.a.1.1		1	3.2	odd	2
147.2.e.b.67.1		2	21.5	even	6
147.2.e.b.79.1		2	21.17	even	6
147.2.e.c.67.1		2	21.2	odd	6
147.2.e.c.79.1		2	21.11	odd	6
336.2.a.a.1.1		1	84.83	odd	2
441.2.a.f.1.1		1	1.1	even	1	trivial
441.2.e.a.226.1		2	7.3	odd	6
441.2.e.a.361.1		2	7.5	odd	6
441.2.e.b.226.1		2	7.4	even	3
441.2.e.b.361.1		2	7.2	even	3
525.2.a.d.1.1		1	105.104	even	2
525.2.d.a.274.1		2	105.62	odd	4
525.2.d.a.274.2		2	105.83	odd	4
567.2.f.b.190.1		2	63.34	odd	6
567.2.f.b.379.1		2	63.13	odd	6
567.2.f.g.190.1		2	63.20	even	6
567.2.f.g.379.1		2	63.41	even	6
1008.2.a.l.1.1		1	28.27	even	2
1344.2.a.g.1.1		1	168.125	even	2
1344.2.a.s.1.1		1	168.83	odd	2
1575.2.a.c.1.1		1	35.34	odd	2
1575.2.d.a.1324.1		2	35.13	even	4
1575.2.d.a.1324.2		2	35.27	even	4
2352.2.a.v.1.1		1	12.11	even	2
2352.2.q.e.961.1		2	84.11	even	6
2352.2.q.e.1537.1		2	84.23	even	6
2352.2.q.x.961.1		2	84.59	odd	6
2352.2.q.x.1537.1		2	84.47	odd	6
2541.2.a.j.1.1		1	231.230	odd	2
3549.2.a.c.1.1		1	273.272	even	2
3675.2.a.n.1.1		1	15.14	odd	2
4032.2.a.h.1.1		1	56.13	odd	2
4032.2.a.k.1.1		1	56.27	even	2
5376.2.c.l.2689.1		2	336.83	odd	4
5376.2.c.l.2689.2		2	336.251	odd	4
5376.2.c.r.2689.1		2	336.293	even	4
5376.2.c.r.2689.2		2	336.125	even	4
6069.2.a.b.1.1		1	357.356	even	2
7056.2.a.p.1.1		1	4.3	odd	2
7581.2.a.d.1.1		1	399.398	odd	2
7623.2.a.g.1.1		1	77.76	even	2
8400.2.a.bn.1.1		1	420.419	odd	2
9408.2.a.m.1.1		1	24.11	even	2
9408.2.a.bv.1.1		1	24.5	odd	2