Embedded newform 441.2.a.a.1.1

• Label: 441.2.a.a.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

\(f(q)\)

\(=\)

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

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−1.41421	−0.707107	−	0.707107i	\(-0.750000\pi\)
			−0.707107	+	0.707107i	\(0.750000\pi\)
\(3\)	0	0
			\(5\)	−2.00000	−0.894427	−0.447214	−	0.894427i	\(-0.647584\pi\)
−0.447214	+	0.894427i				\(0.647584\pi\)
\(7\)	0	0
			\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
0.301511	+	0.953463i				\(0.402509\pi\)
\(13\)	−1.00000	−0.277350	−0.138675	−	0.990338i	\(-0.544284\pi\)
			−0.138675	+	0.990338i	\(0.544284\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	−1.00000	−0.229416	−0.114708	−	0.993399i	\(-0.536593\pi\)
			−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−4.00000	−0.742781	−0.371391	−	0.928477i	\(-0.621119\pi\)
			−0.371391	+	0.928477i	\(0.621119\pi\)
\(31\)	−9.00000	−1.61645	−0.808224	−	0.588875i	\(-0.799571\pi\)
			−0.808224	+	0.588875i	\(0.799571\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	5.00000	0.762493	0.381246	−	0.924473i	\(-0.375495\pi\)
			0.381246	+	0.924473i	\(0.375495\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.a.a.1.1		1
3.2	odd	2		147.2.a.b.1.1		1
4.3	odd	2		7056.2.a.m.1.1		1
7.2	even	3		441.2.e.e.361.1		2
7.3	odd	6		63.2.e.b.37.1		2
7.4	even	3		441.2.e.e.226.1		2
7.5	odd	6		63.2.e.b.46.1		2
7.6	odd	2		441.2.a.b.1.1		1
12.11	even	2		2352.2.a.w.1.1		1
15.14	odd	2		3675.2.a.c.1.1		1
21.2	odd	6		147.2.e.a.67.1		2
21.5	even	6		21.2.e.a.4.1	✓	2
21.11	odd	6		147.2.e.a.79.1		2
21.17	even	6		21.2.e.a.16.1	yes	2
21.20	even	2		147.2.a.c.1.1		1
24.5	odd	2		9408.2.a.bz.1.1		1
24.11	even	2		9408.2.a.k.1.1		1
28.3	even	6		1008.2.s.d.289.1		2
28.19	even	6		1008.2.s.d.865.1		2
28.27	even	2		7056.2.a.bp.1.1		1
63.5	even	6		567.2.h.f.298.1		2
63.31	odd	6		567.2.g.f.541.1		2
63.38	even	6		567.2.h.f.352.1		2
63.40	odd	6		567.2.h.a.298.1		2
63.47	even	6		567.2.g.a.109.1		2
63.52	odd	6		567.2.h.a.352.1		2
63.59	even	6		567.2.g.a.541.1		2
63.61	odd	6		567.2.g.f.109.1		2
84.11	even	6		2352.2.q.c.961.1		2
84.23	even	6		2352.2.q.c.1537.1		2
84.47	odd	6		336.2.q.f.193.1		2
84.59	odd	6		336.2.q.f.289.1		2
84.83	odd	2		2352.2.a.d.1.1		1
105.17	odd	12		525.2.r.e.499.1		4
105.38	odd	12		525.2.r.e.499.2		4
105.47	odd	12		525.2.r.e.424.2		4
105.59	even	6		525.2.i.e.226.1		2
105.68	odd	12		525.2.r.e.424.1		4
105.89	even	6		525.2.i.e.151.1		2
105.104	even	2		3675.2.a.a.1.1		1
168.5	even	6		1344.2.q.m.193.1		2
168.59	odd	6		1344.2.q.c.961.1		2
168.83	odd	2		9408.2.a.cv.1.1		1
168.101	even	6		1344.2.q.m.961.1		2
168.125	even	2		9408.2.a.bg.1.1		1
168.131	odd	6		1344.2.q.c.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.e.a.4.1	✓	2	21.5	even	6
21.2.e.a.16.1	yes	2	21.17	even	6
63.2.e.b.37.1		2	7.3	odd	6
63.2.e.b.46.1		2	7.5	odd	6
147.2.a.b.1.1		1	3.2	odd	2
147.2.a.c.1.1		1	21.20	even	2
147.2.e.a.67.1		2	21.2	odd	6
147.2.e.a.79.1		2	21.11	odd	6
336.2.q.f.193.1		2	84.47	odd	6
336.2.q.f.289.1		2	84.59	odd	6
441.2.a.a.1.1		1	1.1	even	1	trivial
441.2.a.b.1.1		1	7.6	odd	2
441.2.e.e.226.1		2	7.4	even	3
441.2.e.e.361.1		2	7.2	even	3
525.2.i.e.151.1		2	105.89	even	6
525.2.i.e.226.1		2	105.59	even	6
525.2.r.e.424.1		4	105.68	odd	12
525.2.r.e.424.2		4	105.47	odd	12
525.2.r.e.499.1		4	105.17	odd	12
525.2.r.e.499.2		4	105.38	odd	12
567.2.g.a.109.1		2	63.47	even	6
567.2.g.a.541.1		2	63.59	even	6
567.2.g.f.109.1		2	63.61	odd	6
567.2.g.f.541.1		2	63.31	odd	6
567.2.h.a.298.1		2	63.40	odd	6
567.2.h.a.352.1		2	63.52	odd	6
567.2.h.f.298.1		2	63.5	even	6
567.2.h.f.352.1		2	63.38	even	6
1008.2.s.d.289.1		2	28.3	even	6
1008.2.s.d.865.1		2	28.19	even	6
1344.2.q.c.193.1		2	168.131	odd	6
1344.2.q.c.961.1		2	168.59	odd	6
1344.2.q.m.193.1		2	168.5	even	6
1344.2.q.m.961.1		2	168.101	even	6
2352.2.a.d.1.1		1	84.83	odd	2
2352.2.a.w.1.1		1	12.11	even	2
2352.2.q.c.961.1		2	84.11	even	6
2352.2.q.c.1537.1		2	84.23	even	6
3675.2.a.a.1.1		1	105.104	even	2
3675.2.a.c.1.1		1	15.14	odd	2
7056.2.a.m.1.1		1	4.3	odd	2
7056.2.a.bp.1.1		1	28.27	even	2
9408.2.a.k.1.1		1	24.11	even	2
9408.2.a.bg.1.1		1	168.125	even	2
9408.2.a.bz.1.1		1	24.5	odd	2
9408.2.a.cv.1.1		1	168.83	odd	2