Embedded newform 4410.2.a.u.1.1

• Label: 4410.2.a.u.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 210)
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
\(11\)	5.00000	1.50756				0.753778	−	0.657129i	\(-0.228229\pi\)
			0.753778	+	0.657129i	\(0.228229\pi\)
\(13\)	5.00000	1.38675	0.693375	−	0.720577i	\(-0.256123\pi\)
			0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	−4.00000	−0.970143	−0.485071	−	0.874475i	\(-0.661206\pi\)
			−0.485071	+	0.874475i	\(0.661206\pi\)
\(19\)	7.00000	1.60591	0.802955	−	0.596040i	\(-0.203260\pi\)
			0.802955	+	0.596040i	\(0.203260\pi\)
\(23\)	−1.00000	−0.208514	−0.104257	−	0.994550i	\(-0.533247\pi\)
			−0.104257	+	0.994550i	\(0.533247\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	2.00000	0.359211	0.179605	−	0.983739i	\(-0.442518\pi\)
			0.179605	+	0.983739i	\(0.442518\pi\)
\(37\)	1.00000	0.164399	0.0821995	−	0.996616i	\(-0.473806\pi\)
			0.0821995	+	0.996616i	\(0.473806\pi\)
\(41\)	5.00000	0.780869	0.390434	−	0.920631i	\(-0.372325\pi\)
			0.390434	+	0.920631i	\(0.372325\pi\)
\(43\)	12.0000	1.82998	0.914991	−	0.403473i	\(-0.132197\pi\)
			0.914991	+	0.403473i	\(0.132197\pi\)
\(47\)	−11.0000	−1.60451	−0.802257	−	0.596978i	\(-0.796368\pi\)
			−0.802257	+	0.596978i	\(0.796368\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.u.1.1		1
3.2	odd	2		1470.2.a.o.1.1		1
7.3	odd	6		630.2.k.g.541.1		2
7.5	odd	6		630.2.k.g.361.1		2
7.6	odd	2		4410.2.a.j.1.1		1
15.14	odd	2		7350.2.a.a.1.1		1
21.2	odd	6		1470.2.i.e.361.1		2
21.5	even	6		210.2.i.b.151.1	yes	2
21.11	odd	6		1470.2.i.e.961.1		2
21.17	even	6		210.2.i.b.121.1	✓	2
21.20	even	2		1470.2.a.l.1.1		1
84.47	odd	6		1680.2.bg.d.1201.1		2
84.59	odd	6		1680.2.bg.d.961.1		2
105.17	odd	12		1050.2.o.g.499.1		4
105.38	odd	12		1050.2.o.g.499.2		4
105.47	odd	12		1050.2.o.g.949.2		4
105.59	even	6		1050.2.i.p.751.1		2
105.68	odd	12		1050.2.o.g.949.1		4
105.89	even	6		1050.2.i.p.151.1		2
105.104	even	2		7350.2.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.b.121.1	✓	2	21.17	even	6
210.2.i.b.151.1	yes	2	21.5	even	6
630.2.k.g.361.1		2	7.5	odd	6
630.2.k.g.541.1		2	7.3	odd	6
1050.2.i.p.151.1		2	105.89	even	6
1050.2.i.p.751.1		2	105.59	even	6
1050.2.o.g.499.1		4	105.17	odd	12
1050.2.o.g.499.2		4	105.38	odd	12
1050.2.o.g.949.1		4	105.68	odd	12
1050.2.o.g.949.2		4	105.47	odd	12
1470.2.a.l.1.1		1	21.20	even	2
1470.2.a.o.1.1		1	3.2	odd	2
1470.2.i.e.361.1		2	21.2	odd	6
1470.2.i.e.961.1		2	21.11	odd	6
1680.2.bg.d.961.1		2	84.59	odd	6
1680.2.bg.d.1201.1		2	84.47	odd	6
4410.2.a.j.1.1		1	7.6	odd	2
4410.2.a.u.1.1		1	1.1	even	1	trivial
7350.2.a.a.1.1		1	15.14	odd	2
7350.2.a.u.1.1		1	105.104	even	2