Embedded newform 4410.2.a.z.1.1

• Label: 4410.2.a.z.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 30)
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
\(11\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−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	\(-0.598656\pi\)
			−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	4410.2.a.z.1.1		1
3.2	odd	2		1470.2.a.d.1.1		1
7.6	odd	2		90.2.a.c.1.1		1
15.14	odd	2		7350.2.a.ct.1.1		1
21.2	odd	6		1470.2.i.q.361.1		2
21.5	even	6		1470.2.i.o.361.1		2
21.11	odd	6		1470.2.i.q.961.1		2
21.17	even	6		1470.2.i.o.961.1		2
21.20	even	2		30.2.a.a.1.1	✓	1
28.27	even	2		720.2.a.j.1.1		1
35.13	even	4		450.2.c.b.199.1		2
35.27	even	4		450.2.c.b.199.2		2
35.34	odd	2		450.2.a.d.1.1		1
56.13	odd	2		2880.2.a.a.1.1		1
56.27	even	2		2880.2.a.q.1.1		1
63.13	odd	6		810.2.e.b.541.1		2
63.20	even	6		810.2.e.l.271.1		2
63.34	odd	6		810.2.e.b.271.1		2
63.41	even	6		810.2.e.l.541.1		2
84.83	odd	2		240.2.a.b.1.1		1
105.62	odd	4		150.2.c.a.49.1		2
105.83	odd	4		150.2.c.a.49.2		2
105.104	even	2		150.2.a.b.1.1		1
140.27	odd	4		3600.2.f.i.2449.2		2
140.83	odd	4		3600.2.f.i.2449.1		2
140.139	even	2		3600.2.a.f.1.1		1
168.83	odd	2		960.2.a.p.1.1		1
168.125	even	2		960.2.a.e.1.1		1
231.230	odd	2		3630.2.a.w.1.1		1
273.83	odd	4		5070.2.b.k.1351.2		2
273.125	odd	4		5070.2.b.k.1351.1		2
273.272	even	2		5070.2.a.w.1.1		1
336.83	odd	4		3840.2.k.f.1921.1		2
336.125	even	4		3840.2.k.y.1921.2		2
336.251	odd	4		3840.2.k.f.1921.2		2
336.293	even	4		3840.2.k.y.1921.1		2
357.356	even	2		8670.2.a.g.1.1		1
420.83	even	4		1200.2.f.e.49.1		2
420.167	even	4		1200.2.f.e.49.2		2
420.419	odd	2		1200.2.a.k.1.1		1
840.83	even	4		4800.2.f.w.3649.2		2
840.293	odd	4		4800.2.f.p.3649.1		2
840.419	odd	2		4800.2.a.d.1.1		1
840.587	even	4		4800.2.f.w.3649.1		2
840.629	even	2		4800.2.a.cq.1.1		1
840.797	odd	4		4800.2.f.p.3649.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	21.20	even	2
90.2.a.c.1.1		1	7.6	odd	2
150.2.a.b.1.1		1	105.104	even	2
150.2.c.a.49.1		2	105.62	odd	4
150.2.c.a.49.2		2	105.83	odd	4
240.2.a.b.1.1		1	84.83	odd	2
450.2.a.d.1.1		1	35.34	odd	2
450.2.c.b.199.1		2	35.13	even	4
450.2.c.b.199.2		2	35.27	even	4
720.2.a.j.1.1		1	28.27	even	2
810.2.e.b.271.1		2	63.34	odd	6
810.2.e.b.541.1		2	63.13	odd	6
810.2.e.l.271.1		2	63.20	even	6
810.2.e.l.541.1		2	63.41	even	6
960.2.a.e.1.1		1	168.125	even	2
960.2.a.p.1.1		1	168.83	odd	2
1200.2.a.k.1.1		1	420.419	odd	2
1200.2.f.e.49.1		2	420.83	even	4
1200.2.f.e.49.2		2	420.167	even	4
1470.2.a.d.1.1		1	3.2	odd	2
1470.2.i.o.361.1		2	21.5	even	6
1470.2.i.o.961.1		2	21.17	even	6
1470.2.i.q.361.1		2	21.2	odd	6
1470.2.i.q.961.1		2	21.11	odd	6
2880.2.a.a.1.1		1	56.13	odd	2
2880.2.a.q.1.1		1	56.27	even	2
3600.2.a.f.1.1		1	140.139	even	2
3600.2.f.i.2449.1		2	140.83	odd	4
3600.2.f.i.2449.2		2	140.27	odd	4
3630.2.a.w.1.1		1	231.230	odd	2
3840.2.k.f.1921.1		2	336.83	odd	4
3840.2.k.f.1921.2		2	336.251	odd	4
3840.2.k.y.1921.1		2	336.293	even	4
3840.2.k.y.1921.2		2	336.125	even	4
4410.2.a.z.1.1		1	1.1	even	1	trivial
4800.2.a.d.1.1		1	840.419	odd	2
4800.2.a.cq.1.1		1	840.629	even	2
4800.2.f.p.3649.1		2	840.293	odd	4
4800.2.f.p.3649.2		2	840.797	odd	4
4800.2.f.w.3649.1		2	840.587	even	4
4800.2.f.w.3649.2		2	840.83	even	4
5070.2.a.w.1.1		1	273.272	even	2
5070.2.b.k.1351.1		2	273.125	odd	4
5070.2.b.k.1351.2		2	273.83	odd	4
7350.2.a.ct.1.1		1	15.14	odd	2
8670.2.a.g.1.1		1	357.356	even	2