Embedded newform 1200.4.a.o.1.1

• Label: 1200.4.a.o.1.1
• Level: $1200$
• Weight: $4$
• Character: 1200.1
• Self dual: yes
• Analytic conductor: $70.802$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(70.8022920069\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1200.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +20.0000 q^{7} +9.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−3.00000	−0.577350
\(5\)	0	0
			\(7\)	20.0000	1.07990	0.539949	−	0.841698i	\(-0.318443\pi\)
0.539949	+	0.841698i				\(0.318443\pi\)
\(11\)	24.0000	0.657843	0.328921	−	0.944357i	\(-0.393315\pi\)
			0.328921	+	0.944357i	\(0.393315\pi\)
\(13\)	−74.0000	−1.57876	−0.789381	−	0.613904i	\(-0.789598\pi\)
			−0.789381	+	0.613904i	\(0.789598\pi\)
\(17\)	−54.0000	−0.770407	−0.385204	−	0.922832i	\(-0.625869\pi\)
			−0.385204	+	0.922832i	\(0.625869\pi\)
\(19\)	124.000	1.49724	0.748620	−	0.663000i	\(-0.230717\pi\)
			0.748620	+	0.663000i	\(0.230717\pi\)
\(23\)	−120.000	−1.08790	−0.543951	−	0.839117i	\(-0.683072\pi\)
			−0.543951	+	0.839117i	\(0.683072\pi\)
\(29\)	−78.0000	−0.499456	−0.249728	−	0.968316i	\(-0.580341\pi\)
			−0.249728	+	0.968316i	\(0.580341\pi\)
\(31\)	−200.000	−1.15874	−0.579372	−	0.815063i	\(-0.696702\pi\)
			−0.579372	+	0.815063i	\(0.696702\pi\)
\(37\)	70.0000	0.311025	0.155513	−	0.987834i	\(-0.450297\pi\)
			0.155513	+	0.987834i	\(0.450297\pi\)
\(41\)	330.000	1.25701	0.628504	−	0.777806i	\(-0.283668\pi\)
			0.628504	+	0.777806i	\(0.283668\pi\)
\(43\)	92.0000	0.326276	0.163138	−	0.986603i	\(-0.447838\pi\)
			0.163138	+	0.986603i	\(0.447838\pi\)
\(47\)	−24.0000	−0.0744843	−0.0372421	−	0.999306i	\(-0.511857\pi\)
			−0.0372421	+	0.999306i	\(0.511857\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.o.1.1		1
4.3	odd	2		75.4.a.a.1.1		1
5.2	odd	4		1200.4.f.m.49.2		2
5.3	odd	4		1200.4.f.m.49.1		2
5.4	even	2		240.4.a.f.1.1		1
12.11	even	2		225.4.a.g.1.1		1
15.14	odd	2		720.4.a.r.1.1		1
20.3	even	4		75.4.b.a.49.2		2
20.7	even	4		75.4.b.a.49.1		2
20.19	odd	2		15.4.a.b.1.1	✓	1
40.19	odd	2		960.4.a.bi.1.1		1
40.29	even	2		960.4.a.l.1.1		1
60.23	odd	4		225.4.b.d.199.1		2
60.47	odd	4		225.4.b.d.199.2		2
60.59	even	2		45.4.a.b.1.1		1
140.139	even	2		735.4.a.i.1.1		1
180.59	even	6		405.4.e.k.136.1		2
180.79	odd	6		405.4.e.d.271.1		2
180.119	even	6		405.4.e.k.271.1		2
180.139	odd	6		405.4.e.d.136.1		2
220.219	even	2		1815.4.a.a.1.1		1
420.419	odd	2		2205.4.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	20.19	odd	2
45.4.a.b.1.1		1	60.59	even	2
75.4.a.a.1.1		1	4.3	odd	2
75.4.b.a.49.1		2	20.7	even	4
75.4.b.a.49.2		2	20.3	even	4
225.4.a.g.1.1		1	12.11	even	2
225.4.b.d.199.1		2	60.23	odd	4
225.4.b.d.199.2		2	60.47	odd	4
240.4.a.f.1.1		1	5.4	even	2
405.4.e.d.136.1		2	180.139	odd	6
405.4.e.d.271.1		2	180.79	odd	6
405.4.e.k.136.1		2	180.59	even	6
405.4.e.k.271.1		2	180.119	even	6
720.4.a.r.1.1		1	15.14	odd	2
735.4.a.i.1.1		1	140.139	even	2
960.4.a.l.1.1		1	40.29	even	2
960.4.a.bi.1.1		1	40.19	odd	2
1200.4.a.o.1.1		1	1.1	even	1	trivial
1200.4.f.m.49.1		2	5.3	odd	4
1200.4.f.m.49.2		2	5.2	odd	4
1815.4.a.a.1.1		1	220.219	even	2
2205.4.a.c.1.1		1	420.419	odd	2