Embedded newform 2400.4.a.b.1.1

• Label: 2400.4.a.b.1.1
• Level: $2400$
• Weight: $4$
• Character: 2400.1
• Self dual: yes
• Analytic conductor: $141.605$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(141.604584014\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} -12.0000 q^{7} +9.00000 q^{9} -20.0000 q^{11} +58.0000 q^{13} +70.0000 q^{17} -92.0000 q^{19} +36.0000 q^{21} -112.000 q^{23} -27.0000 q^{27} +66.0000 q^{29} -108.000 q^{31} +60.0000 q^{33} +58.0000 q^{37} -174.000 q^{39} +66.0000 q^{41} +388.000 q^{43} +408.000 q^{47} -199.000 q^{49} -210.000 q^{51} -474.000 q^{53} +276.000 q^{57} -540.000 q^{59} +14.0000 q^{61} -108.000 q^{63} +276.000 q^{67} +336.000 q^{69} -96.0000 q^{71} +790.000 q^{73} +240.000 q^{77} +308.000 q^{79} +81.0000 q^{81} +1036.00 q^{83} -198.000 q^{87} +1210.00 q^{89} -696.000 q^{91} +324.000 q^{93} -1426.00 q^{97} -180.000 q^{99} +O(q^{100})\)

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
			\(7\)	−12.0000	−0.647939	−0.323970	−	0.946068i	\(-0.605018\pi\)
−0.323970	+	0.946068i				\(0.605018\pi\)
\(11\)	−20.0000	−0.548202	−0.274101	−	0.961701i	\(-0.588380\pi\)
			−0.274101	+	0.961701i	\(0.588380\pi\)
\(13\)	58.0000	1.23741	0.618704	−	0.785624i	\(-0.287658\pi\)
			0.618704	+	0.785624i	\(0.287658\pi\)
\(17\)	70.0000	0.998676	0.499338	−	0.866407i	\(-0.333577\pi\)
			0.499338	+	0.866407i	\(0.333577\pi\)
\(19\)	−92.0000	−1.11086	−0.555428	−	0.831565i	\(-0.687445\pi\)
			−0.555428	+	0.831565i	\(0.687445\pi\)
\(23\)	−112.000	−1.01537	−0.507687	−	0.861541i	\(-0.669499\pi\)
			−0.507687	+	0.861541i	\(0.669499\pi\)
\(29\)	66.0000	0.422617	0.211308	−	0.977419i	\(-0.432228\pi\)
			0.211308	+	0.977419i	\(0.432228\pi\)
\(31\)	−108.000	−0.625722	−0.312861	−	0.949799i	\(-0.601287\pi\)
			−0.312861	+	0.949799i	\(0.601287\pi\)
\(37\)	58.0000	0.257707	0.128853	−	0.991664i	\(-0.458870\pi\)
			0.128853	+	0.991664i	\(0.458870\pi\)
\(41\)	66.0000	0.251402	0.125701	−	0.992068i	\(-0.459882\pi\)
			0.125701	+	0.992068i	\(0.459882\pi\)
\(43\)	388.000	1.37603	0.688017	−	0.725695i	\(-0.258482\pi\)
			0.688017	+	0.725695i	\(0.258482\pi\)
\(47\)	408.000	1.26623	0.633116	−	0.774057i	\(-0.281776\pi\)
			0.633116	+	0.774057i	\(0.281776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.b.1.1		1
4.3	odd	2		2400.4.a.u.1.1		1
5.4	even	2		480.4.a.h.1.1	yes	1
15.14	odd	2		1440.4.a.q.1.1		1
20.19	odd	2		480.4.a.a.1.1	✓	1
40.19	odd	2		960.4.a.be.1.1		1
40.29	even	2		960.4.a.p.1.1		1
60.59	even	2		1440.4.a.l.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.a.1.1	✓	1	20.19	odd	2
480.4.a.h.1.1	yes	1	5.4	even	2
960.4.a.p.1.1		1	40.29	even	2
960.4.a.be.1.1		1	40.19	odd	2
1440.4.a.l.1.1		1	60.59	even	2
1440.4.a.q.1.1		1	15.14	odd	2
2400.4.a.b.1.1		1	1.1	even	1	trivial
2400.4.a.u.1.1		1	4.3	odd	2