Embedded newform 1440.4.a.l.1.1

• Label: 1440.4.a.l.1.1
• Level: $1440$
• Weight: $4$
• Character: 1440.1
• Self dual: yes
• Analytic conductor: $84.963$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(84.9627504083\)
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	\(\chi\)	\(=\)	1440.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} -12.0000 q^{7} -20.0000 q^{11} -58.0000 q^{13} +70.0000 q^{17} +92.0000 q^{19} +112.000 q^{23} +25.0000 q^{25} -66.0000 q^{29} +108.000 q^{31} -60.0000 q^{35} -58.0000 q^{37} -66.0000 q^{41} +388.000 q^{43} -408.000 q^{47} -199.000 q^{49} -474.000 q^{53} -100.000 q^{55} -540.000 q^{59} +14.0000 q^{61} -290.000 q^{65} +276.000 q^{67} -96.0000 q^{71} -790.000 q^{73} +240.000 q^{77} -308.000 q^{79} -1036.00 q^{83} +350.000 q^{85} -1210.00 q^{89} +696.000 q^{91} +460.000 q^{95} +1426.00 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	5.00000	0.447214
			\(7\)	−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	\(-0.712342\pi\)
			−0.618704	+	0.785624i	\(0.712342\pi\)
\(17\)	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	\(-0.312555\pi\)
			0.555428	+	0.831565i	\(0.312555\pi\)
\(23\)	112.000	1.01537	0.507687	−	0.861541i	\(-0.330501\pi\)
			0.507687	+	0.861541i	\(0.330501\pi\)
\(29\)	−66.0000	−0.422617	−0.211308	−	0.977419i	\(-0.567772\pi\)
			−0.211308	+	0.977419i	\(0.567772\pi\)
\(31\)	108.000	0.625722	0.312861	−	0.949799i	\(-0.398713\pi\)
			0.312861	+	0.949799i	\(0.398713\pi\)
\(37\)	−58.0000	−0.257707	−0.128853	−	0.991664i	\(-0.541130\pi\)
			−0.128853	+	0.991664i	\(0.541130\pi\)
\(41\)	−66.0000	−0.251402	−0.125701	−	0.992068i	\(-0.540118\pi\)
			−0.125701	+	0.992068i	\(0.540118\pi\)
\(43\)	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	\(-0.718224\pi\)
			−0.633116	+	0.774057i	\(0.718224\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.a.1.1	✓	1	3.2	odd	2
480.4.a.h.1.1	yes	1	12.11	even	2
960.4.a.p.1.1		1	24.11	even	2
960.4.a.be.1.1		1	24.5	odd	2
1440.4.a.l.1.1		1	1.1	even	1	trivial
1440.4.a.q.1.1		1	4.3	odd	2
2400.4.a.b.1.1		1	60.59	even	2
2400.4.a.u.1.1		1	15.14	odd	2