Embedded newform 1440.4.a.e.1.1

• Label: 1440.4.a.e.1.1
• Level: $1440$
• Weight: $4$
• Character: 1440.1
• Self dual: yes
• Analytic conductor: $84.963$
• Analytic rank: $0$
• 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:	\(0\)
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

\(f(q)\)

\(=\)

\(q-5.00000 q^{5} -4.00000 q^{7} -40.0000 q^{11} -90.0000 q^{13} +70.0000 q^{17} +40.0000 q^{19} -108.000 q^{23} +25.0000 q^{25} -166.000 q^{29} -40.0000 q^{31} +20.0000 q^{35} -130.000 q^{37} +310.000 q^{41} -268.000 q^{43} +556.000 q^{47} -327.000 q^{49} +370.000 q^{53} +200.000 q^{55} -240.000 q^{59} -130.000 q^{61} +450.000 q^{65} +876.000 q^{67} +840.000 q^{71} +250.000 q^{73} +160.000 q^{77} -880.000 q^{79} +188.000 q^{83} -350.000 q^{85} +726.000 q^{89} +360.000 q^{91} -200.000 q^{95} -1550.00 q^{97} +O(q^{100})\)

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
			\(7\)	−4.00000	−0.215980	−0.107990	−	0.994152i	\(-0.534441\pi\)
−0.107990	+	0.994152i				\(0.534441\pi\)
\(11\)	−40.0000	−1.09640	−0.548202	−	0.836346i	\(-0.684688\pi\)
			−0.548202	+	0.836346i	\(0.684688\pi\)
\(13\)	−90.0000	−1.92012	−0.960058	−	0.279801i	\(-0.909732\pi\)
			−0.960058	+	0.279801i	\(0.909732\pi\)
\(17\)	70.0000	0.998676	0.499338	−	0.866407i	\(-0.333577\pi\)
			0.499338	+	0.866407i	\(0.333577\pi\)
\(19\)	40.0000	0.482980	0.241490	−	0.970403i	\(-0.422364\pi\)
			0.241490	+	0.970403i	\(0.422364\pi\)
\(23\)	−108.000	−0.979111	−0.489556	−	0.871972i	\(-0.662841\pi\)
			−0.489556	+	0.871972i	\(0.662841\pi\)
\(29\)	−166.000	−1.06295	−0.531473	−	0.847075i	\(-0.678361\pi\)
			−0.531473	+	0.847075i	\(0.678361\pi\)
\(31\)	−40.0000	−0.231749	−0.115874	−	0.993264i	\(-0.536967\pi\)
			−0.115874	+	0.993264i	\(0.536967\pi\)
\(37\)	−130.000	−0.577618	−0.288809	−	0.957387i	\(-0.593259\pi\)
			−0.288809	+	0.957387i	\(0.593259\pi\)
\(41\)	310.000	1.18083	0.590413	−	0.807101i	\(-0.298965\pi\)
			0.590413	+	0.807101i	\(0.298965\pi\)
\(43\)	−268.000	−0.950456	−0.475228	−	0.879863i	\(-0.657634\pi\)
			−0.475228	+	0.879863i	\(0.657634\pi\)
\(47\)	556.000	1.72555	0.862776	−	0.505587i	\(-0.168724\pi\)
			0.862776	+	0.505587i	\(0.168724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.e.1.1		1
3.2	odd	2		480.4.a.d.1.1	✓	1
4.3	odd	2		1440.4.a.f.1.1		1
12.11	even	2		480.4.a.k.1.1	yes	1
15.14	odd	2		2400.4.a.s.1.1		1
24.5	odd	2		960.4.a.w.1.1		1
24.11	even	2		960.4.a.f.1.1		1
60.59	even	2		2400.4.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.d.1.1	✓	1	3.2	odd	2
480.4.a.k.1.1	yes	1	12.11	even	2
960.4.a.f.1.1		1	24.11	even	2
960.4.a.w.1.1		1	24.5	odd	2
1440.4.a.e.1.1		1	1.1	even	1	trivial
1440.4.a.f.1.1		1	4.3	odd	2
2400.4.a.d.1.1		1	60.59	even	2
2400.4.a.s.1.1		1	15.14	odd	2