Embedded newform 1440.4.a.d.1.1

• Label: 1440.4.a.d.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

\(f(q)\)

\(=\)

\(q-5.00000 q^{5} -12.0000 q^{7} +24.0000 q^{11} +38.0000 q^{13} +6.00000 q^{17} +104.000 q^{19} -100.000 q^{23} +25.0000 q^{25} -230.000 q^{29} -56.0000 q^{31} +60.0000 q^{35} +190.000 q^{37} -202.000 q^{41} -148.000 q^{43} -124.000 q^{47} -199.000 q^{49} -206.000 q^{53} -120.000 q^{55} +128.000 q^{59} +190.000 q^{61} -190.000 q^{65} -204.000 q^{67} +440.000 q^{71} +1210.00 q^{73} -288.000 q^{77} +816.000 q^{79} +1412.00 q^{83} -30.0000 q^{85} +214.000 q^{89} -456.000 q^{91} -520.000 q^{95} +1202.00 q^{97} +O(q^{100})\)

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				\(0.605018\pi\)
\(11\)	24.0000	0.657843	0.328921	−	0.944357i	\(-0.393315\pi\)
			0.328921	+	0.944357i	\(0.393315\pi\)
\(13\)	38.0000	0.810716	0.405358	−	0.914158i	\(-0.367147\pi\)
			0.405358	+	0.914158i	\(0.367147\pi\)
\(17\)	6.00000	0.0856008	0.0428004	−	0.999084i	\(-0.486372\pi\)
			0.0428004	+	0.999084i	\(0.486372\pi\)
\(19\)	104.000	1.25575	0.627875	−	0.778314i	\(-0.283925\pi\)
			0.627875	+	0.778314i	\(0.283925\pi\)
\(23\)	−100.000	−0.906584	−0.453292	−	0.891362i	\(-0.649751\pi\)
			−0.453292	+	0.891362i	\(0.649751\pi\)
\(29\)	−230.000	−1.47276	−0.736378	−	0.676570i	\(-0.763465\pi\)
			−0.736378	+	0.676570i	\(0.763465\pi\)
\(31\)	−56.0000	−0.324448	−0.162224	−	0.986754i	\(-0.551867\pi\)
			−0.162224	+	0.986754i	\(0.551867\pi\)
\(37\)	190.000	0.844211	0.422106	−	0.906547i	\(-0.361291\pi\)
			0.422106	+	0.906547i	\(0.361291\pi\)
\(41\)	−202.000	−0.769441	−0.384721	−	0.923033i	\(-0.625702\pi\)
			−0.384721	+	0.923033i	\(0.625702\pi\)
\(43\)	−148.000	−0.524879	−0.262439	−	0.964948i	\(-0.584527\pi\)
			−0.262439	+	0.964948i	\(0.584527\pi\)
\(47\)	−124.000	−0.384835	−0.192418	−	0.981313i	\(-0.561633\pi\)
			−0.192418	+	0.981313i	\(0.561633\pi\)

(See \(a_n\) instead)

Twists

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

    

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