Embedded newform 1260.4.a.i.1.1

• Label: 1260.4.a.i.1.1
• Level: $1260$
• Weight: $4$
• Character: 1260.1
• Self dual: yes
• Analytic conductor: $74.342$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1260.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(74.3424066072\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1260.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} -7.00000 q^{7} +7.00000 q^{11} -23.0000 q^{13} +25.0000 q^{17} -62.0000 q^{19} +86.0000 q^{23} +25.0000 q^{25} +29.0000 q^{29} -12.0000 q^{31} -35.0000 q^{35} -150.000 q^{37} -204.000 q^{41} -178.000 q^{43} -33.0000 q^{47} +49.0000 q^{49} -452.000 q^{53} +35.0000 q^{55} -120.000 q^{59} +920.000 q^{61} -115.000 q^{65} -300.000 q^{67} -520.000 q^{71} +370.000 q^{73} -49.0000 q^{77} -1013.00 q^{79} +636.000 q^{83} +125.000 q^{85} -292.000 q^{89} +161.000 q^{91} -310.000 q^{95} -1381.00 q^{97} +O(q^{100})\)

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
			\(7\)	−7.00000	−0.377964
\(11\)	7.00000	0.191871				0.0959354	−	0.995388i	\(-0.469416\pi\)
			0.0959354	+	0.995388i	\(0.469416\pi\)
\(13\)	−23.0000	−0.490696	−0.245348	−	0.969435i	\(-0.578902\pi\)
			−0.245348	+	0.969435i	\(0.578902\pi\)
\(17\)	25.0000	0.356670	0.178335	−	0.983970i	\(-0.442929\pi\)
			0.178335	+	0.983970i	\(0.442929\pi\)
\(19\)	−62.0000	−0.748620	−0.374310	−	0.927304i	\(-0.622120\pi\)
			−0.374310	+	0.927304i	\(0.622120\pi\)
\(23\)	86.0000	0.779663	0.389831	−	0.920886i	\(-0.372533\pi\)
			0.389831	+	0.920886i	\(0.372533\pi\)
\(29\)	29.0000	0.185695	0.0928477	−	0.995680i	\(-0.470403\pi\)
			0.0928477	+	0.995680i	\(0.470403\pi\)
\(31\)	−12.0000	−0.0695246	−0.0347623	−	0.999396i	\(-0.511067\pi\)
			−0.0347623	+	0.999396i	\(0.511067\pi\)
\(37\)	−150.000	−0.666482	−0.333241	−	0.942842i	\(-0.608142\pi\)
			−0.333241	+	0.942842i	\(0.608142\pi\)
\(41\)	−204.000	−0.777060	−0.388530	−	0.921436i	\(-0.627017\pi\)
			−0.388530	+	0.921436i	\(0.627017\pi\)
\(43\)	−178.000	−0.631273	−0.315637	−	0.948880i	\(-0.602218\pi\)
			−0.315637	+	0.948880i	\(0.602218\pi\)
\(47\)	−33.0000	−0.102416	−0.0512079	−	0.998688i	\(-0.516307\pi\)
			−0.0512079	+	0.998688i	\(0.516307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.4.a.i.1.1		1
3.2	odd	2		140.4.a.d.1.1	✓	1
12.11	even	2		560.4.a.h.1.1		1
15.2	even	4		700.4.e.i.449.1		2
15.8	even	4		700.4.e.i.449.2		2
15.14	odd	2		700.4.a.g.1.1		1
21.2	odd	6		980.4.i.i.361.1		2
21.5	even	6		980.4.i.k.361.1		2
21.11	odd	6		980.4.i.i.961.1		2
21.17	even	6		980.4.i.k.961.1		2
21.20	even	2		980.4.a.g.1.1		1
24.5	odd	2		2240.4.a.s.1.1		1
24.11	even	2		2240.4.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.a.d.1.1	✓	1	3.2	odd	2
560.4.a.h.1.1		1	12.11	even	2
700.4.a.g.1.1		1	15.14	odd	2
700.4.e.i.449.1		2	15.2	even	4
700.4.e.i.449.2		2	15.8	even	4
980.4.a.g.1.1		1	21.20	even	2
980.4.i.i.361.1		2	21.2	odd	6
980.4.i.i.961.1		2	21.11	odd	6
980.4.i.k.361.1		2	21.5	even	6
980.4.i.k.961.1		2	21.17	even	6
1260.4.a.i.1.1		1	1.1	even	1	trivial
2240.4.a.s.1.1		1	24.5	odd	2
2240.4.a.u.1.1		1	24.11	even	2