Embedded newform 7360.2.a.t.1.1

• Label: 7360.2.a.t.1.1
• Level: $7360$
• Weight: $2$
• Character: 7360.1
• Self dual: yes
• Analytic conductor: $58.770$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(58.7698958877\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	7360.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -2.00000 q^{11} +5.00000 q^{13} -1.00000 q^{15} -4.00000 q^{17} +2.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +7.00000 q^{31} -2.00000 q^{33} +2.00000 q^{37} +5.00000 q^{39} -9.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -9.00000 q^{47} -7.00000 q^{49} -4.00000 q^{51} +6.00000 q^{53} +2.00000 q^{55} +2.00000 q^{57} -2.00000 q^{61} -5.00000 q^{65} +2.00000 q^{67} -1.00000 q^{69} -1.00000 q^{71} +1.00000 q^{73} +1.00000 q^{75} -14.0000 q^{79} +1.00000 q^{81} +4.00000 q^{85} +3.00000 q^{87} +16.0000 q^{89} +7.00000 q^{93} -2.00000 q^{95} -4.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	−1.00000	−0.447214
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	−2.00000	−0.603023	−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	5.00000	1.38675	0.693375	−	0.720577i	\(-0.256123\pi\)
			0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	−4.00000	−0.970143	−0.485071	−	0.874475i	\(-0.661206\pi\)
			−0.485071	+	0.874475i	\(0.661206\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	−1.00000	−0.208514
			\(29\)	3.00000	0.557086	0.278543	−	0.960424i	\(-0.410149\pi\)
0.278543	+	0.960424i				\(0.410149\pi\)
\(31\)	7.00000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−9.00000	−1.40556	−0.702782	−	0.711405i	\(-0.748059\pi\)
			−0.702782	+	0.711405i	\(0.748059\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−9.00000	−1.31278	−0.656392	−	0.754420i	\(-0.727918\pi\)
			−0.656392	+	0.754420i	\(0.727918\pi\)
\(53\)	6.00000	0.824163	0.412082	−	0.911147i	\(-0.364802\pi\)
			0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−2.00000	−0.256074	−0.128037	−	0.991769i	\(-0.540868\pi\)
			−0.128037	+	0.991769i	\(0.540868\pi\)
\(67\)	2.00000	0.244339	0.122169	−	0.992509i	\(-0.461015\pi\)
			0.122169	+	0.992509i	\(0.461015\pi\)
\(71\)	−1.00000	−0.118678	−0.0593391	−	0.998238i	\(-0.518899\pi\)
			−0.0593391	+	0.998238i	\(0.518899\pi\)
\(73\)	1.00000	0.117041	0.0585206	−	0.998286i	\(-0.481362\pi\)
			0.0585206	+	0.998286i	\(0.481362\pi\)
\(79\)	−14.0000	−1.57512	−0.787562	−	0.616236i	\(-0.788657\pi\)
			−0.787562	+	0.616236i	\(0.788657\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	16.0000	1.69600	0.847998	−	0.529999i	\(-0.177808\pi\)
			0.847998	+	0.529999i	\(0.177808\pi\)
\(97\)	−4.00000	−0.406138	−0.203069	−	0.979164i	\(-0.565092\pi\)
			−0.203069	+	0.979164i	\(0.565092\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.t.1.1		1
4.3	odd	2		7360.2.a.g.1.1		1
8.3	odd	2		1840.2.a.g.1.1		1
8.5	even	2		920.2.a.b.1.1	✓	1
24.5	odd	2		8280.2.a.e.1.1		1
40.13	odd	4		4600.2.e.h.4049.1		2
40.19	odd	2		9200.2.a.n.1.1		1
40.29	even	2		4600.2.a.j.1.1		1
40.37	odd	4		4600.2.e.h.4049.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.b.1.1	✓	1	8.5	even	2
1840.2.a.g.1.1		1	8.3	odd	2
4600.2.a.j.1.1		1	40.29	even	2
4600.2.e.h.4049.1		2	40.13	odd	4
4600.2.e.h.4049.2		2	40.37	odd	4
7360.2.a.g.1.1		1	4.3	odd	2
7360.2.a.t.1.1		1	1.1	even	1	trivial
8280.2.a.e.1.1		1	24.5	odd	2
9200.2.a.n.1.1		1	40.19	odd	2