Embedded newform 7440.2.a.o.1.1

• Label: 7440.2.a.o.1.1
• Level: $7440$
• Weight: $2$
• Character: 7440.1
• Self dual: yes
• Analytic conductor: $59.409$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(59.4086991038\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	7440.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} -6.00000 q^{17} +4.00000 q^{19} -4.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +1.00000 q^{31} -4.00000 q^{33} +4.00000 q^{35} +10.0000 q^{37} -2.00000 q^{39} -6.00000 q^{41} +12.0000 q^{43} +1.00000 q^{45} +9.00000 q^{49} +6.00000 q^{51} -2.00000 q^{53} +4.00000 q^{55} -4.00000 q^{57} +8.00000 q^{59} +6.00000 q^{61} +4.00000 q^{63} +2.00000 q^{65} -8.00000 q^{67} +12.0000 q^{71} +6.00000 q^{73} -1.00000 q^{75} +16.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -6.00000 q^{85} +6.00000 q^{87} +6.00000 q^{89} +8.00000 q^{91} -1.00000 q^{93} +4.00000 q^{95} -6.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(5\)	1.00000	0.447214
			\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
0.755929	+	0.654654i				\(0.227186\pi\)
\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	1.00000	0.179605
			\(37\)	10.0000	1.64399	0.821995	−	0.569495i	\(-0.192861\pi\)
0.821995	+	0.569495i				\(0.192861\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	12.0000	1.82998	0.914991	−	0.403473i	\(-0.132197\pi\)
			0.914991	+	0.403473i	\(0.132197\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	−2.00000	−0.274721	−0.137361	−	0.990521i	\(-0.543862\pi\)
			−0.137361	+	0.990521i	\(0.543862\pi\)
\(59\)	8.00000	1.04151	0.520756	−	0.853706i	\(-0.325650\pi\)
			0.520756	+	0.853706i	\(0.325650\pi\)
\(61\)	6.00000	0.768221	0.384111	−	0.923287i	\(-0.374508\pi\)
			0.384111	+	0.923287i	\(0.374508\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	12.0000	1.42414	0.712069	−	0.702109i	\(-0.247758\pi\)
			0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	6.00000	0.702247	0.351123	−	0.936329i	\(-0.385800\pi\)
			0.351123	+	0.936329i	\(0.385800\pi\)
\(79\)	8.00000	0.900070	0.450035	−	0.893011i	\(-0.351411\pi\)
			0.450035	+	0.893011i	\(0.351411\pi\)
\(83\)	−12.0000	−1.31717	−0.658586	−	0.752506i	\(-0.728845\pi\)
			−0.658586	+	0.752506i	\(0.728845\pi\)
\(89\)	6.00000	0.635999	0.317999	−	0.948091i	\(-0.396989\pi\)
			0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	−6.00000	−0.609208	−0.304604	−	0.952479i	\(-0.598524\pi\)
			−0.304604	+	0.952479i	\(0.598524\pi\)

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	7440.2.a.o.1.1		1
4.3	odd	2		465.2.a.a.1.1	✓	1
12.11	even	2		1395.2.a.c.1.1		1
20.3	even	4		2325.2.c.c.1024.2		2
20.7	even	4		2325.2.c.c.1024.1		2
20.19	odd	2		2325.2.a.i.1.1		1
60.59	even	2		6975.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.a.1.1	✓	1	4.3	odd	2
1395.2.a.c.1.1		1	12.11	even	2
2325.2.a.i.1.1		1	20.19	odd	2
2325.2.c.c.1024.1		2	20.7	even	4
2325.2.c.c.1024.2		2	20.3	even	4
6975.2.a.g.1.1		1	60.59	even	2
7440.2.a.o.1.1		1	1.1	even	1	trivial