Embedded newform 2790.2.a.j.1.1

• Label: 2790.2.a.j.1.1
• Level: $2790$
• Weight: $2$
• Character: 2790.1
• Self dual: yes
• Analytic conductor: $22.278$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(22.2782621639\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2790.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +6.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{20} -6.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +8.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -6.00000 q^{37} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} +2.00000 q^{46} -4.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +6.00000 q^{53} +6.00000 q^{55} -8.00000 q^{58} +4.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +4.00000 q^{68} +8.00000 q^{71} -4.00000 q^{73} +6.00000 q^{74} -4.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} +4.00000 q^{85} -4.00000 q^{86} -6.00000 q^{88} +2.00000 q^{89} -2.00000 q^{92} +4.00000 q^{94} +14.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0
\(5\)	1.00000	0.447214
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	6.00000	1.80907	0.904534	−	0.426401i	\(-0.140219\pi\)
			0.904534	+	0.426401i	\(0.140219\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	4.00000	0.970143	0.485071	−	0.874475i	\(-0.338794\pi\)
			0.485071	+	0.874475i	\(0.338794\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−2.00000	−0.417029	−0.208514	−	0.978019i	\(-0.566863\pi\)
			−0.208514	+	0.978019i	\(0.566863\pi\)
\(29\)	8.00000	1.48556	0.742781	−	0.669534i	\(-0.233506\pi\)
			0.742781	+	0.669534i	\(0.233506\pi\)
\(31\)	1.00000	0.179605
			\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
−0.493197	+	0.869918i				\(0.664172\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)
\(53\)	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	\(0.5\pi\)
\(61\)	4.00000	0.512148	0.256074	−	0.966657i	\(-0.417571\pi\)
			0.256074	+	0.966657i	\(0.417571\pi\)
\(67\)	−4.00000	−0.488678	−0.244339	−	0.969690i	\(-0.578571\pi\)
			−0.244339	+	0.969690i	\(0.578571\pi\)
\(71\)	8.00000	0.949425	0.474713	−	0.880141i	\(-0.342552\pi\)
			0.474713	+	0.880141i	\(0.342552\pi\)
\(73\)	−4.00000	−0.468165	−0.234082	−	0.972217i	\(-0.575209\pi\)
			−0.234082	+	0.972217i	\(0.575209\pi\)
\(79\)	−4.00000	−0.450035	−0.225018	−	0.974355i	\(-0.572244\pi\)
			−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	2.00000	0.212000	0.106000	−	0.994366i	\(-0.466196\pi\)
			0.106000	+	0.994366i	\(0.466196\pi\)
\(97\)	14.0000	1.42148	0.710742	−	0.703452i	\(-0.248359\pi\)
			0.710742	+	0.703452i	\(0.248359\pi\)

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	2790.2.a.j.1.1		1
3.2	odd	2		930.2.a.l.1.1	✓	1
12.11	even	2		7440.2.a.r.1.1		1
15.2	even	4		4650.2.d.a.3349.2		2
15.8	even	4		4650.2.d.a.3349.1		2
15.14	odd	2		4650.2.a.r.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.l.1.1	✓	1	3.2	odd	2
2790.2.a.j.1.1		1	1.1	even	1	trivial
4650.2.a.r.1.1		1	15.14	odd	2
4650.2.d.a.3349.1		2	15.8	even	4
4650.2.d.a.3349.2		2	15.2	even	4
7440.2.a.r.1.1		1	12.11	even	2