Embedded newform 2420.4.a.e.1.1

• Label: 2420.4.a.e.1.1
• Level: $2420$
• Weight: $4$
• Character: 2420.1
• Self dual: yes
• Analytic conductor: $142.785$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2420.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(142.784622214\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2420.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{3} -5.00000 q^{5} +19.0000 q^{7} -2.00000 q^{9} +62.0000 q^{13} -25.0000 q^{15} -19.0000 q^{17} +131.000 q^{19} +95.0000 q^{21} +138.000 q^{23} +25.0000 q^{25} -145.000 q^{27} +79.0000 q^{29} +217.000 q^{31} -95.0000 q^{35} -91.0000 q^{37} +310.000 q^{39} -158.000 q^{41} -120.000 q^{43} +10.0000 q^{45} -546.000 q^{47} +18.0000 q^{49} -95.0000 q^{51} -439.000 q^{53} +655.000 q^{57} +290.000 q^{59} +373.000 q^{61} -38.0000 q^{63} -310.000 q^{65} +728.000 q^{67} +690.000 q^{69} -709.000 q^{71} -850.000 q^{73} +125.000 q^{75} +1194.00 q^{79} -671.000 q^{81} -58.0000 q^{83} +95.0000 q^{85} +395.000 q^{87} +753.000 q^{89} +1178.00 q^{91} +1085.00 q^{93} -655.000 q^{95} +1228.00 q^{97} +O(q^{100})\)

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
			\(3\)	5.00000	0.962250	0.481125	−	0.876652i	\(-0.340228\pi\)
0.481125	+	0.876652i				\(0.340228\pi\)
\(5\)	−5.00000	−0.447214
			\(7\)	19.0000	1.02590	0.512952	−	0.858417i	\(-0.328552\pi\)
0.512952	+	0.858417i				\(0.328552\pi\)
\(11\)	0	0
			\(13\)	62.0000	1.32275	0.661373	−	0.750057i	\(-0.269974\pi\)
0.661373	+	0.750057i				\(0.269974\pi\)
\(17\)	−19.0000	−0.271069	−0.135535	−	0.990773i	\(-0.543275\pi\)
			−0.135535	+	0.990773i	\(0.543275\pi\)
\(19\)	131.000	1.58176	0.790881	−	0.611971i	\(-0.209623\pi\)
			0.790881	+	0.611971i	\(0.209623\pi\)
\(23\)	138.000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	79.0000	0.505860	0.252930	−	0.967485i	\(-0.418606\pi\)
			0.252930	+	0.967485i	\(0.418606\pi\)
\(31\)	217.000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	−91.0000	−0.404333	−0.202166	−	0.979351i	\(-0.564798\pi\)
			−0.202166	+	0.979351i	\(0.564798\pi\)
\(41\)	−158.000	−0.601840	−0.300920	−	0.953649i	\(-0.597294\pi\)
			−0.300920	+	0.953649i	\(0.597294\pi\)
\(43\)	−120.000	−0.425577	−0.212789	−	0.977098i	\(-0.568255\pi\)
			−0.212789	+	0.977098i	\(0.568255\pi\)
\(47\)	−546.000	−1.69452	−0.847258	−	0.531181i	\(-0.821748\pi\)
			−0.847258	+	0.531181i	\(0.821748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.4.a.e.1.1		1
11.10	odd	2		220.4.a.b.1.1	✓	1
33.32	even	2		1980.4.a.d.1.1		1
44.43	even	2		880.4.a.e.1.1		1
55.32	even	4		1100.4.b.d.749.1		2
55.43	even	4		1100.4.b.d.749.2		2
55.54	odd	2		1100.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.b.1.1	✓	1	11.10	odd	2
880.4.a.e.1.1		1	44.43	even	2
1100.4.a.b.1.1		1	55.54	odd	2
1100.4.b.d.749.1		2	55.32	even	4
1100.4.b.d.749.2		2	55.43	even	4
1980.4.a.d.1.1		1	33.32	even	2
2420.4.a.e.1.1		1	1.1	even	1	trivial