Embedded newform 120.4.w.c.53.22

• Label: 120.4.w.c.53.22
• Level: $120$
• Weight: $4$
• Character: 120.53
• Analytic conductor: $7.080$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.08022920069\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.22
Character	\(\chi\)	\(=\)	120.53
Dual form			120.4.w.c.77.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32844 + 2.49705i) q^{2} +(5.19531 + 0.0934179i) q^{3} +(-4.47048 - 6.63436i) q^{4} +(5.77473 - 9.57353i) q^{5} +(-7.13494 + 12.8488i) q^{6} +(-15.4977 + 15.4977i) q^{7} +(22.5051 - 2.34964i) q^{8} +(26.9825 + 0.970671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{6} - 144 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(41\)	\(61\)	\(97\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32844	+	2.49705i	−0.469675	+	0.882839i
							\(3\)	5.19531	+	0.0934179i	0.999838	+	0.0179783i
\(5\)	5.77473	−	9.57353i	0.516507	−	0.856283i
							\(7\)	−15.4977	+	15.4977i	−0.836799	+	0.836799i	−0.988436	−	0.151637i	\(-0.951546\pi\)
0.151637	+	0.988436i	\(0.451546\pi\)
\(11\)	24.0882			0.660259			0.330130	−	0.943936i	\(-0.392908\pi\)
							0.330130	+	0.943936i	\(0.392908\pi\)
\(13\)	64.6828	−	64.6828i	1.37998	−	1.37998i	0.535360	−	0.844624i	\(-0.320176\pi\)
							0.844624	−	0.535360i	\(-0.179824\pi\)
\(17\)	69.5236	+	69.5236i	0.991880	+	0.991880i	0.999967	−	0.00808721i	\(-0.00257427\pi\)
							−0.00808721	+	0.999967i	\(0.502574\pi\)
\(19\)	0.230558			0.00278387			0.00139194	−	0.999999i	\(-0.499557\pi\)
							0.00139194	+	0.999999i	\(0.499557\pi\)
\(23\)	23.1362	−	23.1362i	0.209749	−	0.209749i	−0.594412	−	0.804161i	\(-0.702615\pi\)
							0.804161	+	0.594412i	\(0.202615\pi\)
\(29\)			110.723i			0.708991i	0.935058	+	0.354496i	\(0.115347\pi\)
							−0.935058	+	0.354496i	\(0.884653\pi\)
\(31\)	−210.319			−1.21853			−0.609266	−	0.792966i	\(-0.708536\pi\)
							−0.609266	+	0.792966i	\(0.708536\pi\)
\(37\)	−16.5677	−	16.5677i	−0.0736140	−	0.0736140i	0.669341	−	0.742955i	\(-0.266576\pi\)
							−0.742955	+	0.669341i	\(0.766576\pi\)
\(41\)			28.8516i			0.109899i	0.998489	+	0.0549495i	\(0.0174998\pi\)
							−0.998489	+	0.0549495i	\(0.982500\pi\)
\(43\)	−66.7330	+	66.7330i	−0.236667	+	0.236667i	−0.815469	−	0.578801i	\(-0.803521\pi\)
							0.578801	+	0.815469i	\(0.303521\pi\)
\(47\)	−171.055	−	171.055i	−0.530872	−	0.530872i	0.389960	−	0.920832i	\(-0.372489\pi\)
							−0.920832	+	0.389960i	\(0.872489\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.4.w.c.53.22	yes	128
3.2	odd	2	inner	120.4.w.c.53.43	yes	128
5.2	odd	4	inner	120.4.w.c.77.11	yes	128
8.5	even	2	inner	120.4.w.c.53.54	yes	128
15.2	even	4	inner	120.4.w.c.77.54	yes	128
24.5	odd	2	inner	120.4.w.c.53.11	✓	128
40.37	odd	4	inner	120.4.w.c.77.43	yes	128
120.77	even	4	inner	120.4.w.c.77.22	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.w.c.53.11	✓	128	24.5	odd	2	inner
120.4.w.c.53.22	yes	128	1.1	even	1	trivial
120.4.w.c.53.43	yes	128	3.2	odd	2	inner
120.4.w.c.53.54	yes	128	8.5	even	2	inner
120.4.w.c.77.11	yes	128	5.2	odd	4	inner
120.4.w.c.77.22	yes	128	120.77	even	4	inner
120.4.w.c.77.43	yes	128	40.37	odd	4	inner
120.4.w.c.77.54	yes	128	15.2	even	4	inner