Embedded newform 120.4.w.c.53.53

• Label: 120.4.w.c.53.53
• 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			53.53
Character	\(\chi\)	\(=\)	120.53
Dual form			120.4.w.c.77.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.43671 - 1.43613i) q^{2} +(-4.07292 + 3.22666i) q^{3} +(3.87507 - 6.99884i) q^{4} +(9.99090 - 5.01815i) q^{5} +(-5.29062 + 13.7117i) q^{6} +(-4.98039 + 4.98039i) q^{7} +(-0.608813 - 22.6192i) q^{8} +(6.17736 - 26.2838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{6} - 144 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.43671	−	1.43613i	0.861506	−	0.507748i
							\(3\)	−4.07292	+	3.22666i	−0.783834	+	0.620971i
\(5\)	9.99090	−	5.01815i	0.893614	−	0.448837i
							\(7\)	−4.98039	+	4.98039i	−0.268916	+	0.268916i	−0.828663	−	0.559747i	\(-0.810898\pi\)
0.559747	+	0.828663i	\(0.310898\pi\)
\(11\)	46.5334			1.27549			0.637743	−	0.770249i	\(-0.279868\pi\)
							0.637743	+	0.770249i	\(0.279868\pi\)
\(13\)	29.2455	−	29.2455i	0.623941	−	0.623941i	−0.322596	−	0.946537i	\(-0.604555\pi\)
							0.946537	+	0.322596i	\(0.104555\pi\)
\(17\)	−41.6034	−	41.6034i	−0.593547	−	0.593547i	0.345041	−	0.938588i	\(-0.387865\pi\)
							−0.938588	+	0.345041i	\(0.887865\pi\)
\(19\)	79.4302			0.959081			0.479541	−	0.877520i	\(-0.340803\pi\)
							0.479541	+	0.877520i	\(0.340803\pi\)
\(23\)	−57.9982	+	57.9982i	−0.525803	+	0.525803i	−0.919318	−	0.393515i	\(-0.871259\pi\)
							0.393515	+	0.919318i	\(0.371259\pi\)
\(29\)			153.930i			0.985655i	0.870127	+	0.492828i	\(0.164037\pi\)
							−0.870127	+	0.492828i	\(0.835963\pi\)
\(31\)	−52.3964			−0.303570			−0.151785	−	0.988414i	\(-0.548502\pi\)
							−0.151785	+	0.988414i	\(0.548502\pi\)
\(37\)	−172.501	−	172.501i	−0.766458	−	0.766458i	0.211023	−	0.977481i	\(-0.432320\pi\)
							−0.977481	+	0.211023i	\(0.932320\pi\)
\(41\)			376.458i			1.43397i	0.697087	+	0.716987i	\(0.254479\pi\)
							−0.697087	+	0.716987i	\(0.745521\pi\)
\(43\)	−376.943	+	376.943i	−1.33682	+	1.33682i	−0.437700	+	0.899121i	\(0.644207\pi\)
							−0.899121	+	0.437700i	\(0.855793\pi\)
\(47\)	−0.836257	−	0.836257i	−0.00259533	−	0.00259533i	0.705808	−	0.708403i	\(-0.250584\pi\)
							−0.708403	+	0.705808i	\(0.750584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.4.w.c.53.53	yes	128
3.2	odd	2	inner	120.4.w.c.53.12	✓	128
5.2	odd	4	inner	120.4.w.c.77.44	yes	128
8.5	even	2	inner	120.4.w.c.53.21	yes	128
15.2	even	4	inner	120.4.w.c.77.21	yes	128
24.5	odd	2	inner	120.4.w.c.53.44	yes	128
40.37	odd	4	inner	120.4.w.c.77.12	yes	128
120.77	even	4	inner	120.4.w.c.77.53	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.w.c.53.12	✓	128	3.2	odd	2	inner
120.4.w.c.53.21	yes	128	8.5	even	2	inner
120.4.w.c.53.44	yes	128	24.5	odd	2	inner
120.4.w.c.53.53	yes	128	1.1	even	1	trivial
120.4.w.c.77.12	yes	128	40.37	odd	4	inner
120.4.w.c.77.21	yes	128	15.2	even	4	inner
120.4.w.c.77.44	yes	128	5.2	odd	4	inner
120.4.w.c.77.53	yes	128	120.77	even	4	inner