Embedded newform 120.2.w.c.53.7

• Label: 120.2.w.c.53.7
• Level: $120$
• Weight: $2$
• Character: 120.53
• Analytic conductor: $0.958$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.958204824255\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.7
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.250043 - 1.39193i) q^{2} +(-1.40091 + 1.01856i) q^{3} +(-1.87496 + 0.696087i) q^{4} +(-2.23305 - 0.116202i) q^{5} +(1.76805 + 1.69529i) q^{6} +(-2.29041 + 2.29041i) q^{7} +(1.43773 + 2.43576i) q^{8} +(0.925085 - 2.85381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.250043	−	1.39193i	−0.176807	−	0.984246i
							\(3\)	−1.40091	+	1.01856i	−0.808814	+	0.588064i
\(5\)	−2.23305	−	0.116202i	−0.998649	−	0.0519669i
							\(7\)	−2.29041	+	2.29041i	−0.865694	+	0.865694i	−0.991992	−	0.126298i	\(-0.959690\pi\)
0.126298	+	0.991992i	\(0.459690\pi\)
\(11\)	−2.28378			−0.688586			−0.344293	−	0.938862i	\(-0.611881\pi\)
							−0.344293	+	0.938862i	\(0.611881\pi\)
\(13\)	−1.05635	+	1.05635i	−0.292977	+	0.292977i	−0.838255	−	0.545278i	\(-0.816424\pi\)
							0.545278	+	0.838255i	\(0.316424\pi\)
\(17\)	−3.04391	−	3.04391i	−0.738256	−	0.738256i	0.233984	−	0.972240i	\(-0.424824\pi\)
							−0.972240	+	0.233984i	\(0.924824\pi\)
\(19\)	3.36831			0.772744			0.386372	−	0.922343i	\(-0.373728\pi\)
							0.386372	+	0.922343i	\(0.373728\pi\)
\(23\)	−3.68785	+	3.68785i	−0.768969	+	0.768969i	−0.977925	−	0.208956i	\(-0.932994\pi\)
							0.208956	+	0.977925i	\(0.432994\pi\)
\(29\)		−	2.71461i		−	0.504091i	−0.967715	−	0.252045i	\(-0.918897\pi\)
							0.967715	−	0.252045i	\(-0.0811032\pi\)
\(31\)	−6.49196			−1.16599			−0.582995	−	0.812475i	\(-0.698119\pi\)
							−0.582995	+	0.812475i	\(0.698119\pi\)
\(37\)	−2.31197	−	2.31197i	−0.380085	−	0.380085i	0.491048	−	0.871133i	\(-0.336614\pi\)
							−0.871133	+	0.491048i	\(0.836614\pi\)
\(41\)			10.8056i			1.68754i	0.536702	+	0.843772i	\(0.319670\pi\)
							−0.536702	+	0.843772i	\(0.680330\pi\)
\(43\)	1.16384	−	1.16384i	0.177484	−	0.177484i	−0.612774	−	0.790258i	\(-0.709946\pi\)
							0.790258	+	0.612774i	\(0.209946\pi\)
\(47\)	1.83768	+	1.83768i	0.268053	+	0.268053i	0.828315	−	0.560262i	\(-0.189300\pi\)
							−0.560262	+	0.828315i	\(0.689300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.2.w.c.53.7	yes	32
3.2	odd	2	inner	120.2.w.c.53.10	yes	32
4.3	odd	2		480.2.bi.c.113.13		32
5.2	odd	4	inner	120.2.w.c.77.15	yes	32
5.3	odd	4		600.2.w.j.557.2		32
5.4	even	2		600.2.w.j.293.10		32
8.3	odd	2		480.2.bi.c.113.4		32
8.5	even	2	inner	120.2.w.c.53.2	✓	32
12.11	even	2		480.2.bi.c.113.12		32
15.2	even	4	inner	120.2.w.c.77.2	yes	32
15.8	even	4		600.2.w.j.557.15		32
15.14	odd	2		600.2.w.j.293.7		32
20.7	even	4		480.2.bi.c.17.5		32
24.5	odd	2	inner	120.2.w.c.53.15	yes	32
24.11	even	2		480.2.bi.c.113.5		32
40.13	odd	4		600.2.w.j.557.7		32
40.27	even	4		480.2.bi.c.17.12		32
40.29	even	2		600.2.w.j.293.15		32
40.37	odd	4	inner	120.2.w.c.77.10	yes	32
60.47	odd	4		480.2.bi.c.17.4		32
120.29	odd	2		600.2.w.j.293.2		32
120.53	even	4		600.2.w.j.557.10		32
120.77	even	4	inner	120.2.w.c.77.7	yes	32
120.107	odd	4		480.2.bi.c.17.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.2	✓	32	8.5	even	2	inner
120.2.w.c.53.7	yes	32	1.1	even	1	trivial
120.2.w.c.53.10	yes	32	3.2	odd	2	inner
120.2.w.c.53.15	yes	32	24.5	odd	2	inner
120.2.w.c.77.2	yes	32	15.2	even	4	inner
120.2.w.c.77.7	yes	32	120.77	even	4	inner
120.2.w.c.77.10	yes	32	40.37	odd	4	inner
120.2.w.c.77.15	yes	32	5.2	odd	4	inner
480.2.bi.c.17.4		32	60.47	odd	4
480.2.bi.c.17.5		32	20.7	even	4
480.2.bi.c.17.12		32	40.27	even	4
480.2.bi.c.17.13		32	120.107	odd	4
480.2.bi.c.113.4		32	8.3	odd	2
480.2.bi.c.113.5		32	24.11	even	2
480.2.bi.c.113.12		32	12.11	even	2
480.2.bi.c.113.13		32	4.3	odd	2
600.2.w.j.293.2		32	120.29	odd	2
600.2.w.j.293.7		32	15.14	odd	2
600.2.w.j.293.10		32	5.4	even	2
600.2.w.j.293.15		32	40.29	even	2
600.2.w.j.557.2		32	5.3	odd	4
600.2.w.j.557.7		32	40.13	odd	4
600.2.w.j.557.10		32	120.53	even	4
600.2.w.j.557.15		32	15.8	even	4