Embedded newform 120.2.w.c.77.15

• Label: 120.2.w.c.77.15
• Level: $120$
• Weight: $2$
• Character: 120.77
• 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			77.15
Character	\(\chi\)	\(=\)	120.77
Dual form			120.2.w.c.53.15

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39193	−	0.250043i	0.984246	−	0.176807i
							\(3\)	1.01856	+	1.40091i	0.588064	+	0.808814i
\(5\)	−2.23305	+	0.116202i	−0.998649	+	0.0519669i
							\(7\)	−2.29041	−	2.29041i	−0.865694	−	0.865694i	0.126298	−	0.991992i	\(-0.459690\pi\)
−0.991992	+	0.126298i	\(0.959690\pi\)
\(11\)	−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	\(-0.183576\pi\)
							−0.545278	+	0.838255i	\(0.683576\pi\)
\(17\)	3.04391	−	3.04391i	0.738256	−	0.738256i	−0.233984	−	0.972240i	\(-0.575176\pi\)
							0.972240	+	0.233984i	\(0.0751764\pi\)
\(19\)	−3.36831			−0.772744			−0.386372	−	0.922343i	\(-0.626272\pi\)
							−0.386372	+	0.922343i	\(0.626272\pi\)
\(23\)	3.68785	+	3.68785i	0.768969	+	0.768969i	0.977925	−	0.208956i	\(-0.0670065\pi\)
							−0.208956	+	0.977925i	\(0.567006\pi\)
\(29\)			2.71461i			0.504091i	0.967715	+	0.252045i	\(0.0811032\pi\)
							−0.967715	+	0.252045i	\(0.918897\pi\)
\(31\)	−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	\(-0.663386\pi\)
							0.871133	+	0.491048i	\(0.163386\pi\)
\(41\)			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	\(-0.290054\pi\)
							−0.790258	+	0.612774i	\(0.790054\pi\)
\(47\)	−1.83768	+	1.83768i	−0.268053	+	0.268053i	−0.828315	−	0.560262i	\(-0.810700\pi\)
							0.560262	+	0.828315i	\(0.310700\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.2	✓	32	40.13	odd	4	inner
120.2.w.c.53.7	yes	32	5.3	odd	4	inner
120.2.w.c.53.10	yes	32	15.8	even	4	inner
120.2.w.c.53.15	yes	32	120.53	even	4	inner
120.2.w.c.77.2	yes	32	3.2	odd	2	inner
120.2.w.c.77.7	yes	32	24.5	odd	2	inner
120.2.w.c.77.10	yes	32	8.5	even	2	inner
120.2.w.c.77.15	yes	32	1.1	even	1	trivial
480.2.bi.c.17.4		32	12.11	even	2
480.2.bi.c.17.5		32	4.3	odd	2
480.2.bi.c.17.12		32	8.3	odd	2
480.2.bi.c.17.13		32	24.11	even	2
480.2.bi.c.113.4		32	40.3	even	4
480.2.bi.c.113.5		32	120.83	odd	4
480.2.bi.c.113.12		32	60.23	odd	4
480.2.bi.c.113.13		32	20.3	even	4
600.2.w.j.293.2		32	120.77	even	4
600.2.w.j.293.7		32	15.2	even	4
600.2.w.j.293.10		32	5.2	odd	4
600.2.w.j.293.15		32	40.37	odd	4
600.2.w.j.557.2		32	5.4	even	2
600.2.w.j.557.7		32	40.29	even	2
600.2.w.j.557.10		32	120.29	odd	2
600.2.w.j.557.15		32	15.14	odd	2