Embedded newform 120.2.w.c.77.8

• Label: 120.2.w.c.77.8
• 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			77.8
Character	\(\chi\)	\(=\)	120.77
Dual form			120.2.w.c.53.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0941764 + 1.41107i) q^{2} +(-0.416519 + 1.68122i) q^{3} +(-1.98226 - 0.265780i) q^{4} +(1.62104 + 1.54020i) q^{5} +(-2.33310 - 0.746071i) q^{6} +(-0.361989 - 0.361989i) q^{7} +(0.561717 - 2.77209i) q^{8} +(-2.65302 - 1.40052i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0941764	+	1.41107i	−0.0665928	+	0.997780i
							\(3\)	−0.416519	+	1.68122i	−0.240477	+	0.970655i
\(5\)	1.62104	+	1.54020i	0.724951	+	0.688801i
							\(7\)	−0.361989	−	0.361989i	−0.136819	−	0.136819i	0.635380	−	0.772199i	\(-0.280843\pi\)
−0.772199	+	0.635380i	\(0.780843\pi\)
\(11\)	−2.63380			−0.794119			−0.397060	−	0.917793i	\(-0.629969\pi\)
							−0.397060	+	0.917793i	\(0.629969\pi\)
\(13\)	3.49376	+	3.49376i	0.968995	+	0.968995i	0.999534	−	0.0305386i	\(-0.00972224\pi\)
							−0.0305386	+	0.999534i	\(0.509722\pi\)
\(17\)	3.61339	−	3.61339i	0.876376	−	0.876376i	−0.116782	−	0.993158i	\(-0.537258\pi\)
							0.993158	+	0.116782i	\(0.0372578\pi\)
\(19\)	−0.672266			−0.154228			−0.0771142	−	0.997022i	\(-0.524571\pi\)
							−0.0771142	+	0.997022i	\(0.524571\pi\)
\(23\)	4.31851	+	4.31851i	0.900472	+	0.900472i	0.995477	−	0.0950052i	\(-0.0302868\pi\)
							−0.0950052	+	0.995477i	\(0.530287\pi\)
\(29\)		−	4.76080i		−	0.884058i	−0.897001	−	0.442029i	\(-0.854259\pi\)
							0.897001	−	0.442029i	\(-0.145741\pi\)
\(31\)	3.73793			0.671352			0.335676	−	0.941978i	\(-0.391035\pi\)
							0.335676	+	0.941978i	\(0.391035\pi\)
\(37\)	−2.82150	+	2.82150i	−0.463851	+	0.463851i	−0.899915	−	0.436064i	\(-0.856372\pi\)
							0.436064	+	0.899915i	\(0.356372\pi\)
\(41\)		−	4.10027i		−	0.640355i	−0.947358	−	0.320177i	\(-0.896257\pi\)
							0.947358	−	0.320177i	\(-0.103743\pi\)
\(43\)	−7.57996	−	7.57996i	−1.15593	−	1.15593i	−0.985342	−	0.170591i	\(-0.945432\pi\)
							−0.170591	−	0.985342i	\(-0.554568\pi\)
\(47\)	0.987537	−	0.987537i	0.144047	−	0.144047i	−0.631406	−	0.775453i	\(-0.717522\pi\)
							0.775453	+	0.631406i	\(0.217522\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.2.w.c.77.8	yes	32
3.2	odd	2	inner	120.2.w.c.77.9	yes	32
4.3	odd	2		480.2.bi.c.17.10		32
5.2	odd	4		600.2.w.j.293.1		32
5.3	odd	4	inner	120.2.w.c.53.16	yes	32
5.4	even	2		600.2.w.j.557.9		32
8.3	odd	2		480.2.bi.c.17.7		32
8.5	even	2	inner	120.2.w.c.77.1	yes	32
12.11	even	2		480.2.bi.c.17.2		32
15.2	even	4		600.2.w.j.293.16		32
15.8	even	4	inner	120.2.w.c.53.1	✓	32
15.14	odd	2		600.2.w.j.557.8		32
20.3	even	4		480.2.bi.c.113.15		32
24.5	odd	2	inner	120.2.w.c.77.16	yes	32
24.11	even	2		480.2.bi.c.17.15		32
40.3	even	4		480.2.bi.c.113.2		32
40.13	odd	4	inner	120.2.w.c.53.9	yes	32
40.29	even	2		600.2.w.j.557.16		32
40.37	odd	4		600.2.w.j.293.8		32
60.23	odd	4		480.2.bi.c.113.7		32
120.29	odd	2		600.2.w.j.557.1		32
120.53	even	4	inner	120.2.w.c.53.8	yes	32
120.77	even	4		600.2.w.j.293.9		32
120.83	odd	4		480.2.bi.c.113.10		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.1	✓	32	15.8	even	4	inner
120.2.w.c.53.8	yes	32	120.53	even	4	inner
120.2.w.c.53.9	yes	32	40.13	odd	4	inner
120.2.w.c.53.16	yes	32	5.3	odd	4	inner
120.2.w.c.77.1	yes	32	8.5	even	2	inner
120.2.w.c.77.8	yes	32	1.1	even	1	trivial
120.2.w.c.77.9	yes	32	3.2	odd	2	inner
120.2.w.c.77.16	yes	32	24.5	odd	2	inner
480.2.bi.c.17.2		32	12.11	even	2
480.2.bi.c.17.7		32	8.3	odd	2
480.2.bi.c.17.10		32	4.3	odd	2
480.2.bi.c.17.15		32	24.11	even	2
480.2.bi.c.113.2		32	40.3	even	4
480.2.bi.c.113.7		32	60.23	odd	4
480.2.bi.c.113.10		32	120.83	odd	4
480.2.bi.c.113.15		32	20.3	even	4
600.2.w.j.293.1		32	5.2	odd	4
600.2.w.j.293.8		32	40.37	odd	4
600.2.w.j.293.9		32	120.77	even	4
600.2.w.j.293.16		32	15.2	even	4
600.2.w.j.557.1		32	120.29	odd	2
600.2.w.j.557.8		32	15.14	odd	2
600.2.w.j.557.9		32	5.4	even	2
600.2.w.j.557.16		32	40.29	even	2