Embedded newform 124.3.l.a.35.10

• Label: 124.3.l.a.35.10
• Level: $124$
• Weight: $3$
• Character: 124.35
• Analytic conductor: $3.379$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.37875527807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			35.10
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07058 + 1.68934i) q^{2} +(-5.00352 - 1.62574i) q^{3} +(-1.70773 - 3.61713i) q^{4} -7.77294 q^{5} +(8.10307 - 6.71216i) q^{6} +(3.31847 + 4.56749i) q^{7} +(7.93882 + 0.987476i) q^{8} +(15.1110 + 10.9788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07058	+	1.68934i	−0.535288	+	0.844670i
							\(3\)	−5.00352	−	1.62574i	−1.66784	−	0.541914i	−0.685347	−	0.728217i	\(-0.740349\pi\)
−0.982492	+	0.186303i	\(0.940349\pi\)
\(5\)	−7.77294			−1.55459			−0.777294	−	0.629138i	\(-0.783408\pi\)
							−0.777294	+	0.629138i	\(0.783408\pi\)
\(7\)	3.31847	+	4.56749i	0.474068	+	0.652498i	0.977351	−	0.211623i	\(-0.0678749\pi\)
							−0.503284	+	0.864121i	\(0.667875\pi\)
\(11\)	3.68122	+	5.06676i	0.334656	+	0.460615i	0.942871	−	0.333158i	\(-0.108114\pi\)
							−0.608215	+	0.793772i	\(0.708114\pi\)
\(13\)	6.51039	−	20.0369i	0.500800	−	1.54130i	−0.306920	−	0.951735i	\(-0.599298\pi\)
							0.807719	−	0.589567i	\(-0.200702\pi\)
\(17\)	−5.35700	−	3.89209i	−0.315117	−	0.228946i	0.418972	−	0.907999i	\(-0.362391\pi\)
							−0.734089	+	0.679053i	\(0.762391\pi\)
\(19\)	−9.29022	+	3.01858i	−0.488959	+	0.158872i	−0.543112	−	0.839660i	\(-0.682754\pi\)
							0.0541531	+	0.998533i	\(0.482754\pi\)
\(23\)	5.10644	−	7.02841i	0.222019	−	0.305583i	−0.683448	−	0.729999i	\(-0.739521\pi\)
							0.905467	+	0.424416i	\(0.139521\pi\)
\(29\)	16.0704	+	49.4595i	0.554151	+	1.70550i	0.698176	+	0.715926i	\(0.253995\pi\)
							−0.144026	+	0.989574i	\(0.546005\pi\)
\(31\)	30.7374	−	4.02667i	0.991528	−	0.129893i
							\(37\)	1.56659			0.0423402			0.0211701	−	0.999776i	\(-0.493261\pi\)
0.0211701	+	0.999776i	\(0.493261\pi\)
\(41\)	6.45842	+	19.8770i	0.157522	+	0.484804i	0.998408	−	0.0564091i	\(-0.0179651\pi\)
							−0.840885	+	0.541213i	\(0.817965\pi\)
\(43\)	74.2781	−	24.1344i	1.72740	−	0.561266i	0.734329	−	0.678794i	\(-0.237497\pi\)
							0.993069	+	0.117529i	\(0.0374972\pi\)
\(47\)	36.5782	+	11.8850i	0.778259	+	0.252872i	0.671097	−	0.741370i	\(-0.265824\pi\)
							0.107162	+	0.994242i	\(0.465824\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.10	✓	120
4.3	odd	2	inner	124.3.l.a.35.27	yes	120
31.8	even	5	inner	124.3.l.a.39.27	yes	120
124.39	odd	10	inner	124.3.l.a.39.10	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.10	✓	120	1.1	even	1	trivial
124.3.l.a.35.27	yes	120	4.3	odd	2	inner
124.3.l.a.39.10	yes	120	124.39	odd	10	inner
124.3.l.a.39.27	yes	120	31.8	even	5	inner