Embedded newform 124.3.l.a.35.4

• Label: 124.3.l.a.35.4
• 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			35.4
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84215 + 0.778768i) q^{2} +(-2.10823 - 0.685006i) q^{3} +(2.78704 - 2.86922i) q^{4} +0.107615 q^{5} +(4.41714 - 0.379938i) q^{6} +(2.89081 + 3.97885i) q^{7} +(-2.89970 + 7.45599i) q^{8} +(-3.30575 - 2.40177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84215	+	0.778768i	−0.921076	+	0.389384i
							\(3\)	−2.10823	−	0.685006i	−0.702744	−	0.228335i	−0.0642181	−	0.997936i	\(-0.520455\pi\)
−0.638526	+	0.769601i	\(0.720455\pi\)
\(5\)	0.107615			0.0215230			0.0107615	−	0.999942i	\(-0.496574\pi\)
							0.0107615	+	0.999942i	\(0.496574\pi\)
\(7\)	2.89081	+	3.97885i	0.412972	+	0.568408i	0.963940	−	0.266119i	\(-0.0857413\pi\)
							−0.550968	+	0.834526i	\(0.685741\pi\)
\(11\)	−7.40271	−	10.1889i	−0.672973	−	0.926268i	0.326850	−	0.945076i	\(-0.394013\pi\)
							−0.999823	+	0.0188081i	\(0.994013\pi\)
\(13\)	−7.81513	+	24.0525i	−0.601164	+	1.85019i	−0.0798889	+	0.996804i	\(0.525457\pi\)
							−0.521275	+	0.853389i	\(0.674543\pi\)
\(17\)	−22.3152	−	16.2130i	−1.31266	−	0.953703i	−0.999993	−	0.00384085i	\(-0.998777\pi\)
							−0.312668	−	0.949863i	\(-0.601223\pi\)
\(19\)	−22.4089	+	7.28109i	−1.17941	+	0.383215i	−0.832149	−	0.554552i	\(-0.812890\pi\)
							−0.347266	+	0.937767i	\(0.612890\pi\)
\(23\)	10.9891	−	15.1252i	0.477787	−	0.657618i	−0.500291	−	0.865858i	\(-0.666773\pi\)
							0.978078	+	0.208240i	\(0.0667735\pi\)
\(29\)	7.76856	+	23.9092i	0.267881	+	0.824454i	0.991016	+	0.133746i	\(0.0427007\pi\)
							−0.723134	+	0.690707i	\(0.757299\pi\)
\(31\)	−20.1638	+	23.5461i	−0.650446	+	0.759552i
							\(37\)	2.85137			0.0770641			0.0385321	−	0.999257i	\(-0.487732\pi\)
0.0385321	+	0.999257i	\(0.487732\pi\)
\(41\)	−3.04010	−	9.35648i	−0.0741489	−	0.228207i	0.907112	−	0.420888i	\(-0.138282\pi\)
							−0.981261	+	0.192682i	\(0.938282\pi\)
\(43\)	62.7078	−	20.3750i	1.45832	−	0.473837i	0.530764	−	0.847520i	\(-0.321905\pi\)
							0.927556	+	0.373683i	\(0.121905\pi\)
\(47\)	−33.2881	−	10.8160i	−0.708258	−	0.230127i	−0.0673330	−	0.997731i	\(-0.521449\pi\)
							−0.640925	+	0.767604i	\(0.721449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.4	✓	120
4.3	odd	2	inner	124.3.l.a.35.28	yes	120
31.8	even	5	inner	124.3.l.a.39.28	yes	120
124.39	odd	10	inner	124.3.l.a.39.4	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.4	✓	120	1.1	even	1	trivial
124.3.l.a.35.28	yes	120	4.3	odd	2	inner
124.3.l.a.39.4	yes	120	124.39	odd	10	inner
124.3.l.a.39.28	yes	120	31.8	even	5	inner