Embedded newform 124.3.l.a.35.19

• Label: 124.3.l.a.35.19
• 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			35.19
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761810 + 1.84923i) q^{2} +(2.99163 + 0.972039i) q^{3} +(-2.83929 + 2.81752i) q^{4} +0.689285 q^{5} +(0.481530 + 6.27271i) q^{6} +(2.85110 + 3.92420i) q^{7} +(-7.37324 - 3.10408i) q^{8} +(0.723827 + 0.525891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761810	+	1.84923i	0.380905	+	0.924614i
							\(3\)	2.99163	+	0.972039i	0.997209	+	0.324013i	0.761750	−	0.647871i	\(-0.224341\pi\)
0.235459	+	0.971884i	\(0.424341\pi\)
\(5\)	0.689285			0.137857			0.0689285	−	0.997622i	\(-0.478042\pi\)
							0.0689285	+	0.997622i	\(0.478042\pi\)
\(7\)	2.85110	+	3.92420i	0.407300	+	0.560601i	0.962557	−	0.271078i	\(-0.0873802\pi\)
							−0.555257	+	0.831679i	\(0.687380\pi\)
\(11\)	4.97256	+	6.84414i	0.452051	+	0.622195i	0.972837	−	0.231493i	\(-0.0743609\pi\)
							−0.520786	+	0.853688i	\(0.674361\pi\)
\(13\)	2.39936	−	7.38448i	0.184566	−	0.568037i	−0.815374	−	0.578934i	\(-0.803469\pi\)
							0.999941	+	0.0108972i	\(0.00346876\pi\)
\(17\)	−5.38316	−	3.91110i	−0.316657	−	0.230065i	0.418091	−	0.908405i	\(-0.362699\pi\)
							−0.734748	+	0.678341i	\(0.762699\pi\)
\(19\)	7.92091	−	2.57366i	0.416890	−	0.135456i	−0.0930587	−	0.995661i	\(-0.529664\pi\)
							0.509949	+	0.860205i	\(0.329664\pi\)
\(23\)	15.2666	−	21.0126i	0.663765	−	0.913593i	−0.335834	−	0.941921i	\(-0.609018\pi\)
							0.999599	+	0.0283277i	\(0.00901818\pi\)
\(29\)	3.36838	+	10.3668i	0.116151	+	0.357476i	0.992185	−	0.124773i	\(-0.0398201\pi\)
							−0.876034	+	0.482249i	\(0.839820\pi\)
\(31\)	26.5378	+	16.0233i	0.856057	+	0.516882i
							\(37\)	8.89980			0.240535			0.120268	−	0.992742i	\(-0.461625\pi\)
0.120268	+	0.992742i	\(0.461625\pi\)
\(41\)	−8.58535	−	26.4230i	−0.209399	−	0.644463i	−0.999504	−	0.0314919i	\(-0.989974\pi\)
							0.790105	−	0.612971i	\(-0.210026\pi\)
\(43\)	31.8992	−	10.3647i	0.741843	−	0.241039i	0.0863751	−	0.996263i	\(-0.472472\pi\)
							0.655467	+	0.755223i	\(0.272472\pi\)
\(47\)	−14.5439	−	4.72560i	−0.309445	−	0.100545i	0.150178	−	0.988659i	\(-0.452015\pi\)
							−0.459623	+	0.888114i	\(0.652015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.19	yes	120
4.3	odd	2	inner	124.3.l.a.35.18	✓	120
31.8	even	5	inner	124.3.l.a.39.18	yes	120
124.39	odd	10	inner	124.3.l.a.39.19	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.18	✓	120	4.3	odd	2	inner
124.3.l.a.35.19	yes	120	1.1	even	1	trivial
124.3.l.a.39.18	yes	120	31.8	even	5	inner
124.3.l.a.39.19	yes	120	124.39	odd	10	inner