Embedded newform 124.3.l.a.35.5

• Label: 124.3.l.a.35.5
• 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			35.5
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73407 + 0.996495i) q^{2} +(3.94056 + 1.28037i) q^{3} +(2.01400 - 3.45598i) q^{4} -5.71773 q^{5} +(-8.10909 + 1.70651i) q^{6} +(5.97197 + 8.21971i) q^{7} +(-0.0485371 + 7.99985i) q^{8} +(6.60755 + 4.80067i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73407	+	0.996495i	−0.867035	+	0.498248i
							\(3\)	3.94056	+	1.28037i	1.31352	+	0.426789i	0.880265	−	0.474482i	\(-0.157365\pi\)
0.433256	+	0.901271i	\(0.357365\pi\)
\(5\)	−5.71773			−1.14355			−0.571773	−	0.820412i	\(-0.693744\pi\)
							−0.571773	+	0.820412i	\(0.693744\pi\)
\(7\)	5.97197	+	8.21971i	0.853139	+	1.17424i	0.983162	+	0.182734i	\(0.0584946\pi\)
							−0.130024	+	0.991511i	\(0.541505\pi\)
\(11\)	4.70599	+	6.47723i	0.427817	+	0.588840i	0.967450	−	0.253061i	\(-0.0814372\pi\)
							−0.539633	+	0.841900i	\(0.681437\pi\)
\(13\)	−0.342897	+	1.05533i	−0.0263767	+	0.0811792i	−0.963378	−	0.268146i	\(-0.913589\pi\)
							0.937002	+	0.349325i	\(0.113589\pi\)
\(17\)	12.0970	+	8.78896i	0.711586	+	0.516997i	0.883685	−	0.468082i	\(-0.155055\pi\)
							−0.172099	+	0.985080i	\(0.555055\pi\)
\(19\)	16.3293	−	5.30571i	0.859436	−	0.279248i	0.154043	−	0.988064i	\(-0.450770\pi\)
							0.705393	+	0.708816i	\(0.250770\pi\)
\(23\)	−21.1395	+	29.0961i	−0.919110	+	1.26505i	0.0448479	+	0.998994i	\(0.485720\pi\)
							−0.963958	+	0.266053i	\(0.914280\pi\)
\(29\)	−10.0429	−	30.9087i	−0.346305	−	1.06582i	−0.960881	−	0.276961i	\(-0.910673\pi\)
							0.614576	−	0.788858i	\(-0.289327\pi\)
\(31\)	−15.3143	−	26.9531i	−0.494010	−	0.869456i
							\(37\)	55.6605			1.50434			0.752170	−	0.658970i	\(-0.229007\pi\)
0.752170	+	0.658970i	\(0.229007\pi\)
\(41\)	−23.5499	−	72.4792i	−0.574388	−	1.76779i	−0.638253	−	0.769827i	\(-0.720342\pi\)
							0.0638643	−	0.997959i	\(-0.479658\pi\)
\(43\)	46.1850	−	15.0064i	1.07407	−	0.348987i	0.281999	−	0.959415i	\(-0.409003\pi\)
							0.792072	+	0.610428i	\(0.209003\pi\)
\(47\)	−2.49440	−	0.810481i	−0.0530724	−	0.0172443i	0.282360	−	0.959308i	\(-0.408883\pi\)
							−0.335433	+	0.942064i	\(0.608883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.5	✓	120
4.3	odd	2	inner	124.3.l.a.35.30	yes	120
31.8	even	5	inner	124.3.l.a.39.30	yes	120
124.39	odd	10	inner	124.3.l.a.39.5	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.5	✓	120	1.1	even	1	trivial
124.3.l.a.35.30	yes	120	4.3	odd	2	inner
124.3.l.a.39.5	yes	120	124.39	odd	10	inner
124.3.l.a.39.30	yes	120	31.8	even	5	inner