Embedded newform 124.3.l.a.35.7

• Label: 124.3.l.a.35.7
• 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			35.7
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48076 + 1.34437i) q^{2} +(0.177366 + 0.0576297i) q^{3} +(0.385317 - 3.98140i) q^{4} +0.656973 q^{5} +(-0.340113 + 0.153110i) q^{6} +(-4.86752 - 6.69957i) q^{7} +(4.78192 + 6.41352i) q^{8} +(-7.25302 - 5.26962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48076	+	1.34437i	−0.740381	+	0.672187i
							\(3\)	0.177366	+	0.0576297i	0.0591220	+	0.0192099i	0.338429	−	0.940992i	\(-0.390105\pi\)
−0.279307	+	0.960202i	\(0.590105\pi\)
\(5\)	0.656973			0.131395			0.0656973	−	0.997840i	\(-0.479073\pi\)
							0.0656973	+	0.997840i	\(0.479073\pi\)
\(7\)	−4.86752	−	6.69957i	−0.695360	−	0.957081i	−0.999989	−	0.00461015i	\(-0.998533\pi\)
							0.304629	−	0.952471i	\(-0.401467\pi\)
\(11\)	−1.34717	−	1.85422i	−0.122470	−	0.168566i	0.743380	−	0.668870i	\(-0.233222\pi\)
							−0.865850	+	0.500304i	\(0.833222\pi\)
\(13\)	5.43844	−	16.7378i	0.418341	−	1.28752i	−0.490887	−	0.871224i	\(-0.663327\pi\)
							0.909228	−	0.416299i	\(-0.136673\pi\)
\(17\)	2.66019	+	1.93274i	0.156482	+	0.113691i	0.663271	−	0.748379i	\(-0.269168\pi\)
							−0.506789	+	0.862070i	\(0.669168\pi\)
\(19\)	24.5683	−	7.98272i	1.29307	−	0.420143i	0.419903	−	0.907569i	\(-0.362064\pi\)
							0.873164	+	0.487426i	\(0.162064\pi\)
\(23\)	6.23088	−	8.57607i	0.270908	−	0.372873i	−0.651788	−	0.758401i	\(-0.725981\pi\)
							0.922696	+	0.385528i	\(0.125981\pi\)
\(29\)	−7.05491	−	21.7128i	−0.243273	−	0.748717i	−0.995916	−	0.0902877i	\(-0.971221\pi\)
							0.752643	−	0.658429i	\(-0.228779\pi\)
\(31\)	16.4019	+	26.3055i	0.529094	+	0.848563i
							\(37\)	−52.8832			−1.42928			−0.714638	−	0.699495i	\(-0.753408\pi\)
−0.714638	+	0.699495i	\(0.753408\pi\)
\(41\)	2.67045	+	8.21880i	0.0651329	+	0.200458i	0.978327	−	0.207067i	\(-0.0663918\pi\)
							−0.913194	+	0.407525i	\(0.866392\pi\)
\(43\)	−64.3548	+	20.9101i	−1.49662	+	0.486282i	−0.939031	−	0.343833i	\(-0.888274\pi\)
							−0.557592	+	0.830115i	\(0.688274\pi\)
\(47\)	8.73489	+	2.83814i	0.185849	+	0.0603859i	0.400463	−	0.916313i	\(-0.368849\pi\)
							−0.214614	+	0.976699i	\(0.568849\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.7	✓	120
4.3	odd	2	inner	124.3.l.a.35.29	yes	120
31.8	even	5	inner	124.3.l.a.39.29	yes	120
124.39	odd	10	inner	124.3.l.a.39.7	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.7	✓	120	1.1	even	1	trivial
124.3.l.a.35.29	yes	120	4.3	odd	2	inner
124.3.l.a.39.7	yes	120	124.39	odd	10	inner
124.3.l.a.39.29	yes	120	31.8	even	5	inner