Embedded newform 124.3.l.a.35.9

• Label: 124.3.l.a.35.9
• 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			35.9
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18822 - 1.60877i) q^{2} +(-1.95963 - 0.636724i) q^{3} +(-1.17627 + 3.82314i) q^{4} +7.70420 q^{5} +(1.30413 + 3.90916i) q^{6} +(-7.26049 - 9.99321i) q^{7} +(7.54822 - 2.65037i) q^{8} +(-3.84641 - 2.79458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18822	−	1.60877i	−0.594109	−	0.804384i
							\(3\)	−1.95963	−	0.636724i	−0.653211	−	0.212241i	−0.0363817	−	0.999338i	\(-0.511583\pi\)
−0.616829	+	0.787097i	\(0.711583\pi\)
\(5\)	7.70420			1.54084			0.770420	−	0.637536i	\(-0.220046\pi\)
							0.770420	+	0.637536i	\(0.220046\pi\)
\(7\)	−7.26049	−	9.99321i	−1.03721	−	1.42760i	−0.899391	−	0.437145i	\(-0.855990\pi\)
							−0.137822	−	0.990457i	\(-0.544010\pi\)
\(11\)	3.67882	+	5.06346i	0.334438	+	0.460315i	0.942807	−	0.333340i	\(-0.108176\pi\)
							−0.608368	+	0.793655i	\(0.708176\pi\)
\(13\)	0.848856	−	2.61251i	0.0652966	−	0.200962i	−0.913085	−	0.407769i	\(-0.866307\pi\)
							0.978382	+	0.206806i	\(0.0663070\pi\)
\(17\)	−26.5728	−	19.3063i	−1.56311	−	1.13566i	−0.933403	−	0.358829i	\(-0.883176\pi\)
							−0.629704	−	0.776835i	\(-0.716824\pi\)
\(19\)	−16.9494	+	5.50720i	−0.892074	+	0.289853i	−0.718962	−	0.695050i	\(-0.755382\pi\)
							−0.173112	+	0.984902i	\(0.555382\pi\)
\(23\)	15.4631	−	21.2832i	0.672310	−	0.925356i	−0.327500	−	0.944851i	\(-0.606206\pi\)
							0.999810	+	0.0194956i	\(0.00620603\pi\)
\(29\)	−5.09188	−	15.6712i	−0.175582	−	0.540386i	0.824077	−	0.566477i	\(-0.191694\pi\)
							−0.999660	+	0.0260908i	\(0.991694\pi\)
\(31\)	21.1923	−	22.6249i	0.683624	−	0.729834i
							\(37\)	24.6625			0.666555			0.333278	−	0.942829i	\(-0.391845\pi\)
0.333278	+	0.942829i	\(0.391845\pi\)
\(41\)	6.18086	+	19.0227i	0.150753	+	0.463969i	0.997706	−	0.0676987i	\(-0.0215657\pi\)
							−0.846953	+	0.531667i	\(0.821566\pi\)
\(43\)	33.7727	−	10.9734i	0.785412	−	0.255196i	0.111263	−	0.993791i	\(-0.464510\pi\)
							0.674149	+	0.738595i	\(0.264510\pi\)
\(47\)	−8.15268	−	2.64897i	−0.173461	−	0.0563610i	0.220999	−	0.975274i	\(-0.429068\pi\)
							−0.394460	+	0.918913i	\(0.629068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.9	✓	120
4.3	odd	2	inner	124.3.l.a.35.16	yes	120
31.8	even	5	inner	124.3.l.a.39.16	yes	120
124.39	odd	10	inner	124.3.l.a.39.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.9	✓	120	1.1	even	1	trivial
124.3.l.a.35.16	yes	120	4.3	odd	2	inner
124.3.l.a.39.9	yes	120	124.39	odd	10	inner
124.3.l.a.39.16	yes	120	31.8	even	5	inner