Embedded newform 124.4.p.a.3.8

• Label: 124.4.p.a.3.8
• Level: $124$
• Weight: $4$
• Character: 124.3
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46001 - 1.39582i) q^{2} +(-0.479513 + 4.56226i) q^{3} +(4.10335 + 6.86750i) q^{4} +(0.676684 - 1.17205i) q^{5} +(7.54773 - 10.5539i) q^{6} +(-5.59922 - 26.3423i) q^{7} +(-0.508472 - 22.6217i) q^{8} +(5.82568 + 1.23829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46001	−	1.39582i	−0.869747	−	0.493499i
							\(3\)	−0.479513	+	4.56226i	−0.0922823	+	0.878008i	0.846243	+	0.532797i	\(0.178859\pi\)
−0.938525	+	0.345211i	\(0.887807\pi\)
\(5\)	0.676684	−	1.17205i	0.0605245	−	0.104831i	−0.834176	−	0.551499i	\(-0.814056\pi\)
							0.894700	+	0.446668i	\(0.147389\pi\)
\(7\)	−5.59922	−	26.3423i	−0.302330	−	1.42235i	−0.822733	−	0.568428i	\(-0.807552\pi\)
							0.520404	−	0.853920i	\(-0.325782\pi\)
\(11\)	4.42197	+	4.91110i	0.121207	+	0.134614i	0.800694	−	0.599074i	\(-0.204464\pi\)
							−0.679487	+	0.733687i	\(0.737798\pi\)
\(13\)	32.1302	+	72.1657i	0.685487	+	1.53963i	0.834629	+	0.550812i	\(0.185682\pi\)
							−0.149143	+	0.988816i	\(0.547651\pi\)
\(17\)	45.9023	+	41.3306i	0.654878	+	0.589655i	0.928116	−	0.372291i	\(-0.121428\pi\)
							−0.273238	+	0.961947i	\(0.588094\pi\)
\(19\)	58.3529	−	131.063i	0.704583	−	1.58252i	−0.104298	−	0.994546i	\(-0.533259\pi\)
							0.808881	−	0.587973i	\(-0.200074\pi\)
\(23\)	31.3830	+	96.5870i	0.284514	+	0.875643i	0.986544	+	0.163496i	\(0.0522771\pi\)
							−0.702030	+	0.712147i	\(0.747723\pi\)
\(29\)	−103.736	−	142.781i	−0.664254	−	0.914268i	0.335358	−	0.942091i	\(-0.391143\pi\)
							−0.999613	+	0.0278228i	\(0.991143\pi\)
\(31\)	126.163	+	117.788i	0.730951	+	0.682430i
							\(37\)	292.012	−	168.593i	1.29747	−	0.749097i	0.317507	−	0.948256i	\(-0.397154\pi\)
0.979967	+	0.199159i	\(0.0638211\pi\)
\(41\)	35.3910	+	336.723i	0.134808	+	1.28262i	0.827535	+	0.561415i	\(0.189743\pi\)
							−0.692726	+	0.721201i	\(0.743591\pi\)
\(43\)	141.796	+	63.1315i	0.502875	+	0.223894i	0.642465	−	0.766315i	\(-0.277912\pi\)
							−0.139590	+	0.990209i	\(0.544578\pi\)
\(47\)	64.8281	−	89.2283i	0.201195	−	0.276921i	−0.696483	−	0.717573i	\(-0.745253\pi\)
							0.897678	+	0.440652i	\(0.145253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.8	✓	368
4.3	odd	2	inner	124.4.p.a.3.26	yes	368
31.21	odd	30	inner	124.4.p.a.83.26	yes	368
124.83	even	30	inner	124.4.p.a.83.8	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.8	✓	368	1.1	even	1	trivial
124.4.p.a.3.26	yes	368	4.3	odd	2	inner
124.4.p.a.83.8	yes	368	124.83	even	30	inner
124.4.p.a.83.26	yes	368	31.21	odd	30	inner