Embedded newform 124.4.j.a.15.20

• Label: 124.4.j.a.15.20
• Level: $124$
• Weight: $4$
• Character: 124.15
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			15.20
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.618335 - 2.76001i) q^{2} +(2.24034 + 1.62770i) q^{3} +(-7.23532 + 3.41322i) q^{4} -9.14807 q^{5} +(3.10720 - 7.18983i) q^{6} +(31.6620 + 10.2876i) q^{7} +(13.8944 + 17.8591i) q^{8} +(-5.97375 - 18.3853i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.618335	−	2.76001i	−0.218614	−	0.975811i
							\(3\)	2.24034	+	1.62770i	0.431154	+	0.313251i	0.782110	−	0.623140i	\(-0.214143\pi\)
−0.350956	+	0.936392i	\(0.614143\pi\)
\(5\)	−9.14807			−0.818228			−0.409114	−	0.912483i	\(-0.634162\pi\)
							−0.409114	+	0.912483i	\(0.634162\pi\)
\(7\)	31.6620	+	10.2876i	1.70959	+	0.555479i	0.990265	−	0.139196i	\(-0.0444518\pi\)
							0.719324	+	0.694675i	\(0.244452\pi\)
\(11\)	19.0110	−	58.5099i	0.521094	−	1.60376i	−0.250818	−	0.968034i	\(-0.580700\pi\)
							0.771912	−	0.635729i	\(-0.219300\pi\)
\(13\)	7.35227	−	10.1195i	0.156858	−	0.215896i	−0.723354	−	0.690477i	\(-0.757400\pi\)
							0.880212	+	0.474581i	\(0.157400\pi\)
\(17\)	58.1218	−	18.8849i	0.829212	−	0.269427i	0.136499	−	0.990640i	\(-0.456415\pi\)
							0.692714	+	0.721213i	\(0.256415\pi\)
\(19\)	36.7819	+	50.6260i	0.444124	+	0.611284i	0.971122	−	0.238582i	\(-0.0766827\pi\)
							−0.526999	+	0.849866i	\(0.676683\pi\)
\(23\)	−3.40693	−	10.4854i	−0.0308867	−	0.0950594i	0.934425	−	0.356161i	\(-0.115914\pi\)
							−0.965311	+	0.261101i	\(0.915914\pi\)
\(29\)	34.6120	+	47.6393i	0.221630	+	0.305048i	0.905324	−	0.424721i	\(-0.139628\pi\)
							−0.683694	+	0.729769i	\(0.739628\pi\)
\(31\)	159.765	−	65.3146i	0.925636	−	0.378415i
							\(37\)			17.2706i			0.0767370i	0.999264	+	0.0383685i	\(0.0122161\pi\)
−0.999264	+	0.0383685i	\(0.987784\pi\)
\(41\)	206.295	−	149.882i	0.785803	−	0.570919i	−0.120912	−	0.992663i	\(-0.538582\pi\)
							0.906715	+	0.421744i	\(0.138582\pi\)
\(43\)	−201.398	+	146.324i	−0.714255	+	0.518936i	−0.884543	−	0.466458i	\(-0.845530\pi\)
							0.170289	+	0.985394i	\(0.445530\pi\)
\(47\)	−24.9822	+	34.3851i	−0.0775326	+	0.106715i	−0.846022	−	0.533148i	\(-0.821009\pi\)
							0.768489	+	0.639863i	\(0.221009\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.20	✓	184
4.3	odd	2	inner	124.4.j.a.15.39	yes	184
31.29	odd	10	inner	124.4.j.a.91.39	yes	184
124.91	even	10	inner	124.4.j.a.91.20	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.20	✓	184	1.1	even	1	trivial
124.4.j.a.15.39	yes	184	4.3	odd	2	inner
124.4.j.a.91.20	yes	184	124.91	even	10	inner
124.4.j.a.91.39	yes	184	31.29	odd	10	inner