Embedded newform 124.4.j.a.15.19

• Label: 124.4.j.a.15.19
• 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			15.19
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.861403 + 2.69406i) q^{2} +(-7.87115 - 5.71872i) q^{3} +(-6.51597 - 4.64135i) q^{4} -13.1003 q^{5} +(22.1869 - 16.2793i) q^{6} +(-25.7893 - 8.37945i) q^{7} +(18.1170 - 13.5564i) q^{8} +(20.9077 + 64.3474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.861403	+	2.69406i	−0.304552	+	0.952496i
							\(3\)	−7.87115	−	5.71872i	−1.51480	−	1.10057i	−0.963989	−	0.265941i	\(-0.914317\pi\)
−0.550814	−	0.834628i	\(-0.685683\pi\)
\(5\)	−13.1003			−1.17173			−0.585865	−	0.810409i	\(-0.699245\pi\)
							−0.585865	+	0.810409i	\(0.699245\pi\)
\(7\)	−25.7893	−	8.37945i	−1.39249	−	0.452448i	−0.485736	−	0.874105i	\(-0.661448\pi\)
							−0.906755	+	0.421658i	\(0.861448\pi\)
\(11\)	3.64051	−	11.2043i	0.0997869	−	0.307112i	−0.888685	−	0.458519i	\(-0.848380\pi\)
							0.988472	+	0.151406i	\(0.0483802\pi\)
\(13\)	11.4894	−	15.8137i	0.245121	−	0.337380i	−0.668674	−	0.743556i	\(-0.733138\pi\)
							0.913795	+	0.406175i	\(0.133138\pi\)
\(17\)	−44.2197	+	14.3678i	−0.630873	+	0.204983i	−0.606962	−	0.794731i	\(-0.707612\pi\)
							−0.0239114	+	0.999714i	\(0.507612\pi\)
\(19\)	37.9717	+	52.2635i	0.458490	+	0.631057i	0.974195	−	0.225709i	\(-0.0724700\pi\)
							−0.515705	+	0.856766i	\(0.672470\pi\)
\(23\)	−60.1424	−	185.099i	−0.545242	−	1.67808i	−0.720414	−	0.693544i	\(-0.756048\pi\)
							0.175172	−	0.984538i	\(-0.443952\pi\)
\(29\)	−25.1424	−	34.6055i	−0.160994	−	0.221589i	0.720898	−	0.693042i	\(-0.243730\pi\)
							−0.881891	+	0.471453i	\(0.843730\pi\)
\(31\)	−124.205	+	119.850i	−0.719609	+	0.694379i
							\(37\)			123.287i			0.547790i	0.961760	+	0.273895i	\(0.0883121\pi\)
−0.961760	+	0.273895i	\(0.911688\pi\)
\(41\)	34.8174	−	25.2963i	0.132624	−	0.0963566i	−0.519496	−	0.854473i	\(-0.673880\pi\)
							0.652119	+	0.758116i	\(0.273880\pi\)
\(43\)	369.991	−	268.814i	1.31217	−	0.953344i	0.312171	−	0.950026i	\(-0.398944\pi\)
							0.999994	−	0.00331845i	\(-0.00105630\pi\)
\(47\)	−232.259	+	319.677i	−0.720819	+	0.992122i	0.278677	+	0.960385i	\(0.410104\pi\)
							−0.999496	+	0.0317376i	\(0.989896\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.19	yes	184
4.3	odd	2	inner	124.4.j.a.15.1	✓	184
31.29	odd	10	inner	124.4.j.a.91.1	yes	184
124.91	even	10	inner	124.4.j.a.91.19	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.1	✓	184	4.3	odd	2	inner
124.4.j.a.15.19	yes	184	1.1	even	1	trivial
124.4.j.a.91.1	yes	184	31.29	odd	10	inner
124.4.j.a.91.19	yes	184	124.91	even	10	inner