Embedded newform 124.4.j.a.15.6

• Label: 124.4.j.a.15.6
• 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			15.6
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63900 + 1.01768i) q^{2} +(-7.46913 - 5.42664i) q^{3} +(5.92867 - 5.37130i) q^{4} +5.42895 q^{5} +(25.2336 + 6.71977i) q^{6} +(16.7763 + 5.45096i) q^{7} +(-10.1795 + 20.2083i) q^{8} +(17.9960 + 55.3861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63900	+	1.01768i	−0.933028	+	0.359803i
							\(3\)	−7.46913	−	5.42664i	−1.43743	−	1.04436i	−0.988571	−	0.150757i	\(-0.951829\pi\)
−0.448864	−	0.893600i	\(-0.648171\pi\)
\(5\)	5.42895			0.485580			0.242790	−	0.970079i	\(-0.421937\pi\)
							0.242790	+	0.970079i	\(0.421937\pi\)
\(7\)	16.7763	+	5.45096i	0.905836	+	0.294324i	0.724644	−	0.689124i	\(-0.242004\pi\)
							0.181193	+	0.983448i	\(0.442004\pi\)
\(11\)	18.3297	−	56.4130i	0.502419	−	1.54629i	−0.302648	−	0.953102i	\(-0.597871\pi\)
							0.805067	−	0.593184i	\(-0.202129\pi\)
\(13\)	−15.1873	+	20.9036i	−0.324016	+	0.445970i	−0.939688	−	0.342033i	\(-0.888885\pi\)
							0.615672	+	0.788002i	\(0.288885\pi\)
\(17\)	91.9485	−	29.8759i	1.31181	−	0.426233i	0.432136	−	0.901808i	\(-0.357760\pi\)
							0.879675	+	0.475575i	\(0.157760\pi\)
\(19\)	−46.5474	−	64.0670i	−0.562037	−	0.773577i	0.429547	−	0.903045i	\(-0.358673\pi\)
							−0.991584	+	0.129467i	\(0.958673\pi\)
\(23\)	12.2332	+	37.6501i	0.110905	+	0.341330i	0.991071	−	0.133336i	\(-0.0425690\pi\)
							−0.880166	+	0.474666i	\(0.842569\pi\)
\(29\)	−15.2213	−	20.9504i	−0.0974666	−	0.134151i	0.757498	−	0.652837i	\(-0.226421\pi\)
							−0.854965	+	0.518686i	\(0.826421\pi\)
\(31\)	−159.986	−	64.7731i	−0.926913	−	0.375277i
							\(37\)		−	393.562i		−	1.74868i	−0.485315	−	0.874340i	\(-0.661295\pi\)
0.485315	−	0.874340i	\(-0.338705\pi\)
\(41\)	216.405	−	157.228i	0.824312	−	0.598898i	−0.0936322	−	0.995607i	\(-0.529848\pi\)
							0.917944	+	0.396709i	\(0.129848\pi\)
\(43\)	−207.204	+	150.542i	−0.734843	+	0.533895i	−0.891092	−	0.453823i	\(-0.850060\pi\)
							0.156249	+	0.987718i	\(0.450060\pi\)
\(47\)	−18.9907	+	26.1385i	−0.0589379	+	0.0811210i	−0.837469	−	0.546486i	\(-0.815965\pi\)
							0.778531	+	0.627607i	\(0.215965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.6	✓	184
4.3	odd	2	inner	124.4.j.a.15.13	yes	184
31.29	odd	10	inner	124.4.j.a.91.13	yes	184
124.91	even	10	inner	124.4.j.a.91.6	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.6	✓	184	1.1	even	1	trivial
124.4.j.a.15.13	yes	184	4.3	odd	2	inner
124.4.j.a.91.6	yes	184	124.91	even	10	inner
124.4.j.a.91.13	yes	184	31.29	odd	10	inner