Embedded newform 124.4.j.a.15.9

• Label: 124.4.j.a.15.9
• 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			15.9
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41107 + 1.47877i) q^{2} +(3.63646 + 2.64204i) q^{3} +(3.62648 - 7.13083i) q^{4} +11.6709 q^{5} +(-12.6747 - 0.992650i) q^{6} +(10.8099 + 3.51236i) q^{7} +(1.80118 + 22.5556i) q^{8} +(-2.10001 - 6.46315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41107	+	1.47877i	−0.852440	+	0.522824i
							\(3\)	3.63646	+	2.64204i	0.699837	+	0.508461i	0.879879	−	0.475197i	\(-0.157623\pi\)
−0.180042	+	0.983659i	\(0.557623\pi\)
\(5\)	11.6709			1.04388			0.521939	−	0.852983i	\(-0.325209\pi\)
							0.521939	+	0.852983i	\(0.325209\pi\)
\(7\)	10.8099	+	3.51236i	0.583682	+	0.189650i	0.585949	−	0.810348i	\(-0.300722\pi\)
							−0.00226752	+	0.999997i	\(0.500722\pi\)
\(11\)	16.7953	−	51.6908i	0.460363	−	1.41685i	−0.404360	−	0.914600i	\(-0.632506\pi\)
							0.864722	−	0.502250i	\(-0.167494\pi\)
\(13\)	33.8501	−	46.5906i	0.722179	−	0.993994i	−0.277270	−	0.960792i	\(-0.589430\pi\)
							0.999449	−	0.0332016i	\(-0.0105703\pi\)
\(17\)	−66.8438	+	21.7189i	−0.953647	+	0.309859i	−0.744197	−	0.667960i	\(-0.767168\pi\)
							−0.209451	+	0.977819i	\(0.567168\pi\)
\(19\)	73.2698	+	100.847i	0.884697	+	1.21768i	0.975098	+	0.221775i	\(0.0711851\pi\)
							−0.0904011	+	0.995905i	\(0.528815\pi\)
\(23\)	41.8105	+	128.679i	0.379047	+	1.16659i	0.940707	+	0.339219i	\(0.110163\pi\)
							−0.561660	+	0.827368i	\(0.689837\pi\)
\(29\)	−41.6966	−	57.3905i	−0.266996	−	0.367488i	0.654377	−	0.756168i	\(-0.272931\pi\)
							−0.921373	+	0.388681i	\(0.872931\pi\)
\(31\)	11.2318	+	172.235i	0.0650739	+	0.997880i
							\(37\)			163.878i			0.728144i	0.931371	+	0.364072i	\(0.118614\pi\)
−0.931371	+	0.364072i	\(0.881386\pi\)
\(41\)	−287.201	+	208.664i	−1.09398	+	0.794825i	−0.980067	−	0.198666i	\(-0.936339\pi\)
							−0.113915	+	0.993491i	\(0.536339\pi\)
\(43\)	264.982	−	192.521i	0.939753	−	0.682770i	−0.00860840	−	0.999963i	\(-0.502740\pi\)
							0.948361	+	0.317193i	\(0.102740\pi\)
\(47\)	33.4911	−	46.0966i	0.103940	−	0.143061i	−0.753878	−	0.657014i	\(-0.771819\pi\)
							0.857819	+	0.513953i	\(0.171819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.9	✓	184
4.3	odd	2	inner	124.4.j.a.15.11	yes	184
31.29	odd	10	inner	124.4.j.a.91.11	yes	184
124.91	even	10	inner	124.4.j.a.91.9	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.9	✓	184	1.1	even	1	trivial
124.4.j.a.15.11	yes	184	4.3	odd	2	inner
124.4.j.a.91.9	yes	184	124.91	even	10	inner
124.4.j.a.91.11	yes	184	31.29	odd	10	inner