Embedded newform 124.4.j.a.15.16

• Label: 124.4.j.a.15.16
• 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			15.16
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56253 + 2.35765i) q^{2} +(3.17557 + 2.30719i) q^{3} +(-3.11699 - 7.36779i) q^{4} -16.6602 q^{5} +(-10.4015 + 3.88182i) q^{6} +(3.04000 + 0.987756i) q^{7} +(22.2410 + 4.16365i) q^{8} +(-3.58232 - 11.0253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56253	+	2.35765i	−0.552438	+	0.833554i
							\(3\)	3.17557	+	2.30719i	0.611139	+	0.444018i	0.849815	−	0.527081i	\(-0.176713\pi\)
−0.238676	+	0.971099i	\(0.576713\pi\)
\(5\)	−16.6602			−1.49013			−0.745066	−	0.666991i	\(-0.767582\pi\)
							−0.745066	+	0.666991i	\(0.767582\pi\)
\(7\)	3.04000	+	0.987756i	0.164145	+	0.0533338i	0.389937	−	0.920842i	\(-0.372497\pi\)
							−0.225792	+	0.974176i	\(0.572497\pi\)
\(11\)	7.59546	−	23.3764i	0.208193	−	0.640751i	−0.791375	−	0.611332i	\(-0.790634\pi\)
							0.999567	−	0.0294193i	\(-0.00936581\pi\)
\(13\)	18.8082	−	25.8873i	0.401266	−	0.552296i	−0.559795	−	0.828631i	\(-0.689120\pi\)
							0.961061	+	0.276336i	\(0.0891201\pi\)
\(17\)	22.3729	−	7.26940i	0.319190	−	0.103711i	−0.145040	−	0.989426i	\(-0.546331\pi\)
							0.464230	+	0.885715i	\(0.346331\pi\)
\(19\)	−68.0365	−	93.6442i	−0.821508	−	1.13071i	−0.989445	−	0.144911i	\(-0.953711\pi\)
							0.167937	−	0.985798i	\(-0.446289\pi\)
\(23\)	−10.4256	−	32.0868i	−0.0945170	−	0.290894i	0.892610	−	0.450829i	\(-0.148871\pi\)
							−0.987128	+	0.159935i	\(0.948871\pi\)
\(29\)	106.397	+	146.443i	0.681290	+	0.937716i	0.999948	−	0.0101540i	\(-0.00323219\pi\)
							−0.318658	+	0.947870i	\(0.603232\pi\)
\(31\)	−132.794	−	110.257i	−0.769374	−	0.638799i
							\(37\)		−	277.140i		−	1.23139i	−0.787984	−	0.615696i	\(-0.788875\pi\)
0.787984	−	0.615696i	\(-0.211125\pi\)
\(41\)	−91.1434	+	66.2195i	−0.347176	+	0.252238i	−0.747683	−	0.664056i	\(-0.768834\pi\)
							0.400508	+	0.916293i	\(0.368834\pi\)
\(43\)	−5.16782	+	3.75464i	−0.0183276	+	0.0133157i	−0.596911	−	0.802307i	\(-0.703606\pi\)
							0.578584	+	0.815623i	\(0.303606\pi\)
\(47\)	−298.824	+	411.296i	−0.927404	+	1.27646i	0.0334590	+	0.999440i	\(0.489348\pi\)
							−0.960863	+	0.277023i	\(0.910652\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.16	yes	184
4.3	odd	2	inner	124.4.j.a.15.4	✓	184
31.29	odd	10	inner	124.4.j.a.91.4	yes	184
124.91	even	10	inner	124.4.j.a.91.16	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.4	✓	184	4.3	odd	2	inner
124.4.j.a.15.16	yes	184	1.1	even	1	trivial
124.4.j.a.91.4	yes	184	31.29	odd	10	inner
124.4.j.a.91.16	yes	184	124.91	even	10	inner