Embedded newform 124.4.j.a.15.12

• Label: 124.4.j.a.15.12
• 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			15.12
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78541 - 2.19370i) q^{2} +(-5.33512 - 3.87619i) q^{3} +(-1.62460 + 7.83331i) q^{4} -18.0276 q^{5} +(1.02221 + 18.6242i) q^{6} +(-11.4547 - 3.72185i) q^{7} +(20.0845 - 10.4218i) q^{8} +(5.09519 + 15.6814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78541	−	2.19370i	−0.631239	−	0.775589i
							\(3\)	−5.33512	−	3.87619i	−1.02674	−	0.745973i	−0.0590898	−	0.998253i	\(-0.518820\pi\)
−0.967654	+	0.252279i	\(0.918820\pi\)
\(5\)	−18.0276			−1.61244			−0.806221	−	0.591615i	\(-0.798491\pi\)
							−0.806221	+	0.591615i	\(0.798491\pi\)
\(7\)	−11.4547	−	3.72185i	−0.618495	−	0.200961i	−0.0170227	−	0.999855i	\(-0.505419\pi\)
							−0.601472	+	0.798894i	\(0.705419\pi\)
\(11\)	3.99548	−	12.2968i	0.109516	−	0.337057i	−0.881247	−	0.472655i	\(-0.843296\pi\)
							0.990764	+	0.135598i	\(0.0432956\pi\)
\(13\)	−26.4785	+	36.4445i	−0.564909	+	0.777530i	−0.991940	−	0.126707i	\(-0.959559\pi\)
							0.427032	+	0.904237i	\(0.359559\pi\)
\(17\)	99.3865	−	32.2926i	1.41793	−	0.460712i	0.502984	−	0.864296i	\(-0.332235\pi\)
							0.914943	+	0.403583i	\(0.132235\pi\)
\(19\)	−89.6675	−	123.417i	−1.08269	−	1.49020i	−0.856523	−	0.516109i	\(-0.827380\pi\)
							−0.226169	−	0.974088i	\(-0.572620\pi\)
\(23\)	18.2687	+	56.2253i	0.165621	+	0.509730i	0.999082	−	0.0428495i	\(-0.0136436\pi\)
							−0.833460	+	0.552579i	\(0.813644\pi\)
\(29\)	45.5614	+	62.7098i	0.291743	+	0.401549i	0.929579	−	0.368623i	\(-0.120170\pi\)
							−0.637837	+	0.770172i	\(0.720170\pi\)
\(31\)	125.574	−	118.415i	0.727541	−	0.686064i
							\(37\)			234.257i			1.04085i	0.853907	+	0.520426i	\(0.174227\pi\)
−0.853907	+	0.520426i	\(0.825773\pi\)
\(41\)	−122.561	+	89.0456i	−0.466848	+	0.339185i	−0.796212	−	0.605018i	\(-0.793166\pi\)
							0.329363	+	0.944203i	\(0.393166\pi\)
\(43\)	−182.788	+	132.803i	−0.648254	+	0.470984i	−0.862676	−	0.505757i	\(-0.831213\pi\)
							0.214422	+	0.976741i	\(0.431213\pi\)
\(47\)	−171.716	+	236.347i	−0.532923	+	0.733505i	−0.987572	−	0.157165i	\(-0.949764\pi\)
							0.454650	+	0.890670i	\(0.349764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.12	✓	184
4.3	odd	2	inner	124.4.j.a.15.31	yes	184
31.29	odd	10	inner	124.4.j.a.91.31	yes	184
124.91	even	10	inner	124.4.j.a.91.12	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.12	✓	184	1.1	even	1	trivial
124.4.j.a.15.31	yes	184	4.3	odd	2	inner
124.4.j.a.91.12	yes	184	124.91	even	10	inner
124.4.j.a.91.31	yes	184	31.29	odd	10	inner