Embedded newform 124.4.j.a.15.7

• Label: 124.4.j.a.15.7
• 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			15.7
Character	\(\chi\)	\(=\)	124.15
Dual form			124.4.j.a.91.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61727 - 1.07234i) q^{2} +(-2.61926 - 1.90300i) q^{3} +(5.70016 + 5.61321i) q^{4} -5.30736 q^{5} +(4.81463 + 7.78942i) q^{6} +(21.9191 + 7.12195i) q^{7} +(-8.89956 - 20.8038i) q^{8} +(-5.10436 - 15.7096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61727	−	1.07234i	−0.925343	−	0.379130i
							\(3\)	−2.61926	−	1.90300i	−0.504077	−	0.366233i	0.306495	−	0.951872i	\(-0.400844\pi\)
−0.810572	+	0.585639i	\(0.800844\pi\)
\(5\)	−5.30736			−0.474704			−0.237352	−	0.971424i	\(-0.576280\pi\)
							−0.237352	+	0.971424i	\(0.576280\pi\)
\(7\)	21.9191	+	7.12195i	1.18352	+	0.384549i	0.833673	−	0.552258i	\(-0.186234\pi\)
							0.349847	+	0.936807i	\(0.386234\pi\)
\(11\)	6.36489	−	19.5891i	0.174462	−	0.536940i	−0.825146	−	0.564919i	\(-0.808907\pi\)
							0.999609	+	0.0279793i	\(0.00890724\pi\)
\(13\)	−0.555126	+	0.764065i	−0.0118434	+	0.0163010i	−0.814898	−	0.579605i	\(-0.803207\pi\)
							0.803054	+	0.595906i	\(0.203207\pi\)
\(17\)	−95.2901	+	30.9616i	−1.35948	+	0.441723i	−0.895871	−	0.444314i	\(-0.853447\pi\)
							−0.463614	+	0.886037i	\(0.653447\pi\)
\(19\)	−14.6364	−	20.1453i	−0.176727	−	0.243244i	0.711459	−	0.702728i	\(-0.248035\pi\)
							−0.888187	+	0.459483i	\(0.848035\pi\)
\(23\)	−40.3960	−	124.326i	−0.366224	−	1.12712i	−0.949211	−	0.314640i	\(-0.898116\pi\)
							0.582987	−	0.812482i	\(-0.301884\pi\)
\(29\)	−82.3666	−	113.368i	−0.527417	−	0.725927i	0.459317	−	0.888272i	\(-0.348094\pi\)
							−0.986734	+	0.162345i	\(0.948094\pi\)
\(31\)	−172.422	+	7.85047i	−0.998965	+	0.0454834i
							\(37\)		−	206.595i		−	0.917947i	−0.888450	−	0.458974i	\(-0.848217\pi\)
0.888450	−	0.458974i	\(-0.151783\pi\)
\(41\)	−300.860	+	218.588i	−1.14601	+	0.832626i	−0.987945	−	0.154803i	\(-0.950526\pi\)
							−0.158066	+	0.987429i	\(0.550526\pi\)
\(43\)	325.499	−	236.489i	1.15438	−	0.838703i	0.165319	−	0.986240i	\(-0.447135\pi\)
							0.989056	+	0.147538i	\(0.0471347\pi\)
\(47\)	144.714	−	199.182i	0.449122	−	0.618164i	−0.523087	−	0.852280i	\(-0.675220\pi\)
							0.972209	+	0.234116i	\(0.0752195\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.j.a.15.7	✓	184
4.3	odd	2	inner	124.4.j.a.15.25	yes	184
31.29	odd	10	inner	124.4.j.a.91.25	yes	184
124.91	even	10	inner	124.4.j.a.91.7	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.j.a.15.7	✓	184	1.1	even	1	trivial
124.4.j.a.15.25	yes	184	4.3	odd	2	inner
124.4.j.a.91.7	yes	184	124.91	even	10	inner
124.4.j.a.91.25	yes	184	31.29	odd	10	inner