Embedded newform 124.5.l.a.35.20

• Label: 124.5.l.a.35.20
• Level: $124$
• Weight: $5$
• Character: 124.35
• Analytic conductor: $12.818$
• Analytic rank: $0$
• Dimension: $248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8178754224\)
Analytic rank:	\(0\)
Dimension:	\(248\)
Relative dimension:	\(62\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			35.20
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22430 + 3.32453i) q^{2} +(3.03403 + 0.985817i) q^{3} +(-6.10499 - 14.7895i) q^{4} -15.2943 q^{5} +(-10.0260 + 7.89398i) q^{6} +(44.5004 + 61.2496i) q^{7} +(62.7474 + 12.6000i) q^{8} +(-57.2969 - 41.6286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22430	+	3.32453i	−0.556075	+	0.831132i
							\(3\)	3.03403	+	0.985817i	0.337115	+	0.109535i	0.472682	−	0.881233i	\(-0.343286\pi\)
−0.135568	+	0.990768i	\(0.543286\pi\)
\(5\)	−15.2943			−0.611771			−0.305885	−	0.952068i	\(-0.598952\pi\)
							−0.305885	+	0.952068i	\(0.598952\pi\)
\(7\)	44.5004	+	61.2496i	0.908172	+	1.24999i	0.967787	+	0.251770i	\(0.0810125\pi\)
							−0.0596156	+	0.998221i	\(0.518987\pi\)
\(11\)	37.9189	+	52.1909i	0.313379	+	0.431330i	0.936431	−	0.350851i	\(-0.114108\pi\)
							−0.623052	+	0.782180i	\(0.714108\pi\)
\(13\)	−13.4555	+	41.4117i	−0.0796183	+	0.245040i	−0.982941	−	0.183922i	\(-0.941121\pi\)
							0.903323	+	0.428962i	\(0.141121\pi\)
\(17\)	−262.298	−	190.570i	−0.907604	−	0.659413i	0.0328036	−	0.999462i	\(-0.489556\pi\)
							−0.940408	+	0.340049i	\(0.889556\pi\)
\(19\)	−439.638	+	142.847i	−1.21783	+	0.395698i	−0.846293	−	0.532718i	\(-0.821171\pi\)
							−0.371541	+	0.928416i	\(0.621171\pi\)
\(23\)	−94.3620	+	129.878i	−0.178378	+	0.245516i	−0.888838	−	0.458221i	\(-0.848487\pi\)
							0.710460	+	0.703737i	\(0.248487\pi\)
\(29\)	−125.504	−	386.263i	−0.149232	−	0.459290i	0.848299	−	0.529518i	\(-0.177627\pi\)
							−0.997531	+	0.0702282i	\(0.977627\pi\)
\(31\)	940.565	−	197.127i	0.978735	−	0.205127i
							\(37\)	−2218.44			−1.62048			−0.810240	−	0.586098i	\(-0.800663\pi\)
−0.810240	+	0.586098i	\(0.800663\pi\)
\(41\)	−42.7705	−	131.634i	−0.0254435	−	0.0783070i	0.937528	−	0.347909i	\(-0.113108\pi\)
							−0.962972	+	0.269601i	\(0.913108\pi\)
\(43\)	−1907.16	+	619.675i	−1.03146	+	0.335140i	−0.775366	−	0.631512i	\(-0.782434\pi\)
							−0.256090	+	0.966653i	\(0.582434\pi\)
\(47\)	−2461.61	−	799.827i	−1.11436	−	0.362076i	−0.306745	−	0.951792i	\(-0.599240\pi\)
							−0.807611	+	0.589715i	\(0.799240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.20	✓	248
4.3	odd	2	inner	124.5.l.a.35.56	yes	248
31.8	even	5	inner	124.5.l.a.39.56	yes	248
124.39	odd	10	inner	124.5.l.a.39.20	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.20	✓	248	1.1	even	1	trivial
124.5.l.a.35.56	yes	248	4.3	odd	2	inner
124.5.l.a.39.20	yes	248	124.39	odd	10	inner
124.5.l.a.39.56	yes	248	31.8	even	5	inner