Embedded newform 124.5.l.a.35.19

• Label: 124.5.l.a.35.19
• 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			35.19
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29841 - 3.27373i) q^{2} +(7.29605 + 2.37063i) q^{3} +(-5.43464 + 15.0487i) q^{4} -40.8790 q^{5} +(-9.00849 - 29.3340i) q^{6} +(39.2063 + 53.9628i) q^{7} +(61.7566 - 16.7966i) q^{8} +(-17.9179 - 13.0181i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29841	−	3.27373i	−0.574602	−	0.818433i
							\(3\)	7.29605	+	2.37063i	0.810673	+	0.263403i	0.684883	−	0.728653i	\(-0.259853\pi\)
0.125790	+	0.992057i	\(0.459853\pi\)
\(5\)	−40.8790			−1.63516			−0.817580	−	0.575815i	\(-0.804685\pi\)
							−0.817580	+	0.575815i	\(0.804685\pi\)
\(7\)	39.2063	+	53.9628i	0.800129	+	1.10128i	0.992772	+	0.120014i	\(0.0382938\pi\)
							−0.192644	+	0.981269i	\(0.561706\pi\)
\(11\)	−58.2051	−	80.1124i	−0.481034	−	0.662086i	0.497669	−	0.867367i	\(-0.334189\pi\)
							−0.978703	+	0.205281i	\(0.934189\pi\)
\(13\)	103.014	−	317.044i	0.609549	−	1.87600i	0.147718	−	0.989029i	\(-0.452807\pi\)
							0.461830	−	0.886968i	\(-0.347193\pi\)
\(17\)	140.906	+	102.374i	0.487563	+	0.354235i	0.804246	−	0.594296i	\(-0.202569\pi\)
							−0.316684	+	0.948531i	\(0.602569\pi\)
\(19\)	477.638	−	155.194i	1.32310	−	0.429901i	0.439540	−	0.898223i	\(-0.355141\pi\)
							0.883558	+	0.468322i	\(0.155141\pi\)
\(23\)	239.885	−	330.173i	0.453468	−	0.624145i	−0.519670	−	0.854367i	\(-0.673945\pi\)
							0.973138	+	0.230222i	\(0.0739451\pi\)
\(29\)	−296.656	−	913.014i	−0.352742	−	1.08563i	−0.957307	−	0.289073i	\(-0.906653\pi\)
							0.604565	−	0.796556i	\(-0.293347\pi\)
\(31\)	151.508	+	948.982i	0.157657	+	0.987494i
							\(37\)	−925.532			−0.676065			−0.338032	−	0.941135i	\(-0.609761\pi\)
−0.338032	+	0.941135i	\(0.609761\pi\)
\(41\)	−129.870	−	399.699i	−0.0772576	−	0.237774i	0.904968	−	0.425480i	\(-0.139895\pi\)
							−0.982225	+	0.187706i	\(0.939895\pi\)
\(43\)	1453.25	−	472.188i	0.785963	−	0.255375i	0.111579	−	0.993756i	\(-0.464409\pi\)
							0.674384	+	0.738381i	\(0.264409\pi\)
\(47\)	−3270.10	−	1062.52i	−1.48035	−	0.480996i	−0.546135	−	0.837697i	\(-0.683902\pi\)
							−0.934219	+	0.356701i	\(0.883902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.19	✓	248
4.3	odd	2	inner	124.5.l.a.35.30	yes	248
31.8	even	5	inner	124.5.l.a.39.30	yes	248
124.39	odd	10	inner	124.5.l.a.39.19	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.19	✓	248	1.1	even	1	trivial
124.5.l.a.35.30	yes	248	4.3	odd	2	inner
124.5.l.a.39.19	yes	248	124.39	odd	10	inner
124.5.l.a.39.30	yes	248	31.8	even	5	inner