Embedded newform 124.2.p.a.115.14

• Label: 124.2.p.a.115.14
• Level: $124$
• Weight: $2$
• Character: 124.115
• Analytic conductor: $0.990$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.990144985064\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			115.14
Character	\(\chi\)	\(=\)	124.115
Dual form			124.2.p.a.55.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36737 - 0.360989i) q^{2} +(-3.00083 - 0.637845i) q^{3} +(1.73937 - 0.987206i) q^{4} +(-1.91049 - 3.30907i) q^{5} +(-4.33348 + 0.211096i) q^{6} +(0.138213 + 0.310431i) q^{7} +(2.02199 - 1.97777i) q^{8} +(5.85747 + 2.60791i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36737	−	0.360989i	0.966873	−	0.255257i
							\(3\)	−3.00083	−	0.637845i	−1.73253	−	0.368260i	−0.769706	−	0.638399i	\(-0.779597\pi\)
−0.962821	+	0.270139i	\(0.912930\pi\)
\(5\)	−1.91049	−	3.30907i	−0.854397	−	1.47986i	−0.877203	−	0.480119i	\(-0.840593\pi\)
							0.0228063	−	0.999740i	\(-0.492740\pi\)
\(7\)	0.138213	+	0.310431i	0.0522396	+	0.117332i	0.937780	−	0.347231i	\(-0.112878\pi\)
							−0.885540	+	0.464563i	\(0.846211\pi\)
\(11\)	0.0721963	−	0.686902i	0.0217680	−	0.207109i	−0.978232	−	0.207515i	\(-0.933463\pi\)
							1.00000		0.000405919i	\(0.000129208\pi\)
\(13\)	2.99094	+	2.69305i	0.829538	+	0.746919i	0.969984	−	0.243169i	\(-0.0781870\pi\)
							−0.140446	+	0.990088i	\(0.544854\pi\)
\(17\)	−3.78912	+	0.398253i	−0.918997	+	0.0965905i	−0.552201	−	0.833711i	\(-0.686212\pi\)
							−0.366796	+	0.930301i	\(0.619545\pi\)
\(19\)	2.35953	−	2.12453i	0.541312	−	0.487400i	−0.352518	−	0.935805i	\(-0.614674\pi\)
							0.893830	+	0.448405i	\(0.148008\pi\)
\(23\)	3.13881	−	2.28048i	0.654487	−	0.475512i	−0.210310	−	0.977635i	\(-0.567447\pi\)
							0.864797	+	0.502122i	\(0.167447\pi\)
\(29\)	1.11907	+	0.363608i	0.207806	+	0.0675204i	0.411071	−	0.911604i	\(-0.365155\pi\)
							−0.203264	+	0.979124i	\(0.565155\pi\)
\(31\)	4.61999	+	3.10735i	0.829776	+	0.558097i
							\(37\)	−2.02841	−	1.17110i	−0.333468	−	0.192528i	0.323912	−	0.946087i	\(-0.395002\pi\)
−0.657380	+	0.753559i	\(0.728335\pi\)
\(41\)	3.99073	−	0.848255i	0.623247	−	0.132475i	0.114545	−	0.993418i	\(-0.463459\pi\)
							0.508702	+	0.860943i	\(0.330126\pi\)
\(43\)	−0.595076	−	0.660899i	−0.0907483	−	0.100786i	0.696060	−	0.717983i	\(-0.254935\pi\)
							−0.786808	+	0.617197i	\(0.788268\pi\)
\(47\)	−2.48414	+	0.807148i	−0.362350	+	0.117735i	−0.484533	−	0.874773i	\(-0.661010\pi\)
							0.122183	+	0.992508i	\(0.461010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.p.a.115.14	yes	112
4.3	odd	2	inner	124.2.p.a.115.4	yes	112
31.24	odd	30	inner	124.2.p.a.55.4	✓	112
124.55	even	30	inner	124.2.p.a.55.14	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.p.a.55.4	✓	112	31.24	odd	30	inner
124.2.p.a.55.14	yes	112	124.55	even	30	inner
124.2.p.a.115.4	yes	112	4.3	odd	2	inner
124.2.p.a.115.14	yes	112	1.1	even	1	trivial