Embedded newform 124.2.p.a.11.11

• Label: 124.2.p.a.11.11
• Level: $124$
• Weight: $2$
• Character: 124.11
• 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			11.11
Character	\(\chi\)	\(=\)	124.11
Dual form			124.2.p.a.79.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.699365 - 1.22918i) q^{2} +(0.963066 + 1.06959i) q^{3} +(-1.02178 - 1.71929i) q^{4} +(1.81486 + 3.14343i) q^{5} +(1.98826 - 0.435748i) q^{6} +(-0.437436 + 0.0459763i) q^{7} +(-2.82792 + 0.0535383i) q^{8} +(0.0970520 - 0.923388i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.699365	−	1.22918i	0.494526	−	0.869163i
							\(3\)	0.963066	+	1.06959i	0.556026	+	0.617530i	0.953978	−	0.299876i	\(-0.0969455\pi\)
−0.397952	+	0.917406i	\(0.630279\pi\)
\(5\)	1.81486	+	3.14343i	0.811629	+	1.40578i	0.911723	+	0.410805i	\(0.134752\pi\)
							−0.100094	+	0.994978i	\(0.531914\pi\)
\(7\)	−0.437436	+	0.0459763i	−0.165335	+	0.0173774i	−0.186834	−	0.982391i	\(-0.559823\pi\)
							0.0214992	+	0.999769i	\(0.493156\pi\)
\(11\)	−5.34230	−	2.37855i	−1.61076	−	0.717159i	−0.613404	−	0.789769i	\(-0.710200\pi\)
							−0.997360	+	0.0726107i	\(0.976867\pi\)
\(13\)	−0.889051	−	4.18266i	−0.246578	−	1.16006i	−0.910904	−	0.412617i	\(-0.864615\pi\)
							0.664326	−	0.747443i	\(-0.268719\pi\)
\(17\)	1.56256	+	3.50956i	0.378975	+	0.851193i	0.997840	+	0.0656949i	\(0.0209264\pi\)
							−0.618864	+	0.785498i	\(0.712407\pi\)
\(19\)	0.203508	−	0.957430i	0.0466880	−	0.219650i	−0.948619	−	0.316422i	\(-0.897519\pi\)
							0.995307	+	0.0967722i	\(0.0308518\pi\)
\(23\)	0.628519	+	0.456646i	0.131055	+	0.0952173i	0.651382	−	0.758750i	\(-0.274190\pi\)
							−0.520326	+	0.853968i	\(0.674190\pi\)
\(29\)	2.90944	−	0.945333i	0.540269	−	0.175544i	−0.0261553	−	0.999658i	\(-0.508326\pi\)
							0.566424	+	0.824114i	\(0.308326\pi\)
\(31\)	1.12364	+	5.45320i	0.201811	+	0.979424i
							\(37\)	−1.34352	−	0.775684i	−0.220874	−	0.127522i	0.385481	−	0.922716i	\(-0.374036\pi\)
−0.606355	+	0.795194i	\(0.707369\pi\)
\(41\)	−4.90079	+	5.44288i	−0.765375	+	0.850035i	−0.992297	−	0.123878i	\(-0.960467\pi\)
							0.226923	+	0.973913i	\(0.427134\pi\)
\(43\)	−2.95649	−	0.628421i	−0.450860	−	0.0958332i	−0.0231152	−	0.999733i	\(-0.507358\pi\)
							−0.427745	+	0.903900i	\(0.640692\pi\)
\(47\)	11.2410	+	3.65241i	1.63966	+	0.532759i	0.976463	−	0.215683i	\(-0.0691978\pi\)
							0.663200	+	0.748442i	\(0.269198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.p.a.11.11	yes	112
4.3	odd	2	inner	124.2.p.a.11.2	✓	112
31.17	odd	30	inner	124.2.p.a.79.2	yes	112
124.79	even	30	inner	124.2.p.a.79.11	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.p.a.11.2	✓	112	4.3	odd	2	inner
124.2.p.a.11.11	yes	112	1.1	even	1	trivial
124.2.p.a.79.2	yes	112	31.17	odd	30	inner
124.2.p.a.79.11	yes	112	124.79	even	30	inner