Embedded newform 124.3.n.a.7.11

• Label: 124.3.n.a.7.11
• Level: $124$
• Weight: $3$
• Character: 124.7
• Analytic conductor: $3.379$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.37875527807\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			7.11
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.864102 + 1.80370i) q^{2} +(0.286400 + 1.34741i) q^{3} +(-2.50666 - 3.11716i) q^{4} +(3.60353 - 6.24150i) q^{5} +(-2.67780 - 0.647717i) q^{6} +(1.82861 - 4.10713i) q^{7} +(7.78842 - 1.82771i) q^{8} +(6.48843 - 2.88883i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.864102	+	1.80370i	−0.432051	+	0.901849i
							\(3\)	0.286400	+	1.34741i	0.0954668	+	0.449136i	0.999753	+	0.0222098i	\(0.00707018\pi\)
−0.904287	+	0.426926i	\(0.859596\pi\)
\(5\)	3.60353	−	6.24150i	0.720707	−	1.24830i	−0.240010	−	0.970770i	\(-0.577151\pi\)
							0.960717	−	0.277530i	\(-0.0895158\pi\)
\(7\)	1.82861	−	4.10713i	0.261230	−	0.586732i	−0.734545	−	0.678560i	\(-0.762604\pi\)
							0.995775	+	0.0918277i	\(0.0292709\pi\)
\(11\)	−2.82843	+	0.297280i	−0.257130	+	0.0270255i	−0.232216	−	0.972664i	\(-0.574598\pi\)
							−0.0249136	+	0.999690i	\(0.507931\pi\)
\(13\)	−5.65373	−	6.27910i	−0.434902	−	0.483008i	0.485358	−	0.874316i	\(-0.338689\pi\)
							−0.920260	+	0.391308i	\(0.872023\pi\)
\(17\)	−1.84738	+	17.5766i	−0.108669	+	1.03392i	0.795269	+	0.606256i	\(0.207329\pi\)
							−0.903939	+	0.427662i	\(0.859337\pi\)
\(19\)	4.87546	+	4.38988i	0.256603	+	0.231046i	0.787384	−	0.616462i	\(-0.211435\pi\)
							−0.530781	+	0.847509i	\(0.678101\pi\)
\(23\)	21.0135	−	28.9225i	0.913629	−	1.25750i	−0.0522837	−	0.998632i	\(-0.516650\pi\)
							0.965912	−	0.258870i	\(-0.0833500\pi\)
\(29\)	−4.96383	−	15.2771i	−0.171167	−	0.526797i	0.828271	−	0.560328i	\(-0.189325\pi\)
							−0.999438	+	0.0335310i	\(0.989325\pi\)
\(31\)	21.2828	+	22.5398i	0.686543	+	0.727089i
							\(37\)	1.24723	+	2.16026i	0.0337089	+	0.0583855i	0.882388	−	0.470523i	\(-0.155935\pi\)
−0.848679	+	0.528909i	\(0.822601\pi\)
\(41\)	−30.4704	−	6.47667i	−0.743179	−	0.157968i	−0.179265	−	0.983801i	\(-0.557372\pi\)
							−0.563914	+	0.825833i	\(0.690705\pi\)
\(43\)	23.9726	+	21.5850i	0.557501	+	0.501976i	0.899025	−	0.437897i	\(-0.144276\pi\)
							−0.341524	+	0.939873i	\(0.610943\pi\)
\(47\)	−65.5151	−	21.2871i	−1.39394	−	0.452918i	−0.486712	−	0.873562i	\(-0.661804\pi\)
							−0.907226	+	0.420644i	\(0.861804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.11	✓	240
4.3	odd	2	inner	124.3.n.a.7.26	yes	240
31.9	even	15	inner	124.3.n.a.71.26	yes	240
124.71	odd	30	inner	124.3.n.a.71.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.11	✓	240	1.1	even	1	trivial
124.3.n.a.7.26	yes	240	4.3	odd	2	inner
124.3.n.a.71.11	yes	240	124.71	odd	30	inner
124.3.n.a.71.26	yes	240	31.9	even	15	inner