Embedded newform 124.4.p.a.3.5

• Label: 124.4.p.a.3.5
• Level: $124$
• Weight: $4$
• Character: 124.3
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.70584 + 0.823655i) q^{2} +(0.572364 - 5.44568i) q^{3} +(6.64318 - 4.45737i) q^{4} +(9.37792 - 16.2430i) q^{5} +(2.93664 + 15.2066i) q^{6} +(-6.65861 - 31.3263i) q^{7} +(-14.3041 + 17.5326i) q^{8} +(-2.91787 - 0.620213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.70584	+	0.823655i	−0.956660	+	0.291206i
							\(3\)	0.572364	−	5.44568i	0.110152	−	1.04802i	−0.790197	−	0.612853i	\(-0.790022\pi\)
0.900349	−	0.435169i	\(-0.143311\pi\)
\(5\)	9.37792	−	16.2430i	0.838786	−	1.45282i	−0.0521238	−	0.998641i	\(-0.516599\pi\)
							0.890910	−	0.454180i	\(-0.150068\pi\)
\(7\)	−6.65861	−	31.3263i	−0.359531	−	1.69146i	−0.671205	−	0.741272i	\(-0.734223\pi\)
							0.311674	−	0.950189i	\(-0.399110\pi\)
\(11\)	7.92834	+	8.80532i	0.217317	+	0.241355i	0.841940	−	0.539572i	\(-0.181414\pi\)
							−0.624623	+	0.780927i	\(0.714747\pi\)
\(13\)	29.3976	+	66.0280i	0.627186	+	1.40868i	0.895373	+	0.445316i	\(0.146909\pi\)
							−0.268187	+	0.963367i	\(0.586425\pi\)
\(17\)	−36.4960	−	32.8611i	−0.520681	−	0.468823i	0.366387	−	0.930463i	\(-0.380595\pi\)
							−0.887068	+	0.461639i	\(0.847261\pi\)
\(19\)	−9.80253	+	22.0169i	−0.118361	+	0.265843i	−0.963004	−	0.269487i	\(-0.913146\pi\)
							0.844643	+	0.535330i	\(0.179813\pi\)
\(23\)	8.08421	+	24.8806i	0.0732901	+	0.225564i	0.980991	−	0.194055i	\(-0.0621639\pi\)
							−0.907701	+	0.419618i	\(0.862164\pi\)
\(29\)	163.496	+	225.033i	1.04691	+	1.44095i	0.891457	+	0.453105i	\(0.149684\pi\)
							0.155453	+	0.987843i	\(0.450316\pi\)
\(31\)	48.8144	−	165.554i	0.282817	−	0.959174i
							\(37\)	−14.6205	+	8.44112i	−0.0649618	+	0.0375057i	−0.532129	−	0.846663i	\(-0.678608\pi\)
0.467167	+	0.884169i	\(0.345275\pi\)
\(41\)	27.4413	+	261.086i	0.104527	+	0.994509i	0.913549	+	0.406730i	\(0.133331\pi\)
							−0.809022	+	0.587779i	\(0.800002\pi\)
\(43\)	−29.9199	−	13.3212i	−0.106110	−	0.0472433i	0.352995	−	0.935625i	\(-0.385163\pi\)
							−0.459105	+	0.888382i	\(0.651830\pi\)
\(47\)	−175.618	+	241.718i	−0.545033	+	0.750173i	−0.989328	−	0.145707i	\(-0.953454\pi\)
							0.444295	+	0.895880i	\(0.353454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.5	✓	368
4.3	odd	2	inner	124.4.p.a.3.14	yes	368
31.21	odd	30	inner	124.4.p.a.83.14	yes	368
124.83	even	30	inner	124.4.p.a.83.5	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.5	✓	368	1.1	even	1	trivial
124.4.p.a.3.14	yes	368	4.3	odd	2	inner
124.4.p.a.83.5	yes	368	124.83	even	30	inner
124.4.p.a.83.14	yes	368	31.21	odd	30	inner