Embedded newform 124.4.p.a.3.20

• Label: 124.4.p.a.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.737328 - 2.73063i) q^{2} +(-0.901200 + 8.57435i) q^{3} +(-6.91270 + 4.02674i) q^{4} +(8.05240 - 13.9472i) q^{5} +(24.0779 - 3.86126i) q^{6} +(-1.43706 - 6.76081i) q^{7} +(16.0925 + 15.9070i) q^{8} +(-46.2973 - 9.84079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.737328	−	2.73063i	−0.260685	−	0.965424i
							\(3\)	−0.901200	+	8.57435i	−0.173436	+	1.65013i	0.468564	+	0.883429i	\(0.344771\pi\)
−0.642000	+	0.766704i	\(0.721895\pi\)
\(5\)	8.05240	−	13.9472i	0.720229	−	1.24747i	−0.240679	−	0.970605i	\(-0.577370\pi\)
							0.960908	−	0.276868i	\(-0.0892965\pi\)
\(7\)	−1.43706	−	6.76081i	−0.0775937	−	0.365050i	0.922170	−	0.386785i	\(-0.126414\pi\)
							−0.999764	+	0.0217350i	\(0.993081\pi\)
\(11\)	37.2067	+	41.3222i	1.01984	+	1.13265i	0.991108	+	0.133060i	\(0.0424802\pi\)
							0.0287317	+	0.999587i	\(0.490853\pi\)
\(13\)	13.2387	+	29.7346i	0.282442	+	0.634376i	0.997934	−	0.0642452i	\(-0.0204640\pi\)
							−0.715492	+	0.698621i	\(0.753797\pi\)
\(17\)	76.4353	+	68.8226i	1.09049	+	0.981879i	0.999902	−	0.0140102i	\(-0.00445972\pi\)
							0.0905848	+	0.995889i	\(0.471126\pi\)
\(19\)	−65.7840	+	147.753i	−0.794310	+	1.78405i	−0.200033	+	0.979789i	\(0.564105\pi\)
							−0.594277	+	0.804260i	\(0.702562\pi\)
\(23\)	−28.6962	−	88.3178i	−0.260155	−	0.800675i	−0.992770	−	0.120032i	\(-0.961700\pi\)
							0.732615	−	0.680643i	\(-0.238300\pi\)
\(29\)	87.3985	+	120.294i	0.559638	+	0.770275i	0.991280	−	0.131769i	\(-0.0420659\pi\)
							−0.431643	+	0.902045i	\(0.642066\pi\)
\(31\)	77.7082	−	154.118i	0.450219	−	0.892918i
							\(37\)	97.7756	−	56.4508i	0.434438	−	0.250823i	−0.266797	−	0.963753i	\(-0.585965\pi\)
0.701236	+	0.712930i	\(0.252632\pi\)
\(41\)	0.730611	+	6.95130i	0.00278298	+	0.0264783i	0.995825	−	0.0912800i	\(-0.0290958\pi\)
							−0.993042	+	0.117758i	\(0.962429\pi\)
\(43\)	49.5705	+	22.0702i	0.175801	+	0.0782715i	0.492750	−	0.870171i	\(-0.335992\pi\)
							−0.316949	+	0.948443i	\(0.602658\pi\)
\(47\)	144.213	−	198.493i	0.447568	−	0.616024i	−0.524305	−	0.851531i	\(-0.675675\pi\)
							0.971873	+	0.235506i	\(0.0756748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.20	✓	368
4.3	odd	2	inner	124.4.p.a.3.39	yes	368
31.21	odd	30	inner	124.4.p.a.83.39	yes	368
124.83	even	30	inner	124.4.p.a.83.20	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.20	✓	368	1.1	even	1	trivial
124.4.p.a.3.39	yes	368	4.3	odd	2	inner
124.4.p.a.83.20	yes	368	124.83	even	30	inner
124.4.p.a.83.39	yes	368	31.21	odd	30	inner