Embedded newform 124.4.p.a.3.4

• Label: 124.4.p.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.80221 + 0.384244i) q^{2} +(0.781206 - 7.43268i) q^{3} +(7.70471 - 2.15346i) q^{4} +(-1.74753 + 3.02682i) q^{5} +(0.666864 + 21.1281i) q^{6} +(6.94622 + 32.6794i) q^{7} +(-20.7627 + 8.99494i) q^{8} +(-28.2244 - 5.99928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.80221	+	0.384244i	−0.990729	+	0.135851i
							\(3\)	0.781206	−	7.43268i	0.150343	−	1.43042i	−0.615878	−	0.787842i	\(-0.711198\pi\)
0.766221	−	0.642577i	\(-0.222135\pi\)
\(5\)	−1.74753	+	3.02682i	−0.156304	+	0.270727i	−0.933533	−	0.358491i	\(-0.883291\pi\)
							0.777229	+	0.629218i	\(0.216625\pi\)
\(7\)	6.94622	+	32.6794i	0.375060	+	1.76452i	0.609892	+	0.792484i	\(0.291213\pi\)
							−0.234832	+	0.972036i	\(0.575454\pi\)
\(11\)	26.3035	+	29.2130i	0.720982	+	0.800732i	0.986567	−	0.163356i	\(-0.0522318\pi\)
							−0.265585	+	0.964087i	\(0.585565\pi\)
\(13\)	16.5551	+	37.1834i	0.353197	+	0.793294i	0.999542	+	0.0302486i	\(0.00962991\pi\)
							−0.646345	+	0.763045i	\(0.723703\pi\)
\(17\)	−51.3489	−	46.2348i	−0.732585	−	0.659622i	0.215911	−	0.976413i	\(-0.430728\pi\)
							−0.948495	+	0.316791i	\(0.897395\pi\)
\(19\)	34.1739	−	76.7558i	0.412633	−	0.926789i	−0.580977	−	0.813920i	\(-0.697329\pi\)
							0.993610	−	0.112869i	\(-0.0360040\pi\)
\(23\)	58.2596	+	179.305i	0.528172	+	1.62555i	0.757956	+	0.652306i	\(0.226198\pi\)
							−0.229783	+	0.973242i	\(0.573802\pi\)
\(29\)	−50.8699	−	70.0164i	−0.325735	−	0.448335i	0.614473	−	0.788938i	\(-0.289369\pi\)
							−0.940207	+	0.340603i	\(0.889369\pi\)
\(31\)	−69.6873	+	157.907i	−0.403749	+	0.914870i
							\(37\)	94.4331	−	54.5210i	0.419587	−	0.242249i	−0.275314	−	0.961354i	\(-0.588782\pi\)
0.694901	+	0.719106i	\(0.255448\pi\)
\(41\)	−40.1427	−	381.933i	−0.152908	−	1.45483i	−0.754646	−	0.656132i	\(-0.772191\pi\)
							0.601738	−	0.798694i	\(-0.294475\pi\)
\(43\)	318.770	+	141.926i	1.13051	+	0.503336i	0.884783	−	0.466002i	\(-0.154306\pi\)
							0.245729	+	0.969339i	\(0.420973\pi\)
\(47\)	204.639	−	281.661i	0.635098	−	0.874137i	−0.363244	−	0.931694i	\(-0.618331\pi\)
							0.998342	+	0.0575567i	\(0.0183310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.4	✓	368
4.3	odd	2	inner	124.4.p.a.3.16	yes	368
31.21	odd	30	inner	124.4.p.a.83.16	yes	368
124.83	even	30	inner	124.4.p.a.83.4	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.4	✓	368	1.1	even	1	trivial
124.4.p.a.3.16	yes	368	4.3	odd	2	inner
124.4.p.a.83.4	yes	368	124.83	even	30	inner
124.4.p.a.83.16	yes	368	31.21	odd	30	inner