Embedded newform 124.4.p.a.3.17

• Label: 124.4.p.a.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20752 - 2.55771i) q^{2} +(-0.00192290 + 0.0182952i) q^{3} +(-5.08377 + 6.17699i) q^{4} +(6.27992 - 10.8771i) q^{5} +(0.0491158 - 0.0171736i) q^{6} +(-1.11632 - 5.25185i) q^{7} +(21.9377 + 5.54396i) q^{8} +(26.4097 + 5.61355i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20752	−	2.55771i	−0.426924	−	0.904287i
							\(3\)	−0.00192290	+	0.0182952i	−0.000370063	+	0.00352091i	−0.994705	−	0.102768i	\(-0.967230\pi\)
0.994335	+	0.106289i	\(0.0338968\pi\)
\(5\)	6.27992	−	10.8771i	0.561693	−	0.972881i	−0.435656	−	0.900113i	\(-0.643483\pi\)
							0.997349	−	0.0727678i	\(-0.0231832\pi\)
\(7\)	−1.11632	−	5.25185i	−0.0602754	−	0.283573i	0.937679	−	0.347503i	\(-0.112970\pi\)
							−0.997954	+	0.0639291i	\(0.979637\pi\)
\(11\)	−42.5473	−	47.2535i	−1.16623	−	1.29522i	−0.947616	−	0.319412i	\(-0.896515\pi\)
							−0.218610	−	0.975812i	\(-0.570152\pi\)
\(13\)	10.5916	+	23.7892i	0.225968	+	0.507533i	0.990576	−	0.136963i	\(-0.0437341\pi\)
							−0.764608	+	0.644496i	\(0.777067\pi\)
\(17\)	−100.653	−	90.6285i	−1.43600	−	1.29298i	−0.890588	−	0.454811i	\(-0.849707\pi\)
							−0.545411	−	0.838169i	\(-0.683626\pi\)
\(19\)	−9.69085	+	21.7660i	−0.117012	+	0.262814i	−0.962551	−	0.271100i	\(-0.912613\pi\)
							0.845539	+	0.533914i	\(0.179279\pi\)
\(23\)	−17.3665	−	53.4485i	−0.157442	−	0.484555i	0.840959	−	0.541100i	\(-0.181992\pi\)
							−0.998400	+	0.0565443i	\(0.981992\pi\)
\(29\)	−47.9702	−	66.0253i	−0.307167	−	0.422779i	0.627328	−	0.778755i	\(-0.284148\pi\)
							−0.934495	+	0.355976i	\(0.884148\pi\)
\(31\)	−145.967	+	92.1121i	−0.845692	+	0.533672i
							\(37\)	113.619	−	65.5982i	0.504835	−	0.291467i	−0.225873	−	0.974157i	\(-0.572523\pi\)
0.730708	+	0.682690i	\(0.239190\pi\)
\(41\)	−34.9664	−	332.683i	−0.133191	−	1.26723i	−0.833151	−	0.553046i	\(-0.813466\pi\)
							0.699960	−	0.714182i	\(-0.253201\pi\)
\(43\)	357.458	+	159.151i	1.26772	+	0.564424i	0.926760	−	0.375655i	\(-0.122582\pi\)
							0.340957	+	0.940079i	\(0.389249\pi\)
\(47\)	195.483	−	269.059i	0.606683	−	0.835027i	−0.389617	−	0.920977i	\(-0.627392\pi\)
							0.996299	+	0.0859498i	\(0.0273925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.17	✓	368
4.3	odd	2	inner	124.4.p.a.3.35	yes	368
31.21	odd	30	inner	124.4.p.a.83.35	yes	368
124.83	even	30	inner	124.4.p.a.83.17	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.17	✓	368	1.1	even	1	trivial
124.4.p.a.3.35	yes	368	4.3	odd	2	inner
124.4.p.a.83.17	yes	368	124.83	even	30	inner
124.4.p.a.83.35	yes	368	31.21	odd	30	inner