Embedded newform 124.3.n.a.7.9

• Label: 124.3.n.a.7.9
• 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			7.9
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15685 + 1.63147i) q^{2} +(-0.354741 - 1.66892i) q^{3} +(-1.32341 - 3.77473i) q^{4} +(-1.19665 + 2.07266i) q^{5} +(3.13318 + 1.35194i) q^{6} +(-1.19159 + 2.67635i) q^{7} +(7.68935 + 2.20769i) q^{8} +(5.56244 - 2.47656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15685	+	1.63147i	−0.578424	+	0.815736i
							\(3\)	−0.354741	−	1.66892i	−0.118247	−	0.556308i	−0.996888	−	0.0788258i	\(-0.974883\pi\)
0.878642	−	0.477482i	\(-0.158450\pi\)
\(5\)	−1.19665	+	2.07266i	−0.239330	+	0.414531i	−0.960522	−	0.278203i	\(-0.910261\pi\)
							0.721192	+	0.692735i	\(0.243594\pi\)
\(7\)	−1.19159	+	2.67635i	−0.170227	+	0.382336i	−0.978433	−	0.206564i	\(-0.933772\pi\)
							0.808206	+	0.588899i	\(0.200439\pi\)
\(11\)	18.3534	−	1.92902i	1.66849	−	0.175366i	0.777343	−	0.629077i	\(-0.216567\pi\)
							0.891149	+	0.453711i	\(0.149900\pi\)
\(13\)	10.5406	+	11.7065i	0.810817	+	0.900503i	0.996626	−	0.0820827i	\(-0.0261572\pi\)
							−0.185809	+	0.982586i	\(0.559490\pi\)
\(17\)	2.97531	−	28.3081i	0.175018	−	1.66519i	−0.456424	−	0.889762i	\(-0.650870\pi\)
							0.631442	−	0.775423i	\(-0.282463\pi\)
\(19\)	10.2032	+	9.18696i	0.537008	+	0.483524i	0.892433	−	0.451181i	\(-0.148997\pi\)
							−0.355424	+	0.934705i	\(0.615664\pi\)
\(23\)	−22.4588	+	30.9118i	−0.976468	+	1.34399i	−0.0377571	+	0.999287i	\(0.512021\pi\)
							−0.938711	+	0.344706i	\(0.887979\pi\)
\(29\)	−8.75075	−	26.9320i	−0.301750	−	0.928691i	−0.980870	−	0.194664i	\(-0.937638\pi\)
							0.679120	−	0.734027i	\(-0.262362\pi\)
\(31\)	30.9465	+	1.82011i	0.998275	+	0.0587133i
							\(37\)	−21.5640	−	37.3500i	−0.582812	−	1.00946i	−0.995144	−	0.0984264i	\(-0.968619\pi\)
0.412332	−	0.911033i	\(-0.364714\pi\)
\(41\)	13.9313	+	2.96118i	0.339787	+	0.0722239i	0.374645	−	0.927168i	\(-0.377765\pi\)
							−0.0348581	+	0.999392i	\(0.511098\pi\)
\(43\)	−11.2594	−	10.1380i	−0.261846	−	0.235768i	0.527741	−	0.849405i	\(-0.323039\pi\)
							−0.789588	+	0.613637i	\(0.789706\pi\)
\(47\)	25.2771	+	8.21303i	0.537811	+	0.174745i	0.565313	−	0.824877i	\(-0.308755\pi\)
							−0.0275025	+	0.999622i	\(0.508755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.9	✓	240
4.3	odd	2	inner	124.3.n.a.7.27	yes	240
31.9	even	15	inner	124.3.n.a.71.27	yes	240
124.71	odd	30	inner	124.3.n.a.71.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.9	✓	240	1.1	even	1	trivial
124.3.n.a.7.27	yes	240	4.3	odd	2	inner
124.3.n.a.71.9	yes	240	124.71	odd	30	inner
124.3.n.a.71.27	yes	240	31.9	even	15	inner