Embedded newform 126.4.h.a.67.1

• Label: 126.4.h.a.67.1
• Level: $126$
• Weight: $4$
• Character: 126.67
• Analytic conductor: $7.434$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.43424066072\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.1
Character	\(\chi\)	\(=\)	126.67
Dual form			126.4.h.a.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-5.18577 + 0.328291i) q^{3} +(-2.00000 - 3.46410i) q^{4} +10.1344 q^{5} +(4.61715 - 9.31031i) q^{6} +(2.82404 + 18.3037i) q^{7} +8.00000 q^{8} +(26.7844 - 3.40489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - q^{3} - 48 q^{4} - 20 q^{5} + 4 q^{6} - 16 q^{7} + 192 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	−5.18577	+	0.328291i	−0.998002	+	0.0631797i
\(5\)	10.1344			0.906444										0.453222	−	0.891398i	\(-0.350274\pi\)
							0.453222	+	0.891398i	\(0.350274\pi\)
\(7\)	2.82404	+	18.3037i	0.152484	+	0.988306i
							\(11\)	−38.1092			−1.04458			−0.522289	−	0.852768i	\(-0.674922\pi\)
−0.522289	+	0.852768i	\(0.674922\pi\)
\(13\)	−1.46126	+	2.53098i	−0.0311755	+	0.0539976i	−0.881192	−	0.472758i	\(-0.843258\pi\)
							0.850017	+	0.526756i	\(0.176592\pi\)
\(17\)	−56.0159	+	97.0224i	−0.799168	+	1.38420i	0.120991	+	0.992654i	\(0.461393\pi\)
							−0.920159	+	0.391546i	\(0.871940\pi\)
\(19\)	−39.2887	−	68.0500i	−0.474391	−	0.821670i	0.525179	−	0.850992i	\(-0.323999\pi\)
							−0.999570	+	0.0293220i	\(0.990665\pi\)
\(23\)	−147.024			−1.33290			−0.666450	−	0.745549i	\(-0.732187\pi\)
							−0.666450	+	0.745549i	\(0.732187\pi\)
\(29\)	137.710	+	238.521i	0.881798	+	1.52732i	0.849340	+	0.527846i	\(0.177000\pi\)
							0.0324583	+	0.999473i	\(0.489666\pi\)
\(31\)	−51.1672	−	88.6242i	−0.296448	−	0.513464i	0.678872	−	0.734256i	\(-0.262469\pi\)
							−0.975321	+	0.220793i	\(0.929136\pi\)
\(37\)	−25.1233	−	43.5149i	−0.111628	−	0.193346i	0.804799	−	0.593548i	\(-0.202273\pi\)
							−0.916427	+	0.400202i	\(0.868940\pi\)
\(41\)	1.16281	−	2.01405i	0.00442929	−	0.00767176i	−0.863802	−	0.503831i	\(-0.831923\pi\)
							0.868232	+	0.496159i	\(0.165257\pi\)
\(43\)	184.064	+	318.808i	0.652778	+	1.13064i	0.982446	+	0.186547i	\(0.0597297\pi\)
							−0.329668	+	0.944097i	\(0.606937\pi\)
\(47\)	−199.792	+	346.050i	−0.620056	+	1.07397i	0.369419	+	0.929263i	\(0.379557\pi\)
							−0.989475	+	0.144705i	\(0.953777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.4.h.a.67.1	yes	24
3.2	odd	2		378.4.h.b.361.4		24
7.2	even	3		126.4.e.b.121.8	yes	24
9.2	odd	6		378.4.e.a.235.9		24
9.7	even	3		126.4.e.b.25.8	✓	24
21.2	odd	6		378.4.e.a.37.9		24
63.2	odd	6		378.4.h.b.289.4		24
63.16	even	3	inner	126.4.h.a.79.1	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.e.b.25.8	✓	24	9.7	even	3
126.4.e.b.121.8	yes	24	7.2	even	3
126.4.h.a.67.1	yes	24	1.1	even	1	trivial
126.4.h.a.79.1	yes	24	63.16	even	3	inner
378.4.e.a.37.9		24	21.2	odd	6
378.4.e.a.235.9		24	9.2	odd	6
378.4.h.b.289.4		24	63.2	odd	6
378.4.h.b.361.4		24	3.2	odd	2