Embedded newform 126.4.e.b.121.8

• Label: 126.4.e.b.121.8
• Level: $126$
• Weight: $4$
• Character: 126.121
• Analytic conductor: $7.434$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.e (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			121.8
Character	\(\chi\)	\(=\)	126.121
Dual form			126.4.e.b.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +(2.30858 - 4.65516i) q^{3} +4.00000 q^{4} +(-5.06718 - 8.77661i) q^{5} +(4.61715 - 9.31031i) q^{6} +(14.4394 + 11.5975i) q^{7} +8.00000 q^{8} +(-16.3409 - 21.4936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{2} + 2 q^{3} + 96 q^{4} + 10 q^{5} + 4 q^{6} + 17 q^{7} + 192 q^{8} + 32 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000			0.707107
							\(3\)	2.30858	−	4.65516i	0.444286	−	0.895885i
\(5\)	−5.06718	−	8.77661i	−0.453222	−	0.785004i								0.545362	−	0.838201i	\(-0.316392\pi\)
							−0.998584	+	0.0531971i	\(0.983059\pi\)
\(7\)	14.4394	+	11.5975i	0.779656	+	0.626208i
							\(11\)	19.0546	−	33.0036i	0.522289	−	0.904632i	−0.477374	−	0.878700i	\(-0.658411\pi\)
0.999664	−	0.0259317i	\(-0.00825523\pi\)
\(13\)	−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	−0.920159	−	0.391546i	\(-0.871940\pi\)
							0.120991	−	0.992654i	\(-0.461393\pi\)
\(19\)	−39.2887	+	68.0500i	−0.474391	+	0.821670i	−0.999570	−	0.0293220i	\(-0.990665\pi\)
							0.525179	+	0.850992i	\(0.323999\pi\)
\(23\)	73.5122	+	127.327i	0.666450	+	1.15433i	0.978890	+	0.204388i	\(0.0655205\pi\)
							−0.312440	+	0.949938i	\(0.601146\pi\)
\(29\)	137.710	+	238.521i	0.881798	+	1.52732i	0.849340	+	0.527846i	\(0.177000\pi\)
							0.0324583	+	0.999473i	\(0.489666\pi\)
\(31\)	102.334			0.592897			0.296448	−	0.955049i	\(-0.404198\pi\)
							0.296448	+	0.955049i	\(0.404198\pi\)
\(37\)	−25.1233	+	43.5149i	−0.111628	+	0.193346i	−0.916427	−	0.400202i	\(-0.868940\pi\)
							0.804799	+	0.593548i	\(0.202273\pi\)
\(41\)	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	\(0.606937\pi\)
\(47\)	399.584			1.24011			0.620056	−	0.784558i	\(-0.287110\pi\)
							0.620056	+	0.784558i	\(0.287110\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.e.b.25.8	✓	24	63.25	even	3	inner
126.4.e.b.121.8	yes	24	1.1	even	1	trivial
126.4.h.a.67.1	yes	24	7.4	even	3
126.4.h.a.79.1	yes	24	9.7	even	3
378.4.e.a.37.9		24	3.2	odd	2
378.4.e.a.235.9		24	63.11	odd	6
378.4.h.b.289.4		24	9.2	odd	6
378.4.h.b.361.4		24	21.11	odd	6