Newform orbit 1728.2.z.a

• Label: 1728.2.z.a
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.z
• Analytic conductor: $13.798$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.z (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
143.1		0			0			0		−3.73424	−	1.00059i		0		1.68236	−	2.91393i		0			0			0
143.2		0			0			0		−3.72190	−	0.997280i		0		0.481387	−	0.833787i		0			0			0
143.3		0			0			0		−2.80938	−	0.752772i		0		1.02581	−	1.77675i		0			0			0
143.4		0			0			0		−2.31044	−	0.619079i		0		−2.51270	+	4.35213i		0			0			0
143.5		0			0			0		−2.03779	−	0.546024i		0		0.0638076	−	0.110518i		0			0			0
143.6		0			0			0		−1.76649	−	0.473330i		0		−1.40613	+	2.43549i		0			0			0
143.7		0			0			0		−1.20583	−	0.323102i		0		−0.140266	+	0.242948i		0			0			0
143.8		0			0			0		−1.15827	−	0.310357i		0		−0.356047	+	0.616691i		0			0			0
143.9		0			0			0		−1.05401	−	0.282421i		0		1.93586	−	3.35301i		0			0			0
143.10		0			0			0		−0.664471	−	0.178044i		0		−0.645693	+	1.11837i		0			0			0
143.11		0			0			0		−0.289971	−	0.0776974i		0		0.374023	−	0.647827i		0			0			0
143.12		0			0			0		0.170993	+	0.0458174i		0		−1.17432	+	2.03397i		0			0			0
143.13		0			0			0		0.769670	+	0.206232i		0		2.17574	−	3.76849i		0			0			0
143.14		0			0			0		0.923380	+	0.247419i		0		1.93471	−	3.35102i		0			0			0
143.15		0			0			0		1.17929	+	0.315990i		0		−1.93802	+	3.35676i		0			0			0
143.16		0			0			0		1.94452	+	0.521033i		0		0.322227	−	0.558114i		0			0			0
143.17		0			0			0		2.39818	+	0.642590i		0		1.93190	−	3.34616i		0			0			0
143.18		0			0			0		2.70956	+	0.726024i		0		0.00424642	−	0.00735502i		0			0			0
143.19		0			0			0		2.78704	+	0.746784i		0		−1.16672	+	2.02082i		0			0			0
143.20		0			0			0		2.83365	+	0.759273i		0		−1.41719	+	2.45465i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
16.f	odd	4	1	inner
144.u	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.2.z.a		88
3.b	odd	2	1		576.2.y.a		88
4.b	odd	2	1		432.2.v.a		88
9.c	even	3	1		576.2.y.a		88
9.d	odd	6	1	inner	1728.2.z.a		88
12.b	even	2	1		144.2.u.a	✓	88
16.e	even	4	1		432.2.v.a		88
16.f	odd	4	1	inner	1728.2.z.a		88
36.f	odd	6	1		144.2.u.a	✓	88
36.h	even	6	1		432.2.v.a		88
48.i	odd	4	1		144.2.u.a	✓	88
48.k	even	4	1		576.2.y.a		88
144.u	even	12	1	inner	1728.2.z.a		88
144.v	odd	12	1		576.2.y.a		88
144.w	odd	12	1		432.2.v.a		88
144.x	even	12	1		144.2.u.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.u.a	✓	88	12.b	even	2	1
144.2.u.a	✓	88	36.f	odd	6	1
144.2.u.a	✓	88	48.i	odd	4	1
144.2.u.a	✓	88	144.x	even	12	1
432.2.v.a		88	4.b	odd	2	1
432.2.v.a		88	16.e	even	4	1
432.2.v.a		88	36.h	even	6	1
432.2.v.a		88	144.w	odd	12	1
576.2.y.a		88	3.b	odd	2	1
576.2.y.a		88	9.c	even	3	1
576.2.y.a		88	48.k	even	4	1
576.2.y.a		88	144.v	odd	12	1
1728.2.z.a		88	1.a	even	1	1	trivial
1728.2.z.a		88	9.d	odd	6	1	inner
1728.2.z.a		88	16.f	odd	4	1	inner
1728.2.z.a		88	144.u	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1728, [\chi])\).