Space of modular forms of level 7056, weight 2, and Character 7056.h

• Label: 7056.2.h
• Level: $7056$
• Weight: $2$
• Character orbit: 7056.h
• Rep. character: $\chi_{7056}(4607,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $15$
• Sturm bound: $2688$
• Trace bound: $83$

Defining parameters

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(2688\)
Trace bound:			\(83\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(61\), \(83\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(7056, [\chi])\).

	Total	New	Old
Modular forms	1440	82	1358
Cusp forms	1248	82	1166
Eisenstein series	192	0	192

Trace form

\( 82 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(7056, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
7056.2.h.a	$7056$	$2$	7056.h	12.b	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{5}-6q^{13}-5\beta q^{17}+3q^{25}+\cdots\)
7056.2.h.b	$7056$	$2$	7056.h	12.b	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{5}-4q^{13}-\beta q^{17}-13q^{25}+\cdots\)
7056.2.h.c	$7056$	$2$	7056.h	12.b	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{5}+6q^{13}-5\beta q^{17}+3q^{25}+\cdots\)
7056.2.h.d	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}-\beta _{1}q^{11}-q^{13}+2\beta _{2}q^{17}+\cdots\)
7056.2.h.e	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-2\zeta_{8}q^{5}-5\zeta_{8}^{3}q^{13}-4\zeta_{8}q^{17}+\cdots\)
7056.2.h.f	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{5}-\beta _{2}q^{11}-2\beta _{1}q^{17}-2\beta _{3}q^{19}+\cdots\)
7056.2.h.g	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{5}-\beta _{2}q^{11}-2\beta _{1}q^{17}-2\beta _{3}q^{19}+\cdots\)
7056.2.h.h	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{8}q^{5}+\zeta_{8}^{3}q^{13}-\zeta_{8}q^{17}+q^{25}+\cdots\)
7056.2.h.i	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{1}q^{11}+q^{13}+2\beta _{2}q^{17}+\cdots\)
7056.2.h.j	$7056$	$2$	7056.h	12.b	$2$	$4$	$4$	$56.342$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{2}q^{5}+\zeta_{8}^{3}q^{11}+4q^{13}+\zeta_{8}^{2}q^{17}+\cdots\)
7056.2.h.k	$7056$	$2$	7056.h	12.b	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{5}+(\zeta_{24}^{4}+\zeta_{24}^{7})q^{11}+\cdots\)
7056.2.h.l	$7056$	$2$	7056.h	12.b	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{24}q^{5}-3\zeta_{24}^{5}q^{11}+(2\zeta_{24}^{3}+\zeta_{24}^{5}+\cdots)q^{13}+\cdots\)
7056.2.h.m	$7056$	$2$	7056.h	12.b	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{24}q^{5}+3\zeta_{24}^{5}q^{11}+(2\zeta_{24}^{3}+\zeta_{24}^{5}+\cdots)q^{13}+\cdots\)
7056.2.h.n	$7056$	$2$	7056.h	12.b	$2$	$12$	$12$	$56.342$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+\beta _{3}q^{11}+\beta _{5}q^{13}+(\beta _{4}-\beta _{9}+\cdots)q^{17}+\cdots\)
7056.2.h.o	$7056$	$2$	7056.h	12.b	$2$	$12$	$12$	$56.342$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+\beta _{3}q^{11}-\beta _{5}q^{13}+(\beta _{4}-\beta _{9}+\cdots)q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(7056, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(7056, [\chi]) \cong \)