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

• Label: 7056.2.b
• Level: $7056$
• Weight: $2$
• Character orbit: 7056.b
• Rep. character: $\chi_{7056}(1567,\cdot)$
• Character field: $\Q$
• Dimension: $100$
• Newform subspaces: $26$
• Sturm bound: $2688$
• Trace bound: $53$

Defining parameters

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(2688\)
Trace bound:			\(53\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(17\), \(19\), \(31\), \(53\)

Dimensions

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

	Total	New	Old
Modular forms	1440	100	1340
Cusp forms	1248	100	1148
Eisenstein series	192	0	192

Trace form

\( 100 q - 124 q^{25} + 24 q^{29} - 8 q^{37} - 24 q^{53} + 24 q^{65} - 72 q^{85}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
7056.2.b.a	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{5}-2\beta q^{11}-3\beta q^{13}+4\beta q^{17}+\cdots\)
7056.2.b.b	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					112.2.p.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}+\beta q^{11}-3\beta q^{17}-7 q^{19}+\cdots\)
7056.2.b.c	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					1008.2.cs.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{13}-7 q^{19}+5 q^{25}+7 q^{31}+\cdots\)
7056.2.b.d	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{5}-2\beta q^{11}-\beta q^{13}-5 q^{19}+\cdots\)
7056.2.b.e	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}+\beta q^{11}+2\beta q^{17}-2 q^{19}+\cdots\)
7056.2.b.f	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}+3\beta q^{11}+4\beta q^{13}-2\beta q^{17}+\cdots\)
7056.2.b.g	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					1008.2.cs.g	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{13}-q^{19}+5 q^{25}-11 q^{31}+\cdots\)
7056.2.b.h	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					1008.2.cs.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{13}+q^{19}+5 q^{25}+11 q^{31}+\cdots\)
7056.2.b.i	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}-\beta q^{11}+2\beta q^{17}+2 q^{19}+\cdots\)
7056.2.b.j	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}-3\beta q^{11}+4\beta q^{13}-2\beta q^{17}+\cdots\)
7056.2.b.k	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{5}+2\beta q^{11}-\beta q^{13}+5 q^{19}+\cdots\)
7056.2.b.l	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					336.2.bl.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{5}+2\beta q^{11}-3\beta q^{13}+4\beta q^{17}+\cdots\)
7056.2.b.m	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	None					112.2.p.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{5}-\beta q^{11}-3\beta q^{17}+7 q^{19}+\cdots\)
7056.2.b.n	$7056$	$2$	7056.b	28.d	$2$	$2$	$2$	$56.342$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					1008.2.cs.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{13}+7 q^{19}+5 q^{25}-7 q^{31}+\cdots\)
7056.2.b.o	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					1008.2.cs.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+\beta _{2}q^{11}+2\beta _{1}q^{13}-4q^{19}+\cdots\)
7056.2.b.p	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2 + \sqrt{2}})\)	\(\Q(\sqrt{-1}) \)		✓			7056.2.b.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(2\beta _{1}+\beta _{3})q^{5}+(-2\beta _{1}-3\beta _{3})q^{13}+\cdots\)
7056.2.b.q	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2 + \sqrt{2}})\)	\(\Q(\sqrt{-1}) \)					784.2.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2})q^{13}+(\beta _{1}-\beta _{2}+\cdots)q^{17}+\cdots\)
7056.2.b.r	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2 + \sqrt{2}})\)	\(\Q(\sqrt{-1}) \)		✓			7056.2.b.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(2\beta _{1}+\beta _{3})q^{5}+(2\beta _{1}+3\beta _{3})q^{13}+\cdots\)
7056.2.b.s	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					112.2.p.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{3}q^{11}+2\beta _{2}q^{13}+3\beta _{2}q^{17}+\cdots\)
7056.2.b.t	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2 + \sqrt{2}})\)	None					2352.2.b.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(-2\beta _{1}+2\beta _{3})q^{11}+(2\beta _{1}+\cdots)q^{13}+\cdots\)
7056.2.b.u	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2 + \sqrt{2}})\)	None					2352.2.b.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(2\beta _{1}-2\beta _{3})q^{11}+(2\beta _{1}-\beta _{3})q^{13}+\cdots\)
7056.2.b.v	$7056$	$2$	7056.b	28.d	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					1008.2.cs.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{2}q^{11}-2\beta _{1}q^{13}+4q^{19}+\cdots\)
7056.2.b.w	$7056$	$2$	7056.b	28.d	$2$	$8$	$8$	$56.342$	8.0.339738624.1	None					2352.2.b.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{4}+\beta _{6})q^{5}+(-\beta _{2}-\beta _{4}+\beta _{6}+\cdots)q^{11}+\cdots\)
7056.2.b.x	$7056$	$2$	7056.b	28.d	$2$	$8$	$8$	$56.342$	8.0.339738624.1	None					2352.2.b.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{4}+\beta _{6})q^{5}+(\beta _{2}+\beta _{4}-\beta _{6})q^{11}+\cdots\)
7056.2.b.y	$7056$	$2$	7056.b	28.d	$2$	$8$	$8$	$56.342$	8.0.339738624.1	None					784.2.f.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}-\beta _{4})q^{5}+(-\beta _{6}+\beta _{7})q^{11}+(-\beta _{2}+\cdots)q^{13}+\cdots\)
7056.2.b.z	$7056$	$2$	7056.b	28.d	$2$	$16$	$16$	$56.342$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		7056.2.b.z	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{15}q^{5}-\beta _{9}q^{11}+\beta _{8}q^{13}+(-2\beta _{6}+\cdots)q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(7056, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1764, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2352, [\chi])\)\(^{\oplus 2}\)