Space of modular forms of level 3528, weight 2, and trivial character

• Label: 3528.2.a
• Level: $3528$
• Weight: $2$
• Character orbit: 3528.a
• Rep. character: $\chi_{3528}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $39$
• Sturm bound: $1344$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 39 \)
Sturm bound:			\(1344\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3528))\).

	Total	New	Old
Modular forms	736	51	685
Cusp forms	609	51	558
Eisenstein series	127	0	127

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(-\)	\(-\)	$+$	\(7\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(24\)
Minus space			\(-\)	\(27\)

Trace form

\( 51 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
3528.2.a.a	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{13}+4q^{17}+7q^{19}-4q^{23}+\cdots\)
3528.2.a.b	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-2q^{17}+2q^{19}-8q^{23}+11q^{25}+\cdots\)
3528.2.a.c	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+3q^{13}+4q^{17}-7q^{19}+4q^{23}+\cdots\)
3528.2.a.d	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
3528.2.a.e	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
3528.2.a.f	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+6q^{11}-3q^{13}-4q^{17}-5q^{19}+\cdots\)
3528.2.a.g	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+6q^{11}+6q^{13}+2q^{17}-4q^{19}+\cdots\)
3528.2.a.h	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-5q^{11}-2q^{13}+6q^{17}-2q^{19}+\cdots\)
3528.2.a.i	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{11}-4q^{13}+4q^{19}-8q^{23}+\cdots\)
3528.2.a.j	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{11}+6q^{13}-5q^{17}-q^{19}+\cdots\)
3528.2.a.k	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{11}-2q^{13}+3q^{17}-5q^{19}+\cdots\)
3528.2.a.l	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+5q^{11}+2q^{13}+6q^{17}+2q^{19}+\cdots\)
3528.2.a.m	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{13}+4q^{17}+4q^{19}-4q^{23}+\cdots\)
3528.2.a.n	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{13}-4q^{17}-4q^{19}-4q^{23}+\cdots\)
3528.2.a.o	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-5q^{11}+2q^{13}-6q^{17}+2q^{19}+\cdots\)
3528.2.a.p	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{11}-6q^{13}+5q^{17}+q^{19}+\cdots\)
3528.2.a.q	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{11}+4q^{13}-4q^{19}-8q^{23}+\cdots\)
3528.2.a.r	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{11}+2q^{13}-3q^{17}+5q^{19}+\cdots\)
3528.2.a.s	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+5q^{11}-2q^{13}-6q^{17}-2q^{19}+\cdots\)
3528.2.a.t	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-6q^{11}+6q^{13}-2q^{17}-4q^{19}+\cdots\)
3528.2.a.u	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{11}-2q^{13}+6q^{17}+4q^{19}+\cdots\)
3528.2.a.v	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-6q^{13}-2q^{17}-4q^{19}+4q^{23}+\cdots\)
3528.2.a.w	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{13}+6q^{17}+4q^{19}+4q^{23}+\cdots\)
3528.2.a.x	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{11}-2q^{13}-6q^{17}-8q^{19}+\cdots\)
3528.2.a.y	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+6q^{11}+3q^{13}+4q^{17}+5q^{19}+\cdots\)
3528.2.a.z	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{13}-4q^{17}+7q^{19}+4q^{23}+\cdots\)
3528.2.a.ba	$3528$	$2$	3528.a	1.a	$1$	$1$	$1$	$28.171$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+3q^{13}-4q^{17}-7q^{19}-4q^{23}+\cdots\)
3528.2.a.bb	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(-2+2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bc	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(2+2\beta )q^{11}+3\beta q^{13}+\cdots\)
3528.2.a.bd	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{11}+(-3+\beta )q^{13}-4q^{17}+\cdots\)
3528.2.a.be	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}-6q^{11}+4\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bf	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{11}-2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bg	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{13}-\beta q^{17}-4q^{23}-3q^{25}+\cdots\)
3528.2.a.bh	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{13}-\beta q^{17}+4q^{23}-3q^{25}+\cdots\)
3528.2.a.bi	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+2q^{11}+2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bj	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+4q^{11}+\beta q^{13}-2\beta q^{17}+\cdots\)
3528.2.a.bk	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{11}+(3-\beta )q^{13}+4q^{17}+\cdots\)
3528.2.a.bl	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-2-2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bm	$3528$	$2$	3528.a	1.a	$1$	$2$	$2$	$28.171$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(2-2\beta )q^{11}+3\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3528))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3528)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\)\(^{\oplus 2}\)