Space of modular forms of level 3300, weight 2, and trivial character

• Label: 3300.2.a
• Level: $3300$
• Weight: $2$
• Character orbit: 3300.a
• Rep. character: $\chi_{3300}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $24$
• Sturm bound: $1440$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1440\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3300))\).

	Total	New	Old
Modular forms	756	30	726
Cusp forms	685	30	655
Eisenstein series	71	0	71

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

\(2\)	\(3\)	\(5\)	\(11\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(2\)
Plus space				\(+\)	\(10\)
Minus space				\(-\)	\(20\)

Trace form

\( 30 q + 30 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3300))\) 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	5	11
3300.2.a.a	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
3300.2.a.b	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3300.2.a.c	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+q^{11}+4q^{13}-8q^{19}+\cdots\)
3300.2.a.d	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-q^{11}-q^{13}+2q^{17}+\cdots\)
3300.2.a.e	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-q^{11}+4q^{13}-3q^{17}+\cdots\)
3300.2.a.f	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
3300.2.a.g	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
3300.2.a.h	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
3300.2.a.i	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
3300.2.a.j	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
3300.2.a.k	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
3300.2.a.l	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3300.2.a.m	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-q^{11}-4q^{13}+3q^{17}+\cdots\)
3300.2.a.n	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-q^{11}+q^{13}-2q^{17}+\cdots\)
3300.2.a.o	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+q^{11}-4q^{13}-8q^{19}+\cdots\)
3300.2.a.p	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+q^{11}+4q^{13}+2q^{17}+\cdots\)
3300.2.a.q	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
3300.2.a.r	$3300$	$2$	3300.a	1.a	$1$	$1$	$1$	$26.351$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
3300.2.a.s	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}+q^{11}+(-1+\beta )q^{13}+\cdots\)
3300.2.a.t	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{7}+q^{9}-q^{11}+(-3+\cdots)q^{13}+\cdots\)
3300.2.a.u	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{7}+q^{9}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
3300.2.a.v	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{7}+q^{9}-q^{11}+(1+\cdots)q^{13}+\cdots\)
3300.2.a.w	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{7}+q^{9}+q^{11}+(1+\cdots)q^{13}+\cdots\)
3300.2.a.x	$3300$	$2$	3300.a	1.a	$1$	$2$	$2$	$26.351$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}+q^{11}+\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1650))\)\(^{\oplus 2}\)