Space of modular forms of level 8100, weight 2, and trivial character

• Label: 8100.2.a
• Level: $8100$
• Weight: $2$
• Character orbit: 8100.a
• Rep. character: $\chi_{8100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $31$
• Sturm bound: $3240$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(3240\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8100))\).

	Total	New	Old
Modular forms	1728	76	1652
Cusp forms	1513	76	1437
Eisenstein series	215	0	215

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(19\)
\(-\)	\(+\)	\(-\)	$+$	\(18\)
\(-\)	\(-\)	\(+\)	$+$	\(17\)
\(-\)	\(-\)	\(-\)	$-$	\(22\)
Plus space			\(+\)	\(35\)
Minus space			\(-\)	\(41\)

Trace form

\( 76 q + 2 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8100))\) 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
8100.2.a.a	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-6q^{11}-5q^{13}+3q^{17}+2q^{19}+\cdots\)
8100.2.a.b	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{11}+4q^{13}+6q^{17}-7q^{19}+\cdots\)
8100.2.a.c	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-2q^{17}-3q^{19}+4q^{23}+\cdots\)
8100.2.a.d	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}+2q^{17}-3q^{19}-4q^{23}+\cdots\)
8100.2.a.e	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+3q^{11}+4q^{13}-6q^{17}-7q^{19}+\cdots\)
8100.2.a.f	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+6q^{11}-5q^{13}-3q^{17}+2q^{19}+\cdots\)
8100.2.a.g	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{11}+q^{13}+6q^{17}-4q^{19}+\cdots\)
8100.2.a.h	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{13}-6q^{17}+2q^{19}-3q^{23}+\cdots\)
8100.2.a.i	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{13}+6q^{17}+2q^{19}+3q^{23}+\cdots\)
8100.2.a.j	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+3q^{11}+q^{13}-6q^{17}-4q^{19}+\cdots\)
8100.2.a.k	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}+2q^{17}-3q^{19}-4q^{23}+\cdots\)
8100.2.a.l	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{11}-2q^{17}-3q^{19}+4q^{23}+\cdots\)
8100.2.a.m	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{11}+4q^{13}+5q^{19}-6q^{23}+\cdots\)
8100.2.a.n	$8100$	$2$	8100.a	1.a	$1$	$1$	$1$	$64.679$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+3q^{11}+4q^{13}+5q^{19}+6q^{23}+\cdots\)
8100.2.a.o	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+(-2-\beta )q^{11}+(3+\beta )q^{13}+\cdots\)
8100.2.a.p	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+(2+\beta )q^{11}+(3+\beta )q^{13}+\cdots\)
8100.2.a.q	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+(-2-\beta )q^{11}+(-3-\beta )q^{13}+\cdots\)
8100.2.a.r	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+(2+\beta )q^{11}+(-3-\beta )q^{13}+\cdots\)
8100.2.a.s	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}-\beta q^{11}+(-2-2\beta )q^{13}+\cdots\)
8100.2.a.t	$8100$	$2$	8100.a	1.a	$1$	$2$	$2$	$64.679$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+\beta q^{11}+(-2-2\beta )q^{13}+\cdots\)
8100.2.a.u	$8100$	$2$	8100.a	1.a	$1$	$3$	$3$	$64.679$	3.3.564.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{7}+\beta _{2}q^{11}+(-2-\beta _{2})q^{13}+\cdots\)
8100.2.a.v	$8100$	$2$	8100.a	1.a	$1$	$3$	$3$	$64.679$	3.3.564.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{7}-\beta _{2}q^{11}+(-2-\beta _{2})q^{13}+\cdots\)
8100.2.a.w	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{7}-\beta _{3}q^{11}+\beta _{2}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
8100.2.a.x	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	4.4.3981.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{7}+(-1-\beta _{2})q^{11}-\beta _{1}q^{13}+\cdots\)
8100.2.a.y	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	4.4.3981.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{7}+(1+\beta _{2})q^{11}-\beta _{1}q^{13}+(2+\cdots)q^{17}+\cdots\)
8100.2.a.z	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	4.4.3981.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{7}+(-1-\beta _{2})q^{11}+\beta _{1}q^{13}+\cdots\)
8100.2.a.ba	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	4.4.3981.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{7}+(1+\beta _{2})q^{11}+\beta _{1}q^{13}+(-2+\cdots)q^{17}+\cdots\)
8100.2.a.bb	$8100$	$2$	8100.a	1.a	$1$	$4$	$4$	$64.679$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{7}+\beta _{3}q^{11}-\beta _{2}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
8100.2.a.bc	$8100$	$2$	8100.a	1.a	$1$	$6$	$6$	$64.679$	6.6.1207701504.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{7}+\beta _{2}q^{11}+(-\beta _{1}+\beta _{4})q^{13}+\cdots\)
8100.2.a.bd	$8100$	$2$	8100.a	1.a	$1$	$6$	$6$	$64.679$	6.6.1207701504.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{7}-\beta _{2}q^{11}+(-\beta _{1}+\beta _{4})q^{13}+\cdots\)
8100.2.a.be	$8100$	$2$	8100.a	1.a	$1$	$8$	$8$	$64.679$	8.8.\(\cdots\).1	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{7}+(-\beta _{2}+\beta _{7})q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8100))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8100)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2700))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\)\(^{\oplus 2}\)