Space of modular forms of level 3900, weight 2, and trivial character

• Label: 3900.2.a
• Level: $3900$
• Weight: $2$
• Character orbit: 3900.a
• Rep. character: $\chi_{3900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $24$
• Sturm bound: $1680$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1680\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3900))\).

	Total	New	Old
Modular forms	876	38	838
Cusp forms	805	38	767
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.

\(2\)	\(3\)	\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(48\)	\(0\)	\(48\)		\(43\)	\(0\)	\(43\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(60\)	\(0\)	\(60\)		\(54\)	\(0\)	\(54\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(59\)	\(0\)	\(59\)		\(53\)	\(0\)	\(53\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(55\)	\(0\)	\(55\)		\(49\)	\(0\)	\(49\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(63\)	\(0\)	\(63\)		\(57\)	\(0\)	\(57\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(51\)	\(0\)	\(51\)		\(45\)	\(0\)	\(45\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(52\)	\(0\)	\(52\)		\(46\)	\(0\)	\(46\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(56\)	\(0\)	\(56\)		\(50\)	\(0\)	\(50\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(51\)	\(3\)	\(48\)		\(48\)	\(3\)	\(45\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(57\)	\(6\)	\(51\)		\(54\)	\(6\)	\(48\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(55\)	\(6\)	\(49\)		\(52\)	\(6\)	\(46\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(53\)	\(4\)	\(49\)		\(50\)	\(4\)	\(46\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(57\)	\(3\)	\(54\)		\(54\)	\(3\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(51\)	\(6\)	\(45\)		\(48\)	\(6\)	\(42\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(53\)	\(6\)	\(47\)		\(50\)	\(6\)	\(44\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(55\)	\(4\)	\(51\)		\(52\)	\(4\)	\(48\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(430\)	\(19\)	\(411\)		\(395\)	\(19\)	\(376\)		\(35\)	\(0\)	\(35\)
Minus space				\(-\)		\(446\)	\(19\)	\(427\)		\(410\)	\(19\)	\(391\)		\(36\)	\(0\)	\(36\)

Trace form

38 * q + 38 * q^9 + 4 * q^11 + 2 * q^13 + 4 * q^17 - 8 * q^19 - 4 * q^21 - 8 * q^23 + 12 * q^29 - 4 * q^33 - 4 * q^37 + 12 * q^41 - 16 * q^43 - 28 * q^47 + 54 * q^49 + 4 * q^53 + 4 * q^57 - 12 * q^59 + 20 * q^61 + 8 * q^67 + 8 * q^69 - 20 * q^71 + 20 * q^73 - 16 * q^79 + 38 * q^81 - 12 * q^83 - 8 * q^87 - 12 * q^89 + 12 * q^93 + 20 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\) 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					A-L signs				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5	13
3900.2.a.a	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				156.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}-q^{13}+6q^{17}+\cdots\)
3900.2.a.b	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓	✓			3900.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-3q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.c	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.h.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-3q^{11}-q^{13}+7q^{17}+\cdots\)
3900.2.a.d	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+3q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.e	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-6q^{11}+q^{13}+\cdots\)
3900.2.a.f	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓	✓			3900.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}+q^{11}+q^{13}+3q^{17}+\cdots\)
3900.2.a.g	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.h.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}+3q^{11}+q^{13}+\cdots\)
3900.2.a.h	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓	✓			3900.2.a.f	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}+q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.i	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}+q^{11}-q^{13}+3q^{17}+\cdots\)
3900.2.a.j	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.h.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
3900.2.a.k	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.h.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-3q^{11}+q^{13}-7q^{17}+\cdots\)
3900.2.a.l	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓	✓			3900.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-3q^{11}+q^{13}+3q^{17}+\cdots\)
3900.2.a.m	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				156.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3900.2.a.n	$3900$	$2$	3900.a	1.a	$1$	$1$	$1$	$31.142$	\(\Q\)	None	✓				780.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
3900.2.a.o	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{6}) \)	None	✓		✓		780.2.h.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+(2-\beta )q^{11}+q^{13}+\cdots\)
3900.2.a.p	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{17}) \)	None	✓				780.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
3900.2.a.q	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{2}) \)	None	✓				780.2.h.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+2\beta )q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
3900.2.a.r	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{2}) \)	None	✓		✓		780.2.h.d	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+2\beta )q^{7}+q^{9}+\beta q^{11}+\cdots\)
3900.2.a.s	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{73}) \)	None	✓				780.2.a.e	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}+\beta q^{11}+q^{13}+\cdots\)
3900.2.a.t	$3900$	$2$	3900.a	1.a	$1$	$2$	$2$	$31.142$	\(\Q(\sqrt{6}) \)	None	✓				780.2.h.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+(2+\beta )q^{11}-q^{13}+\cdots\)
3900.2.a.u	$3900$	$2$	3900.a	1.a	$1$	$3$	$3$	$31.142$	3.3.3028.1	None	✓	✓	✓		3900.2.a.u	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1})q^{7}+q^{9}+(2+\beta _{2})q^{11}+\cdots\)
3900.2.a.v	$3900$	$2$	3900.a	1.a	$1$	$3$	$3$	$31.142$	3.3.1940.1	None	✓	✓	✓		3900.2.a.v	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{1}-\beta _{2})q^{7}+q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
3900.2.a.w	$3900$	$2$	3900.a	1.a	$1$	$3$	$3$	$31.142$	3.3.1940.1	None	✓	✓	✓		3900.2.a.v	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3900.2.a.x	$3900$	$2$	3900.a	1.a	$1$	$3$	$3$	$31.142$	3.3.3028.1	None	✓	✓	✓		3900.2.a.u	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{1})q^{7}+q^{9}+(2+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3900)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\)\(^{\oplus 2}\)