Space of modular forms of level 1900, weight 2, and trivial character

• Label: 1900.2.a
• Level: $1900$
• Weight: $2$
• Character orbit: 1900.a
• Rep. character: $\chi_{1900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $11$
• Sturm bound: $600$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(600\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1900))\).

	Total	New	Old
Modular forms	318	29	289
Cusp forms	283	29	254
Eisenstein series	35	0	35

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

\(2\)	\(5\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(36\)	\(0\)	\(36\)		\(31\)	\(0\)	\(31\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(45\)	\(0\)	\(45\)		\(39\)	\(0\)	\(39\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(42\)	\(0\)	\(42\)		\(36\)	\(0\)	\(36\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(39\)	\(0\)	\(39\)		\(33\)	\(0\)	\(33\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(39\)	\(7\)	\(32\)		\(36\)	\(7\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(39\)	\(6\)	\(33\)		\(36\)	\(6\)	\(30\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(39\)	\(7\)	\(32\)		\(36\)	\(7\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(39\)	\(9\)	\(30\)		\(36\)	\(9\)	\(27\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(153\)	\(13\)	\(140\)		\(136\)	\(13\)	\(123\)		\(17\)	\(0\)	\(17\)
Minus space			\(-\)		\(165\)	\(16\)	\(149\)		\(147\)	\(16\)	\(131\)		\(18\)	\(0\)	\(18\)

Trace form

29 * q - 2 * q^3 + 3 * q^7 + 29 * q^9 - q^11 + 8 * q^13 - q^17 + q^19 + 2 * q^21 + 8 * q^23 + 16 * q^27 - 18 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 4 * q^39 - 10 * q^41 - 9 * q^43 - 7 * q^47 + 36 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^57 - 14 * q^59 - 3 * q^61 - 29 * q^63 + 8 * q^67 + 16 * q^69 + 22 * q^71 + 3 * q^73 - q^77 - 8 * q^79 + 41 * q^81 + 4 * q^83 - 20 * q^87 + 20 * q^89 - 32 * q^91 - 32 * q^93 + 24 * q^97 + 51 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\) 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	5	19
1900.2.a.a	$1900$	$2$	1900.a	1.a	$1$	$1$	$1$	$15.172$	\(\Q\)	None	✓				380.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}-6q^{13}-2q^{17}+\cdots\)
1900.2.a.b	$1900$	$2$	1900.a	1.a	$1$	$1$	$1$	$15.172$	\(\Q\)	None	✓				76.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{7}+q^{9}+5q^{11}+4q^{13}+\cdots\)
1900.2.a.c	$1900$	$2$	1900.a	1.a	$1$	$1$	$1$	$15.172$	\(\Q\)	None	✓				380.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}-4q^{11}+4q^{13}-6q^{17}+\cdots\)
1900.2.a.d	$1900$	$2$	1900.a	1.a	$1$	$2$	$2$	$15.172$	\(\Q(\sqrt{3}) \)	None	✓				380.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2q^{7}+(1-2\beta )q^{9}+\cdots\)
1900.2.a.e	$1900$	$2$	1900.a	1.a	$1$	$2$	$2$	$15.172$	\(\Q(\sqrt{2}) \)	None	✓				380.2.a.c	$1$	$0$	\(0\)	\(4\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(2-2\beta )q^{7}+(3+4\beta )q^{9}+\cdots\)
1900.2.a.f	$1900$	$2$	1900.a	1.a	$1$	$3$	$3$	$15.172$	3.3.257.1	None	✓	✓			1900.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1})q^{7}+(1+\cdots)q^{9}+\cdots\)
1900.2.a.g	$1900$	$2$	1900.a	1.a	$1$	$3$	$3$	$15.172$	3.3.321.1	None	✓	✓	✓		1900.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
1900.2.a.h	$1900$	$2$	1900.a	1.a	$1$	$3$	$3$	$15.172$	3.3.257.1	None	✓	✓	✓		1900.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1})q^{7}+(1-2\beta _{1}+\cdots)q^{9}+\cdots\)
1900.2.a.i	$1900$	$2$	1900.a	1.a	$1$	$3$	$3$	$15.172$	3.3.321.1	None	✓	✓			1900.2.a.g	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-2\beta _{1}-\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
1900.2.a.j	$1900$	$2$	1900.a	1.a	$1$	$4$	$4$	$15.172$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓		✓		380.2.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
1900.2.a.k	$1900$	$2$	1900.a	1.a	$1$	$6$	$6$	$15.172$	6.6.56310016.1	None	✓		✓		380.2.c.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-\beta _{4}q^{7}+(2-\beta _{1}+\beta _{5})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1900)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 2}\)