Space of modular forms of level 2075, weight 4, and trivial character

• Label: 2075.4.a
• Level: $2075$
• Weight: $4$
• Character orbit: 2075.a
• Rep. character: $\chi_{2075}(1,\cdot)$
• Character field: $\Q$
• Dimension: $390$
• Newform subspaces: $14$
• Sturm bound: $840$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 2075 = 5^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2075.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(840\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2075))\).

	Total	New	Old
Modular forms	636	390	246
Cusp forms	624	390	234
Eisenstein series	12	0	12

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

\(5\)	\(83\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(101\)
\(+\)	\(-\)	\(-\)	\(83\)
\(-\)	\(+\)	\(-\)	\(97\)
\(-\)	\(-\)	\(+\)	\(109\)
Plus space		\(+\)	\(210\)
Minus space		\(-\)	\(180\)

Trace form

\( 390 q - 4 q^{2} + 4 q^{3} + 1574 q^{4} - 30 q^{6} + 6 q^{7} + 42 q^{8} + 3496 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	83
2075.4.a.a	$2075$	$4$	2075.a	1.a	$1$	$1$	$1$	$122.429$	\(\Q\)	None	✓	✓			2075.4.a.a	$1$	$1$	\(-4\)	\(-7\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-7q^{3}+8q^{4}+28q^{6}+22q^{9}+\cdots\)
2075.4.a.b	$2075$	$4$	2075.a	1.a	$1$	$1$	$1$	$122.429$	\(\Q\)	None	✓	✓			2075.4.a.a	$1$	$1$	\(4\)	\(7\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+7q^{3}+8q^{4}+28q^{6}+22q^{9}+\cdots\)
2075.4.a.c	$2075$	$4$	2075.a	1.a	$1$	$7$	$7$	$122.429$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				83.4.a.a	$1$	$0$	\(5\)	\(11\)	\(0\)	\(46\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{4})q^{2}+(1-\beta _{1}-\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots\)
2075.4.a.d	$2075$	$4$	2075.a	1.a	$1$	$13$	$13$	$122.429$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				83.4.a.b	$1$	$1$	\(-5\)	\(-7\)	\(0\)	\(-52\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{8})q^{3}+(6+\beta _{2})q^{4}+\cdots\)
2075.4.a.e	$2075$	$4$	2075.a	1.a	$1$	$15$	$15$	$122.429$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				415.4.a.a	$1$	$1$	\(3\)	\(12\)	\(0\)	\(37\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(2+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2075.4.a.f	$2075$	$4$	2075.a	1.a	$1$	$20$	$20$	$122.429$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				415.4.a.b	$1$	$1$	\(-5\)	\(-12\)	\(0\)	\(-31\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{5})q^{3}+(3+\beta _{2})q^{4}+\cdots\)
2075.4.a.g	$2075$	$4$	2075.a	1.a	$1$	$21$	$21$	$122.429$		None	✓				415.4.a.c	$1$	$0$	\(5\)	\(12\)	\(0\)	\(11\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
2075.4.a.h	$2075$	$4$	2075.a	1.a	$1$	$26$	$26$	$122.429$		None	✓				415.4.a.d	$1$	$0$	\(-7\)	\(-12\)	\(0\)	\(-5\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
2075.4.a.i	$2075$	$4$	2075.a	1.a	$1$	$34$	$34$	$122.429$		None	✓	✓	✓		2075.4.a.i	$1$	$1$	\(-3\)	\(-13\)	\(0\)	\(-6\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
2075.4.a.j	$2075$	$4$	2075.a	1.a	$1$	$34$	$34$	$122.429$		None	✓	✓			2075.4.a.i	$1$	$1$	\(3\)	\(13\)	\(0\)	\(6\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
2075.4.a.k	$2075$	$4$	2075.a	1.a	$1$	$47$	$47$	$122.429$		None	✓	✓			2075.4.a.k	$1$	$0$	\(-1\)	\(6\)	\(0\)	\(6\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
2075.4.a.l	$2075$	$4$	2075.a	1.a	$1$	$47$	$47$	$122.429$		None	✓	✓	✓		2075.4.a.k	$1$	$0$	\(1\)	\(-6\)	\(0\)	\(-6\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
2075.4.a.m	$2075$	$4$	2075.a	1.a	$1$	$62$	$62$	$122.429$		None	✓		✓		415.4.b.a	$1$	$1$	\(-10\)	\(-18\)	\(0\)	\(-98\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
2075.4.a.n	$2075$	$4$	2075.a	1.a	$1$	$62$	$62$	$122.429$		None	✓		✓		415.4.b.a	$1$	$0$	\(10\)	\(18\)	\(0\)	\(98\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2075))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2075)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(83))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(415))\)\(^{\oplus 2}\)