Space of modular forms of level 250, weight 4, and trivial character

• Label: 250.4.a
• Level: $250$
• Weight: $4$
• Character orbit: 250.a
• Rep. character: $\chi_{250}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $8$
• Sturm bound: $150$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 250 = 2 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	250.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(150\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(250))\).

	Total	New	Old
Modular forms	123	24	99
Cusp forms	103	24	79
Eisenstein series	20	0	20

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

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(33\)	\(6\)	\(27\)		\(28\)	\(6\)	\(22\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(28\)	\(6\)	\(22\)		\(23\)	\(6\)	\(17\)		\(5\)	\(0\)	\(5\)
\(-\)	\(+\)	\(-\)		\(31\)	\(4\)	\(27\)		\(26\)	\(4\)	\(22\)		\(5\)	\(0\)	\(5\)
\(-\)	\(-\)	\(+\)		\(31\)	\(8\)	\(23\)		\(26\)	\(8\)	\(18\)		\(5\)	\(0\)	\(5\)
Plus space		\(+\)		\(64\)	\(14\)	\(50\)		\(54\)	\(14\)	\(40\)		\(10\)	\(0\)	\(10\)
Minus space		\(-\)		\(59\)	\(10\)	\(49\)		\(49\)	\(10\)	\(39\)		\(10\)	\(0\)	\(10\)

Trace form

24 * q + 96 * q^4 - 4 * q^6 + 198 * q^9 - 22 * q^11 + 48 * q^14 + 384 * q^16 + 610 * q^19 + 548 * q^21 - 16 * q^24 + 76 * q^26 + 110 * q^29 - 772 * q^31 - 72 * q^34 + 792 * q^36 - 1424 * q^39 - 92 * q^41 - 88 * q^44 + 16 * q^46 - 1608 * q^49 + 2428 * q^51 + 2000 * q^54 + 192 * q^56 + 1570 * q^59 - 3742 * q^61 + 1536 * q^64 - 88 * q^66 + 716 * q^69 + 328 * q^71 - 172 * q^74 + 2440 * q^76 + 1720 * q^79 + 2704 * q^81 + 2192 * q^84 - 2724 * q^86 - 760 * q^89 - 6412 * q^91 + 248 * q^94 - 64 * q^96 - 5394 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(250))\) 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
250.4.a.a	$250$	$4$	250.a	1.a	$1$	$2$	$2$	$14.750$	\(\Q(\sqrt{5}) \)	None	✓	✓			250.4.a.a	$1$	$1$	\(-4\)	\(1\)	\(0\)	\(28\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}+(15+\cdots)q^{7}+\cdots\)
250.4.a.b	$250$	$4$	250.a	1.a	$1$	$2$	$2$	$14.750$	\(\Q(\sqrt{5}) \)	None	✓	✓			250.4.a.b	$1$	$0$	\(-4\)	\(11\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(4+3\beta )q^{3}+4q^{4}+(-8-6\beta )q^{6}+\cdots\)
250.4.a.c	$250$	$4$	250.a	1.a	$1$	$2$	$2$	$14.750$	\(\Q(\sqrt{5}) \)	None	✓	✓			250.4.a.b	$1$	$1$	\(4\)	\(-11\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-4-3\beta )q^{3}+4q^{4}+(-8+\cdots)q^{6}+\cdots\)
250.4.a.d	$250$	$4$	250.a	1.a	$1$	$2$	$2$	$14.750$	\(\Q(\sqrt{5}) \)	None	✓	✓			250.4.a.a	$1$	$1$	\(4\)	\(-1\)	\(0\)	\(-28\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-\beta q^{3}+4q^{4}-2\beta q^{6}+(-15+\cdots)q^{7}+\cdots\)
250.4.a.e	$250$	$4$	250.a	1.a	$1$	$4$	$4$	$14.750$	\(\Q(\sqrt{30 +2 \sqrt{5}})\)	None	✓	✓	✓		250.4.a.e	$1$	$1$	\(-8\)	\(-8\)	\(0\)	\(-34\)		$+$	$-$	$-$	$5^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2-\beta _{2})q^{3}+4q^{4}+(4+2\beta _{2}+\cdots)q^{6}+\cdots\)
250.4.a.f	$250$	$4$	250.a	1.a	$1$	$4$	$4$	$14.750$	\(\Q(\sqrt{110 -6 \sqrt{5}})\)	None	✓	✓	✓		250.4.a.f	$1$	$0$	\(-8\)	\(-3\)	\(0\)	\(-9\)		$+$	$+$	$+$	$5^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1-\beta _{3})q^{3}+4q^{4}+(2+2\beta _{3})q^{6}+\cdots\)
250.4.a.g	$250$	$4$	250.a	1.a	$1$	$4$	$4$	$14.750$	\(\Q(\sqrt{110 -6 \sqrt{5}})\)	None	✓	✓			250.4.a.f	$1$	$0$	\(8\)	\(3\)	\(0\)	\(9\)		$-$	$-$	$+$	$5^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1+\beta _{3})q^{3}+4q^{4}+(2+2\beta _{3})q^{6}+\cdots\)
250.4.a.h	$250$	$4$	250.a	1.a	$1$	$4$	$4$	$14.750$	\(\Q(\sqrt{30 +2 \sqrt{5}})\)	None	✓	✓			250.4.a.e	$1$	$0$	\(8\)	\(8\)	\(0\)	\(34\)		$-$	$-$	$+$	$5^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2-\beta _{2})q^{3}+4q^{4}+(4-2\beta _{2}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(250))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(250)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 2}\)