Space of modular forms of level 1445, weight 4, and trivial character

• Label: 1445.4.a
• Level: $1445$
• Weight: $4$
• Character orbit: 1445.a
• Rep. character: $\chi_{1445}(1,\cdot)$
• Character field: $\Q$
• Dimension: $271$
• Newform subspaces: $26$
• Sturm bound: $612$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(612\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1445))\).

	Total	New	Old
Modular forms	478	271	207
Cusp forms	442	271	171
Eisenstein series	36	0	36

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

\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(72\)
\(+\)	\(-\)	$-$	\(64\)
\(-\)	\(+\)	$-$	\(63\)
\(-\)	\(-\)	$+$	\(72\)
Plus space		\(+\)	\(144\)
Minus space		\(-\)	\(127\)

Trace form

\( 271 q - 2 q^{3} + 1100 q^{4} - 5 q^{5} - 60 q^{6} + 6 q^{7} + 48 q^{8} + 2499 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	17
1445.4.a.a	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(-2\)	\(5\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-2q^{3}+8q^{4}+5q^{5}+8q^{6}+\cdots\)
1445.4.a.b	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$0$	\(-3\)	\(-2\)	\(-5\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-2q^{3}+q^{4}-5q^{5}+6q^{6}+\cdots\)
1445.4.a.c	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$0$	\(-3\)	\(2\)	\(5\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+2q^{3}+q^{4}+5q^{5}-6q^{6}+\cdots\)
1445.4.a.d	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$2$	\(-1\)	\(-8\)	\(-5\)	\(-14\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-8q^{3}-7q^{4}-5q^{5}+8q^{6}+\cdots\)
1445.4.a.e	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(8\)	\(5\)	\(14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+8q^{3}-7q^{4}+5q^{5}-8q^{6}+\cdots\)
1445.4.a.f	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(-10\)	\(-5\)	\(22\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-10q^{3}+q^{4}-5q^{5}-30q^{6}+\cdots\)
1445.4.a.g	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$1$	\(3\)	\(5\)	\(5\)	\(22\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+5q^{3}+q^{4}+5q^{5}+15q^{6}+\cdots\)
1445.4.a.h	$1445$	$4$	1445.a	1.a	$1$	$1$	$1$	$85.258$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(7\)	\(-5\)	\(22\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+7q^{3}+q^{4}-5q^{5}+21q^{6}+\cdots\)
1445.4.a.i	$1445$	$4$	1445.a	1.a	$1$	$2$	$2$	$85.258$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-4\)	\(2\)	\(-10\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(1-\beta )q^{3}+(-1-4\beta )q^{4}+\cdots\)
1445.4.a.j	$1445$	$4$	1445.a	1.a	$1$	$3$	$3$	$85.258$	3.3.1304.1	None	✓	$1$	$1$	\(-6\)	\(4\)	\(15\)	\(-8\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
1445.4.a.k	$1445$	$4$	1445.a	1.a	$1$	$3$	$3$	$85.258$	3.3.568.1	None	✓	$1$	$1$	\(3\)	\(-9\)	\(15\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{2}+(-4+3\beta _{1})q^{3}+\cdots\)
1445.4.a.l	$1445$	$4$	1445.a	1.a	$1$	$5$	$5$	$85.258$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(-25\)	\(-10\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+\beta _{2}q^{3}+(6-2\beta _{1}+\beta _{3})q^{4}+\cdots\)
1445.4.a.m	$1445$	$4$	1445.a	1.a	$1$	$7$	$7$	$85.258$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-35\)	\(11\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1445.4.a.n	$1445$	$4$	1445.a	1.a	$1$	$7$	$7$	$85.258$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(35\)	\(-11\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1445.4.a.o	$1445$	$4$	1445.a	1.a	$1$	$8$	$8$	$85.258$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-18\)	\(40\)	\(-38\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2-\beta _{3})q^{3}+(6+\beta _{2})q^{4}+\cdots\)
1445.4.a.p	$1445$	$4$	1445.a	1.a	$1$	$8$	$8$	$85.258$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(18\)	\(-40\)	\(38\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{3})q^{3}+(6+\beta _{2})q^{4}+\cdots\)
1445.4.a.q	$1445$	$4$	1445.a	1.a	$1$	$8$	$8$	$85.258$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(-2\)	\(40\)	\(32\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(6+\beta _{2})q^{4}+\cdots\)
1445.4.a.r	$1445$	$4$	1445.a	1.a	$1$	$8$	$8$	$85.258$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(2\)	\(-40\)	\(-32\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{6}q^{3}+(6+\beta _{2})q^{4}+\cdots\)
1445.4.a.s	$1445$	$4$	1445.a	1.a	$1$	$18$	$18$	$85.258$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-12\)	\(90\)	\(-56\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
1445.4.a.t	$1445$	$4$	1445.a	1.a	$1$	$18$	$18$	$85.258$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(12\)	\(-90\)	\(56\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{7})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
1445.4.a.u	$1445$	$4$	1445.a	1.a	$1$	$21$	$21$	$85.258$		None	✓	$1$	$1$	\(-6\)	\(-18\)	\(105\)	\(-42\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
1445.4.a.v	$1445$	$4$	1445.a	1.a	$1$	$21$	$21$	$85.258$		None	✓	$1$	$1$	\(-6\)	\(18\)	\(-105\)	\(42\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
1445.4.a.w	$1445$	$4$	1445.a	1.a	$1$	$27$	$27$	$85.258$		None	✓	$1$	$0$	\(6\)	\(-18\)	\(-135\)	\(-42\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1445.4.a.x	$1445$	$4$	1445.a	1.a	$1$	$27$	$27$	$85.258$		None	✓	$1$	$0$	\(6\)	\(18\)	\(135\)	\(42\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
1445.4.a.y	$1445$	$4$	1445.a	1.a	$1$	$36$	$36$	$85.258$		None	✓	$1$	$1$	\(8\)	\(-24\)	\(-180\)	\(-112\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
1445.4.a.z	$1445$	$4$	1445.a	1.a	$1$	$36$	$36$	$85.258$		None	✓	$1$	$0$	\(8\)	\(24\)	\(180\)	\(112\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1445))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1445)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 2}\)