Space of modular forms of level 725, weight 6, and trivial character

• Label: 725.6.a
• Level: $725$
• Weight: $6$
• Character orbit: 725.a
• Rep. character: $\chi_{725}(1,\cdot)$
• Character field: $\Q$
• Dimension: $221$
• Newform subspaces: $12$
• Sturm bound: $450$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(450\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(725))\).

	Total	New	Old
Modular forms	380	221	159
Cusp forms	368	221	147
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.

\(5\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(51\)
\(+\)	\(-\)	\(-\)	\(54\)
\(-\)	\(+\)	\(-\)	\(61\)
\(-\)	\(-\)	\(+\)	\(55\)
Plus space		\(+\)	\(106\)
Minus space		\(-\)	\(115\)

Trace form

221 * q + 4 * q^2 - 6 * q^3 + 3484 * q^4 + 188 * q^6 + 212 * q^7 - 54 * q^8 + 17611 * q^9 + 290 * q^11 + 1562 * q^12 - 52 * q^13 - 780 * q^14 + 56492 * q^16 - 1454 * q^17 + 2154 * q^18 + 8108 * q^19 - 6916 * q^21 + 1812 * q^22 + 3920 * q^23 - 12612 * q^24 + 1706 * q^26 + 3078 * q^27 + 8160 * q^28 - 2523 * q^29 - 4542 * q^31 + 14646 * q^32 - 8126 * q^33 - 39452 * q^34 + 325424 * q^36 + 40410 * q^37 + 12152 * q^38 - 9290 * q^39 - 47590 * q^41 + 10388 * q^42 - 37654 * q^43 + 32138 * q^44 + 7544 * q^46 + 6710 * q^47 - 55270 * q^48 + 565293 * q^49 + 123632 * q^51 - 40220 * q^52 + 53344 * q^53 + 104852 * q^54 - 175824 * q^56 - 38092 * q^57 + 3364 * q^58 - 114708 * q^59 - 7482 * q^61 - 2020 * q^62 - 71708 * q^63 + 927772 * q^64 + 29302 * q^66 + 32832 * q^67 + 140032 * q^68 + 420204 * q^69 + 158968 * q^71 + 272868 * q^72 + 41618 * q^73 + 136568 * q^74 + 288148 * q^76 + 116156 * q^77 - 138360 * q^78 + 38562 * q^79 + 1567393 * q^81 + 281848 * q^82 - 113748 * q^83 + 134540 * q^84 - 499548 * q^86 + 45414 * q^87 - 54068 * q^88 - 124458 * q^89 + 175700 * q^91 - 462236 * q^92 - 49474 * q^93 + 406852 * q^94 - 160020 * q^96 - 73910 * q^97 + 350896 * q^98 - 649576 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(725))\) 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	29
725.6.a.a	$725$	$6$	725.a	1.a	$1$	$4$	$4$	$116.278$	4.4.3257317.1	None	✓				29.6.a.a	$1$	$1$	\(0\)	\(28\)	\(0\)	\(208\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(7-\beta _{1}-\beta _{2})q^{3}+(2-3\beta _{1}+\cdots)q^{4}+\cdots\)
725.6.a.b	$725$	$6$	725.a	1.a	$1$	$7$	$7$	$116.278$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				29.6.a.b	$1$	$0$	\(-4\)	\(-26\)	\(0\)	\(-184\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-4-\beta _{5})q^{3}+(23+\cdots)q^{4}+\cdots\)
725.6.a.c	$725$	$6$	725.a	1.a	$1$	$11$	$11$	$116.278$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				145.6.a.b	$1$	$1$	\(9\)	\(-20\)	\(0\)	\(18\)		$+$	$+$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(11+\beta _{2}+\cdots)q^{4}+\cdots\)
725.6.a.d	$725$	$6$	725.a	1.a	$1$	$11$	$11$	$116.278$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				145.6.a.a	$1$	$0$	\(15\)	\(52\)	\(0\)	\(468\)		$+$	$-$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(5-\beta _{4})q^{3}+(17+3\beta _{1}+\cdots)q^{4}+\cdots\)
725.6.a.e	$725$	$6$	725.a	1.a	$1$	$13$	$13$	$116.278$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				145.6.a.d	$1$	$1$	\(-9\)	\(-38\)	\(0\)	\(-316\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-3+\beta _{5})q^{3}+(23+\cdots)q^{4}+\cdots\)
725.6.a.f	$725$	$6$	725.a	1.a	$1$	$13$	$13$	$116.278$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				145.6.a.c	$1$	$0$	\(-7\)	\(-2\)	\(0\)	\(18\)		$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{4}q^{3}+(2^{4}+\beta _{2}+\cdots)q^{4}+\cdots\)
725.6.a.g	$725$	$6$	725.a	1.a	$1$	$23$	$23$	$116.278$		None	✓	✓			725.6.a.g	$1$	$1$	\(-8\)	\(-74\)	\(0\)	\(-494\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
725.6.a.h	$725$	$6$	725.a	1.a	$1$	$23$	$23$	$116.278$		None	✓	✓	✓		725.6.a.h	$1$	$1$	\(0\)	\(-34\)	\(0\)	\(-290\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
725.6.a.i	$725$	$6$	725.a	1.a	$1$	$23$	$23$	$116.278$		None	✓	✓			725.6.a.h	$1$	$0$	\(0\)	\(34\)	\(0\)	\(290\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
725.6.a.j	$725$	$6$	725.a	1.a	$1$	$23$	$23$	$116.278$		None	✓	✓	✓		725.6.a.g	$1$	$0$	\(8\)	\(74\)	\(0\)	\(494\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
725.6.a.k	$725$	$6$	725.a	1.a	$1$	$32$	$32$	$116.278$		None	✓		✓		145.6.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
725.6.a.l	$725$	$6$	725.a	1.a	$1$	$38$	$38$	$116.278$		None	✓		✓		145.6.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(725))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(725)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 2}\)