Space of modular forms of level 725, weight 2, and trivial character

• Label: 725.2.a
• Level: $725$
• Weight: $2$
• Character orbit: 725.a
• Rep. character: $\chi_{725}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $12$
• Sturm bound: $150$
• Trace bound: $4$

Defining parameters

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.a (trivial)
Character field:			$\Q$
Newform subspaces:			$12$
Sturm bound:			$150$
Trace bound:			$4$
Distinguishing $T_p$:			$2$, $3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(725))$.

	Total	New	Old
Modular forms	80	45	35
Cusp forms	69	45	24
Eisenstein series	11	0	11

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
$+$	$+$	$+$	$9$
$+$	$-$	$-$	$12$
$-$	$+$	$-$	$15$
$-$	$-$	$+$	$9$
Plus space		$+$	$18$
Minus space		$-$	$27$

Trace form

45 * q + q^2 + 2 * q^3 + 43 * q^4 - 2 * q^6 + 4 * q^7 - 3 * q^8 + 51 * q^9 - 6 * q^11 + 6 * q^12 + 12 * q^13 - 12 * q^14 + 23 * q^16 + 10 * q^17 + 15 * q^18 + 4 * q^19 + 20 * q^21 - 14 * q^22 - 16 * q^23 - 6 * q^24 - 36 * q^26 - 10 * q^27 + 12 * q^28 - 3 * q^29 + 2 * q^31 - 15 * q^32 + 10 * q^33 - 6 * q^34 + 29 * q^36 + 2 * q^37 + 24 * q^38 - 18 * q^39 - 6 * q^41 - 24 * q^42 + 2 * q^43 - 30 * q^44 - 52 * q^46 - 10 * q^47 + 6 * q^48 + 45 * q^49 + 16 * q^51 + 24 * q^52 - 16 * q^53 - 50 * q^54 - 48 * q^56 - 4 * q^57 + q^58 - 4 * q^59 - 10 * q^61 - 14 * q^62 + 4 * q^63 - q^64 + 6 * q^66 - 16 * q^67 + 26 * q^68 - 20 * q^69 - 8 * q^71 + 39 * q^72 + 18 * q^73 - 2 * q^74 - 28 * q^76 - 12 * q^77 + 2 * q^78 + 18 * q^79 + 105 * q^81 - 42 * q^82 - 20 * q^83 + 60 * q^84 + 6 * q^86 + 6 * q^87 - 50 * q^88 - 2 * q^89 + 20 * q^91 - 24 * q^92 - 26 * q^93 + 26 * q^94 - 30 * q^96 + 42 * q^97 - 67 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(725))$ 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
725.2.a.a	$725$	$2$	725.a	1.a	$1$	$1$	$1$	$5.789$	$\Q$	None	✓				145.2.a.a	$1$	$1$	$1$	$0$	$0$	$2$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+2q^{7}-3q^{8}-3q^{9}-6q^{11}+\cdots$
725.2.a.b	$725$	$2$	725.a	1.a	$1$	$2$	$2$	$5.789$	$\Q(\sqrt{2})$	None	✓				29.2.a.a	$1$	$0$	$2$	$-2$	$0$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(-1-\beta )q^{3}+(1+2\beta )q^{4}+\cdots$
725.2.a.c	$725$	$2$	725.a	1.a	$1$	$2$	$2$	$5.789$	$\Q(\sqrt{2})$	None	✓				145.2.a.b	$1$	$0$	$2$	$4$	$0$	$4$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+2q^{3}+(1+2\beta )q^{4}+(2+\cdots)q^{6}+\cdots$
725.2.a.d	$725$	$2$	725.a	1.a	$1$	$3$	$3$	$5.789$	3.3.148.1	None	✓				145.2.a.d	$1$	$0$	$-3$	$2$	$0$	$2$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+\cdots$
725.2.a.e	$725$	$2$	725.a	1.a	$1$	$3$	$3$	$5.789$	3.3.148.1	None	✓				145.2.a.c	$1$	$1$	$-1$	$-2$	$0$	$-4$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots$
725.2.a.f	$725$	$2$	725.a	1.a	$1$	$4$	$4$	$5.789$	$\Q(\zeta_{24})^+$	None	✓				145.2.b.b	$2$	$1$	$0$	$0$	$0$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+\beta _{2}q^{4}+\cdots$
725.2.a.g	$725$	$2$	725.a	1.a	$1$	$4$	$4$	$5.789$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓				145.2.b.a	$2$	$0$	$0$	$0$	$0$	$0$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+\beta _{1}q^{3}+q^{4}+(-2-\beta _{3})q^{6}+\cdots$
725.2.a.h	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.240881.1	None	✓	✓	✓		725.2.a.h	$1$	$1$	$-2$	$-6$	$0$	$-6$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots$
725.2.a.i	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.294577.1	None	✓	✓	✓		725.2.a.i	$1$	$1$	$0$	$-6$	$0$	$-10$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$
725.2.a.j	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.294577.1	None	✓	✓			725.2.a.i	$1$	$0$	$0$	$6$	$0$	$10$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$
725.2.a.k	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.240881.1	None	✓	✓	✓		725.2.a.h	$1$	$0$	$2$	$6$	$0$	$6$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots$
725.2.a.l	$725$	$2$	725.a	1.a	$1$	$6$	$6$	$5.789$	6.6.337383424.1	None	✓		✓		145.2.b.c	$2$	$0$	$0$	$0$	$0$	$0$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(\beta _{1}-\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(725))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(725)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(145))$$^{\oplus 2}$