Space of modular forms of level 625, weight 2, and Character 625.d

• Label: 625.2.d
• Level: $625$
• Weight: $2$
• Character orbit: 625.d
• Rep. character: $\chi_{625}(126,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $136$
• Newform subspaces: $17$
• Sturm bound: $125$
• Trace bound: $6$

Defining parameters

Level:	$N$	$=$	$625 = 5^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	625.d (of order $5$ and degree $4$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$25$
Character field:			$\Q(\zeta_{5})$
Newform subspaces:			$17$
Sturm bound:			$125$
Trace bound:			$6$
Distinguishing $T_p$:			$2$, $3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(625, [\chi])$.

	Total	New	Old
Modular forms	308	184	124
Cusp forms	188	136	52
Eisenstein series	120	48	72

Trace form

136 * q - 28 * q^4 + 12 * q^6 - 22 * q^9 + 12 * q^11 + 24 * q^14 - 4 * q^16 + 12 * q^21 - 40 * q^24 - 68 * q^26 - 20 * q^29 + 12 * q^31 + 14 * q^34 - 24 * q^36 - 24 * q^39 - 8 * q^41 - 16 * q^44 + 12 * q^46 - 8 * q^49 - 48 * q^51 + 60 * q^54 - 20 * q^56 - 8 * q^61 + 32 * q^64 - 36 * q^66 + 36 * q^69 - 28 * q^71 - 36 * q^74 - 80 * q^76 - 14 * q^81 + 104 * q^84 + 12 * q^86 - 10 * q^89 + 12 * q^91 - 106 * q^94 - 78 * q^96 - 64 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(625, [\chi])$ 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
625.2.d.a	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					125.2.a.a	$2$	$0$	$-3$	$-4$	$0$	$12$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-1+\zeta_{10}^{3})q^{2}+(-1+\zeta_{10}-\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.b	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					25.2.d.a	$2$	$0$	$-3$	$1$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-1+\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$
625.2.d.c	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None		✓			625.2.a.a	$2$	$0$	$-3$	$1$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-1+\zeta_{10}^{3})q^{2}+(1-\zeta_{10}-2\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.d	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					125.2.a.a	$2$	$1$	$-2$	$-1$	$0$	$-12$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\cdots)q^{3}+\cdots$
625.2.d.e	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None		✓			625.2.a.a	$2$	$0$	$-2$	$4$	$0$	$-2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+(1-\zeta_{10}+\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.f	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None		✓			625.2.a.a	$2$	$0$	$2$	$-4$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}-\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.g	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					125.2.a.a	$2$	$0$	$2$	$1$	$0$	$12$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(1-\zeta_{10}-2\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.h	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					25.2.d.a	$2$	$0$	$3$	$-1$	$0$	$-2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(1-\zeta_{10}^{3})q^{2}-\zeta_{10}^{3}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$
625.2.d.i	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None		✓			625.2.a.a	$2$	$0$	$3$	$-1$	$0$	$-2$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(1-\zeta_{10}^{3})q^{2}+(-1+\zeta_{10}+2\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.j	$625$	$2$	625.d	25.d	$5$	$4$	$1$	$4.991$	$\Q(\zeta_{10})$	None					125.2.a.a	$2$	$0$	$3$	$4$	$0$	$-12$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(1-\zeta_{10}^{3})q^{2}+(1-\zeta_{10}+\zeta_{10}^{3})q^{3}+\cdots$
625.2.d.k	$625$	$2$	625.d	25.d	$5$	$8$	$2$	$4.991$	8.0.484000000.9	None					125.2.a.c	$4$	$0$	$0$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(\beta _{1}-\beta _{6})q^{2}+(-\beta _{1}+\beta _{6}+\beta _{7})q^{3}+\cdots$
625.2.d.l	$625$	$2$	625.d	25.d	$5$	$8$	$2$	$4.991$	8.0.484000000.9	None					125.2.a.c	$4$	$0$	$0$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-\beta _{1}-\beta _{4})q^{2}-\beta _{1}q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots$
625.2.d.m	$625$	$2$	625.d	25.d	$5$	$16$	$4$	$4.991$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓			625.2.a.e	$2$	$0$	$-5$	$0$	$0$	$20$	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	$q+(\beta _{1}+\beta _{3}+\beta _{4}-\beta _{6})q^{2}+(\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots$
625.2.d.n	$625$	$2$	625.d	25.d	$5$	$16$	$4$	$4.991$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓			625.2.a.e	$2$	$0$	$0$	$-5$	$0$	$20$	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	$q+(-\beta _{5}-\beta _{6}+\beta _{7}-\beta _{8}-\beta _{12})q^{2}+\cdots$
625.2.d.o	$625$	$2$	625.d	25.d	$5$	$16$	$4$	$4.991$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None					25.2.e.a	$4$	$0$	$0$	$0$	$0$	$0$	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	$q+(-\beta _{1}-\beta _{11}-\beta _{13})q^{2}+(-\beta _{6}+\beta _{11}+\cdots)q^{3}+\cdots$
625.2.d.p	$625$	$2$	625.d	25.d	$5$	$16$	$4$	$4.991$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓			625.2.a.e	$2$	$0$	$0$	$5$	$0$	$-20$	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	$q+(\beta _{5}+\beta _{6}-\beta _{7}+\beta _{8}+\beta _{12})q^{2}+(1+\cdots)q^{3}+\cdots$
625.2.d.q	$625$	$2$	625.d	25.d	$5$	$16$	$4$	$4.991$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓			625.2.a.e	$2$	$0$	$5$	$0$	$0$	$-20$	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	$q+(1+\beta _{3}-\beta _{5}-\beta _{7})q^{2}+(\beta _{5}-\beta _{15})q^{3}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(625, [\chi])$ into lower level spaces

$S_{2}^{\mathrm{old}}(625, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(25, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(125, [\chi])$$^{\oplus 2}$