Space of modular forms of level 325, weight 2, and Character 325.e

• Label: 325.2.e
• Level: $325$
• Weight: $2$
• Character orbit: 325.e
• Rep. character: $\chi_{325}(126,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $40$
• Newform subspaces: $5$
• Sturm bound: $70$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$325 = 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	325.e (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$13$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$5$
Sturm bound:			$70$
Trace bound:			$3$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(325, [\chi])$.

	Total	New	Old
Modular forms	80	52	28
Cusp forms	56	40	16
Eisenstein series	24	12	12

Trace form

40 * q + 2 * q^3 - 18 * q^4 - 10 * q^6 + 2 * q^7 + 12 * q^8 - 14 * q^9 - 6 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 - 18 * q^16 + 10 * q^17 - 8 * q^18 - 4 * q^19 - 24 * q^21 - 4 * q^22 + 6 * q^23 - 8 * q^24 + 20 * q^27 - 10 * q^28 - 12 * q^29 - 32 * q^31 - 2 * q^32 - 14 * q^33 + 80 * q^34 + 32 * q^36 + 6 * q^37 - 24 * q^38 + 12 * q^39 - 10 * q^41 - 4 * q^42 - 14 * q^43 - 8 * q^44 + 10 * q^46 - 24 * q^47 + 12 * q^48 - 12 * q^49 - 18 * q^52 - 40 * q^53 - 26 * q^54 + 2 * q^56 - 12 * q^57 + 32 * q^58 + 16 * q^59 - 4 * q^61 + 4 * q^62 + 12 * q^63 - 8 * q^64 + 124 * q^66 - 6 * q^67 + 34 * q^68 - 30 * q^71 + 12 * q^72 + 56 * q^73 - 2 * q^74 - 32 * q^76 - 4 * q^77 - 12 * q^78 + 40 * q^79 - 24 * q^81 - 20 * q^82 + 8 * q^83 - 4 * q^84 - 112 * q^86 - 18 * q^87 + 2 * q^88 + 16 * q^89 - 30 * q^91 + 20 * q^92 - 8 * q^93 + 10 * q^94 + 84 * q^96 - 22 * q^97 - 12 * q^98 + 40 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(325, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
325.2.e.a	$325$	$2$	325.e	13.c	$3$	$4$	$2$	$2.595$	$\Q(\sqrt{-3}, \sqrt{13})$	None					65.2.e.b	$2$	$0$	$-1$	$2$	$0$	$-2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
325.2.e.b	$325$	$2$	325.e	13.c	$3$	$4$	$2$	$2.595$	$\Q(\sqrt{-3}, \sqrt{5})$	None					65.2.e.a	$2$	$0$	$1$	$0$	$0$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(1-2\beta _{1}+\beta _{3})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
325.2.e.c	$325$	$2$	325.e	13.c	$3$	$10$	$5$	$2.595$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		✓			325.2.e.c	$2$	$0$	$0$	$-3$	$0$	$2$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(-1-\beta _{4}+\beta _{6}+\beta _{8})q^{3}+\cdots$
325.2.e.d	$325$	$2$	325.e	13.c	$3$	$10$	$5$	$2.595$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		✓			325.2.e.c	$2$	$0$	$0$	$3$	$0$	$-2$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{1}q^{2}+(1+\beta _{4}-\beta _{6}-\beta _{8})q^{3}+(-\beta _{6}+\cdots)q^{4}+\cdots$
325.2.e.e	$325$	$2$	325.e	13.c	$3$	$12$	$6$	$2.595$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None			✓		65.2.n.a	$4$	$0$	$0$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{6}q^{2}+(\beta _{6}-\beta _{11})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(325, [\chi])$ into lower level spaces

$S_{2}^{\mathrm{old}}(325, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(65, [\chi])$$^{\oplus 2}$