Space of modular forms of level 315, weight 2, and trivial character

• Label: 315.2.a
• Level: $315$
• Weight: $2$
• Character orbit: 315.a
• Rep. character: $\chi_{315}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $96$
• Trace bound: $2$

Defining parameters

Level:	$N$	$=$	$315 = 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	315.a (trivial)
Character field:			$\Q$
Newform subspaces:			$6$
Sturm bound:			$96$
Trace bound:			$2$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(315))$.

	Total	New	Old
Modular forms	56	10	46
Cusp forms	41	10	31
Eisenstein series	15	0	15

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

$3$	$5$	$7$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$4$	$2$	$2$		$3$	$2$	$1$		$1$	$0$	$1$
$+$	$+$	$-$	$-$		$8$	$0$	$8$		$6$	$0$	$6$		$2$	$0$	$2$
$+$	$-$	$+$	$-$		$10$	$2$	$8$		$8$	$2$	$6$		$2$	$0$	$2$
$+$	$-$	$-$	$+$		$6$	$0$	$6$		$4$	$0$	$4$		$2$	$0$	$2$
$-$	$+$	$+$	$-$		$7$	$2$	$5$		$5$	$2$	$3$		$2$	$0$	$2$
$-$	$+$	$-$	$+$		$7$	$1$	$6$		$5$	$1$	$4$		$2$	$0$	$2$
$-$	$-$	$+$	$+$		$7$	$0$	$7$		$5$	$0$	$5$		$2$	$0$	$2$
$-$	$-$	$-$	$-$		$7$	$3$	$4$		$5$	$3$	$2$		$2$	$0$	$2$
Plus space			$+$		$24$	$3$	$21$		$17$	$3$	$14$		$7$	$0$	$7$
Minus space			$-$		$32$	$7$	$25$		$24$	$7$	$17$		$8$	$0$	$8$

Trace form

10 * q + 12 * q^4 - 2 * q^7 + 12 * q^8 + 4 * q^10 - 2 * q^11 - 4 * q^13 - 2 * q^14 + 16 * q^16 + 4 * q^17 - 8 * q^19 - 20 * q^22 - 8 * q^23 + 10 * q^25 + 20 * q^26 - 6 * q^28 + 2 * q^29 + 12 * q^31 + 4 * q^32 - 28 * q^34 + 4 * q^35 - 8 * q^37 - 32 * q^38 + 20 * q^41 - 12 * q^43 - 12 * q^44 + 32 * q^46 - 20 * q^47 + 10 * q^49 - 40 * q^52 - 4 * q^53 + 8 * q^55 - 6 * q^56 + 4 * q^58 + 4 * q^59 + 32 * q^61 - 24 * q^62 - 48 * q^64 + 6 * q^65 - 20 * q^67 + 24 * q^68 - 2 * q^70 - 24 * q^71 - 16 * q^73 - 32 * q^74 + 16 * q^76 + 6 * q^79 + 6 * q^85 + 24 * q^86 - 8 * q^88 + 16 * q^89 + 2 * q^91 - 40 * q^92 - 20 * q^94 + 20 * q^95 - 44 * q^97

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(315))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
315.2.a.a	$315$	$2$	315.a	1.a	$1$	$1$	$1$	$2.515$	$\Q$	None	✓		✓	✓	105.2.a.a	$1$	$1$	$-1$	$0$	$-1$	$1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-q^{5}+q^{7}+3q^{8}+q^{10}+\cdots$
315.2.a.b	$315$	$2$	315.a	1.a	$1$	$1$	$1$	$2.515$	$\Q$	None	✓				35.2.a.a	$1$	$0$	$0$	$0$	$1$	$1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+q^{5}+q^{7}+3q^{11}+5q^{13}+\cdots$
315.2.a.c	$315$	$2$	315.a	1.a	$1$	$2$	$2$	$2.515$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	315.2.a.c	$1$	$1$	$-2$	$0$	$-2$	$-2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-q^{7}+\cdots$
315.2.a.d	$315$	$2$	315.a	1.a	$1$	$2$	$2$	$2.515$	$\Q(\sqrt{5})$	None	✓		✓		105.2.a.b	$1$	$0$	$0$	$0$	$2$	$2$		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{2}+3q^{4}+q^{5}+q^{7}-\beta q^{8}-\beta q^{10}+\cdots$
315.2.a.e	$315$	$2$	315.a	1.a	$1$	$2$	$2$	$2.515$	$\Q(\sqrt{17})$	None	✓		✓	✓	35.2.a.b	$1$	$0$	$1$	$0$	$-2$	$-2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(2+\beta )q^{4}-q^{5}-q^{7}+(4+\beta )q^{8}+\cdots$
315.2.a.f	$315$	$2$	315.a	1.a	$1$	$2$	$2$	$2.515$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	315.2.a.c	$1$	$0$	$2$	$0$	$2$	$-2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(315))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(315)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(63))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(105))$$^{\oplus 2}$