Space of modular forms of level 105, weight 4, and trivial character

• Label: 105.4.a
• Level: $105$
• Weight: $4$
• Character orbit: 105.a
• Rep. character: $\chi_{105}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $7$
• Sturm bound: $64$
• Trace bound: $2$

Defining parameters

Level:	$N$	$=$	$105 = 3 \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	105.a (trivial)
Character field:			$\Q$
Newform subspaces:			$7$
Sturm bound:			$64$
Trace bound:			$2$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(105))$.

	Total	New	Old
Modular forms	52	12	40
Cusp forms	44	12	32
Eisenstein series	8	0	8

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
$+$	$+$	$+$	$+$		$9$	$2$	$7$		$8$	$2$	$6$		$1$	$0$	$1$
$+$	$+$	$-$	$-$		$5$	$0$	$5$		$4$	$0$	$4$		$1$	$0$	$1$
$+$	$-$	$+$	$-$		$5$	$2$	$3$		$4$	$2$	$2$		$1$	$0$	$1$
$+$	$-$	$-$	$+$		$7$	$2$	$5$		$6$	$2$	$4$		$1$	$0$	$1$
$-$	$+$	$+$	$-$		$7$	$2$	$5$		$6$	$2$	$4$		$1$	$0$	$1$
$-$	$+$	$-$	$+$		$7$	$2$	$5$		$6$	$2$	$4$		$1$	$0$	$1$
$-$	$-$	$+$	$+$		$5$	$2$	$3$		$4$	$2$	$2$		$1$	$0$	$1$
$-$	$-$	$-$	$-$		$7$	$0$	$7$		$6$	$0$	$6$		$1$	$0$	$1$
Plus space			$+$		$28$	$8$	$20$		$24$	$8$	$16$		$4$	$0$	$4$
Minus space			$-$		$24$	$4$	$20$		$20$	$4$	$16$		$4$	$0$	$4$

Trace form

12 * q - 4 * q^2 + 56 * q^4 + 12 * q^6 - 28 * q^7 + 36 * q^8 + 108 * q^9 + 60 * q^10 - 40 * q^11 - 32 * q^13 + 140 * q^14 - 60 * q^15 + 328 * q^16 - 64 * q^17 - 36 * q^18 - 104 * q^19 + 72 * q^22 + 248 * q^23 + 324 * q^24 + 300 * q^25 - 160 * q^26 - 140 * q^28 - 240 * q^29 - 120 * q^30 + 288 * q^31 - 596 * q^32 - 192 * q^33 - 928 * q^34 + 504 * q^36 - 56 * q^37 - 648 * q^38 - 24 * q^39 + 420 * q^40 - 80 * q^41 - 168 * q^42 - 416 * q^43 - 432 * q^44 - 1040 * q^46 + 272 * q^47 - 1248 * q^48 + 588 * q^49 - 100 * q^50 + 864 * q^51 - 216 * q^52 + 1824 * q^53 + 108 * q^54 + 360 * q^55 + 1092 * q^56 - 456 * q^57 + 920 * q^58 - 304 * q^59 + 180 * q^60 - 1408 * q^61 - 696 * q^62 - 252 * q^63 - 488 * q^64 - 320 * q^65 + 624 * q^66 - 976 * q^67 - 3088 * q^68 - 1008 * q^69 - 280 * q^70 + 1408 * q^71 + 324 * q^72 + 2592 * q^73 + 688 * q^74 - 3952 * q^76 + 1904 * q^77 - 2856 * q^78 + 2544 * q^79 - 1760 * q^80 + 972 * q^81 - 3096 * q^82 + 2000 * q^83 - 440 * q^85 + 1800 * q^86 + 240 * q^87 + 6872 * q^88 - 3392 * q^89 + 540 * q^90 + 840 * q^91 + 4760 * q^92 + 72 * q^93 - 472 * q^94 + 40 * q^95 + 1644 * q^96 - 3776 * q^97 - 196 * q^98 - 360 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(105))$ 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
105.4.a.a	$105$	$4$	105.a	1.a	$1$	$1$	$1$	$6.195$	$\Q$	None	✓	✓			105.4.a.a	$1$	$0$	$0$	$-3$	$5$	$7$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}-8q^{4}+5q^{5}+7q^{7}+9q^{9}+\cdots$
105.4.a.b	$105$	$4$	105.a	1.a	$1$	$1$	$1$	$6.195$	$\Q$	None	✓	✓			105.4.a.b	$1$	$0$	$5$	$-3$	$5$	$7$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+5q^{2}-3q^{3}+17q^{4}+5q^{5}-15q^{6}+\cdots$
105.4.a.c	$105$	$4$	105.a	1.a	$1$	$2$	$2$	$6.195$	$\Q(\sqrt{17})$	None	✓	✓	✓	✓	105.4.a.c	$1$	$0$	$-7$	$-6$	$-10$	$-14$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-3-\beta )q^{2}-3q^{3}+(5+7\beta )q^{4}+\cdots$
105.4.a.d	$105$	$4$	105.a	1.a	$1$	$2$	$2$	$6.195$	$\Q(\sqrt{5})$	None	✓	✓	✓	✓	105.4.a.d	$1$	$1$	$-4$	$6$	$-10$	$-14$		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(-2-\beta )q^{2}+3q^{3}+(1+4\beta )q^{4}+\cdots$
105.4.a.e	$105$	$4$	105.a	1.a	$1$	$2$	$2$	$6.195$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	105.4.a.e	$1$	$1$	$-2$	$-6$	$10$	$-14$		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}-3q^{3}+(1-2\beta )q^{4}+\cdots$
105.4.a.f	$105$	$4$	105.a	1.a	$1$	$2$	$2$	$6.195$	$\Q(\sqrt{65})$	None	✓	✓	✓	✓	105.4.a.f	$1$	$0$	$1$	$6$	$10$	$-14$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+3q^{3}+(8+\beta )q^{4}+5q^{5}+3\beta q^{6}+\cdots$
105.4.a.g	$105$	$4$	105.a	1.a	$1$	$2$	$2$	$6.195$	$\Q(\sqrt{41})$	None	✓	✓	✓	✓	105.4.a.g	$1$	$0$	$3$	$6$	$-10$	$14$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+3q^{3}+(3+3\beta )q^{4}-5q^{5}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(105))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(105)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 2}$