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

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