Space of modular forms of level 105, weight 10, and trivial character

• Label: 105.10.a
• Level: $105$
• Weight: $10$
• Character orbit: 105.a
• Rep. character: $\chi_{105}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $8$
• Sturm bound: $160$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(160\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(105))\).

	Total	New	Old
Modular forms	148	36	112
Cusp forms	140	36	104
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(16\)
Minus space			\(-\)	\(20\)

Trace form

36 * q - 68 * q^2 + 8872 * q^4 + 6156 * q^6 + 9604 * q^7 - 67932 * q^8 + 236196 * q^9 - 22500 * q^10 + 156280 * q^11 - 188896 * q^13 - 48020 * q^14 + 202500 * q^15 + 1762184 * q^16 + 1076032 * q^17 - 446148 * q^18 - 1591048 * q^19 + 4038744 * q^22 - 2779016 * q^23 + 1495908 * q^24 + 14062500 * q^25 - 7028480 * q^26 + 8115380 * q^28 + 6151920 * q^29 + 3240000 * q^30 + 10777344 * q^31 + 9917228 * q^32 - 7096896 * q^33 - 44552816 * q^34 + 58209192 * q^36 + 7506968 * q^37 - 27185544 * q^38 - 4245048 * q^39 - 21217500 * q^40 - 72213040 * q^41 + 12446784 * q^42 + 19555712 * q^43 + 227853936 * q^44 + 80474960 * q^46 - 164244656 * q^47 - 66684384 * q^48 + 207532836 * q^49 - 26562500 * q^50 - 64291968 * q^51 - 77202072 * q^52 + 88587552 * q^53 + 40389516 * q^54 + 92205000 * q^55 - 39097884 * q^56 + 84448008 * q^57 + 108425560 * q^58 + 82984912 * q^59 + 261832500 * q^60 + 126164416 * q^61 + 1381506888 * q^62 + 63011844 * q^63 + 323960504 * q^64 - 143320000 * q^65 + 161354592 * q^66 - 4004912 * q^67 + 97528816 * q^68 - 394037136 * q^69 + 96040000 * q^70 - 698908576 * q^71 - 445701852 * q^72 - 336562464 * q^73 + 428300096 * q^74 + 1755531184 * q^76 - 412088432 * q^77 + 713654712 * q^78 + 400633968 * q^79 - 1132780000 * q^80 + 1549681956 * q^81 - 1462650072 * q^82 - 365294000 * q^83 + 846815000 * q^85 - 2690823240 * q^86 - 1449245520 * q^87 + 32917816 * q^88 + 969140096 * q^89 - 147622500 * q^90 + 343150920 * q^91 + 4015016920 * q^92 + 2083596696 * q^93 - 101147144 * q^94 + 102605000 * q^95 - 721624788 * q^96 - 3365623552 * q^97 - 392006468 * q^98 + 1025353080 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(105))\) 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
105.10.a.a	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.a	$1$	$1$	\(-41\)	\(-324\)	\(2500\)	\(9604\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-10+\beta _{1})q^{2}-3^{4}q^{3}+(120-21\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.b	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.b	$1$	$0$	\(-20\)	\(-324\)	\(2500\)	\(-9604\)		$+$	$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{1})q^{2}-3^{4}q^{3}+(106-8\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.c	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.c	$1$	$1$	\(-17\)	\(324\)	\(2500\)	\(-9604\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(235+9\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.d	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.d	$1$	$1$	\(-13\)	\(324\)	\(-2500\)	\(9604\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+3^{4}q^{3}+(123+8\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.e	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.e	$1$	$1$	\(5\)	\(-324\)	\(-2500\)	\(-9604\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}-3^{4}q^{3}+(237+3\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.f	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.f	$1$	$0$	\(8\)	\(324\)	\(-2500\)	\(-9604\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+3^{4}q^{3}+(106-\beta _{1}+\beta _{3})q^{4}+\cdots\)
105.10.a.g	$105$	$10$	105.a	1.a	$1$	$6$	$6$	$54.079$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.g	$1$	$0$	\(-16\)	\(-486\)	\(-3750\)	\(14406\)		$+$	$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}-3^{4}q^{3}+(428-2\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.h	$105$	$10$	105.a	1.a	$1$	$6$	$6$	$54.079$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.h	$1$	$0$	\(26\)	\(486\)	\(3750\)	\(14406\)		$-$	$-$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+3^{4}q^{3}+(426+3\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(105))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(105)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)