Space of modular forms of level 104, weight 10, and trivial character

• Label: 104.10.a
• Level: $104$
• Weight: $10$
• Character orbit: 104.a
• Rep. character: $\chi_{104}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $4$
• Sturm bound: $140$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(140\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(104))\).

	Total	New	Old
Modular forms	130	27	103
Cusp forms	122	27	95
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.

\(2\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(31\)	\(6\)	\(25\)		\(29\)	\(6\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(34\)	\(8\)	\(26\)		\(32\)	\(8\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(34\)	\(7\)	\(27\)		\(32\)	\(7\)	\(25\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(31\)	\(6\)	\(25\)		\(29\)	\(6\)	\(23\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(62\)	\(12\)	\(50\)		\(58\)	\(12\)	\(46\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(68\)	\(15\)	\(53\)		\(64\)	\(15\)	\(49\)		\(4\)	\(0\)	\(4\)

Trace form

27 * q + 146 * q^3 + 1250 * q^5 - 9756 * q^7 + 252553 * q^9 - 138196 * q^11 + 28561 * q^13 - 159528 * q^15 - 392692 * q^17 - 1175856 * q^19 + 147292 * q^21 + 2753268 * q^23 + 8805591 * q^25 + 5241074 * q^27 - 5154058 * q^29 - 6869916 * q^31 + 15328428 * q^33 - 21233998 * q^35 + 7594994 * q^37 + 15054874 * q^41 - 11413186 * q^43 - 70013666 * q^45 + 129230396 * q^47 + 46999493 * q^49 + 139064902 * q^51 + 84083842 * q^53 + 107648816 * q^55 + 17725744 * q^57 - 236488264 * q^59 + 28169598 * q^61 + 131121900 * q^63 + 71402500 * q^65 - 434142476 * q^67 + 412461948 * q^69 + 484995460 * q^71 + 318092354 * q^73 + 1084453508 * q^75 + 41526968 * q^77 + 217021628 * q^79 + 2227885739 * q^81 + 1371525812 * q^83 - 680800568 * q^85 + 1783568408 * q^87 + 472572346 * q^89 - 411449766 * q^91 + 1616029536 * q^93 - 4210723500 * q^95 - 1399132538 * q^97 + 2904694208 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(104))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
104.10.a.a	$104$	$10$	104.a	1.a	$1$	$6$	$6$	$53.564$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	104.10.a.a	$1$	$1$	\(0\)	\(-68\)	\(-176\)	\(-9664\)		$-$	$-$	$+$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{3}+(-29+\beta _{2})q^{5}+\cdots\)
104.10.a.b	$104$	$10$	104.a	1.a	$1$	$6$	$6$	$53.564$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	104.10.a.b	$1$	$1$	\(0\)	\(60\)	\(176\)	\(3416\)		$+$	$+$	$+$	$2^{12}\cdot 3$	$\mathrm{SU}(2)$	\(q+(10-\beta _{1})q^{3}+(29-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
104.10.a.c	$104$	$10$	104.a	1.a	$1$	$7$	$7$	$53.564$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	104.10.a.c	$1$	$0$	\(0\)	\(13\)	\(-801\)	\(-1091\)		$-$	$+$	$-$	$2^{19}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-114+\beta _{2})q^{5}+(-157+\cdots)q^{7}+\cdots\)
104.10.a.d	$104$	$10$	104.a	1.a	$1$	$8$	$8$	$53.564$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	104.10.a.d	$1$	$0$	\(0\)	\(141\)	\(2051\)	\(-2417\)		$+$	$-$	$-$	$2^{25}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(18-\beta _{1})q^{3}+(2^{8}+2\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(104))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(104)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)