Space of modular forms of level 15, weight 10, and trivial character

• Label: 15.10.a
• Level: $15$
• Weight: $10$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $20$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(20\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	20	6	14
Cusp forms	16	6	10
Eisenstein series	4	0	4

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(11\)	\(4\)	\(7\)		\(9\)	\(4\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

6 * q + 68 * q^2 + 1538 * q^4 - 3078 * q^6 - 11428 * q^7 + 67932 * q^8 + 39366 * q^9 - 8750 * q^10 - 87076 * q^11 + 41472 * q^12 + 55888 * q^13 + 115452 * q^14 - 101250 * q^15 + 1085330 * q^16 - 420520 * q^17 + 446148 * q^18 + 658912 * q^19 - 905000 * q^20 - 2241756 * q^21 - 7156540 * q^22 + 3122496 * q^23 + 3193506 * q^24 + 2343750 * q^25 - 5892868 * q^26 + 3780524 * q^28 - 6996536 * q^29 - 1620000 * q^30 - 4504264 * q^31 + 17859412 * q^32 - 1195236 * q^33 + 5709836 * q^34 + 15182500 * q^35 + 10090818 * q^36 + 25085112 * q^37 - 2385976 * q^38 - 5547852 * q^39 - 4751250 * q^40 + 24274636 * q^41 - 44096724 * q^42 - 4089728 * q^43 - 102869396 * q^44 - 90064944 * q^46 + 51128264 * q^47 + 65192688 * q^48 + 84594310 * q^49 + 26562500 * q^50 + 62822628 * q^51 - 52677360 * q^52 - 114062224 * q^53 - 20194758 * q^54 - 80472500 * q^55 - 68883420 * q^56 - 2667816 * q^57 - 205991852 * q^58 + 180942428 * q^59 - 130916250 * q^60 + 479495740 * q^61 - 31830936 * q^62 - 74979108 * q^63 + 464210442 * q^64 - 192047500 * q^65 - 57426084 * q^66 - 484354360 * q^67 + 717440536 * q^68 + 5244912 * q^69 + 543322500 * q^70 + 83143192 * q^71 + 445701852 * q^72 - 591042044 * q^73 - 773558108 * q^74 - 849378912 * q^76 + 72064224 * q^77 + 265485600 * q^78 - 274527880 * q^79 + 103030000 * q^80 + 258280326 * q^81 + 1526220184 * q^82 + 804884184 * q^83 - 1559325492 * q^84 - 192897500 * q^85 - 752029576 * q^86 + 37145628 * q^87 - 2168339028 * q^88 - 329760348 * q^89 - 57408750 * q^90 - 871611944 * q^91 - 2807106528 * q^92 - 830087352 * q^93 + 3271510568 * q^94 + 363665000 * q^95 + 2378839914 * q^96 + 1387556460 * q^97 + 5935831732 * q^98 - 571305636 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.10.a.a	$15$	$10$	15.a	1.a	$1$	$1$	$1$	$7.726$	\(\Q\)	None	✓	✓	✓	✓	15.10.a.a	$1$	$1$	\(-4\)	\(81\)	\(625\)	\(-7680\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3^{4}q^{3}-496q^{4}+5^{4}q^{5}+\cdots\)
15.10.a.b	$15$	$10$	15.a	1.a	$1$	$1$	$1$	$7.726$	\(\Q\)	None	✓	✓	✓	✓	15.10.a.b	$1$	$1$	\(22\)	\(-81\)	\(-625\)	\(-5988\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+22q^{2}-3^{4}q^{3}-28q^{4}-5^{4}q^{5}+\cdots\)
15.10.a.c	$15$	$10$	15.a	1.a	$1$	$2$	$2$	$7.726$	\(\Q(\sqrt{4729}) \)	None	✓	✓	✓	✓	15.10.a.c	$1$	$0$	\(19\)	\(162\)	\(-1250\)	\(-11872\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{2}+3^{4}q^{3}+(770-19\beta )q^{4}+\cdots\)
15.10.a.d	$15$	$10$	15.a	1.a	$1$	$2$	$2$	$7.726$	\(\Q(\sqrt{241}) \)	None	✓	✓	✓	✓	15.10.a.d	$1$	$0$	\(31\)	\(-162\)	\(1250\)	\(14112\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(15-\beta )q^{2}-3^{4}q^{3}+(255-31\beta )q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)