Space of modular forms of level 15, weight 8, and trivial character

• Label: 15.8.a
• Level: $15$
• Weight: $8$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(16\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	16	4	12
Cusp forms	12	4	8
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
\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

4 * q - 28 * q^2 + 54 * q^3 + 466 * q^4 - 54 * q^6 + 2264 * q^7 - 2436 * q^8 + 2916 * q^9 + 5250 * q^10 - 2800 * q^11 + 10368 * q^12 - 11488 * q^13 - 20964 * q^14 + 6750 * q^15 + 20338 * q^16 - 32032 * q^17 - 20412 * q^18 - 33712 * q^19 - 41000 * q^20 - 13392 * q^21 - 19644 * q^22 + 170088 * q^23 - 126846 * q^24 + 62500 * q^25 + 347708 * q^26 + 39366 * q^27 - 165748 * q^28 + 477616 * q^29 + 54000 * q^30 + 46832 * q^31 - 1063916 * q^32 + 102816 * q^33 + 54012 * q^34 + 43000 * q^35 + 339714 * q^36 - 153616 * q^37 - 769432 * q^38 - 769500 * q^39 + 666750 * q^40 + 375592 * q^41 + 402732 * q^42 - 421000 * q^43 - 2040596 * q^44 + 922272 * q^46 + 1210520 * q^47 + 1626480 * q^48 + 579204 * q^49 - 437500 * q^50 - 325188 * q^51 - 2146192 * q^52 + 1575488 * q^53 - 39366 * q^54 + 1212000 * q^55 + 5773380 * q^56 - 3136968 * q^57 - 2186604 * q^58 - 2288608 * q^59 - 830250 * q^60 - 4545304 * q^61 + 5573928 * q^62 + 1650456 * q^63 + 10293082 * q^64 - 811000 * q^65 - 2849796 * q^66 - 5036632 * q^67 - 9434504 * q^68 + 49896 * q^69 - 445500 * q^70 - 2625392 * q^71 - 1775844 * q^72 + 2617208 * q^73 - 3041564 * q^74 + 843750 * q^75 - 13374352 * q^76 + 6733248 * q^77 + 15359328 * q^78 + 3505280 * q^79 - 8666000 * q^80 + 2125764 * q^81 + 13003128 * q^82 - 17321256 * q^83 - 7530516 * q^84 + 2631000 * q^85 + 13624760 * q^86 + 5724324 * q^87 + 23149164 * q^88 + 12431928 * q^89 + 3827250 * q^90 - 3365648 * q^91 + 15479424 * q^92 + 4277664 * q^93 - 32795880 * q^94 - 8182000 * q^95 - 34913862 * q^96 + 11522888 * q^97 - 21248492 * q^98 - 2041200 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.8.a.a	$15$	$8$	15.a	1.a	$1$	$1$	$1$	$4.686$	\(\Q\)	None	✓	✓	✓	✓	15.8.a.a	$1$	$1$	\(-22\)	\(27\)	\(-125\)	\(-420\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{2}+3^{3}q^{3}+356q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.b	$15$	$8$	15.a	1.a	$1$	$1$	$1$	$4.686$	\(\Q\)	None	✓	✓	✓	✓	15.8.a.b	$1$	$0$	\(-13\)	\(-27\)	\(-125\)	\(1380\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-13q^{2}-3^{3}q^{3}+41q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.c	$15$	$8$	15.a	1.a	$1$	$2$	$2$	$4.686$	\(\Q(\sqrt{601}) \)	None	✓	✓	✓	✓	15.8.a.c	$1$	$0$	\(7\)	\(54\)	\(250\)	\(1304\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+3^{3}q^{3}+(38-7\beta )q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)