Space of modular forms of level 15, weight 20, and trivial character

• Label: 15.20.a
• Level: $15$
• Weight: $20$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $40$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(40\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	40	12	28
Cusp forms	36	12	24
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
\(+\)	\(+\)	\(+\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(10\)	\(3\)	\(7\)		\(9\)	\(3\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(10\)	\(4\)	\(6\)		\(9\)	\(4\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(21\)	\(7\)	\(14\)		\(19\)	\(7\)	\(12\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(19\)	\(5\)	\(14\)		\(17\)	\(5\)	\(12\)		\(2\)	\(0\)	\(2\)

Trace form

12 * q + 1828 * q^2 + 39366 * q^3 + 1212402 * q^4 + 25469802 * q^6 - 5156904 * q^7 + 755634876 * q^8 + 4649045868 * q^9 + 2144531250 * q^10 + 22444900192 * q^11 + 30958682112 * q^12 + 97171285488 * q^13 - 222654354180 * q^14 + 76886718750 * q^15 - 166970369742 * q^16 - 799714831328 * q^17 + 708204653892 * q^18 - 2715173155488 * q^19 + 1342484375000 * q^20 + 5648312412000 * q^21 + 19716390128484 * q^22 + 1240179614472 * q^23 + 31143866507586 * q^24 + 45776367187500 * q^25 - 41613182096996 * q^26 + 15251194969974 * q^27 - 7086316219572 * q^28 - 349283021628496 * q^29 + 39366000000000 * q^30 - 87363909503040 * q^31 + 318399592776596 * q^32 + 243765400925664 * q^33 + 453725938988604 * q^34 + 419989953125000 * q^35 + 469709375704578 * q^36 - 332161196519424 * q^37 + 2809750710946792 * q^38 + 390168974542356 * q^39 + 1724218136718750 * q^40 + 662909216270360 * q^41 - 4825596170060532 * q^42 - 10020683304972840 * q^43 + 19711254798330092 * q^44 + 18849189965512320 * q^46 + 9887089860276280 * q^47 + 7050194086420080 * q^48 + 4712200658405964 * q^49 + 6973266601562500 * q^50 + 14474932522954236 * q^51 + 145506973520918832 * q^52 - 89359976436409568 * q^53 + 9867523145573178 * q^54 + 57782123156250000 * q^55 - 207952787404480380 * q^56 + 37828484781348168 * q^57 - 235354019854023756 * q^58 - 226238075734647152 * q^59 + 120616578246093750 * q^60 - 131476454092634664 * q^61 + 363615908844228072 * q^62 - 1997890269406056 * q^63 - 389029926364328454 * q^64 + 396989202671875000 * q^65 - 210530584127460708 * q^66 + 741802430478899112 * q^67 - 315623836951057096 * q^68 - 314157056703985080 * q^69 + 132754892414062500 * q^70 - 122935259365244656 * q^71 + 292748433165374364 * q^72 - 750414477287063928 * q^73 - 3414458286837119740 * q^74 + 150169372558593750 * q^75 - 1446244524798817008 * q^76 + 2014962183403953312 * q^77 + 418782094638498432 * q^78 - 3148598674399345680 * q^79 + 2092714573343750000 * q^80 + 1801135623563989452 * q^81 + 8240721234672777912 * q^82 - 3874959522484719624 * q^83 - 3659985923820164820 * q^84 + 1809076836234375000 * q^85 + 3448720514316983992 * q^86 + 5413146010969975236 * q^87 + 8895533838747285996 * q^88 - 12590711245939546488 * q^89 + 830835345550781250 * q^90 - 6884704454626723920 * q^91 + 5326212477750048576 * q^92 - 4723749151661721744 * q^93 - 10489217876788940424 * q^94 - 2932180569406250000 * q^95 + 358612158029287770 * q^96 + 6588335019646327992 * q^97 + 49254748499054655572 * q^98 + 8695614207940833888 * q^99

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.20.a.a	$15$	$20$	15.a	1.a	$1$	$2$	$2$	$34.323$	\(\Q(\sqrt{129}) \)	None	✓	✓	✓	✓	15.20.a.a	$1$	$1$	\(152\)	\(-39366\)	\(3906250\)	\(-736176\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(76+\beta )q^{2}-3^{9}q^{3}+(-351328+\cdots)q^{4}+\cdots\)
15.20.a.b	$15$	$20$	15.a	1.a	$1$	$3$	$3$	$34.323$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	15.20.a.b	$1$	$0$	\(115\)	\(-59049\)	\(-5859375\)	\(-145324276\)		$+$	$+$	$+$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(38+\beta _{1})q^{2}-3^{9}q^{3}+(174188-139\beta _{1}+\cdots)q^{4}+\cdots\)
15.20.a.c	$15$	$20$	15.a	1.a	$1$	$3$	$3$	$34.323$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	15.20.a.c	$1$	$1$	\(250\)	\(59049\)	\(-5859375\)	\(35228396\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(83+\beta _{1})q^{2}+3^{9}q^{3}+(-86738+\cdots)q^{4}+\cdots\)
15.20.a.d	$15$	$20$	15.a	1.a	$1$	$4$	$4$	$34.323$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	15.20.a.d	$1$	$0$	\(1311\)	\(78732\)	\(7812500\)	\(105675152\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}\cdot 5^{2}\cdot 13$	$\mathrm{SU}(2)$	\(q+(328-\beta _{1})q^{2}+3^{9}q^{3}+(413212+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)