Properties

Label 15.12.a
Level 15
Weight 12
Character orbit a
Rep. character \(\chi_{15}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 4
Sturm bound 24
Trace bound 2

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 15.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(15))\).

Total New Old
Modular forms 24 8 16
Cusp forms 20 8 12
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.

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(3\)

Trace form

\( 8q - 92q^{2} + 486q^{3} + 2098q^{4} + 11178q^{6} + 10296q^{7} + 36156q^{8} + 472392q^{9} + O(q^{10}) \) \( 8q - 92q^{2} + 486q^{3} + 2098q^{4} + 11178q^{6} + 10296q^{7} + 36156q^{8} + 472392q^{9} - 68750q^{10} + 362368q^{11} + 1492992q^{12} - 1732192q^{13} + 3055740q^{14} + 1518750q^{15} - 2528078q^{16} + 15690832q^{17} - 5432508q^{18} + 6097088q^{19} + 29975000q^{20} - 14881320q^{21} + 80817764q^{22} - 54036408q^{23} - 34419006q^{24} + 78125000q^{25} - 42340964q^{26} + 28697814q^{27} - 206985012q^{28} - 344119264q^{29} + 48600000q^{30} - 244730640q^{31} + 392770196q^{32} - 1145016q^{33} - 1039380484q^{34} + 51425000q^{35} + 123884802q^{36} + 411385216q^{37} + 925664872q^{38} - 207359676q^{39} + 10518750q^{40} + 50518160q^{41} - 39765492q^{42} + 3382732760q^{43} + 993772268q^{44} + 1792516800q^{46} - 6579674120q^{47} + 486330480q^{48} + 7476115336q^{49} - 898437500q^{50} - 235693476q^{51} - 11457178448q^{52} + 2398000672q^{53} + 660049722q^{54} + 1546300000q^{55} + 6876152580q^{56} - 421367832q^{57} - 15688092236q^{58} + 11641748272q^{59} + 906693750q^{60} - 1339937296q^{61} - 14092986648q^{62} + 607968504q^{63} - 22506773446q^{64} + 3813400000q^{65} + 3967032348q^{66} + 17395245672q^{67} - 16371169096q^{68} - 18746409960q^{69} + 14610862500q^{70} + 17460704816q^{71} + 2134975644q^{72} + 13234384592q^{73} - 73427688700q^{74} + 4746093750q^{75} + 91790636368q^{76} + 39494634432q^{77} - 6583379328q^{78} - 29704597920q^{79} + 6801350000q^{80} + 27894275208q^{81} - 7271225288q^{82} + 29154140856q^{83} - 38819245140q^{84} + 44187850000q^{85} + 230761968568q^{86} + 16697096676q^{87} - 159023820564q^{88} - 22145809872q^{89} - 4059618750q^{90} - 121971765360q^{91} - 185727975744q^{92} + 104018918256q^{93} + 205545366584q^{94} + 6142450000q^{95} + 21391744410q^{96} - 385635691568q^{97} - 216115979308q^{98} + 21397468032q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
15.12.a.a \(1\) \(11.525\) \(\Q\) None \(-56\) \(-243\) \(3125\) \(27984\) \(+\) \(-\) \(q-56q^{2}-3^{5}q^{3}+1088q^{4}+5^{5}q^{5}+\cdots\)
15.12.a.b \(2\) \(11.525\) \(\Q(\sqrt{1609}) \) None \(-22\) \(486\) \(-6250\) \(-10864\) \(-\) \(+\) \(q+(-11-\beta )q^{2}+3^{5}q^{3}+(-318+22\beta )q^{4}+\cdots\)
15.12.a.c \(2\) \(11.525\) \(\Q(\sqrt{1801}) \) None \(-13\) \(-486\) \(-6250\) \(7784\) \(+\) \(+\) \(q+(-6-\beta )q^{2}-3^{5}q^{3}+(-1562+13\beta )q^{4}+\cdots\)
15.12.a.d \(3\) \(11.525\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-1\) \(729\) \(9375\) \(-14608\) \(-\) \(-\) \(q-\beta _{1}q^{2}+3^{5}q^{3}+(1585+3\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)