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

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

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

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