Properties

Label 15.12.a
Level $15$
Weight $12$
Character orbit 15.a
Rep. character $\chi_{15}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
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: \(\Q\)
Newform subspaces: \( 4 \)
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

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

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
15.12.a.a 15.a 1.a $1$ $11.525$ \(\Q\) None \(-56\) \(-243\) \(3125\) \(27984\) $+$ $-$ $\mathrm{SU}(2)$ \(q-56q^{2}-3^{5}q^{3}+1088q^{4}+5^{5}q^{5}+\cdots\)
15.12.a.b 15.a 1.a $2$ $11.525$ \(\Q(\sqrt{1609}) \) None \(-22\) \(486\) \(-6250\) \(-10864\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-11-\beta )q^{2}+3^{5}q^{3}+(-318+22\beta )q^{4}+\cdots\)
15.12.a.c 15.a 1.a $2$ $11.525$ \(\Q(\sqrt{1801}) \) None \(-13\) \(-486\) \(-6250\) \(7784\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{2}-3^{5}q^{3}+(-1562+13\beta )q^{4}+\cdots\)
15.12.a.d 15.a 1.a $3$ $11.525$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-1\) \(729\) \(9375\) \(-14608\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3^{5}q^{3}+(1585+3\beta _{1}+\beta _{2})q^{4}+\cdots\)

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

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