Properties

Label 15.10.a
Level 15
Weight 10
Character orbit a
Rep. character \(\chi_{15}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 4
Sturm bound 20
Trace bound 2

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 15.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(20\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(15))\).

Total New Old
Modular forms 20 6 14
Cusp forms 16 6 10
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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(4\)

Trace form

\(6q \) \(\mathstrut +\mathstrut 68q^{2} \) \(\mathstrut +\mathstrut 1538q^{4} \) \(\mathstrut -\mathstrut 3078q^{6} \) \(\mathstrut -\mathstrut 11428q^{7} \) \(\mathstrut +\mathstrut 67932q^{8} \) \(\mathstrut +\mathstrut 39366q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 68q^{2} \) \(\mathstrut +\mathstrut 1538q^{4} \) \(\mathstrut -\mathstrut 3078q^{6} \) \(\mathstrut -\mathstrut 11428q^{7} \) \(\mathstrut +\mathstrut 67932q^{8} \) \(\mathstrut +\mathstrut 39366q^{9} \) \(\mathstrut -\mathstrut 8750q^{10} \) \(\mathstrut -\mathstrut 87076q^{11} \) \(\mathstrut +\mathstrut 41472q^{12} \) \(\mathstrut +\mathstrut 55888q^{13} \) \(\mathstrut +\mathstrut 115452q^{14} \) \(\mathstrut -\mathstrut 101250q^{15} \) \(\mathstrut +\mathstrut 1085330q^{16} \) \(\mathstrut -\mathstrut 420520q^{17} \) \(\mathstrut +\mathstrut 446148q^{18} \) \(\mathstrut +\mathstrut 658912q^{19} \) \(\mathstrut -\mathstrut 905000q^{20} \) \(\mathstrut -\mathstrut 2241756q^{21} \) \(\mathstrut -\mathstrut 7156540q^{22} \) \(\mathstrut +\mathstrut 3122496q^{23} \) \(\mathstrut +\mathstrut 3193506q^{24} \) \(\mathstrut +\mathstrut 2343750q^{25} \) \(\mathstrut -\mathstrut 5892868q^{26} \) \(\mathstrut +\mathstrut 3780524q^{28} \) \(\mathstrut -\mathstrut 6996536q^{29} \) \(\mathstrut -\mathstrut 1620000q^{30} \) \(\mathstrut -\mathstrut 4504264q^{31} \) \(\mathstrut +\mathstrut 17859412q^{32} \) \(\mathstrut -\mathstrut 1195236q^{33} \) \(\mathstrut +\mathstrut 5709836q^{34} \) \(\mathstrut +\mathstrut 15182500q^{35} \) \(\mathstrut +\mathstrut 10090818q^{36} \) \(\mathstrut +\mathstrut 25085112q^{37} \) \(\mathstrut -\mathstrut 2385976q^{38} \) \(\mathstrut -\mathstrut 5547852q^{39} \) \(\mathstrut -\mathstrut 4751250q^{40} \) \(\mathstrut +\mathstrut 24274636q^{41} \) \(\mathstrut -\mathstrut 44096724q^{42} \) \(\mathstrut -\mathstrut 4089728q^{43} \) \(\mathstrut -\mathstrut 102869396q^{44} \) \(\mathstrut -\mathstrut 90064944q^{46} \) \(\mathstrut +\mathstrut 51128264q^{47} \) \(\mathstrut +\mathstrut 65192688q^{48} \) \(\mathstrut +\mathstrut 84594310q^{49} \) \(\mathstrut +\mathstrut 26562500q^{50} \) \(\mathstrut +\mathstrut 62822628q^{51} \) \(\mathstrut -\mathstrut 52677360q^{52} \) \(\mathstrut -\mathstrut 114062224q^{53} \) \(\mathstrut -\mathstrut 20194758q^{54} \) \(\mathstrut -\mathstrut 80472500q^{55} \) \(\mathstrut -\mathstrut 68883420q^{56} \) \(\mathstrut -\mathstrut 2667816q^{57} \) \(\mathstrut -\mathstrut 205991852q^{58} \) \(\mathstrut +\mathstrut 180942428q^{59} \) \(\mathstrut -\mathstrut 130916250q^{60} \) \(\mathstrut +\mathstrut 479495740q^{61} \) \(\mathstrut -\mathstrut 31830936q^{62} \) \(\mathstrut -\mathstrut 74979108q^{63} \) \(\mathstrut +\mathstrut 464210442q^{64} \) \(\mathstrut -\mathstrut 192047500q^{65} \) \(\mathstrut -\mathstrut 57426084q^{66} \) \(\mathstrut -\mathstrut 484354360q^{67} \) \(\mathstrut +\mathstrut 717440536q^{68} \) \(\mathstrut +\mathstrut 5244912q^{69} \) \(\mathstrut +\mathstrut 543322500q^{70} \) \(\mathstrut +\mathstrut 83143192q^{71} \) \(\mathstrut +\mathstrut 445701852q^{72} \) \(\mathstrut -\mathstrut 591042044q^{73} \) \(\mathstrut -\mathstrut 773558108q^{74} \) \(\mathstrut -\mathstrut 849378912q^{76} \) \(\mathstrut +\mathstrut 72064224q^{77} \) \(\mathstrut +\mathstrut 265485600q^{78} \) \(\mathstrut -\mathstrut 274527880q^{79} \) \(\mathstrut +\mathstrut 103030000q^{80} \) \(\mathstrut +\mathstrut 258280326q^{81} \) \(\mathstrut +\mathstrut 1526220184q^{82} \) \(\mathstrut +\mathstrut 804884184q^{83} \) \(\mathstrut -\mathstrut 1559325492q^{84} \) \(\mathstrut -\mathstrut 192897500q^{85} \) \(\mathstrut -\mathstrut 752029576q^{86} \) \(\mathstrut +\mathstrut 37145628q^{87} \) \(\mathstrut -\mathstrut 2168339028q^{88} \) \(\mathstrut -\mathstrut 329760348q^{89} \) \(\mathstrut -\mathstrut 57408750q^{90} \) \(\mathstrut -\mathstrut 871611944q^{91} \) \(\mathstrut -\mathstrut 2807106528q^{92} \) \(\mathstrut -\mathstrut 830087352q^{93} \) \(\mathstrut +\mathstrut 3271510568q^{94} \) \(\mathstrut +\mathstrut 363665000q^{95} \) \(\mathstrut +\mathstrut 2378839914q^{96} \) \(\mathstrut +\mathstrut 1387556460q^{97} \) \(\mathstrut +\mathstrut 5935831732q^{98} \) \(\mathstrut -\mathstrut 571305636q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.10.a.a \(1\) \(7.726\) \(\Q\) None \(-4\) \(81\) \(625\) \(-7680\) \(-\) \(-\) \(q-4q^{2}+3^{4}q^{3}-496q^{4}+5^{4}q^{5}+\cdots\)
15.10.a.b \(1\) \(7.726\) \(\Q\) None \(22\) \(-81\) \(-625\) \(-5988\) \(+\) \(+\) \(q+22q^{2}-3^{4}q^{3}-28q^{4}-5^{4}q^{5}+\cdots\)
15.10.a.c \(2\) \(7.726\) \(\Q(\sqrt{4729}) \) None \(19\) \(162\) \(-1250\) \(-11872\) \(-\) \(+\) \(q+(10-\beta )q^{2}+3^{4}q^{3}+(770-19\beta )q^{4}+\cdots\)
15.10.a.d \(2\) \(7.726\) \(\Q(\sqrt{241}) \) None \(31\) \(-162\) \(1250\) \(14112\) \(+\) \(-\) \(q+(15-\beta )q^{2}-3^{4}q^{3}+(255-31\beta )q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)