Properties

Label 15.8.a
Level 15
Weight 8
Character orbit a
Rep. character \(\chi_{15}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 3
Sturm bound 16
Trace bound 2

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 15.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(15))\).

Total New Old
Modular forms 16 4 12
Cusp forms 12 4 8
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.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(1\)

Trace form

\(4q \) \(\mathstrut -\mathstrut 28q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 466q^{4} \) \(\mathstrut -\mathstrut 54q^{6} \) \(\mathstrut +\mathstrut 2264q^{7} \) \(\mathstrut -\mathstrut 2436q^{8} \) \(\mathstrut +\mathstrut 2916q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 28q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 466q^{4} \) \(\mathstrut -\mathstrut 54q^{6} \) \(\mathstrut +\mathstrut 2264q^{7} \) \(\mathstrut -\mathstrut 2436q^{8} \) \(\mathstrut +\mathstrut 2916q^{9} \) \(\mathstrut +\mathstrut 5250q^{10} \) \(\mathstrut -\mathstrut 2800q^{11} \) \(\mathstrut +\mathstrut 10368q^{12} \) \(\mathstrut -\mathstrut 11488q^{13} \) \(\mathstrut -\mathstrut 20964q^{14} \) \(\mathstrut +\mathstrut 6750q^{15} \) \(\mathstrut +\mathstrut 20338q^{16} \) \(\mathstrut -\mathstrut 32032q^{17} \) \(\mathstrut -\mathstrut 20412q^{18} \) \(\mathstrut -\mathstrut 33712q^{19} \) \(\mathstrut -\mathstrut 41000q^{20} \) \(\mathstrut -\mathstrut 13392q^{21} \) \(\mathstrut -\mathstrut 19644q^{22} \) \(\mathstrut +\mathstrut 170088q^{23} \) \(\mathstrut -\mathstrut 126846q^{24} \) \(\mathstrut +\mathstrut 62500q^{25} \) \(\mathstrut +\mathstrut 347708q^{26} \) \(\mathstrut +\mathstrut 39366q^{27} \) \(\mathstrut -\mathstrut 165748q^{28} \) \(\mathstrut +\mathstrut 477616q^{29} \) \(\mathstrut +\mathstrut 54000q^{30} \) \(\mathstrut +\mathstrut 46832q^{31} \) \(\mathstrut -\mathstrut 1063916q^{32} \) \(\mathstrut +\mathstrut 102816q^{33} \) \(\mathstrut +\mathstrut 54012q^{34} \) \(\mathstrut +\mathstrut 43000q^{35} \) \(\mathstrut +\mathstrut 339714q^{36} \) \(\mathstrut -\mathstrut 153616q^{37} \) \(\mathstrut -\mathstrut 769432q^{38} \) \(\mathstrut -\mathstrut 769500q^{39} \) \(\mathstrut +\mathstrut 666750q^{40} \) \(\mathstrut +\mathstrut 375592q^{41} \) \(\mathstrut +\mathstrut 402732q^{42} \) \(\mathstrut -\mathstrut 421000q^{43} \) \(\mathstrut -\mathstrut 2040596q^{44} \) \(\mathstrut +\mathstrut 922272q^{46} \) \(\mathstrut +\mathstrut 1210520q^{47} \) \(\mathstrut +\mathstrut 1626480q^{48} \) \(\mathstrut +\mathstrut 579204q^{49} \) \(\mathstrut -\mathstrut 437500q^{50} \) \(\mathstrut -\mathstrut 325188q^{51} \) \(\mathstrut -\mathstrut 2146192q^{52} \) \(\mathstrut +\mathstrut 1575488q^{53} \) \(\mathstrut -\mathstrut 39366q^{54} \) \(\mathstrut +\mathstrut 1212000q^{55} \) \(\mathstrut +\mathstrut 5773380q^{56} \) \(\mathstrut -\mathstrut 3136968q^{57} \) \(\mathstrut -\mathstrut 2186604q^{58} \) \(\mathstrut -\mathstrut 2288608q^{59} \) \(\mathstrut -\mathstrut 830250q^{60} \) \(\mathstrut -\mathstrut 4545304q^{61} \) \(\mathstrut +\mathstrut 5573928q^{62} \) \(\mathstrut +\mathstrut 1650456q^{63} \) \(\mathstrut +\mathstrut 10293082q^{64} \) \(\mathstrut -\mathstrut 811000q^{65} \) \(\mathstrut -\mathstrut 2849796q^{66} \) \(\mathstrut -\mathstrut 5036632q^{67} \) \(\mathstrut -\mathstrut 9434504q^{68} \) \(\mathstrut +\mathstrut 49896q^{69} \) \(\mathstrut -\mathstrut 445500q^{70} \) \(\mathstrut -\mathstrut 2625392q^{71} \) \(\mathstrut -\mathstrut 1775844q^{72} \) \(\mathstrut +\mathstrut 2617208q^{73} \) \(\mathstrut -\mathstrut 3041564q^{74} \) \(\mathstrut +\mathstrut 843750q^{75} \) \(\mathstrut -\mathstrut 13374352q^{76} \) \(\mathstrut +\mathstrut 6733248q^{77} \) \(\mathstrut +\mathstrut 15359328q^{78} \) \(\mathstrut +\mathstrut 3505280q^{79} \) \(\mathstrut -\mathstrut 8666000q^{80} \) \(\mathstrut +\mathstrut 2125764q^{81} \) \(\mathstrut +\mathstrut 13003128q^{82} \) \(\mathstrut -\mathstrut 17321256q^{83} \) \(\mathstrut -\mathstrut 7530516q^{84} \) \(\mathstrut +\mathstrut 2631000q^{85} \) \(\mathstrut +\mathstrut 13624760q^{86} \) \(\mathstrut +\mathstrut 5724324q^{87} \) \(\mathstrut +\mathstrut 23149164q^{88} \) \(\mathstrut +\mathstrut 12431928q^{89} \) \(\mathstrut +\mathstrut 3827250q^{90} \) \(\mathstrut -\mathstrut 3365648q^{91} \) \(\mathstrut +\mathstrut 15479424q^{92} \) \(\mathstrut +\mathstrut 4277664q^{93} \) \(\mathstrut -\mathstrut 32795880q^{94} \) \(\mathstrut -\mathstrut 8182000q^{95} \) \(\mathstrut -\mathstrut 34913862q^{96} \) \(\mathstrut +\mathstrut 11522888q^{97} \) \(\mathstrut -\mathstrut 21248492q^{98} \) \(\mathstrut -\mathstrut 2041200q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.8.a.a \(1\) \(4.686\) \(\Q\) None \(-22\) \(27\) \(-125\) \(-420\) \(-\) \(+\) \(q-22q^{2}+3^{3}q^{3}+356q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.b \(1\) \(4.686\) \(\Q\) None \(-13\) \(-27\) \(-125\) \(1380\) \(+\) \(+\) \(q-13q^{2}-3^{3}q^{3}+41q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.c \(2\) \(4.686\) \(\Q(\sqrt{601}) \) None \(7\) \(54\) \(250\) \(1304\) \(-\) \(-\) \(q+(4-\beta )q^{2}+3^{3}q^{3}+(38-7\beta )q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)