Properties

Label 15.8.a
Level $15$
Weight $8$
Character orbit 15.a
Rep. character $\chi_{15}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

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

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

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