Properties

Label 10.6.a
Level $10$
Weight $6$
Character orbit 10.a
Rep. character $\chi_{10}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $9$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(9\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(10))\).

Total New Old
Modular forms 9 3 6
Cusp forms 5 3 2
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.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3 q - 4 q^{2} + 4 q^{3} + 48 q^{4} - 25 q^{5} + 32 q^{6} - 312 q^{7} - 64 q^{8} + 559 q^{9} + O(q^{10}) \) \( 3 q - 4 q^{2} + 4 q^{3} + 48 q^{4} - 25 q^{5} + 32 q^{6} - 312 q^{7} - 64 q^{8} + 559 q^{9} - 100 q^{10} - 444 q^{11} + 64 q^{12} + 114 q^{13} + 304 q^{14} + 1100 q^{15} + 768 q^{16} + 918 q^{17} - 3892 q^{18} - 1140 q^{19} - 400 q^{20} - 4264 q^{21} + 3312 q^{22} + 3144 q^{23} + 512 q^{24} + 1875 q^{25} + 8392 q^{26} - 5480 q^{27} - 4992 q^{28} + 2490 q^{29} - 5600 q^{30} - 8544 q^{31} - 1024 q^{32} + 24288 q^{33} + 2424 q^{34} - 800 q^{35} + 8944 q^{36} - 16902 q^{37} - 17360 q^{38} - 14872 q^{39} - 1600 q^{40} + 2406 q^{41} + 11392 q^{42} + 13164 q^{43} - 7104 q^{44} + 2675 q^{45} - 48 q^{46} + 44448 q^{47} + 1024 q^{48} - 6429 q^{49} - 2500 q^{50} - 10584 q^{51} + 1824 q^{52} - 44406 q^{53} + 320 q^{54} + 17700 q^{55} + 4864 q^{56} - 33040 q^{57} + 37320 q^{58} + 1380 q^{59} + 17600 q^{60} + 12306 q^{61} - 20768 q^{62} - 42376 q^{63} + 12288 q^{64} - 50150 q^{65} - 87936 q^{66} + 10788 q^{67} + 14688 q^{68} + 146568 q^{69} + 26800 q^{70} + 88056 q^{71} - 62272 q^{72} - 9666 q^{73} + 23464 q^{74} + 2500 q^{75} - 18240 q^{76} - 28464 q^{77} + 112576 q^{78} - 81360 q^{79} - 6400 q^{80} + 28243 q^{81} - 12648 q^{82} - 220476 q^{83} - 68224 q^{84} - 34050 q^{85} - 72128 q^{86} + 233880 q^{87} + 52992 q^{88} - 14130 q^{89} + 30700 q^{90} + 33216 q^{91} + 50304 q^{92} + 117968 q^{93} - 72816 q^{94} + 53500 q^{95} + 8192 q^{96} - 224682 q^{97} + 2652 q^{98} - 328332 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
10.6.a.a 10.a 1.a $1$ $1.604$ \(\Q\) None \(-4\) \(-26\) \(-25\) \(-22\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-26q^{3}+2^{4}q^{4}-5^{2}q^{5}+104q^{6}+\cdots\)
10.6.a.b 10.a 1.a $1$ $1.604$ \(\Q\) None \(-4\) \(24\) \(25\) \(-172\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+24q^{3}+2^{4}q^{4}+5^{2}q^{5}-96q^{6}+\cdots\)
10.6.a.c 10.a 1.a $1$ $1.604$ \(\Q\) None \(4\) \(6\) \(-25\) \(-118\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+6q^{3}+2^{4}q^{4}-5^{2}q^{5}+24q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)