Properties

Label 30.6.a
Level $30$
Weight $6$
Character orbit 30.a
Rep. character $\chi_{30}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $36$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(30))\).

Total New Old
Modular forms 34 2 32
Cusp forms 26 2 24
Eisenstein series 8 0 8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2q + 18q^{3} + 32q^{4} + 196q^{7} + 162q^{9} + O(q^{10}) \) \( 2q + 18q^{3} + 32q^{4} + 196q^{7} + 162q^{9} + 200q^{10} + 732q^{11} + 288q^{12} + 544q^{13} - 528q^{14} + 512q^{16} - 3144q^{17} - 704q^{19} + 1764q^{21} - 2832q^{22} - 6024q^{23} + 1250q^{25} - 3408q^{26} + 1458q^{27} + 3136q^{28} - 7176q^{29} + 1800q^{30} + 6040q^{31} + 6588q^{33} + 5232q^{34} - 3300q^{35} + 2592q^{36} + 5080q^{37} - 5664q^{38} + 4896q^{39} + 3200q^{40} + 12996q^{41} - 4752q^{42} + 9160q^{43} + 11712q^{44} - 9696q^{46} + 12432q^{47} + 4608q^{48} - 5694q^{49} - 28296q^{51} + 8704q^{52} - 1968q^{53} - 17700q^{55} - 8448q^{56} - 6336q^{57} - 34416q^{58} - 6420q^{59} - 61388q^{61} + 17376q^{62} + 15876q^{63} + 8192q^{64} - 21300q^{65} - 25488q^{66} - 32384q^{67} - 50304q^{68} - 54216q^{69} + 19600q^{70} + 86472q^{71} + 98284q^{73} + 110736q^{74} + 11250q^{75} - 11264q^{76} + 118464q^{77} - 30672q^{78} - 83672q^{79} + 13122q^{81} - 22848q^{82} + 46464q^{83} + 28224q^{84} + 32700q^{85} + 84288q^{86} - 64584q^{87} - 45312q^{88} - 64692q^{89} + 16200q^{90} + 109544q^{91} - 96384q^{92} + 54360q^{93} + 139008q^{94} - 35400q^{95} - 39740q^{97} - 103488q^{98} + 59292q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(30))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
30.6.a.a \(1\) \(4.812\) \(\Q\) None \(-4\) \(9\) \(-25\) \(164\) \(+\) \(-\) \(+\) \(q-4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
30.6.a.b \(1\) \(4.812\) \(\Q\) None \(4\) \(9\) \(25\) \(32\) \(-\) \(-\) \(-\) \(q+4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(30)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)