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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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(30))\) 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 3 5
30.6.a.a 30.a 1.a $1$ $4.812$ \(\Q\) None \(-4\) \(9\) \(-25\) \(164\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
30.6.a.b 30.a 1.a $1$ $4.812$ \(\Q\) None \(4\) \(9\) \(25\) \(32\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)

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

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