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Properties

Label 30.4.a
Level $30$
Weight $4$
Character orbit 30.a
Rep. character $\chi_{30}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $24$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(30))\).

Total New Old
Modular forms 22 2 20
Cusp forms 14 2 12
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\(+\)\(2\)
Minus space\(-\)\(0\)

Trace form

\( 2 q + 6 q^{3} + 8 q^{4} + 28 q^{7} + 18 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{3} + 8 q^{4} + 28 q^{7} + 18 q^{9} - 20 q^{10} - 108 q^{11} + 24 q^{12} - 32 q^{13} - 72 q^{14} + 32 q^{16} - 72 q^{17} + 64 q^{19} + 84 q^{21} + 24 q^{22} + 72 q^{23} + 50 q^{25} + 72 q^{26} + 54 q^{27} + 112 q^{28} + 216 q^{29} - 60 q^{30} + 40 q^{31} - 324 q^{33} - 312 q^{34} + 180 q^{35} + 72 q^{36} - 200 q^{37} + 432 q^{38} - 96 q^{39} - 80 q^{40} + 36 q^{41} - 216 q^{42} - 680 q^{43} - 432 q^{44} + 144 q^{46} - 144 q^{47} + 96 q^{48} + 354 q^{49} - 216 q^{51} - 128 q^{52} + 144 q^{53} - 60 q^{55} - 288 q^{56} + 192 q^{57} + 408 q^{58} + 900 q^{59} - 188 q^{61} + 1008 q^{62} + 252 q^{63} + 128 q^{64} - 180 q^{65} + 72 q^{66} + 1408 q^{67} - 288 q^{68} + 216 q^{69} - 280 q^{70} - 648 q^{71} + 268 q^{73} - 936 q^{74} + 150 q^{75} + 256 q^{76} - 1728 q^{77} + 216 q^{78} - 488 q^{79} + 162 q^{81} - 864 q^{82} - 1152 q^{83} + 336 q^{84} + 780 q^{85} + 288 q^{86} + 648 q^{87} + 96 q^{88} - 468 q^{89} - 180 q^{90} - 1096 q^{91} + 288 q^{92} + 120 q^{93} + 1152 q^{94} - 1080 q^{95} - 1340 q^{97} - 2016 q^{98} - 972 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(30))\) 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
30.4.a.a 30.a 1.a $1$ $1.770$ \(\Q\) None \(-2\) \(3\) \(5\) \(32\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
30.4.a.b 30.a 1.a $1$ $1.770$ \(\Q\) None \(2\) \(3\) \(-5\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

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