Properties

Label 48.4.a
Level $48$
Weight $4$
Character orbit 48.a
Rep. character $\chi_{48}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $32$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(48))\).

Total New Old
Modular forms 30 3 27
Cusp forms 18 3 15
Eisenstein series 12 0 12

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

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

Trace form

\( 3 q - 3 q^{3} + 2 q^{5} + 32 q^{7} + 27 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 2 q^{5} + 32 q^{7} + 27 q^{9} - 20 q^{11} - 46 q^{13} + 30 q^{15} - 26 q^{17} - 12 q^{19} - 248 q^{23} + 181 q^{25} - 27 q^{27} - 342 q^{29} + 24 q^{31} - 12 q^{33} + 576 q^{35} + 58 q^{37} + 366 q^{39} + 414 q^{41} - 404 q^{43} + 18 q^{45} - 576 q^{47} - 133 q^{49} - 678 q^{51} + 482 q^{53} + 968 q^{55} - 84 q^{57} + 748 q^{59} - 334 q^{61} + 288 q^{63} - 628 q^{65} - 780 q^{67} - 264 q^{69} - 1288 q^{71} - 754 q^{73} - 1077 q^{75} + 768 q^{77} + 40 q^{79} + 243 q^{81} + 3188 q^{83} + 68 q^{85} + 1206 q^{87} - 66 q^{89} - 1088 q^{91} + 456 q^{93} - 3208 q^{95} + 966 q^{97} - 180 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
48.4.a.a 48.a 1.a $1$ $2.832$ \(\Q\) None 12.4.a.a \(0\) \(-3\) \(-18\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-18q^{5}-8q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
48.4.a.b 48.a 1.a $1$ $2.832$ \(\Q\) None 24.4.a.a \(0\) \(-3\) \(14\) \(24\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+14q^{5}+24q^{7}+9q^{9}+28q^{11}+\cdots\)
48.4.a.c 48.a 1.a $1$ $2.832$ \(\Q\) None 6.4.a.a \(0\) \(3\) \(6\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+6q^{5}+2^{4}q^{7}+9q^{9}-12q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(48)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)