Properties

Label 48.13.g
Level $48$
Weight $13$
Character orbit 48.g
Rep. character $\chi_{48}(31,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $3$
Sturm bound $104$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(104\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(48, [\chi])\).

Total New Old
Modular forms 102 12 90
Cusp forms 90 12 78
Eisenstein series 12 0 12

Trace form

\( 12 q + 30888 q^{5} - 2125764 q^{9} + O(q^{10}) \) \( 12 q + 30888 q^{5} - 2125764 q^{9} - 8834760 q^{13} + 7258680 q^{17} + 75057840 q^{21} + 619612260 q^{25} - 1765403640 q^{29} - 966304080 q^{33} + 120720600 q^{37} + 10244732472 q^{41} - 5471716536 q^{45} + 23307267660 q^{49} - 60434995320 q^{53} + 19439980560 q^{57} - 175710390120 q^{61} + 332241524496 q^{65} + 506726668440 q^{73} - 1143003631680 q^{77} + 376572715308 q^{81} - 1463329887984 q^{85} + 2444968318104 q^{89} - 1744253397360 q^{93} + 3334060903320 q^{97} + O(q^{100}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
48.13.g.a 48.g 4.b $4$ $43.872$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 48.13.g.a \(0\) \(0\) \(-21960\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{5}\beta _{1}q^{3}+(-5490-5\beta _{2})q^{5}+(-5372\beta _{1}+\cdots)q^{7}+\cdots\)
48.13.g.b 48.g 4.b $4$ $43.872$ \(\Q(\sqrt{-3}, \sqrt{-2803})\) None 48.13.g.b \(0\) \(0\) \(22392\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(5598+\beta _{2})q^{5}+(-63\beta _{1}+\cdots)q^{7}+\cdots\)
48.13.g.c 48.g 4.b $4$ $43.872$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 48.13.g.c \(0\) \(0\) \(30456\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3^{5}\beta _{1}q^{3}+(7614-\beta _{2})q^{5}+(-15668\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{13}^{\mathrm{old}}(48, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(48, [\chi]) \simeq \) \(S_{13}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)