Properties

Label 48.13.e
Level $48$
Weight $13$
Character orbit 48.e
Rep. character $\chi_{48}(17,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $5$
Sturm bound $104$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(104\)
Trace bound: \(3\)
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 25 77
Cusp forms 90 23 67
Eisenstein series 12 2 10

Trace form

\( 23 q + q^{3} + 102962 q^{7} - 302681 q^{9} + O(q^{10}) \) \( 23 q + q^{3} + 102962 q^{7} - 302681 q^{9} - 2 q^{13} + 4135936 q^{15} + 27398978 q^{19} + 36362910 q^{21} - 1059968185 q^{25} + 1114602481 q^{27} + 2458372754 q^{31} - 489875200 q^{33} - 3136092962 q^{37} - 1081717070 q^{39} + 12606938402 q^{43} + 4844890624 q^{45} + 40136357349 q^{49} - 20126808064 q^{51} - 12844836864 q^{55} + 68937822478 q^{57} + 114025416382 q^{61} + 31269068562 q^{63} + 66406114562 q^{67} + 54365513216 q^{69} - 186350447282 q^{73} + 429596529233 q^{75} + 341505538130 q^{79} - 240963053513 q^{81} - 135025760256 q^{85} - 39159659520 q^{87} + 530519671492 q^{91} - 927192127682 q^{93} + 415013715118 q^{97} - 1358551601152 q^{99} + O(q^{100}) \)

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.e.a 48.e 3.b $1$ $43.872$ \(\Q\) \(\Q(\sqrt{-3}) \) 3.13.b.a \(0\) \(-729\) \(0\) \(153502\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{6}q^{3}+153502q^{7}+3^{12}q^{9}-9397582q^{13}+\cdots\)
48.13.e.b 48.e 3.b $2$ $43.872$ \(\Q(\sqrt{-26}) \) None 3.13.b.b \(0\) \(1350\) \(0\) \(-80500\) $\mathrm{SU}(2)[C_{2}]$ \(q+(675+3\beta )q^{3}+230\beta q^{5}-40250q^{7}+\cdots\)
48.13.e.c 48.e 3.b $4$ $43.872$ \(\Q(\sqrt{-2}, \sqrt{1009})\) None 6.13.b.a \(0\) \(-780\) \(0\) \(-153080\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-195+\beta _{3})q^{3}+(-18\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
48.13.e.d 48.e 3.b $4$ $43.872$ \(\mathbb{Q}[x]/(x^{4} + \cdots)\) None 12.13.c.a \(0\) \(-300\) \(0\) \(-15800\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-75+\beta _{1})q^{3}+(3\beta _{1}-\beta _{2})q^{5}+(-3950+\cdots)q^{7}+\cdots\)
48.13.e.e 48.e 3.b $12$ $43.872$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 24.13.e.a \(0\) \(460\) \(0\) \(198840\) $\mathrm{SU}(2)[C_{2}]$ \(q+(38-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(16571+\cdots)q^{7}+\cdots\)

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

\( S_{13}^{\mathrm{old}}(48, [\chi]) \simeq \) \(S_{13}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)