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Properties

Label 1638.2.bj
Level $1638$
Weight $2$
Character orbit 1638.bj
Rep. character $\chi_{1638}(127,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $72$
Newform subspaces $9$
Sturm bound $672$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bj (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 9 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1638, [\chi])\).

Total New Old
Modular forms 704 72 632
Cusp forms 640 72 568
Eisenstein series 64 0 64

Trace form

\( 72 q + 36 q^{4} + O(q^{10}) \) \( 72 q + 36 q^{4} - 12 q^{11} + 20 q^{13} - 8 q^{14} - 36 q^{16} - 4 q^{17} - 4 q^{22} - 8 q^{23} - 88 q^{25} + 4 q^{29} + 8 q^{35} - 32 q^{38} + 60 q^{41} - 12 q^{43} + 12 q^{46} + 36 q^{49} + 12 q^{50} + 4 q^{52} + 8 q^{53} + 28 q^{55} - 4 q^{56} + 12 q^{58} - 96 q^{59} - 4 q^{61} - 12 q^{62} - 72 q^{64} - 36 q^{67} + 4 q^{68} - 24 q^{71} + 8 q^{74} + 32 q^{77} + 8 q^{79} + 20 q^{82} + 4 q^{88} + 72 q^{89} - 4 q^{91} - 16 q^{92} - 16 q^{94} + 12 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1638, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1638.2.bj.a 1638.bj 13.e $4$ $13.079$ \(\Q(\zeta_{12})\) None 546.2.s.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{4}+\cdots\)
1638.2.bj.b 1638.bj 13.e $4$ $13.079$ \(\Q(\zeta_{12})\) None 546.2.s.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(1-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
1638.2.bj.c 1638.bj 13.e $4$ $13.079$ \(\Q(\zeta_{12})\) None 182.2.m.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{4}+\cdots\)
1638.2.bj.d 1638.bj 13.e $4$ $13.079$ \(\Q(\zeta_{12})\) None 546.2.s.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(1-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
1638.2.bj.e 1638.bj 13.e $4$ $13.079$ \(\Q(\zeta_{12})\) None 546.2.s.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{4}+\cdots\)
1638.2.bj.f 1638.bj 13.e $8$ $13.079$ 8.0.195105024.2 None 546.2.s.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{2}-\beta _{4}q^{4}+(1-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1638.2.bj.g 1638.bj 13.e $12$ $13.079$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 182.2.m.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{6}q^{2}+\beta _{7}q^{4}+(\beta _{1}-\beta _{5}+\beta _{6}-\beta _{7}+\cdots)q^{5}+\cdots\)
1638.2.bj.h 1638.bj 13.e $16$ $13.079$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1638.2.bj.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{4}q^{2}+(1-\beta _{5})q^{4}-\beta _{9}q^{5}-\beta _{3}q^{7}+\cdots\)
1638.2.bj.i 1638.bj 13.e $16$ $13.079$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1638.2.bj.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+\beta _{5}q^{4}-\beta _{9}q^{5}-\beta _{4}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1638, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1638, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(234, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(273, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(546, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(819, [\chi])\)\(^{\oplus 2}\)