Properties

Label 1300.2.d
Level $1300$
Weight $2$
Character orbit 1300.d
Rep. character $\chi_{1300}(649,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $4$
Sturm bound $420$
Trace bound $13$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(420\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1300, [\chi])\).

Total New Old
Modular forms 228 20 208
Cusp forms 192 20 172
Eisenstein series 36 0 36

Trace form

\( 20 q - 8 q^{9} + O(q^{10}) \) \( 20 q - 8 q^{9} - 20 q^{39} + 16 q^{49} + 48 q^{51} + 8 q^{61} + 12 q^{69} + 4 q^{79} - 76 q^{81} + 48 q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1300.2.d.a 1300.d 65.d $4$ $10.381$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\zeta_{12}q^{3}-\zeta_{12}^{3}q^{7}-q^{9}+3\zeta_{12}^{2}q^{11}+\cdots\)
1300.2.d.b 1300.d 65.d $4$ $10.381$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{7}+2q^{9}+(\zeta_{12}+\cdots)q^{13}+\cdots\)
1300.2.d.c 1300.d 65.d $6$ $10.381$ 6.0.9144576.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{3}q^{7}+(-1+\beta _{3})q^{9}+(\beta _{4}+\cdots)q^{11}+\cdots\)
1300.2.d.d 1300.d 65.d $6$ $10.381$ 6.0.9144576.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(-1+\beta _{3})q^{9}+(-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)