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Properties

Label 1530.2.d
Level $1530$
Weight $2$
Character orbit 1530.d
Rep. character $\chi_{1530}(919,\cdot)$
Character field $\Q$
Dimension $40$
Newform subspaces $10$
Sturm bound $648$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(648\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1530, [\chi])\).

Total New Old
Modular forms 340 40 300
Cusp forms 308 40 268
Eisenstein series 32 0 32

Trace form

\( 40 q - 40 q^{4} - 4 q^{5} + O(q^{10}) \) \( 40 q - 40 q^{4} - 4 q^{5} + 4 q^{10} - 8 q^{11} + 8 q^{14} + 40 q^{16} - 4 q^{19} + 4 q^{20} - 12 q^{25} - 4 q^{26} - 8 q^{29} + 32 q^{31} - 4 q^{34} - 20 q^{35} - 4 q^{40} + 8 q^{41} + 8 q^{44} - 8 q^{46} - 32 q^{49} + 4 q^{50} - 8 q^{55} - 8 q^{56} - 12 q^{59} + 8 q^{61} - 40 q^{64} - 8 q^{65} + 4 q^{70} + 48 q^{71} + 8 q^{74} + 4 q^{76} - 40 q^{79} - 4 q^{80} - 4 q^{85} + 24 q^{86} + 12 q^{89} + 56 q^{91} + 20 q^{94} - 16 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1530, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1530.2.d.a 1530.d 5.b $2$ $12.217$ \(\Q(\sqrt{-1}) \) None 1530.2.d.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-i q^{2}-q^{4}+(i-2)q^{5}+i q^{8}+\cdots\)
1530.2.d.b 1530.d 5.b $2$ $12.217$ \(\Q(\sqrt{-1}) \) None 170.2.c.a \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-i q^{2}-q^{4}+(-2 i-1)q^{5}+i q^{8}+\cdots\)
1530.2.d.c 1530.d 5.b $2$ $12.217$ \(\Q(\sqrt{-1}) \) None 510.2.d.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+i q^{2}-q^{4}+(i+2)q^{5}+4 i q^{7}+\cdots\)
1530.2.d.d 1530.d 5.b $2$ $12.217$ \(\Q(\sqrt{-1}) \) None 1530.2.d.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-i q^{2}-q^{4}+(i+2)q^{5}+i q^{8}+\cdots\)
1530.2.d.e 1530.d 5.b $4$ $12.217$ \(\Q(i, \sqrt{6})\) None 510.2.d.c \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-q^{4}+(-1+\beta _{1}+\beta _{2})q^{5}+\cdots\)
1530.2.d.f 1530.d 5.b $4$ $12.217$ \(\Q(i, \sqrt{5})\) None 510.2.d.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-q^{4}-\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1530.2.d.g 1530.d 5.b $6$ $12.217$ 6.0.5161984.1 None 170.2.c.b \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}-q^{4}+(\beta _{1}+\beta _{3})q^{5}+(\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1530.2.d.h 1530.d 5.b $6$ $12.217$ 6.0.350464.1 None 1530.2.d.h \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-q^{4}-\beta _{2}q^{5}+(-\beta _{2}-\beta _{5})q^{7}+\cdots\)
1530.2.d.i 1530.d 5.b $6$ $12.217$ 6.0.5161984.1 None 510.2.d.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-q^{4}-\beta _{4}q^{5}+(\beta _{1}+\beta _{2}+2\beta _{3}+\cdots)q^{7}+\cdots\)
1530.2.d.j 1530.d 5.b $6$ $12.217$ 6.0.350464.1 None 1530.2.d.h \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-q^{4}+\beta _{2}q^{5}+(-\beta _{2}-\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1530, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1530, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(255, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(510, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(765, [\chi])\)\(^{\oplus 2}\)