Properties

Label 900.4.d
Level $900$
Weight $4$
Character orbit 900.d
Rep. character $\chi_{900}(649,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $11$
Sturm bound $720$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(720\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(900, [\chi])\).

Total New Old
Modular forms 576 22 554
Cusp forms 504 22 482
Eisenstein series 72 0 72

Trace form

\( 22 q + O(q^{10}) \) \( 22 q + 30 q^{11} + 118 q^{19} + 96 q^{29} - 508 q^{31} + 1446 q^{41} - 1830 q^{49} - 648 q^{59} - 460 q^{61} + 2328 q^{71} + 580 q^{79} - 2946 q^{89} - 3968 q^{91} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(900, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
900.4.d.a 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+26iq^{7}-45q^{11}-44iq^{13}-117iq^{17}+\cdots\)
900.4.d.b 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2^{4}iq^{7}-6^{2}q^{11}+5iq^{13}+39iq^{17}+\cdots\)
900.4.d.c 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{7}-6^{2}q^{11}+5iq^{13}-9iq^{17}+\cdots\)
900.4.d.d 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}-30q^{11}+2iq^{13}-45iq^{17}+\cdots\)
900.4.d.e 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+13iq^{7}-6q^{11}+5iq^{13}-78iq^{17}+\cdots\)
900.4.d.f 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+17iq^{7}+19iq^{13}-107q^{19}-19q^{31}+\cdots\)
900.4.d.g 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+37iq^{7}+89iq^{13}+163q^{19}-17^{2}q^{31}+\cdots\)
900.4.d.h 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+14iq^{7}+24q^{11}-35iq^{13}+51iq^{17}+\cdots\)
900.4.d.i 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+30q^{11}+2iq^{13}+45iq^{17}+\cdots\)
900.4.d.j 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+7iq^{7}+54q^{11}-55iq^{13}+18iq^{17}+\cdots\)
900.4.d.k 900.d 5.b $2$ $53.102$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8iq^{7}+60q^{11}+43iq^{13}+9iq^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(900, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)