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Properties

Label 450.3.g
Level $450$
Weight $3$
Character orbit 450.g
Rep. character $\chi_{450}(307,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $30$
Newform subspaces $10$
Sturm bound $270$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 10 \)
Sturm bound: \(270\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(450, [\chi])\).

Total New Old
Modular forms 408 30 378
Cusp forms 312 30 282
Eisenstein series 96 0 96

Trace form

\( 30 q + 2 q^{2} - 20 q^{7} - 4 q^{8} + O(q^{10}) \) \( 30 q + 2 q^{2} - 20 q^{7} - 4 q^{8} - 64 q^{11} - 6 q^{13} - 120 q^{16} + 58 q^{17} + 16 q^{22} - 20 q^{23} + 84 q^{26} + 40 q^{28} + 24 q^{31} - 8 q^{32} + 126 q^{37} - 104 q^{38} - 208 q^{41} - 108 q^{43} - 272 q^{46} + 60 q^{47} - 12 q^{52} + 206 q^{53} + 160 q^{56} + 8 q^{58} + 688 q^{61} - 184 q^{62} - 284 q^{67} - 116 q^{68} + 376 q^{71} + 62 q^{73} - 112 q^{76} - 160 q^{77} + 232 q^{82} + 228 q^{83} - 192 q^{86} - 32 q^{88} - 600 q^{91} - 40 q^{92} + 222 q^{97} - 206 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.3.g.a 450.g 5.c $2$ $12.262$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(-8-8i)q^{7}+\cdots\)
450.3.g.b 450.g 5.c $2$ $12.262$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(-2-2i)q^{7}+\cdots\)
450.3.g.c 450.g 5.c $2$ $12.262$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(3+3i)q^{7}+\cdots\)
450.3.g.d 450.g 5.c $2$ $12.262$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(-8-8i)q^{7}+\cdots\)
450.3.g.e 450.g 5.c $2$ $12.262$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(-3-3i)q^{7}+\cdots\)
450.3.g.f 450.g 5.c $4$ $12.262$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+\cdots\)
450.3.g.g 450.g 5.c $4$ $12.262$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(24\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{2}-2\beta _{2}q^{4}+(6-6\beta _{2}+\cdots)q^{7}+\cdots\)
450.3.g.h 450.g 5.c $4$ $12.262$ \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{2})q^{2}-2\beta _{2}q^{4}+(-6+6\beta _{2}+\cdots)q^{7}+\cdots\)
450.3.g.i 450.g 5.c $4$ $12.262$ \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+(-2+\cdots)q^{8}+\cdots\)
450.3.g.j 450.g 5.c $4$ $12.262$ \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+(4+\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

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