Properties

Label 225.3.g
Level $225$
Weight $3$
Character orbit 225.g
Rep. character $\chi_{225}(82,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $28$
Newform subspaces $7$
Sturm bound $90$
Trace bound $16$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(90\)
Trace bound: \(16\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(225, [\chi])\).

Total New Old
Modular forms 144 32 112
Cusp forms 96 28 68
Eisenstein series 48 4 44

Trace form

\( 28 q - 4 q^{2} + 16 q^{7} + 12 q^{8} + O(q^{10}) \) \( 28 q - 4 q^{2} + 16 q^{7} + 12 q^{8} + 44 q^{11} - 8 q^{13} - 228 q^{16} - 40 q^{17} + 80 q^{22} + 56 q^{23} + 128 q^{26} - 64 q^{28} + 96 q^{31} - 76 q^{32} - 104 q^{37} - 96 q^{38} + 140 q^{41} - 32 q^{43} + 520 q^{46} + 128 q^{47} + 40 q^{52} + 56 q^{53} - 600 q^{56} + 312 q^{58} - 568 q^{61} + 88 q^{62} - 80 q^{67} - 104 q^{68} - 712 q^{71} - 296 q^{73} - 140 q^{76} + 88 q^{77} - 328 q^{82} - 16 q^{83} + 1064 q^{86} + 288 q^{88} + 520 q^{91} + 104 q^{92} + 40 q^{97} - 188 q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.3.g.a 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
225.3.g.b 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+2\beta _{1}q^{2}+8\beta _{2}q^{4}+5\beta _{1}q^{7}+8\beta _{3}q^{8}+\cdots\)
225.3.g.c 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{30})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}+7\beta _{3}q^{8}-61q^{16}+\cdots\)
225.3.g.d 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-4\beta _{2}q^{4}+\beta _{1}q^{7}-3\beta _{3}q^{13}-2^{4}q^{16}+\cdots\)
225.3.g.e 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{4}+4\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots\)
225.3.g.f 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{4}-6\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots\)
225.3.g.g 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(5+5\beta _{2})q^{7}-3\beta _{3}q^{8}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)