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Properties

Label 825.2.k
Level $825$
Weight $2$
Character orbit 825.k
Rep. character $\chi_{825}(518,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $120$
Newform subspaces $12$
Sturm bound $240$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 12 \)
Sturm bound: \(240\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(2\), \(7\), \(13\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(825, [\chi])\).

Total New Old
Modular forms 264 120 144
Cusp forms 216 120 96
Eisenstein series 48 0 48

Trace form

\( 120 q + 2 q^{3} + 16 q^{6} + O(q^{10}) \) \( 120 q + 2 q^{3} + 16 q^{6} - 8 q^{12} + 16 q^{13} - 136 q^{16} + 12 q^{18} + 32 q^{21} - 4 q^{27} - 32 q^{28} + 24 q^{31} - 2 q^{33} - 48 q^{36} + 36 q^{37} + 64 q^{42} + 4 q^{48} - 32 q^{51} - 88 q^{52} - 40 q^{57} - 8 q^{58} + 40 q^{61} - 4 q^{63} + 28 q^{67} - 48 q^{72} - 32 q^{76} + 40 q^{78} + 164 q^{81} - 8 q^{82} - 16 q^{87} + 24 q^{88} - 24 q^{91} - 58 q^{93} - 256 q^{96} - 76 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(825, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
825.2.k.a 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(-4\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.b 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(-4\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.c 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(-4\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.d 825.k 15.e $4$ $6.588$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}-3\beta _{2}q^{6}+\cdots\)
825.2.k.e 825.k 15.e $4$ $6.588$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+3\beta _{2}q^{6}+\cdots\)
825.2.k.f 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(4\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\cdots\)
825.2.k.g 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(4\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots\)
825.2.k.h 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None \(4\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots\)
825.2.k.i 825.k 15.e $16$ $6.588$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{2}-\beta _{4}q^{3}+(-\beta _{1}-\beta _{3}-\beta _{13}+\cdots)q^{4}+\cdots\)
825.2.k.j 825.k 15.e $16$ $6.588$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(2\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{14}q^{3}+(\beta _{1}+\beta _{3}+\beta _{13}+\cdots)q^{4}+\cdots\)
825.2.k.k 825.k 15.e $28$ $6.588$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
825.2.k.l 825.k 15.e $28$ $6.588$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(825, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(825, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(165, [\chi])\)\(^{\oplus 2}\)