Properties

Label 825.2.a
Level $825$
Weight $2$
Character orbit 825.a
Rep. character $\chi_{825}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $14$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(825))\).

Total New Old
Modular forms 132 32 100
Cusp forms 109 32 77
Eisenstein series 23 0 23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(11\)
Minus space\(-\)\(21\)

Trace form

\( 32 q + 2 q^{2} - 2 q^{3} + 36 q^{4} + 4 q^{6} - 4 q^{7} + 18 q^{8} + 32 q^{9} + O(q^{10}) \) \( 32 q + 2 q^{2} - 2 q^{3} + 36 q^{4} + 4 q^{6} - 4 q^{7} + 18 q^{8} + 32 q^{9} - 6 q^{12} + 20 q^{14} + 52 q^{16} + 12 q^{17} + 2 q^{18} - 8 q^{19} - 16 q^{21} - 2 q^{22} + 12 q^{24} - 44 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} + 12 q^{31} + 18 q^{32} - 2 q^{33} + 8 q^{34} + 36 q^{36} - 16 q^{37} - 4 q^{38} - 16 q^{39} - 20 q^{41} + 12 q^{42} + 4 q^{43} + 8 q^{44} - 8 q^{46} + 8 q^{47} + 2 q^{48} + 12 q^{49} + 56 q^{52} + 16 q^{53} + 4 q^{54} + 36 q^{56} - 20 q^{57} + 32 q^{59} - 20 q^{61} - 32 q^{62} - 4 q^{63} + 76 q^{64} + 4 q^{66} - 8 q^{67} - 52 q^{68} + 16 q^{69} + 24 q^{71} + 18 q^{72} + 24 q^{73} + 4 q^{74} - 20 q^{76} - 4 q^{77} + 16 q^{78} - 4 q^{79} + 32 q^{81} - 16 q^{83} - 36 q^{84} + 20 q^{86} + 6 q^{88} + 16 q^{89} - 4 q^{91} - 8 q^{93} - 64 q^{94} - 12 q^{96} - 16 q^{97} - 70 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(825))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 11
825.2.a.a 825.a 1.a $1$ $6.588$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots\)
825.2.a.b 825.a 1.a $1$ $6.588$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{7}+q^{9}-q^{11}+2q^{12}+\cdots\)
825.2.a.c 825.a 1.a $1$ $6.588$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{7}+q^{9}-q^{11}-2q^{12}+\cdots\)
825.2.a.d 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
825.2.a.e 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}-2q^{7}-\beta q^{8}+\cdots\)
825.2.a.f 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
825.2.a.g 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
825.2.a.h 825.a 1.a $3$ $6.588$ 3.3.148.1 None \(-3\) \(3\) \(0\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.2.a.i 825.a 1.a $3$ $6.588$ 3.3.568.1 None \(-2\) \(-3\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+(3+\beta _{2})q^{4}+\cdots\)
825.2.a.j 825.a 1.a $3$ $6.588$ 3.3.148.1 None \(-1\) \(-3\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
825.2.a.k 825.a 1.a $3$ $6.588$ 3.3.148.1 None \(1\) \(-3\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
825.2.a.l 825.a 1.a $3$ $6.588$ 3.3.148.1 None \(1\) \(3\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
825.2.a.m 825.a 1.a $3$ $6.588$ 3.3.568.1 None \(2\) \(3\) \(0\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(3+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)
825.2.a.n 825.a 1.a $3$ $6.588$ 3.3.148.1 None \(3\) \(-3\) \(0\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(825))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(825)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 2}\)