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Properties

Label 325.2.a
Level $325$
Weight $2$
Character orbit 325.a
Rep. character $\chi_{325}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $11$
Sturm bound $70$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(70\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(325))\).

Total New Old
Modular forms 40 19 21
Cusp forms 29 19 10
Eisenstein series 11 0 11

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

\(5\)\(13\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(8\)\(3\)\(5\)\(6\)\(3\)\(3\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(-\)\(11\)\(6\)\(5\)\(8\)\(6\)\(2\)\(3\)\(0\)\(3\)
\(-\)\(+\)\(-\)\(12\)\(6\)\(6\)\(9\)\(6\)\(3\)\(3\)\(0\)\(3\)
\(-\)\(-\)\(+\)\(9\)\(4\)\(5\)\(6\)\(4\)\(2\)\(3\)\(0\)\(3\)
Plus space\(+\)\(17\)\(7\)\(10\)\(12\)\(7\)\(5\)\(5\)\(0\)\(5\)
Minus space\(-\)\(23\)\(12\)\(11\)\(17\)\(12\)\(5\)\(6\)\(0\)\(6\)

Trace form

\( 19 q + 3 q^{2} + 17 q^{4} - 8 q^{6} - 4 q^{7} + 3 q^{8} + 19 q^{9} + 4 q^{12} + q^{13} + 4 q^{14} + 17 q^{16} + 2 q^{17} + 11 q^{18} - 4 q^{19} - 20 q^{21} + 4 q^{22} - 28 q^{24} - 3 q^{26} - 12 q^{27}+ \cdots + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(325))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 13
325.2.a.a 325.a 1.a $1$ $2.595$ \(\Q\) None 325.2.a.a \(-2\) \(-1\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots\)
325.2.a.b 325.a 1.a $1$ $2.595$ \(\Q\) None 325.2.a.b \(0\) \(-1\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+4q^{7}-2q^{9}-6q^{11}+\cdots\)
325.2.a.c 325.a 1.a $1$ $2.595$ \(\Q\) None 325.2.a.b \(0\) \(1\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-4q^{7}-2q^{9}-6q^{11}+\cdots\)
325.2.a.d 325.a 1.a $1$ $2.595$ \(\Q\) None 65.2.a.a \(1\) \(2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}-q^{4}+2q^{6}+4q^{7}-3q^{8}+\cdots\)
325.2.a.e 325.a 1.a $1$ $2.595$ \(\Q\) None 325.2.a.a \(2\) \(1\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
325.2.a.f 325.a 1.a $2$ $2.595$ \(\Q(\sqrt{2}) \) None 325.2.a.f \(-2\) \(0\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
325.2.a.g 325.a 1.a $2$ $2.595$ \(\Q(\sqrt{3}) \) None 65.2.a.c \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1-\beta )q^{3}+q^{4}+(-3-\beta )q^{6}+\cdots\)
325.2.a.h 325.a 1.a $2$ $2.595$ \(\Q(\sqrt{2}) \) None 325.2.a.f \(2\) \(0\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}+(-4+\cdots)q^{6}+\cdots\)
325.2.a.i 325.a 1.a $2$ $2.595$ \(\Q(\sqrt{2}) \) None 65.2.a.b \(2\) \(0\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}+(2+\cdots)q^{6}+\cdots\)
325.2.a.j 325.a 1.a $3$ $2.595$ 3.3.148.1 None 65.2.b.a \(-3\) \(-4\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
325.2.a.k 325.a 1.a $3$ $2.595$ 3.3.148.1 None 65.2.b.a \(3\) \(4\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(325)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)