Properties

Label 693.2.a
Level $693$
Weight $2$
Character orbit 693.a
Rep. character $\chi_{693}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $13$
Sturm bound $192$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(693))\).

Total New Old
Modular forms 104 24 80
Cusp forms 89 24 65
Eisenstein series 15 0 15

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

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(10\)
Minus space\(-\)\(14\)

Trace form

\( 24 q - 2 q^{2} + 30 q^{4} - 6 q^{5} + 6 q^{8} + 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 42 q^{16} - 20 q^{17} + 12 q^{19} + 16 q^{20} - 2 q^{22} + 10 q^{23} - 6 q^{25} - 4 q^{26} - 24 q^{29} + 6 q^{31}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(693))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 11
693.2.a.a 693.a 1.a $1$ $5.534$ \(\Q\) None 77.2.a.c \(-1\) \(0\) \(2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{5}-q^{7}+3q^{8}-2q^{10}+\cdots\)
693.2.a.b 693.a 1.a $1$ $5.534$ \(\Q\) None 77.2.a.b \(0\) \(0\) \(-3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{5}+q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.c 693.a 1.a $1$ $5.534$ \(\Q\) None 77.2.a.a \(0\) \(0\) \(1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}-q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.d 693.a 1.a $1$ $5.534$ \(\Q\) None 231.2.a.a \(1\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{5}+q^{7}-3q^{8}+2q^{10}+\cdots\)
693.2.a.e 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{5}) \) None 693.2.a.e \(-3\) \(0\) \(-2\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{5}+q^{7}+\cdots\)
693.2.a.f 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{5}) \) None 231.2.a.c \(-1\) \(0\) \(-2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+q^{7}+(-1+\cdots)q^{8}+\cdots\)
693.2.a.g 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{13}) \) None 693.2.a.g \(-1\) \(0\) \(-2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}-q^{5}-q^{7}-3q^{8}+\cdots\)
693.2.a.h 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{5}) \) None 77.2.a.d \(0\) \(0\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}+2q^{5}+q^{7}-\beta q^{8}+\cdots\)
693.2.a.i 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{13}) \) None 693.2.a.g \(1\) \(0\) \(2\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+q^{5}-q^{7}+3q^{8}+\cdots\)
693.2.a.j 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{21}) \) None 231.2.a.b \(1\) \(0\) \(-6\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(3+\beta )q^{4}-3q^{5}+q^{7}+(5+\cdots)q^{8}+\cdots\)
693.2.a.k 693.a 1.a $2$ $5.534$ \(\Q(\sqrt{5}) \) None 693.2.a.e \(3\) \(0\) \(2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3\beta q^{4}+q^{5}+q^{7}+(1+\cdots)q^{8}+\cdots\)
693.2.a.l 693.a 1.a $3$ $5.534$ 3.3.229.1 None 231.2.a.e \(-2\) \(0\) \(-4\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
693.2.a.m 693.a 1.a $3$ $5.534$ 3.3.837.1 None 231.2.a.d \(0\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(693))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(693)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)