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

\( 24 q - 2 q^{2} + 30 q^{4} - 6 q^{5} + 6 q^{8} + O(q^{10}) \) \( 24 q - 2 q^{2} + 30 q^{4} - 6 q^{5} + 6 q^{8} + 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 42 q^{16} - 20 q^{17} + 12 q^{19} + 16 q^{20} - 2 q^{22} + 10 q^{23} - 6 q^{25} - 4 q^{26} - 24 q^{29} + 6 q^{31} + 30 q^{32} - 20 q^{34} - 4 q^{35} - 22 q^{37} + 16 q^{38} - 12 q^{40} - 20 q^{41} + 10 q^{44} - 8 q^{46} + 16 q^{47} + 24 q^{49} + 66 q^{50} - 56 q^{52} + 16 q^{53} - 6 q^{55} + 12 q^{56} - 28 q^{58} + 22 q^{59} + 4 q^{61} - 4 q^{62} - 6 q^{64} - 16 q^{65} - 6 q^{67} - 24 q^{68} + 8 q^{70} + 6 q^{71} - 20 q^{73} - 28 q^{74} - 24 q^{76} + 4 q^{77} + 36 q^{79} - 20 q^{80} - 92 q^{82} - 24 q^{83} - 4 q^{85} - 24 q^{86} + 6 q^{88} - 42 q^{89} - 4 q^{92} + 12 q^{94} + 12 q^{95} - 50 q^{97} - 2 q^{98} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 11
693.2.a.a \(1\) \(5.534\) \(\Q\) None \(-1\) \(0\) \(2\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{2}-q^{4}+2q^{5}-q^{7}+3q^{8}-2q^{10}+\cdots\)
693.2.a.b \(1\) \(5.534\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(-\) \(-\) \(q-2q^{4}-3q^{5}+q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.c \(1\) \(5.534\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(-\) \(+\) \(-\) \(q-2q^{4}+q^{5}-q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.d \(1\) \(5.534\) \(\Q\) None \(1\) \(0\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{4}+2q^{5}+q^{7}-3q^{8}+2q^{10}+\cdots\)
693.2.a.e \(2\) \(5.534\) \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(-2\) \(2\) \(+\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{5}+q^{7}+\cdots\)
693.2.a.f \(2\) \(5.534\) \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(-2\) \(2\) \(-\) \(-\) \(+\) \(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+q^{7}+(-1+\cdots)q^{8}+\cdots\)
693.2.a.g \(2\) \(5.534\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta q^{2}+(1+\beta )q^{4}-q^{5}-q^{7}-3q^{8}+\cdots\)
693.2.a.h \(2\) \(5.534\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(4\) \(2\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+3q^{4}+2q^{5}+q^{7}-\beta q^{8}+\cdots\)
693.2.a.i \(2\) \(5.534\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(2\) \(-2\) \(+\) \(+\) \(-\) \(q+\beta q^{2}+(1+\beta )q^{4}+q^{5}-q^{7}+3q^{8}+\cdots\)
693.2.a.j \(2\) \(5.534\) \(\Q(\sqrt{21}) \) None \(1\) \(0\) \(-6\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+(3+\beta )q^{4}-3q^{5}+q^{7}+(5+\cdots)q^{8}+\cdots\)
693.2.a.k \(2\) \(5.534\) \(\Q(\sqrt{5}) \) None \(3\) \(0\) \(2\) \(2\) \(+\) \(-\) \(+\) \(q+(1+\beta )q^{2}+3\beta q^{4}+q^{5}+q^{7}+(1+\cdots)q^{8}+\cdots\)
693.2.a.l \(3\) \(5.534\) 3.3.229.1 None \(-2\) \(0\) \(-4\) \(-3\) \(-\) \(+\) \(-\) \(q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
693.2.a.m \(3\) \(5.534\) 3.3.837.1 None \(0\) \(0\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)

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

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