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Properties

Label 738.2.a
Level $738$
Weight $2$
Character orbit 738.a
Rep. character $\chi_{738}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $13$
Sturm bound $252$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(252\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(738))\).

Total New Old
Modular forms 134 18 116
Cusp forms 119 18 101
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.

\(2\)\(3\)\(41\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(6\)
Minus space\(-\)\(12\)

Trace form

\( 18 q + 18 q^{4} + O(q^{10}) \) \( 18 q + 18 q^{4} + 4 q^{10} + 6 q^{11} - 2 q^{13} + 18 q^{16} - 8 q^{17} + 2 q^{19} - 2 q^{22} + 14 q^{25} + 10 q^{26} + 6 q^{29} + 8 q^{31} + 4 q^{34} - 4 q^{37} + 2 q^{38} + 4 q^{40} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 24 q^{47} + 30 q^{49} + 8 q^{50} - 2 q^{52} + 2 q^{53} - 12 q^{55} - 14 q^{58} - 12 q^{59} + 16 q^{62} + 18 q^{64} + 20 q^{65} - 22 q^{67} - 8 q^{68} - 12 q^{71} - 8 q^{73} + 16 q^{74} + 2 q^{76} + 36 q^{77} - 24 q^{79} - 2 q^{82} - 48 q^{83} - 8 q^{85} - 24 q^{86} - 2 q^{88} + 12 q^{89} + 8 q^{94} - 44 q^{95} + 48 q^{97} - 4 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(738))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 41
738.2.a.a 738.a 1.a $1$ $5.893$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
738.2.a.b 738.a 1.a $1$ $5.893$ \(\Q\) None \(-1\) \(0\) \(-1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
738.2.a.c 738.a 1.a $1$ $5.893$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
738.2.a.d 738.a 1.a $1$ $5.893$ \(\Q\) None \(-1\) \(0\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}+4q^{7}-q^{8}-2q^{10}+\cdots\)
738.2.a.e 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(-3\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}-2q^{7}+q^{8}-3q^{10}+\cdots\)
738.2.a.f 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(-3\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}+2q^{7}+q^{8}-3q^{10}+\cdots\)
738.2.a.g 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
738.2.a.h 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(2\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}-4q^{7}+q^{8}+2q^{10}+\cdots\)
738.2.a.i 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}+2q^{10}+\cdots\)
738.2.a.j 738.a 1.a $1$ $5.893$ \(\Q\) None \(1\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}+2q^{10}+\cdots\)
738.2.a.k 738.a 1.a $2$ $5.893$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2\beta q^{5}+(-2-\beta )q^{7}+\cdots\)
738.2.a.l 738.a 1.a $3$ $5.893$ 3.3.1304.1 None \(-3\) \(0\) \(-3\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1+\beta _{1})q^{5}+(1-\beta _{2})q^{7}+\cdots\)
738.2.a.m 738.a 1.a $3$ $5.893$ 3.3.1304.1 None \(3\) \(0\) \(3\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(738))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(738)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(123))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(246))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(369))\)\(^{\oplus 2}\)