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Properties

Label 735.2.a
Level $735$
Weight $2$
Character orbit 735.a
Rep. character $\chi_{735}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $15$
Sturm bound $224$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(224\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(735))\).

Total New Old
Modular forms 128 28 100
Cusp forms 97 28 69
Eisenstein series 31 0 31

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

\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(7\)
Minus space\(-\)\(21\)

Trace form

\( 28 q + 4 q^{2} + 2 q^{3} + 32 q^{4} - 2 q^{6} + 12 q^{8} + 28 q^{9} + O(q^{10}) \) \( 28 q + 4 q^{2} + 2 q^{3} + 32 q^{4} - 2 q^{6} + 12 q^{8} + 28 q^{9} + 8 q^{11} + 6 q^{12} + 8 q^{13} - 2 q^{15} + 48 q^{16} + 4 q^{18} + 8 q^{20} + 24 q^{22} - 8 q^{23} + 6 q^{24} + 28 q^{25} + 24 q^{26} + 2 q^{27} + 24 q^{29} - 2 q^{30} - 16 q^{31} + 28 q^{32} + 32 q^{36} - 4 q^{37} - 8 q^{38} - 24 q^{39} - 20 q^{43} + 8 q^{44} + 8 q^{46} - 24 q^{47} - 2 q^{48} + 4 q^{50} + 4 q^{51} - 8 q^{52} + 32 q^{53} - 2 q^{54} + 8 q^{55} - 12 q^{57} + 16 q^{58} - 6 q^{60} + 8 q^{61} - 24 q^{62} + 72 q^{64} + 8 q^{65} - 16 q^{66} - 20 q^{67} + 16 q^{68} + 16 q^{71} + 12 q^{72} + 8 q^{73} - 32 q^{74} + 2 q^{75} - 16 q^{76} - 4 q^{78} - 28 q^{79} + 28 q^{81} + 16 q^{82} + 8 q^{83} - 8 q^{85} - 112 q^{86} - 4 q^{87} - 112 q^{88} + 16 q^{89} - 160 q^{92} - 20 q^{93} + 40 q^{94} + 16 q^{95} - 10 q^{96} + 8 q^{97} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\) 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 7
735.2.a.a 735.a 1.a $1$ $5.869$ \(\Q\) None \(-2\) \(-1\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}-q^{5}+2q^{6}+q^{9}+\cdots\)
735.2.a.b 735.a 1.a $1$ $5.869$ \(\Q\) None \(-2\) \(1\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+q^{9}+\cdots\)
735.2.a.c 735.a 1.a $1$ $5.869$ \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+3q^{8}+\cdots\)
735.2.a.d 735.a 1.a $1$ $5.869$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{5}+q^{9}+2q^{12}+q^{13}+\cdots\)
735.2.a.e 735.a 1.a $1$ $5.869$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+q^{9}-2q^{12}-q^{13}+\cdots\)
735.2.a.f 735.a 1.a $1$ $5.869$ \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-3q^{8}+\cdots\)
735.2.a.g 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.h 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(2-2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.i 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}-q^{5}-\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.j 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{5}+\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.k 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+3q^{4}+q^{5}-\beta q^{6}-\beta q^{8}+\cdots\)
735.2.a.l 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.m 735.a 1.a $2$ $5.869$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.n 735.a 1.a $4$ $5.869$ 4.4.4352.1 None \(4\) \(-4\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}-\beta _{3})q^{2}-q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.a.o 735.a 1.a $4$ $5.869$ 4.4.4352.1 None \(4\) \(4\) \(-4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}-\beta _{3})q^{2}+q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(735))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(735)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)