Properties

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

Related objects

Downloads

Learn more about

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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
735.2.a.a \(1\) \(5.869\) \(\Q\) None \(-2\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}-q^{5}+2q^{6}+q^{9}+\cdots\)
735.2.a.b \(1\) \(5.869\) \(\Q\) None \(-2\) \(1\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+q^{9}+\cdots\)
735.2.a.c \(1\) \(5.869\) \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+3q^{8}+\cdots\)
735.2.a.d \(1\) \(5.869\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{4}+q^{5}+q^{9}+2q^{12}+q^{13}+\cdots\)
735.2.a.e \(1\) \(5.869\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{4}-q^{5}+q^{9}-2q^{12}-q^{13}+\cdots\)
735.2.a.f \(1\) \(5.869\) \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-3q^{8}+\cdots\)
735.2.a.g \(2\) \(5.869\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(0\) \(+\) \(-\) \(+\) \(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.h \(2\) \(5.869\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(2-2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.i \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(+\) \(q+\beta q^{2}-q^{3}-q^{5}-\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.j \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+q^{5}+\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.k \(2\) \(5.869\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+q^{3}+3q^{4}+q^{5}-\beta q^{6}-\beta q^{8}+\cdots\)
735.2.a.l \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.m \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.n \(4\) \(5.869\) 4.4.4352.1 None \(4\) \(-4\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q+(1+\beta _{1}-\beta _{3})q^{2}-q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.a.o \(4\) \(5.869\) 4.4.4352.1 None \(4\) \(4\) \(-4\) \(0\) \(-\) \(+\) \(+\) \(q+(1+\beta _{1}-\beta _{3})q^{2}+q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

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

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