Properties

Label 840.2.a
Level $840$
Weight $2$
Character orbit 840.a
Rep. character $\chi_{840}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $11$
Sturm bound $384$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(384\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(840))\).

Total New Old
Modular forms 208 12 196
Cusp forms 177 12 165
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.

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(4\)
Minus space\(-\)\(8\)

Trace form

\( 12q + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{9} + 16q^{23} + 12q^{25} + 24q^{31} + 8q^{37} + 8q^{39} + 16q^{41} - 16q^{43} - 32q^{47} + 12q^{49} + 8q^{51} + 16q^{53} + 8q^{55} + 8q^{57} - 32q^{59} + 16q^{61} - 24q^{71} + 32q^{73} - 16q^{79} + 12q^{81} - 32q^{83} + 8q^{85} + 16q^{87} - 32q^{89} + 8q^{91} + 24q^{93} + 8q^{95} - 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(840))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
840.2.a.a \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.b \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
840.2.a.c \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-2q^{13}+q^{15}+\cdots\)
840.2.a.d \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(1\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
840.2.a.e \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(1\) \(-1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
840.2.a.f \(1\) \(6.707\) \(\Q\) None \(0\) \(-1\) \(1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.g \(1\) \(6.707\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
840.2.a.h \(1\) \(6.707\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-q^{7}+q^{9}+6q^{13}-q^{15}+\cdots\)
840.2.a.i \(1\) \(6.707\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
840.2.a.j \(1\) \(6.707\) \(\Q\) None \(0\) \(1\) \(1\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.k \(2\) \(6.707\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(840)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)