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Properties

Label 700.2.a
Level $700$
Weight $2$
Character orbit 700.a
Rep. character $\chi_{700}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $10$
Sturm bound $240$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(240\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(700))\).

Total New Old
Modular forms 138 10 128
Cusp forms 103 10 93
Eisenstein series 35 0 35

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

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

Trace form

\( 10q - 4q^{3} + 6q^{9} + O(q^{10}) \) \( 10q - 4q^{3} + 6q^{9} + 4q^{13} + 4q^{17} - 8q^{19} - 4q^{27} - 4q^{29} + 4q^{31} + 12q^{33} + 8q^{37} + 12q^{39} - 24q^{41} - 12q^{43} + 4q^{47} + 10q^{49} - 36q^{51} - 4q^{53} - 20q^{57} - 52q^{59} - 24q^{61} + 8q^{63} - 20q^{67} + 36q^{69} + 36q^{71} - 16q^{73} - 8q^{77} - 16q^{79} + 42q^{81} + 16q^{83} + 36q^{87} + 8q^{89} - 8q^{91} + 4q^{93} - 4q^{97} + 36q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(700))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7
700.2.a.a \(1\) \(5.590\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-3q^{3}-q^{7}+6q^{9}+3q^{11}-q^{13}+\cdots\)
700.2.a.b \(1\) \(5.590\) \(\Q\) None \(0\) \(-3\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-3q^{3}+q^{7}+6q^{9}-5q^{11}+3q^{13}+\cdots\)
700.2.a.c \(1\) \(5.590\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q-2q^{3}+q^{7}+q^{9}+3q^{11}-4q^{13}+\cdots\)
700.2.a.d \(1\) \(5.590\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}-q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
700.2.a.e \(1\) \(5.590\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-3q^{9}-5q^{11}+6q^{13}-4q^{17}+\cdots\)
700.2.a.f \(1\) \(5.590\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-3q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
700.2.a.g \(1\) \(5.590\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q+q^{7}-3q^{9}-5q^{11}-6q^{13}+4q^{17}+\cdots\)
700.2.a.h \(1\) \(5.590\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}-3q^{9}+4q^{13}+4q^{17}+4q^{19}+\cdots\)
700.2.a.i \(1\) \(5.590\) \(\Q\) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+2q^{3}-q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
700.2.a.j \(1\) \(5.590\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{3}+q^{7}+6q^{9}+3q^{11}+q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(700)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)