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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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(700))\) 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 5 7
700.2.a.a 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}+3q^{11}-q^{13}+\cdots\)
700.2.a.b 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(-3\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{7}+6q^{9}-5q^{11}+3q^{13}+\cdots\)
700.2.a.c 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}+3q^{11}-4q^{13}+\cdots\)
700.2.a.d 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
700.2.a.e 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}-5q^{11}+6q^{13}-4q^{17}+\cdots\)
700.2.a.f 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
700.2.a.g 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}-5q^{11}-6q^{13}+4q^{17}+\cdots\)
700.2.a.h 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}+4q^{13}+4q^{17}+4q^{19}+\cdots\)
700.2.a.i 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
700.2.a.j 700.a 1.a $1$ $5.590$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}+3q^{11}+q^{13}+\cdots\)

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

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