Properties

Label 400.2.a
Level $400$
Weight $2$
Character orbit 400.a
Rep. character $\chi_{400}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $8$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(400))\).

Total New Old
Modular forms 78 11 67
Cusp forms 43 8 35
Eisenstein series 35 3 32

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

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

Trace form

\( 8 q - 2 q^{3} - 2 q^{7} + 8 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{3} - 2 q^{7} + 8 q^{9} + 8 q^{11} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 10 q^{23} + 4 q^{27} - 8 q^{29} - 12 q^{31} - 8 q^{37} + 20 q^{39} - 8 q^{41} - 18 q^{43} - 2 q^{47} - 12 q^{49} + 24 q^{51} - 8 q^{57} + 4 q^{61} + 14 q^{63} + 10 q^{67} + 4 q^{69} - 4 q^{71} + 4 q^{73} + 16 q^{77} - 32 q^{79} - 4 q^{81} - 10 q^{83} - 12 q^{87} + 12 q^{89} - 28 q^{91} - 8 q^{93} + 12 q^{97} - 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\) 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
400.2.a.a 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(-3\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
400.2.a.b 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
400.2.a.c 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(-2\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
400.2.a.d 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
400.2.a.e 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(0\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{7}-3q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
400.2.a.f 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
400.2.a.g 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(2\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
400.2.a.h 400.a 1.a $1$ $3.194$ \(\Q\) None \(0\) \(3\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)