Properties

Label 400.2.a
Level 400
Weight 2
Character orbit a
Rep. character \(\chi_{400}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 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: \(\Q\)
Newforms: \( 8 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
400.2.a.a \(1\) \(3.194\) \(\Q\) None \(0\) \(-3\) \(0\) \(2\) \(+\) \(+\) \(q-3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
400.2.a.b \(1\) \(3.194\) \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) \(+\) \(-\) \(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
400.2.a.c \(1\) \(3.194\) \(\Q\) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(q-2q^{3}+2q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
400.2.a.d \(1\) \(3.194\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(-\) \(q-q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
400.2.a.e \(1\) \(3.194\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(q-4q^{7}-3q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
400.2.a.f \(1\) \(3.194\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(+\) \(q+q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
400.2.a.g \(1\) \(3.194\) \(\Q\) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(q+2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
400.2.a.h \(1\) \(3.194\) \(\Q\) None \(0\) \(3\) \(0\) \(-2\) \(+\) \(-\) \(q+3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)

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

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