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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))
$5$ 1
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))
$11$ (\( 1 + T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))
$17$ (\( 1 + 5 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))
$23$ (\( 1 + 2 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))
$29$ (\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))
$31$ (\( 1 + 10 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))
$67$ (\( 1 + T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))
$73$ (\( 1 + 3 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 3 T + 73 T^{2} \))
$79$ (\( 1 + 6 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))
$83$ (\( 1 + 13 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 - 13 T + 83 T^{2} \))
$89$ (\( 1 + 9 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))
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