Properties

Label 200.2.a
Level $200$
Weight $2$
Character orbit 200.a
Rep. character $\chi_{200}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(200))\).

Total New Old
Modular forms 42 5 37
Cusp forms 19 5 14
Eisenstein series 23 0 23

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.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5q + 4q^{7} + 11q^{9} + O(q^{10}) \) \( 5q + 4q^{7} + 11q^{9} - 2q^{11} + 2q^{13} - 2q^{17} - 2q^{19} - 4q^{21} - 4q^{23} - 14q^{29} + 12q^{31} - 6q^{37} - 8q^{39} - 8q^{41} + 8q^{43} - 4q^{47} - 3q^{49} - 30q^{51} - 6q^{53} - 12q^{59} - 2q^{61} - 12q^{63} - 8q^{67} + 4q^{69} - 8q^{71} + 6q^{73} + 16q^{77} + 44q^{79} + 5q^{81} + 16q^{83} - 12q^{89} + 40q^{91} + 14q^{97} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(200))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
200.2.a.a \(1\) \(1.597\) \(\Q\) None \(0\) \(-3\) \(0\) \(2\) \(+\) \(-\) \(q-3q^{3}+2q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
200.2.a.b \(1\) \(1.597\) \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) \(-\) \(-\) \(q-2q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
200.2.a.c \(1\) \(1.597\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(+\) \(q+4q^{7}-3q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
200.2.a.d \(1\) \(1.597\) \(\Q\) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
200.2.a.e \(1\) \(1.597\) \(\Q\) None \(0\) \(3\) \(0\) \(-2\) \(-\) \(+\) \(q+3q^{3}-2q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

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

Hecke characteristic polynomials

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