Properties

Label 4031.2.a
Level 4031
Weight 2
Character orbit a
Rep. character \(\chi_{4031}(1,\cdot)\)
Character field \(\Q\)
Dimension 323
Newform subspaces 5
Sturm bound 700
Trace bound 1

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Defining parameters

Level: \( N \) = \( 4031 = 29 \cdot 139 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4031.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(700\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4031))\).

Total New Old
Modular forms 352 323 29
Cusp forms 349 323 26
Eisenstein series 3 0 3

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

\(29\)\(139\)FrickeDim.
\(+\)\(+\)\(+\)\(61\)
\(+\)\(-\)\(-\)\(103\)
\(-\)\(+\)\(-\)\(100\)
\(-\)\(-\)\(+\)\(59\)
Plus space\(+\)\(120\)
Minus space\(-\)\(203\)

Trace form

\( 323q + 3q^{2} + 329q^{4} + 2q^{5} + 12q^{6} + 8q^{7} + 3q^{8} + 331q^{9} + O(q^{10}) \) \( 323q + 3q^{2} + 329q^{4} + 2q^{5} + 12q^{6} + 8q^{7} + 3q^{8} + 331q^{9} + 6q^{10} + 4q^{11} - 8q^{12} + 14q^{13} - 24q^{14} + 4q^{15} + 337q^{16} + 18q^{17} + 19q^{18} + 20q^{19} - 22q^{20} + 32q^{21} + 16q^{22} + 20q^{23} + 40q^{24} + 341q^{25} + 6q^{26} + 12q^{27} + 44q^{28} - 5q^{29} + 60q^{30} + 8q^{31} - 21q^{32} + 12q^{33} + 18q^{34} + 4q^{35} + 357q^{36} + 22q^{37} - 32q^{38} + 20q^{39} + 6q^{40} - 2q^{41} - 36q^{42} + 40q^{43} + 28q^{44} - 58q^{45} + 16q^{46} - 32q^{47} - 48q^{48} + 379q^{49} - 39q^{50} + 8q^{51} + 38q^{52} - 26q^{53} + 36q^{54} + 28q^{55} - 32q^{56} + 40q^{57} + 3q^{58} + 20q^{60} + 50q^{61} + 32q^{62} + 8q^{63} + 405q^{64} + 20q^{65} + 4q^{66} - 16q^{67} + 2q^{68} - 24q^{69} - 36q^{70} - 32q^{71} - 13q^{72} + 34q^{73} + 38q^{74} + 4q^{75} + 60q^{76} + 20q^{77} - 108q^{78} + 40q^{79} + 18q^{80} + 355q^{81} - 90q^{82} - 40q^{83} + 100q^{84} + 60q^{85} + 4q^{86} + 4q^{87} + 72q^{88} + 6q^{89} - 34q^{90} + 48q^{91} - 96q^{92} - 44q^{93} + 28q^{94} + 8q^{95} + 44q^{96} + 70q^{97} + 19q^{98} + 80q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 29 139
4031.2.a.a \(2\) \(32.188\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(4\) \(-2\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1+\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)
4031.2.a.b \(59\) \(32.188\) None \(-5\) \(-6\) \(-5\) \(-10\) \(-\) \(-\)
4031.2.a.c \(61\) \(32.188\) None \(-1\) \(-4\) \(-7\) \(-10\) \(+\) \(+\)
4031.2.a.d \(98\) \(32.188\) None \(6\) \(6\) \(1\) \(12\) \(-\) \(+\)
4031.2.a.e \(103\) \(32.188\) None \(1\) \(2\) \(9\) \(18\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4031))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4031)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(139))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ (\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( ( 1 - 2 T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 2 T + 7 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 6 T + 41 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4} \))
$29$ (\( ( 1 - T )^{2} \))
$31$ (\( 1 - 4 T + 58 T^{2} - 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 2 T^{2} + 1369 T^{4} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))
$43$ (\( 1 + 2 T + 85 T^{2} + 86 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 4 T + 90 T^{2} - 188 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 12 T + 122 T^{2} + 708 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 2 T + 25 T^{2} - 122 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 2 T + 7 T^{2} + 134 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 6 T + 23 T^{2} + 426 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 6 T + 105 T^{2} - 438 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 12 T + 122 T^{2} - 948 T^{3} + 6241 T^{4} \))
$83$ (\( ( 1 + 7 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 28 T + 366 T^{2} + 2492 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 10 T + 201 T^{2} - 970 T^{3} + 9409 T^{4} \))
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