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Properties

Label 8002.2.a
Level 8002
Weight 2
Character orbit a
Rep. character \(\chi_{8002}(1,\cdot)\)
Character field \(\Q\)
Dimension 333
Newform subspaces 7
Sturm bound 2001
Trace bound 3

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Defining parameters

Level: \( N \) = \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8002.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(2001\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8002))\).

Total New Old
Modular forms 1002 333 669
Cusp forms 999 333 666
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.

\(2\)\(4001\)FrickeDim.
\(+\)\(+\)\(+\)\(89\)
\(+\)\(-\)\(-\)\(77\)
\(-\)\(+\)\(-\)\(95\)
\(-\)\(-\)\(+\)\(72\)
Plus space\(+\)\(161\)
Minus space\(-\)\(172\)

Trace form

\( 333q + q^{2} - 2q^{3} + 333q^{4} + 2q^{5} + 2q^{6} - 4q^{7} + q^{8} + 337q^{9} + O(q^{10}) \) \( 333q + q^{2} - 2q^{3} + 333q^{4} + 2q^{5} + 2q^{6} - 4q^{7} + q^{8} + 337q^{9} + 2q^{10} + 6q^{11} - 2q^{12} + 2q^{13} + 8q^{14} + 12q^{15} + 333q^{16} + 6q^{17} + 5q^{18} + 2q^{20} + 8q^{21} - 2q^{22} + 8q^{23} + 2q^{24} + 331q^{25} + 2q^{26} + 16q^{27} - 4q^{28} + 18q^{29} + 12q^{30} - 24q^{31} + q^{32} + 12q^{33} + 6q^{34} + 337q^{36} - 16q^{37} + 4q^{38} + 8q^{39} + 2q^{40} + 2q^{41} + 6q^{43} + 6q^{44} + 2q^{45} - 8q^{46} - 12q^{47} - 2q^{48} + 337q^{49} - q^{50} + 28q^{51} + 2q^{52} + 4q^{53} + 20q^{54} - 4q^{55} + 8q^{56} - 12q^{57} - 6q^{58} + 12q^{59} + 12q^{60} + 6q^{61} + 8q^{62} - 8q^{63} + 333q^{64} + 8q^{65} + 8q^{66} - 10q^{67} + 6q^{68} - 20q^{69} + 12q^{70} - 4q^{71} + 5q^{72} - 6q^{73} + 20q^{74} - 54q^{75} - 36q^{77} - 4q^{78} - 20q^{79} + 2q^{80} + 333q^{81} + 6q^{82} - 32q^{83} + 8q^{84} + 36q^{85} - 6q^{86} - 8q^{87} - 2q^{88} + 2q^{89} + 6q^{90} - 4q^{91} + 8q^{92} - 44q^{93} + 16q^{94} + 2q^{96} + 14q^{97} + 25q^{98} + 38q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 4001
8002.2.a.a \(1\) \(63.896\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
8002.2.a.b \(1\) \(63.896\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}-2q^{11}+4q^{13}+\cdots\)
8002.2.a.c \(1\) \(63.896\) \(\Q\) None \(1\) \(2\) \(-2\) \(0\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{8}+\cdots\)
8002.2.a.d \(69\) \(63.896\) None \(69\) \(-25\) \(-33\) \(-19\) \(-\) \(-\)
8002.2.a.e \(77\) \(63.896\) None \(-77\) \(10\) \(18\) \(21\) \(+\) \(-\)
8002.2.a.f \(89\) \(63.896\) None \(-89\) \(-12\) \(-18\) \(-27\) \(+\) \(+\)
8002.2.a.g \(95\) \(63.896\) None \(95\) \(24\) \(36\) \(21\) \(-\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8002))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8002)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(4001))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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