Properties

Label 4003.2.a
Level 4003
Weight 2
Character orbit a
Rep. character \(\chi_{4003}(1,\cdot)\)
Character field \(\Q\)
Dimension 333
Newform subspaces 3
Sturm bound 667
Trace bound 1

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Defining parameters

Level: \( N \) = \( 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4003.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(667\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4003))\).

Total New Old
Modular forms 334 334 0
Cusp forms 333 333 0
Eisenstein series 1 1 0

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

\(4003\)Dim.
\(+\)\(154\)
\(-\)\(179\)

Trace form

\( 333q - 2q^{3} + 334q^{4} + 2q^{5} - 6q^{6} - 6q^{8} + 325q^{9} + O(q^{10}) \) \( 333q - 2q^{3} + 334q^{4} + 2q^{5} - 6q^{6} - 6q^{8} + 325q^{9} - 2q^{10} + 2q^{11} - 20q^{12} - 4q^{13} - 14q^{14} + 332q^{16} + 8q^{17} - 16q^{18} - 2q^{19} - 6q^{20} - 12q^{21} - 6q^{22} + 2q^{23} - 42q^{24} + 341q^{25} - 4q^{26} - 14q^{27} - 12q^{28} + 6q^{29} - 40q^{30} - 6q^{31} + 8q^{32} + 6q^{33} - 14q^{34} - 4q^{35} + 324q^{36} + 6q^{37} + 14q^{38} - 38q^{39} - 6q^{40} + 10q^{41} - 40q^{42} - 14q^{43} - 6q^{44} + 16q^{45} + 18q^{46} - 18q^{47} - 42q^{48} + 335q^{49} + 14q^{50} - 22q^{51} - 44q^{52} + 48q^{53} - 18q^{54} - 20q^{56} - 54q^{57} - 10q^{58} - 14q^{59} + 12q^{60} + 4q^{61} + 2q^{62} + 10q^{63} + 332q^{64} + 14q^{65} + 32q^{66} - 2q^{67} + 66q^{68} + 24q^{69} - 42q^{70} + 12q^{71} - 50q^{72} + 12q^{73} + 8q^{74} - 32q^{75} - 24q^{76} + 20q^{77} + 26q^{78} + 20q^{79} + 20q^{80} + 285q^{81} + 2q^{82} + 6q^{83} - 38q^{84} + 30q^{85} + 10q^{86} - 8q^{87} - 14q^{88} + 28q^{89} - 32q^{90} - 14q^{91} - 4q^{92} + 44q^{93} + 10q^{94} + 44q^{95} - 136q^{96} - 14q^{97} + 4q^{98} - 34q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 4003
4003.2.a.a \(2\) \(31.964\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(q+\beta q^{2}+\beta q^{3}+\beta q^{5}+2q^{6}-q^{7}+\cdots\)
4003.2.a.b \(152\) \(31.964\) None \(-22\) \(-18\) \(-59\) \(-19\) \(+\)
4003.2.a.c \(179\) \(31.964\) None \(22\) \(16\) \(61\) \(21\) \(-\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} + 4 T^{4} \))
$3$ (\( 1 + 4 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 8 T^{2} + 25 T^{4} \))
$7$ (\( ( 1 + T + 7 T^{2} )^{2} \))
$11$ (\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - 8 T + 34 T^{2} - 104 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 2 T + 3 T^{2} + 34 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 2 T + 7 T^{2} - 38 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 12 T + 74 T^{2} + 276 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 - 4 T - 6 T^{2} - 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 14 T + 115 T^{2} + 518 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 16 T + 138 T^{2} - 656 T^{3} + 1681 T^{4} \))
$43$ (\( ( 1 + 43 T^{2} )^{2} \))
$47$ (\( 1 - 8 T + 60 T^{2} - 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 4 T - 18 T^{2} + 212 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 12 T + 146 T^{2} + 708 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 20 T + 214 T^{2} - 1220 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 14 T + 151 T^{2} - 938 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 + 3 T + 73 T^{2} )^{2} \))
$79$ (\( 1 - 6 T + 95 T^{2} - 474 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T - 25 T^{2} - 498 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 18 T + 251 T^{2} + 1602 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 12 T + 212 T^{2} - 1164 T^{3} + 9409 T^{4} \))
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