Properties

Label 4006.2.a
Level 4006
Weight 2
Character orbit a
Rep. character \(\chi_{4006}(1,\cdot)\)
Character field \(\Q\)
Dimension 166
Newform subspaces 9
Sturm bound 1002
Trace bound 3

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Defining parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(1002\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4006))\).

Total New Old
Modular forms 503 166 337
Cusp forms 500 166 334
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.

\(2\)\(2003\)FrickeDim.
\(+\)\(+\)\(+\)\(40\)
\(+\)\(-\)\(-\)\(43\)
\(-\)\(+\)\(-\)\(47\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(76\)
Minus space\(-\)\(90\)

Trace form

\( 166q + 2q^{3} + 166q^{4} + 4q^{6} - 4q^{7} + 164q^{9} + O(q^{10}) \) \( 166q + 2q^{3} + 166q^{4} + 4q^{6} - 4q^{7} + 164q^{9} - 2q^{10} - 4q^{11} + 2q^{12} - 2q^{13} - 4q^{14} - 4q^{15} + 166q^{16} - 4q^{17} - 2q^{19} + 6q^{22} + 8q^{23} + 4q^{24} + 176q^{25} + 8q^{27} - 4q^{28} - 8q^{29} + 12q^{30} - 4q^{31} + 36q^{33} - 4q^{34} + 4q^{35} + 164q^{36} + 16q^{37} + 4q^{38} - 24q^{39} - 2q^{40} + 8q^{41} + 28q^{42} - 8q^{43} - 4q^{44} - 8q^{47} + 2q^{48} + 150q^{49} - 8q^{50} - 4q^{51} - 2q^{52} - 2q^{53} + 28q^{54} + 4q^{55} - 4q^{56} - 52q^{57} - 6q^{58} - 26q^{59} - 4q^{60} - 8q^{61} - 4q^{62} - 44q^{63} + 166q^{64} - 24q^{65} + 8q^{66} - 36q^{67} - 4q^{68} + 24q^{70} - 28q^{71} - 16q^{73} - 2q^{74} - 2q^{75} - 2q^{76} - 4q^{77} - 24q^{78} + 4q^{79} + 174q^{81} - 20q^{82} - 36q^{83} - 40q^{85} - 2q^{86} + 6q^{88} - 18q^{90} - 80q^{91} + 8q^{92} + 8q^{93} + 4q^{94} - 4q^{95} + 4q^{96} - 20q^{97} - 16q^{98} + 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 2003
4006.2.a.a \(1\) \(31.988\) \(\Q\) None \(-1\) \(0\) \(-3\) \(-2\) \(+\) \(-\) \(q-q^{2}+q^{4}-3q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
4006.2.a.b \(1\) \(31.988\) \(\Q\) None \(1\) \(-2\) \(-3\) \(-2\) \(-\) \(+\) \(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-2q^{7}+\cdots\)
4006.2.a.c \(1\) \(31.988\) \(\Q\) None \(1\) \(1\) \(2\) \(-2\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
4006.2.a.d \(2\) \(31.988\) \(\Q(\sqrt{2}) \) None \(2\) \(-4\) \(2\) \(-4\) \(-\) \(-\) \(q+q^{2}+(-2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots\)
4006.2.a.e \(2\) \(31.988\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-4\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
4006.2.a.f \(31\) \(31.988\) None \(31\) \(-13\) \(-23\) \(-18\) \(-\) \(-\)
4006.2.a.g \(40\) \(31.988\) None \(-40\) \(-1\) \(-23\) \(12\) \(+\) \(+\)
4006.2.a.h \(42\) \(31.988\) None \(-42\) \(0\) \(27\) \(-10\) \(+\) \(-\)
4006.2.a.i \(46\) \(31.988\) None \(46\) \(21\) \(23\) \(26\) \(-\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4006))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4006)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(2003))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 + 4 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 40 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 40 T^{2} - 76 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 28 T^{2} + 529 T^{4} \))(\( 1 + 4 T + 32 T^{2} + 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 58 T^{2} + 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 14 T + 115 T^{2} - 518 T^{3} + 1369 T^{4} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 12 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 4 T + 58 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 172 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 13 T + 47 T^{2} \))(\( 1 - 20 T + 186 T^{2} - 940 T^{3} + 2209 T^{4} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 4 T + 38 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 4 T + 90 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 10 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 14 T + 139 T^{2} + 854 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 3 T^{2} - 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 15 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 14 T + 151 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( 1 + 14 T + 151 T^{2} + 938 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 140 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 16 T + 204 T^{2} + 1136 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 3 T + 73 T^{2} \))(\( 1 + 9 T + 73 T^{2} \))(\( 1 + T + 73 T^{2} \))(\( ( 1 - 7 T + 73 T^{2} )^{2} \))(\( 1 - 6 T + 147 T^{2} - 438 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 5 T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 + 17 T + 79 T^{2} \))(\( 1 - 10 T + 175 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 31 T^{2} + 158 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 15 T + 83 T^{2} \))(\( 1 - T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( ( 1 - T + 83 T^{2} )^{2} \))(\( 1 + 6 T + 103 T^{2} + 498 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 13 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 10 T + 131 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 219 T^{2} - 1246 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 12 T + 228 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 + 24 T + 320 T^{2} + 2328 T^{3} + 9409 T^{4} \))
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