Properties

Label 4009.2.a
Level 4009
Weight 2
Character orbit a
Rep. character \(\chi_{4009}(1,\cdot)\)
Character field \(\Q\)
Dimension 315
Newform subspaces 6
Sturm bound 706
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(706\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4009))\).

Total New Old
Modular forms 354 315 39
Cusp forms 351 315 36
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.

\(19\)\(211\)FrickeDim.
\(+\)\(+\)\(+\)\(75\)
\(+\)\(-\)\(-\)\(84\)
\(-\)\(+\)\(-\)\(82\)
\(-\)\(-\)\(+\)\(74\)
Plus space\(+\)\(149\)
Minus space\(-\)\(166\)

Trace form

\( 315q + q^{2} + 319q^{4} - 12q^{5} - 2q^{7} + 9q^{8} + 311q^{9} + O(q^{10}) \) \( 315q + q^{2} + 319q^{4} - 12q^{5} - 2q^{7} + 9q^{8} + 311q^{9} - 2q^{10} - 2q^{11} - 8q^{12} - 6q^{13} - 20q^{14} - 4q^{15} + 323q^{16} - 4q^{17} + 9q^{18} - 3q^{19} - 30q^{20} - 12q^{21} + 4q^{22} - 4q^{23} - 36q^{24} + 279q^{25} - 2q^{26} - 10q^{29} - 16q^{31} - 11q^{32} - 22q^{34} + 26q^{35} + 295q^{36} - 26q^{37} + 3q^{38} + 40q^{39} + 30q^{40} - 22q^{41} + 4q^{42} + 14q^{43} + 8q^{44} - 64q^{45} + 32q^{46} - 22q^{47} + 44q^{48} + 305q^{49} - 13q^{50} + 12q^{51} - 22q^{52} - 18q^{53} - 20q^{54} - 42q^{55} - 84q^{56} + 4q^{57} + 26q^{58} - 8q^{59} - 8q^{60} - 24q^{61} + 24q^{62} - 22q^{63} + 359q^{64} - 44q^{66} + 4q^{67} - 18q^{68} - 4q^{69} - 4q^{70} - 24q^{71} + 33q^{72} - 40q^{73} - 6q^{74} + 36q^{75} - 3q^{76} + 34q^{77} - 16q^{78} - 16q^{79} - 106q^{80} + 307q^{81} + 22q^{82} - 20q^{83} - 56q^{84} - 26q^{85} - 28q^{87} + 108q^{88} - 70q^{89} + 6q^{90} - 28q^{91} - 56q^{92} + 28q^{93} + 56q^{94} - 12q^{95} - 152q^{96} - 66q^{97} - 43q^{98} - 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 19 211
4009.2.a.a \(1\) \(32.012\) \(\Q\) None \(-1\) \(2\) \(3\) \(0\) \(+\) \(-\) \(q-q^{2}+2q^{3}-q^{4}+3q^{5}-2q^{6}+3q^{8}+\cdots\)
4009.2.a.b \(3\) \(32.012\) \(\Q(\zeta_{14})^+\) None \(2\) \(-2\) \(-3\) \(3\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(-1+\beta _{1})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4009.2.a.c \(71\) \(32.012\) None \(-15\) \(-8\) \(-18\) \(-19\) \(-\) \(-\)
4009.2.a.d \(75\) \(32.012\) None \(-11\) \(-4\) \(-18\) \(-19\) \(+\) \(+\)
4009.2.a.e \(82\) \(32.012\) None \(15\) \(12\) \(9\) \(14\) \(-\) \(+\)
4009.2.a.f \(83\) \(32.012\) None \(11\) \(0\) \(15\) \(19\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4009)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(211))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - 2 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 8 T^{5} + 8 T^{6} \))
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( 1 + 3 T + 11 T^{2} + 17 T^{3} + 55 T^{4} + 75 T^{5} + 125 T^{6} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 - 3 T + 17 T^{2} - 29 T^{3} + 119 T^{4} - 147 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - T + 24 T^{2} - 21 T^{3} + 264 T^{4} - 121 T^{5} + 1331 T^{6} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{3} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 2 T + 50 T^{2} + 67 T^{3} + 850 T^{4} + 578 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + T \))(\( ( 1 - T )^{3} \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 - 2 T + 68 T^{2} - 91 T^{3} + 1564 T^{4} - 1058 T^{5} + 12167 T^{6} \))
$29$ (\( 1 - 3 T + 29 T^{2} \))(\( 1 - 6 T + 92 T^{2} - 335 T^{3} + 2668 T^{4} - 5046 T^{5} + 24389 T^{6} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 5 T + 57 T^{2} - 143 T^{3} + 1767 T^{4} - 4805 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 - 9 T + 47 T^{2} - 217 T^{3} + 1739 T^{4} - 12321 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 7 T + 95 T^{2} + 581 T^{3} + 3895 T^{4} + 11767 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 6 T + 43 T^{2} \))(\( 1 + T + 120 T^{2} + 85 T^{3} + 5160 T^{4} + 1849 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 3 T + 53 T^{2} + 479 T^{3} + 2491 T^{4} + 6627 T^{5} + 103823 T^{6} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 27 T + 395 T^{2} + 3535 T^{3} + 20935 T^{4} + 75843 T^{5} + 148877 T^{6} \))
$59$ (\( 1 - 11 T + 59 T^{2} \))(\( 1 + 2 T + 148 T^{2} + 249 T^{3} + 8732 T^{4} + 6962 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 - 4 T + 179 T^{2} - 480 T^{3} + 10919 T^{4} - 14884 T^{5} + 226981 T^{6} \))
$67$ (\( 1 - 2 T + 67 T^{2} \))(\( 1 - 8 T + 45 T^{2} + 280 T^{3} + 3015 T^{4} - 35912 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 9 T + 71 T^{2} \))(\( 1 + T + 127 T^{2} - 195 T^{3} + 9017 T^{4} + 5041 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 15 T + 73 T^{2} \))(\( 1 - 12 T + 246 T^{2} - 1739 T^{3} + 17958 T^{4} - 63948 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 5 T + 79 T^{2} \))(\( 1 + 33 T + 593 T^{2} + 6475 T^{3} + 46847 T^{4} + 205953 T^{5} + 493039 T^{6} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 + 7 T + 221 T^{2} + 959 T^{3} + 18343 T^{4} + 48223 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 9 T + 89 T^{2} \))(\( 1 - 4 T - 24 T^{2} + 1641 T^{3} - 2136 T^{4} - 31684 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 11 T + 97 T^{2} \))(\( 1 + 7 T + 291 T^{2} + 1309 T^{3} + 28227 T^{4} + 65863 T^{5} + 912673 T^{6} \))
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