Properties

Label 4022.2.a
Level 4022
Weight 2
Character orbit a
Rep. character \(\chi_{4022}(1,\cdot)\)
Character field \(\Q\)
Dimension 168
Newform subspaces 6
Sturm bound 1006
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(1006\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4022))\).

Total New Old
Modular forms 505 168 337
Cusp forms 502 168 334
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.

\(2\)\(2011\)FrickeDim.
\(+\)\(+\)\(+\)\(37\)
\(+\)\(-\)\(-\)\(47\)
\(-\)\(+\)\(-\)\(50\)
\(-\)\(-\)\(+\)\(34\)
Plus space\(+\)\(71\)
Minus space\(-\)\(97\)

Trace form

\( 168q + 4q^{3} + 168q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 174q^{9} + O(q^{10}) \) \( 168q + 4q^{3} + 168q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 174q^{9} - 4q^{10} + 8q^{11} + 4q^{12} - 6q^{13} - 4q^{14} + 4q^{15} + 168q^{16} + 8q^{17} + 12q^{19} + 2q^{20} + 8q^{21} - 6q^{22} - 4q^{23} - 2q^{24} + 170q^{25} + 4q^{26} + 28q^{27} + 4q^{28} + 8q^{29} + 4q^{31} + 20q^{33} - 12q^{34} + 32q^{35} + 174q^{36} + 16q^{37} - 2q^{38} + 12q^{39} - 4q^{40} - 16q^{41} - 8q^{42} + 26q^{43} + 8q^{44} + 26q^{45} - 4q^{46} + 20q^{47} + 4q^{48} + 176q^{49} - 16q^{50} + 24q^{51} - 6q^{52} + 4q^{54} - 4q^{56} + 12q^{57} - 22q^{58} + 4q^{60} + 12q^{61} + 24q^{62} + 40q^{63} + 168q^{64} + 28q^{65} + 12q^{67} + 8q^{68} + 40q^{69} - 4q^{70} + 8q^{71} + 16q^{73} - 26q^{74} + 40q^{75} + 12q^{76} - 16q^{77} + 8q^{78} + 20q^{79} + 2q^{80} + 144q^{81} + 2q^{83} + 8q^{84} - 28q^{85} + 12q^{86} - 16q^{87} - 6q^{88} + 20q^{89} - 36q^{90} + 60q^{91} - 4q^{92} - 40q^{93} + 4q^{94} - 48q^{95} - 2q^{96} - 8q^{97} + 52q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 2011
4022.2.a.a \(1\) \(32.116\) \(\Q\) None \(-1\) \(0\) \(2\) \(-2\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
4022.2.a.b \(3\) \(32.116\) 3.3.169.1 None \(3\) \(-3\) \(1\) \(-1\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4022.2.a.c \(31\) \(32.116\) None \(31\) \(-14\) \(-13\) \(-29\) \(-\) \(-\)
4022.2.a.d \(37\) \(32.116\) None \(-37\) \(-5\) \(-13\) \(-22\) \(+\) \(+\)
4022.2.a.e \(46\) \(32.116\) None \(-46\) \(8\) \(14\) \(28\) \(+\) \(-\)
4022.2.a.f \(50\) \(32.116\) None \(50\) \(18\) \(11\) \(30\) \(-\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4022))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4022)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(2011))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( ( 1 - T )^{3} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{3} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - T + 11 T^{2} - 11 T^{3} + 55 T^{4} - 25 T^{5} + 125 T^{6} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 17 T^{2} + 15 T^{3} + 119 T^{4} + 49 T^{5} + 343 T^{6} \))
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 - 9 T + 47 T^{2} - 173 T^{3} + 517 T^{4} - 1089 T^{5} + 1331 T^{6} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 + 4 T + 27 T^{2} + 64 T^{3} + 351 T^{4} + 676 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 7 T + 37 T^{2} + 165 T^{3} + 629 T^{4} + 2023 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 2 T + 54 T^{2} - 71 T^{3} + 1026 T^{4} - 722 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 5 T + 23 T^{2} \))(\( 1 + 8 T + 86 T^{2} + 373 T^{3} + 1978 T^{4} + 4232 T^{5} + 12167 T^{6} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 20 T + 216 T^{2} + 1425 T^{3} + 6264 T^{4} + 16820 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 5 T + 31 T^{2} \))(\( 1 + 5 T + 19 T^{2} - 85 T^{3} + 589 T^{4} + 4805 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 + 9 T + 125 T^{2} + 641 T^{3} + 4625 T^{4} + 12321 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + T + 41 T^{2} \))(\( 1 - T + 15 T^{2} + 255 T^{3} + 615 T^{4} - 1681 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 11 T + 165 T^{2} + 977 T^{3} + 7095 T^{4} + 20339 T^{5} + 79507 T^{6} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 16 T + 105 T^{2} - 504 T^{3} + 4935 T^{4} - 35344 T^{5} + 103823 T^{6} \))
$53$ (\( 1 - 13 T + 53 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 605 T^{3} + 8374 T^{4} + 16854 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 4 T + 22 T^{2} - 153 T^{3} + 1298 T^{4} + 13924 T^{5} + 205379 T^{6} \))
$61$ (\( 1 - 5 T + 61 T^{2} \))(\( 1 + 144 T^{2} + 65 T^{3} + 8784 T^{4} + 226981 T^{6} \))
$67$ (\( 1 - 5 T + 67 T^{2} \))(\( 1 - 2 T + 16 T^{2} - 497 T^{3} + 1072 T^{4} - 8978 T^{5} + 300763 T^{6} \))
$71$ (\( 1 - 5 T + 71 T^{2} \))(\( 1 + 8 T + 217 T^{2} + 1128 T^{3} + 15407 T^{4} + 40328 T^{5} + 357911 T^{6} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 158 T^{2} + 705 T^{3} + 11534 T^{4} + 42632 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 2 T + 79 T^{2} \))(\( 1 - 25 T + 337 T^{2} - 3325 T^{3} + 26623 T^{4} - 156025 T^{5} + 493039 T^{6} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 + 7 T + 105 T^{2} + 387 T^{3} + 8715 T^{4} + 48223 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 18 T + 89 T^{2} \))(\( 1 + 25 T + 367 T^{2} + 3825 T^{3} + 32663 T^{4} + 198025 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 12 T + 97 T^{2} \))(\( 1 - 20 T + 342 T^{2} - 3365 T^{3} + 33174 T^{4} - 188180 T^{5} + 912673 T^{6} \))
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