Properties

Label 22.4.a
Level 22
Weight 4
Character orbit a
Rep. character \(\chi_{22}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 12
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(12\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(22))\).

Total New Old
Modular forms 11 3 8
Cusp forms 7 3 4
Eisenstein series 4 0 4

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

\(2\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(1\)

Trace form

\( 3q - 2q^{2} - 2q^{3} + 12q^{4} - 8q^{5} + 8q^{6} - 4q^{7} - 8q^{8} - 15q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 2q^{3} + 12q^{4} - 8q^{5} + 8q^{6} - 4q^{7} - 8q^{8} - 15q^{9} + 4q^{10} + 11q^{11} - 8q^{12} - 138q^{13} - 32q^{14} + 186q^{15} + 48q^{16} + 126q^{17} - 74q^{18} - 12q^{19} - 32q^{20} - 140q^{21} + 22q^{22} - 14q^{23} + 32q^{24} + 191q^{25} + 212q^{26} - 170q^{27} - 16q^{28} + 142q^{29} - 384q^{30} - 6q^{31} - 32q^{32} - 110q^{33} - 84q^{34} - 348q^{35} - 60q^{36} - 228q^{37} + 488q^{38} + 288q^{39} + 16q^{40} + 122q^{41} + 240q^{42} + 88q^{43} + 44q^{44} - 494q^{45} + 784q^{46} + 568q^{47} - 32q^{48} - 669q^{49} - 846q^{50} + 884q^{51} - 552q^{52} + 786q^{53} + 128q^{54} - 396q^{55} - 128q^{56} - 176q^{57} - 764q^{58} - 870q^{59} + 744q^{60} - 458q^{61} - 640q^{62} + 656q^{63} + 192q^{64} + 716q^{65} + 264q^{66} + 330q^{67} + 504q^{68} + 554q^{69} + 816q^{70} - 1690q^{71} - 296q^{72} + 770q^{73} - 1180q^{74} - 1484q^{75} - 48q^{76} + 132q^{77} - 640q^{78} - 508q^{79} - 128q^{80} - 501q^{81} + 1628q^{82} + 1616q^{83} - 560q^{84} + 2568q^{85} + 264q^{86} - 392q^{87} + 88q^{88} + 824q^{89} + 1300q^{90} - 448q^{91} - 56q^{92} - 822q^{93} - 560q^{94} - 1480q^{95} + 128q^{96} - 236q^{97} + 366q^{98} + 77q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(22))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
22.4.a.a \(1\) \(1.298\) \(\Q\) None \(-2\) \(-7\) \(-19\) \(14\) \(+\) \(-\) \(q-2q^{2}-7q^{3}+4q^{4}-19q^{5}+14q^{6}+\cdots\)
22.4.a.b \(1\) \(1.298\) \(\Q\) None \(-2\) \(4\) \(14\) \(-8\) \(+\) \(+\) \(q-2q^{2}+4q^{3}+4q^{4}+14q^{5}-8q^{6}+\cdots\)
22.4.a.c \(1\) \(1.298\) \(\Q\) None \(2\) \(1\) \(-3\) \(-10\) \(-\) \(-\) \(q+2q^{2}+q^{3}+4q^{4}-3q^{5}+2q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 - 2 T \))
$3$ (\( 1 + 7 T + 27 T^{2} \))(\( 1 - 4 T + 27 T^{2} \))(\( 1 - T + 27 T^{2} \))
$5$ (\( 1 + 19 T + 125 T^{2} \))(\( 1 - 14 T + 125 T^{2} \))(\( 1 + 3 T + 125 T^{2} \))
$7$ (\( 1 - 14 T + 343 T^{2} \))(\( 1 + 8 T + 343 T^{2} \))(\( 1 + 10 T + 343 T^{2} \))
$11$ (\( 1 - 11 T \))(\( 1 + 11 T \))(\( 1 - 11 T \))
$13$ (\( 1 + 72 T + 2197 T^{2} \))(\( 1 + 50 T + 2197 T^{2} \))(\( 1 + 16 T + 2197 T^{2} \))
$17$ (\( 1 + 46 T + 4913 T^{2} \))(\( 1 - 130 T + 4913 T^{2} \))(\( 1 - 42 T + 4913 T^{2} \))
$19$ (\( 1 + 20 T + 6859 T^{2} \))(\( 1 + 108 T + 6859 T^{2} \))(\( 1 - 116 T + 6859 T^{2} \))
$23$ (\( 1 + 107 T + 12167 T^{2} \))(\( 1 + 96 T + 12167 T^{2} \))(\( 1 - 189 T + 12167 T^{2} \))
$29$ (\( 1 - 120 T + 24389 T^{2} \))(\( 1 - 142 T + 24389 T^{2} \))(\( 1 + 120 T + 24389 T^{2} \))
$31$ (\( 1 - 117 T + 29791 T^{2} \))(\( 1 - 40 T + 29791 T^{2} \))(\( 1 + 163 T + 29791 T^{2} \))
$37$ (\( 1 + 201 T + 50653 T^{2} \))(\( 1 - 382 T + 50653 T^{2} \))(\( 1 + 409 T + 50653 T^{2} \))
$41$ (\( 1 + 228 T + 68921 T^{2} \))(\( 1 + 118 T + 68921 T^{2} \))(\( 1 - 468 T + 68921 T^{2} \))
$43$ (\( 1 + 242 T + 79507 T^{2} \))(\( 1 - 220 T + 79507 T^{2} \))(\( 1 - 110 T + 79507 T^{2} \))
$47$ (\( 1 + 96 T + 103823 T^{2} \))(\( 1 - 520 T + 103823 T^{2} \))(\( 1 - 144 T + 103823 T^{2} \))
$53$ (\( 1 - 458 T + 148877 T^{2} \))(\( 1 - 238 T + 148877 T^{2} \))(\( 1 - 90 T + 148877 T^{2} \))
$59$ (\( 1 - 435 T + 205379 T^{2} \))(\( 1 + 852 T + 205379 T^{2} \))(\( 1 + 453 T + 205379 T^{2} \))
$61$ (\( 1 + 668 T + 226981 T^{2} \))(\( 1 - 190 T + 226981 T^{2} \))(\( 1 - 20 T + 226981 T^{2} \))
$67$ (\( 1 - 439 T + 300763 T^{2} \))(\( 1 + 12 T + 300763 T^{2} \))(\( 1 + 97 T + 300763 T^{2} \))
$71$ (\( 1 + 1113 T + 357911 T^{2} \))(\( 1 + 112 T + 357911 T^{2} \))(\( 1 + 465 T + 357911 T^{2} \))
$73$ (\( 1 + 72 T + 389017 T^{2} \))(\( 1 + 6 T + 389017 T^{2} \))(\( 1 - 848 T + 389017 T^{2} \))
$79$ (\( 1 + 70 T + 493039 T^{2} \))(\( 1 - 304 T + 493039 T^{2} \))(\( 1 + 742 T + 493039 T^{2} \))
$83$ (\( 1 - 358 T + 571787 T^{2} \))(\( 1 - 820 T + 571787 T^{2} \))(\( 1 - 438 T + 571787 T^{2} \))
$89$ (\( 1 - 895 T + 704969 T^{2} \))(\( 1 - 202 T + 704969 T^{2} \))(\( 1 + 273 T + 704969 T^{2} \))
$97$ (\( 1 - 409 T + 912673 T^{2} \))(\( 1 + 1406 T + 912673 T^{2} \))(\( 1 - 761 T + 912673 T^{2} \))
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