Properties

Label 22.6.a
Level 22
Weight 6
Character orbit a
Rep. character \(\chi_{22}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 18
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(22))\).

Total New Old
Modular forms 17 5 12
Cusp forms 13 5 8
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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(3\)

Trace form

\( 5q + 4q^{2} - 20q^{3} + 80q^{4} - 14q^{5} + 80q^{6} - 312q^{7} + 64q^{8} + 885q^{9} + O(q^{10}) \) \( 5q + 4q^{2} - 20q^{3} + 80q^{4} - 14q^{5} + 80q^{6} - 312q^{7} + 64q^{8} + 885q^{9} - 296q^{10} - 121q^{11} - 320q^{12} + 982q^{13} - 704q^{14} - 3024q^{15} + 1280q^{16} - 1110q^{17} + 3892q^{18} - 3476q^{19} - 224q^{20} + 1864q^{21} - 484q^{22} + 328q^{23} + 1280q^{24} + 4495q^{25} - 8200q^{26} - 584q^{27} - 4992q^{28} + 7030q^{29} + 1920q^{30} + 11776q^{31} + 1024q^{32} - 9680q^{33} - 6360q^{34} + 35520q^{35} + 14160q^{36} - 24566q^{37} - 8176q^{38} - 33192q^{39} - 4736q^{40} - 8758q^{41} + 25248q^{42} + 22292q^{43} - 1936q^{44} - 49802q^{45} - 14048q^{46} - 7736q^{47} - 5120q^{48} + 20397q^{49} - 5316q^{50} + 57992q^{51} + 15712q^{52} - 11826q^{53} - 6016q^{54} + 13794q^{55} - 11264q^{56} + 15520q^{57} + 16792q^{58} + 44436q^{59} - 48384q^{60} - 2314q^{61} + 59648q^{62} - 149272q^{63} + 20480q^{64} + 52004q^{65} - 17424q^{66} + 26380q^{67} - 17760q^{68} - 20884q^{69} + 10848q^{70} - 11656q^{71} + 62272q^{72} - 169782q^{73} + 128792q^{74} + 70180q^{75} - 55616q^{76} + 5808q^{77} + 128q^{78} - 92640q^{79} - 3584q^{80} + 333309q^{81} - 129400q^{82} + 91868q^{83} + 29824q^{84} - 92220q^{85} - 167088q^{86} - 190712q^{87} - 7744q^{88} - 24538q^{89} - 426248q^{90} - 31568q^{91} + 5248q^{92} + 444204q^{93} + 269344q^{94} - 125800q^{95} + 20480q^{96} - 155002q^{97} + 53220q^{98} + 85547q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
22.6.a.a \(1\) \(3.528\) \(\Q\) None \(-4\) \(-21\) \(81\) \(98\) \(+\) \(-\) \(q-4q^{2}-21q^{3}+2^{4}q^{4}+3^{4}q^{5}+84q^{6}+\cdots\)
22.6.a.b \(1\) \(3.528\) \(\Q\) None \(-4\) \(1\) \(-51\) \(-166\) \(+\) \(+\) \(q-4q^{2}+q^{3}+2^{4}q^{4}-51q^{5}-4q^{6}+\cdots\)
22.6.a.c \(1\) \(3.528\) \(\Q\) None \(4\) \(-29\) \(-31\) \(-230\) \(-\) \(-\) \(q+4q^{2}-29q^{3}+2^{4}q^{4}-31q^{5}-116q^{6}+\cdots\)
22.6.a.d \(2\) \(3.528\) \(\Q(\sqrt{793}) \) None \(8\) \(29\) \(-13\) \(-14\) \(-\) \(+\) \(q+4q^{2}+(15-\beta )q^{3}+2^{4}q^{4}+(-9+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( ( 1 - 4 T )^{2} \))
$3$ (\( 1 + 21 T + 243 T^{2} \))(\( 1 - T + 243 T^{2} \))(\( 1 + 29 T + 243 T^{2} \))(\( 1 - 29 T + 498 T^{2} - 7047 T^{3} + 59049 T^{4} \))
$5$ (\( 1 - 81 T + 3125 T^{2} \))(\( 1 + 51 T + 3125 T^{2} \))(\( 1 + 31 T + 3125 T^{2} \))(\( 1 + 13 T + 1336 T^{2} + 40625 T^{3} + 9765625 T^{4} \))
$7$ (\( 1 - 98 T + 16807 T^{2} \))(\( 1 + 166 T + 16807 T^{2} \))(\( 1 + 230 T + 16807 T^{2} \))(\( 1 + 14 T + 26526 T^{2} + 235298 T^{3} + 282475249 T^{4} \))
$11$ (\( 1 - 121 T \))(\( 1 + 121 T \))(\( 1 - 121 T \))(\( ( 1 + 121 T )^{2} \))
$13$ (\( 1 - 824 T + 371293 T^{2} \))(\( 1 - 692 T + 371293 T^{2} \))(\( 1 - 112 T + 371293 T^{2} \))(\( 1 + 646 T + 827090 T^{2} + 239855278 T^{3} + 137858491849 T^{4} \))
$17$ (\( 1 - 978 T + 1419857 T^{2} \))(\( 1 + 738 T + 1419857 T^{2} \))(\( 1 + 1142 T + 1419857 T^{2} \))(\( 1 + 208 T - 197762 T^{2} + 295330256 T^{3} + 2015993900449 T^{4} \))
$19$ (\( 1 + 2140 T + 2476099 T^{2} \))(\( 1 - 1424 T + 2476099 T^{2} \))(\( 1 + 612 T + 2476099 T^{2} \))(\( 1 + 2148 T + 5721862 T^{2} + 5318660652 T^{3} + 6131066257801 T^{4} \))
$23$ (\( 1 - 3699 T + 6436343 T^{2} \))(\( 1 + 1779 T + 6436343 T^{2} \))(\( 1 + 1941 T + 6436343 T^{2} \))(\( 1 - 349 T + 12893422 T^{2} - 2246283707 T^{3} + 41426511213649 T^{4} \))
$29$ (\( 1 - 3480 T + 20511149 T^{2} \))(\( 1 + 2064 T + 20511149 T^{2} \))(\( 1 - 1192 T + 20511149 T^{2} \))(\( 1 - 4422 T + 19354042 T^{2} - 90700300878 T^{3} + 420707233300201 T^{4} \))
$31$ (\( 1 + 7813 T + 28629151 T^{2} \))(\( 1 - 6245 T + 28629151 T^{2} \))(\( 1 + 1037 T + 28629151 T^{2} \))(\( 1 - 14381 T + 107391254 T^{2} - 411715820531 T^{3} + 819628286980801 T^{4} \))
$37$ (\( 1 + 13597 T + 69343957 T^{2} \))(\( 1 + 14785 T + 69343957 T^{2} \))(\( 1 - 8083 T + 69343957 T^{2} \))(\( 1 + 4267 T + 138838388 T^{2} + 295890664519 T^{3} + 4808584372417849 T^{4} \))
$41$ (\( 1 - 6492 T + 115856201 T^{2} \))(\( 1 - 5304 T + 115856201 T^{2} \))(\( 1 + 10444 T + 115856201 T^{2} \))(\( 1 + 10110 T + 53425570 T^{2} + 1171306192110 T^{3} + 13422659310152401 T^{4} \))
$43$ (\( 1 - 14234 T + 147008443 T^{2} \))(\( 1 - 17798 T + 147008443 T^{2} \))(\( 1 - 58 T + 147008443 T^{2} \))(\( 1 + 9798 T + 274223662 T^{2} + 1440388724514 T^{3} + 21611482313284249 T^{4} \))
$47$ (\( 1 + 20352 T + 229345007 T^{2} \))(\( 1 + 17184 T + 229345007 T^{2} \))(\( 1 - 8656 T + 229345007 T^{2} \))(\( 1 - 21144 T + 551158750 T^{2} - 4849270828008 T^{3} + 52599132235830049 T^{4} \))
$53$ (\( 1 + 366 T + 418195493 T^{2} \))(\( 1 + 30726 T + 418195493 T^{2} \))(\( 1 + 20318 T + 418195493 T^{2} \))(\( 1 - 39584 T + 1198268902 T^{2} - 16553850394912 T^{3} + 174887470365513049 T^{4} \))
$59$ (\( 1 - 9825 T + 714924299 T^{2} \))(\( 1 + 34989 T + 714924299 T^{2} \))(\( 1 + 21351 T + 714924299 T^{2} \))(\( 1 - 90951 T + 3475885954 T^{2} - 65023079918349 T^{3} + 511116753300641401 T^{4} \))
$61$ (\( 1 - 26132 T + 844596301 T^{2} \))(\( 1 + 45940 T + 844596301 T^{2} \))(\( 1 - 47044 T + 844596301 T^{2} \))(\( 1 + 29550 T + 1852642210 T^{2} + 24957820694550 T^{3} + 713342911662882601 T^{4} \))
$67$ (\( 1 - 17093 T + 1350125107 T^{2} \))(\( 1 - 25343 T + 1350125107 T^{2} \))(\( 1 - 48093 T + 1350125107 T^{2} \))(\( 1 + 64149 T + 3285930058 T^{2} + 86609175488943 T^{3} + 1822837804551761449 T^{4} \))
$71$ (\( 1 + 23583 T + 1804229351 T^{2} \))(\( 1 - 13311 T + 1804229351 T^{2} \))(\( 1 + 24967 T + 1804229351 T^{2} \))(\( 1 - 23583 T + 295186750 T^{2} - 42549140784633 T^{3} + 3255243551009881201 T^{4} \))
$73$ (\( 1 + 35176 T + 2073071593 T^{2} \))(\( 1 + 53260 T + 2073071593 T^{2} \))(\( 1 + 42288 T + 2073071593 T^{2} \))(\( 1 + 39058 T + 2830212410 T^{2} + 80970030279394 T^{3} + 4297625829703557649 T^{4} \))
$79$ (\( 1 + 42490 T + 3077056399 T^{2} \))(\( 1 - 77234 T + 3077056399 T^{2} \))(\( 1 + 72410 T + 3077056399 T^{2} \))(\( 1 + 54974 T + 4805310654 T^{2} + 169158098478626 T^{3} + 9468276082626847201 T^{4} \))
$83$ (\( 1 - 22674 T + 3939040643 T^{2} \))(\( 1 - 55014 T + 3939040643 T^{2} \))(\( 1 + 15806 T + 3939040643 T^{2} \))(\( 1 - 29986 T + 2580164782 T^{2} - 118116072720998 T^{3} + 15516041187205853449 T^{4} \))
$89$ (\( 1 + 17145 T + 5584059449 T^{2} \))(\( 1 - 125415 T + 5584059449 T^{2} \))(\( 1 + 114761 T + 5584059449 T^{2} \))(\( 1 + 18047 T + 11050895752 T^{2} + 100775520876103 T^{3} + 31181719929966183601 T^{4} \))
$97$ (\( 1 + 30727 T + 8587340257 T^{2} \))(\( 1 + 88807 T + 8587340257 T^{2} \))(\( 1 + 5159 T + 8587340257 T^{2} \))(\( 1 + 30309 T + 4031708980 T^{2} + 260273695849413 T^{3} + 73742412689492826049 T^{4} \))
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