Properties

Label 22.8.a
Level 22
Weight 8
Character orbit a
Rep. character \(\chi_{22}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 24
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(22))\).

Total New Old
Modular forms 23 5 18
Cusp forms 19 5 14
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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(2\)

Trace form

\( 5q + 8q^{2} + 28q^{3} + 320q^{4} + 282q^{5} - 928q^{6} + 104q^{7} + 512q^{8} + 5853q^{9} + O(q^{10}) \) \( 5q + 8q^{2} + 28q^{3} + 320q^{4} + 282q^{5} - 928q^{6} + 104q^{7} + 512q^{8} + 5853q^{9} - 5776q^{10} + 1331q^{11} + 1792q^{12} - 7562q^{13} + 28864q^{14} + 11136q^{15} + 20480q^{16} - 62166q^{17} - 21464q^{18} + 75196q^{19} + 18048q^{20} - 172232q^{21} + 10648q^{22} - 9792q^{23} - 59392q^{24} + 109911q^{25} - 2000q^{26} + 355240q^{27} + 6656q^{28} - 238506q^{29} - 83904q^{30} - 681704q^{31} + 32768q^{32} - 149072q^{33} + 258960q^{34} - 93168q^{35} + 374592q^{36} + 361954q^{37} + 141280q^{38} + 2190312q^{39} - 369664q^{40} - 800838q^{41} - 639744q^{42} - 1373532q^{43} + 85184q^{44} + 2233006q^{45} - 239296q^{46} - 252096q^{47} + 114688q^{48} + 535869q^{49} + 1223864q^{50} - 2295304q^{51} - 483968q^{52} + 34278q^{53} - 4066624q^{54} + 1349634q^{55} + 1847296q^{56} - 5983760q^{57} - 2608720q^{58} + 2526420q^{59} + 712704q^{60} + 6110982q^{61} + 1374976q^{62} + 2756408q^{63} + 1310720q^{64} - 5212764q^{65} + 1149984q^{66} + 1641004q^{67} - 3978624q^{68} - 1954036q^{69} + 6615936q^{70} + 2743728q^{71} - 1373696q^{72} - 8208982q^{73} + 3933808q^{74} - 10536284q^{75} + 4812544q^{76} + 1895344q^{77} - 2984128q^{78} - 8421376q^{79} + 1155072q^{80} + 16288077q^{81} + 4059856q^{82} + 2347068q^{83} - 11022848q^{84} + 8022948q^{85} - 8017376q^{86} + 23329960q^{87} + 681472q^{88} + 12976758q^{89} + 9087280q^{90} - 9888048q^{91} - 626688q^{92} - 28008564q^{93} - 3114112q^{94} + 485160q^{95} - 3801088q^{96} - 31697466q^{97} + 5325384q^{98} - 3784033q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(22))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
22.8.a.a \(1\) \(6.872\) \(\Q\) None \(-8\) \(-19\) \(317\) \(-1030\) \(+\) \(-\) \(q-8q^{2}-19q^{3}+2^{6}q^{4}+317q^{5}+\cdots\)
22.8.a.b \(1\) \(6.872\) \(\Q\) None \(-8\) \(91\) \(185\) \(-722\) \(+\) \(+\) \(q-8q^{2}+91q^{3}+2^{6}q^{4}+185q^{5}+\cdots\)
22.8.a.c \(1\) \(6.872\) \(\Q\) None \(8\) \(-21\) \(-551\) \(62\) \(-\) \(+\) \(q+8q^{2}-21q^{3}+2^{6}q^{4}-551q^{5}+\cdots\)
22.8.a.d \(2\) \(6.872\) \(\Q(\sqrt{14881}) \) None \(16\) \(-23\) \(331\) \(1794\) \(-\) \(-\) \(q+8q^{2}+(-11-\beta )q^{3}+2^{6}q^{4}+(165+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T \))(\( 1 + 8 T \))(\( 1 - 8 T \))(\( ( 1 - 8 T )^{2} \))
$3$ (\( 1 + 19 T + 2187 T^{2} \))(\( 1 - 91 T + 2187 T^{2} \))(\( 1 + 21 T + 2187 T^{2} \))(\( 1 + 23 T + 786 T^{2} + 50301 T^{3} + 4782969 T^{4} \))
$5$ (\( 1 - 317 T + 78125 T^{2} \))(\( 1 - 185 T + 78125 T^{2} \))(\( 1 + 551 T + 78125 T^{2} \))(\( 1 - 331 T + 179920 T^{2} - 25859375 T^{3} + 6103515625 T^{4} \))
$7$ (\( 1 + 1030 T + 823543 T^{2} \))(\( 1 + 722 T + 823543 T^{2} \))(\( 1 - 62 T + 823543 T^{2} \))(\( 1 - 1794 T + 1722526 T^{2} - 1477436142 T^{3} + 678223072849 T^{4} \))
$11$ (\( 1 - 1331 T \))(\( 1 + 1331 T \))(\( 1 + 1331 T \))(\( ( 1 - 1331 T )^{2} \))
$13$ (\( 1 + 14676 T + 62748517 T^{2} \))(\( 1 - 11020 T + 62748517 T^{2} \))(\( 1 - 1500 T + 62748517 T^{2} \))(\( 1 + 5406 T + 81002482 T^{2} + 339218482902 T^{3} + 3937376385699289 T^{4} \))
$17$ (\( 1 + 30058 T + 410338673 T^{2} \))(\( 1 + 17210 T + 410338673 T^{2} \))(\( 1 + 29930 T + 410338673 T^{2} \))(\( 1 - 15032 T + 669964558 T^{2} - 6168210932536 T^{3} + 168377826559400929 T^{4} \))
$19$ (\( 1 - 38056 T + 893871739 T^{2} \))(\( 1 + 9288 T + 893871739 T^{2} \))(\( 1 - 29512 T + 893871739 T^{2} \))(\( 1 - 16916 T + 987790358 T^{2} - 15120734336924 T^{3} + 799006685782884121 T^{4} \))
$23$ (\( 1 + 12911 T + 3404825447 T^{2} \))(\( 1 - 22971 T + 3404825447 T^{2} \))(\( 1 - 31499 T + 3404825447 T^{2} \))(\( 1 + 51351 T + 6272862742 T^{2} + 174841191528897 T^{3} + 11592836324538749809 T^{4} \))
$29$ (\( 1 + 90480 T + 17249876309 T^{2} \))(\( 1 - 134272 T + 17249876309 T^{2} \))(\( 1 + 75168 T + 17249876309 T^{2} \))(\( 1 + 207130 T + 43242940618 T^{2} + 3572966879883170 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))
$31$ (\( 1 + 139023 T + 27512614111 T^{2} \))(\( 1 + 287765 T + 27512614111 T^{2} \))(\( 1 + 235845 T + 27512614111 T^{2} \))(\( 1 + 19071 T + 48877201726 T^{2} + 524693063710881 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$37$ (\( 1 - 251511 T + 94931877133 T^{2} \))(\( 1 + 316397 T + 94931877133 T^{2} \))(\( 1 - 75507 T + 94931877133 T^{2} \))(\( 1 - 351333 T + 178095011932 T^{2} - 33352701188768289 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))
$41$ (\( 1 + 318192 T + 194754273881 T^{2} \))(\( 1 + 335968 T + 194754273881 T^{2} \))(\( 1 + 270288 T + 194754273881 T^{2} \))(\( 1 - 123610 T + 154875127858 T^{2} - 24073575794430410 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))
$43$ (\( 1 - 672430 T + 271818611107 T^{2} \))(\( 1 + 858110 T + 271818611107 T^{2} \))(\( 1 + 1028030 T + 271818611107 T^{2} \))(\( 1 + 159822 T - 48316439426 T^{2} + 43442594064342954 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 519096 T + 506623120463 T^{2} \))(\( 1 - 587680 T + 506623120463 T^{2} \))(\( 1 + 771840 T + 506623120463 T^{2} \))(\( 1 - 451160 T + 940360514590 T^{2} - 228568087028087080 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))
$53$ (\( 1 - 773570 T + 1174711139837 T^{2} \))(\( 1 + 244238 T + 1174711139837 T^{2} \))(\( 1 - 765778 T + 1174711139837 T^{2} \))(\( 1 + 1260832 T + 2424528378934 T^{2} + 1481113395862964384 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))
$59$ (\( 1 - 2194167 T + 2488651484819 T^{2} \))(\( 1 + 163287 T + 2488651484819 T^{2} \))(\( 1 + 392007 T + 2488651484819 T^{2} \))(\( 1 - 887547 T + 4412738990098 T^{2} - 2208795159396648993 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))
$61$ (\( 1 - 3163180 T + 3142742836021 T^{2} \))(\( 1 - 2297260 T + 3142742836021 T^{2} \))(\( 1 - 1248460 T + 3142742836021 T^{2} \))(\( 1 + 597918 T + 5828558606962 T^{2} + 1879102511028004278 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))
$67$ (\( 1 + 1293557 T + 6060711605323 T^{2} \))(\( 1 + 3428283 T + 6060711605323 T^{2} \))(\( 1 - 3498133 T + 6060711605323 T^{2} \))(\( 1 - 2864711 T + 9630590949434 T^{2} - 17362187203596456653 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))
$71$ (\( 1 + 1207245 T + 9095120158391 T^{2} \))(\( 1 - 1542953 T + 9095120158391 T^{2} \))(\( 1 - 1101753 T + 9095120158391 T^{2} \))(\( 1 - 1306267 T + 12435173811382 T^{2} - 11880655323940936397 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))
$73$ (\( 1 + 4724772 T + 11047398519097 T^{2} \))(\( 1 - 2216316 T + 11047398519097 T^{2} \))(\( 1 + 1122996 T + 11047398519097 T^{2} \))(\( 1 + 4577530 T + 24133774153130 T^{2} + 50569798143122090410 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))
$79$ (\( 1 + 2638102 T + 19203908986159 T^{2} \))(\( 1 - 1526014 T + 19203908986159 T^{2} \))(\( 1 + 4362946 T + 19203908986159 T^{2} \))(\( 1 + 2946342 T + 29218466187838 T^{2} + 56581283610097680378 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))
$83$ (\( 1 + 4830962 T + 27136050989627 T^{2} \))(\( 1 - 1650370 T + 27136050989627 T^{2} \))(\( 1 + 4437790 T + 27136050989627 T^{2} \))(\( 1 - 9965450 T + 76820028095230 T^{2} - \)\(27\!\cdots\!50\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))
$89$ (\( 1 + 2448233 T + 44231334895529 T^{2} \))(\( 1 - 5760847 T + 44231334895529 T^{2} \))(\( 1 + 521233 T + 44231334895529 T^{2} \))(\( 1 - 10185377 T + 97278081607768 T^{2} - \)\(45\!\cdots\!33\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))
$97$ (\( 1 - 3948601 T + 80798284478113 T^{2} \))(\( 1 + 5750759 T + 80798284478113 T^{2} \))(\( 1 + 2129831 T + 80798284478113 T^{2} \))(\( 1 + 27765477 T + 354321137728852 T^{2} + \)\(22\!\cdots\!01\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))
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