Properties

Label 22.20.a
Level $22$
Weight $20$
Character orbit 22.a
Rep. character $\chi_{22}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $5$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(22))\).

Total New Old
Modular forms 59 15 44
Cusp forms 55 15 40
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
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(4\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(7\)

Trace form

\( 15 q - 512 q^{2} + 70300 q^{3} + 3932160 q^{4} + 473182 q^{5} - 5994496 q^{6} - 272909624 q^{7} - 134217728 q^{8} + 6985184871 q^{9} + O(q^{10}) \) \( 15 q - 512 q^{2} + 70300 q^{3} + 3932160 q^{4} + 473182 q^{5} - 5994496 q^{6} - 272909624 q^{7} - 134217728 q^{8} + 6985184871 q^{9} + 5729321984 q^{10} + 2357947691 q^{11} + 18428723200 q^{12} + 65197479546 q^{13} + 136043978752 q^{14} - 492203318064 q^{15} + 1030792151040 q^{16} + 1232953636830 q^{17} - 2782508009984 q^{18} - 339686323812 q^{19} + 124041822208 q^{20} + 10800486893080 q^{21} + 1207269217792 q^{22} + 620387316664 q^{23} - 1571421159424 q^{24} + 70084668877621 q^{25} + 28001150489600 q^{26} + 15452893797112 q^{27} - 71541620473856 q^{28} - 70719927149702 q^{29} - 245628024520704 q^{30} + 313908087670128 q^{31} - 35184372088832 q^{32} + 304646841677200 q^{33} + 203704329415680 q^{34} - 2214347414419488 q^{35} + 1831124302823424 q^{36} - 1167494422193562 q^{37} + 2006206644119552 q^{38} - 3346128615123144 q^{39} + 1501907382173696 q^{40} + 367615619406446 q^{41} + 10915713403908096 q^{42} - 7592611145620924 q^{43} + 618121839509504 q^{44} + 20371282709960266 q^{45} - 7289134264545280 q^{46} - 16921588804084232 q^{47} + 4830979214540800 q^{48} + 22342882838021751 q^{49} + 3315409345196544 q^{50} - 18334207899536056 q^{51} + 17091128078106624 q^{52} - 27479242669684158 q^{53} - 30258546719260672 q^{54} - 40624104577894926 q^{55} + 35663112765964288 q^{56} + 61799918770337920 q^{57} - 80427252182416384 q^{58} + 246142150612880484 q^{59} - 129028146610569216 q^{60} + 87638974734670490 q^{61} - 39492606055972864 q^{62} - 187731027387508216 q^{63} + 270215977642229760 q^{64} - 310213455009693844 q^{65} + 95050720055199744 q^{66} + 1033012492105945500 q^{67} + 323211398173163520 q^{68} + 1508999055505962452 q^{69} - 148380615994527744 q^{70} - 1577427842689407640 q^{71} - 729417779769245696 q^{72} - 777945936431695346 q^{73} - 66331697487748096 q^{74} + 2526416510281683316 q^{75} - 89046731669372928 q^{76} + 846588144913039056 q^{77} + 476445055609339904 q^{78} + 2032718325815723488 q^{79} + 32516819440893952 q^{80} + 4148021726632331463 q^{81} - 4337545380067189760 q^{82} - 11011053385514110996 q^{83} + 2831282836099563520 q^{84} + 563989022444919228 q^{85} + 5791441246129489920 q^{86} - 24448202171179979384 q^{87} + 316478381828866048 q^{88} + 13058415060908117954 q^{89} + 7421463544043115520 q^{90} - 6809640622659343184 q^{91} + 162630812739567616 q^{92} - 5800141883790299148 q^{93} - 16115443889673515008 q^{94} + 8936652151838471480 q^{95} - 411938628416045056 q^{96} + 24421944790951060946 q^{97} + 15273440476366794240 q^{98} + 4188265814404866887 q^{99} + O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(22))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 11
22.20.a.a 22.a 1.a $1$ $50.340$ \(\Q\) None \(512\) \(-54309\) \(7241785\) \(-192705562\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{9}q^{2}-54309q^{3}+2^{18}q^{4}+7241785q^{5}+\cdots\)
22.20.a.b 22.a 1.a $2$ $50.340$ \(\Q(\sqrt{51151}) \) None \(1024\) \(16974\) \(-2948510\) \(84929956\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{9}q^{2}+(8487+9\beta )q^{3}+2^{18}q^{4}+\cdots\)
22.20.a.c 22.a 1.a $4$ $50.340$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2048\) \(7885\) \(4557609\) \(-208197214\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{9}q^{2}+(1971-\beta _{1})q^{3}+2^{18}q^{4}+\cdots\)
22.20.a.d 22.a 1.a $4$ $50.340$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2048\) \(33119\) \(-9916059\) \(-61113046\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{9}q^{2}+(8280-\beta _{1})q^{3}+2^{18}q^{4}+\cdots\)
22.20.a.e 22.a 1.a $4$ $50.340$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(2048\) \(66631\) \(1538357\) \(104176242\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{9}q^{2}+(16658-\beta _{1})q^{3}+2^{18}q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)