Properties

Label 4.22.a
Level $4$
Weight $22$
Character orbit 4.a
Rep. character $\chi_{4}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $1$
Sturm bound $11$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(11\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 12 2 10
Cusp forms 9 2 7
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.

\(2\)Dim
\(-\)\(2\)

Trace form

\( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + O(q^{10}) \) \( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + 145435963320 q^{11} + 1428900417340 q^{13} + 6819782714352 q^{15} + 1840620576420 q^{17} - 16780743928568 q^{19} - 192357511002048 q^{21} - 319691925426960 q^{23} + 439606295919326 q^{25} + 1772171769223440 q^{27} + 3742111775766492 q^{29} + 112042353462592 q^{31} - 5046260260851360 q^{33} - 39279847462424352 q^{35} - 33362705637547220 q^{37} + 36755051864070192 q^{39} + 175129744323133332 q^{41} + 15346613416528120 q^{43} + 503454557153115324 q^{45} - 684848819288455200 q^{47} - 1267753878311886 q^{49} - 2589533339352286896 q^{51} + 675305394244421580 q^{53} - 1007711578141009200 q^{55} + 7738689471355209120 q^{57} + 1042445250435434904 q^{59} + 9065997829736468764 q^{61} - 13688298514444201200 q^{63} + 7711482582413403048 q^{65} - 30464301046802775320 q^{67} - 30093532090729291584 q^{69} - 8199093502830518064 q^{71} + 25415086659374793940 q^{73} + 101635704862879905048 q^{75} + 38853770260581178560 q^{77} + 121204353225060164896 q^{79} - 136996120065469586382 q^{81} + 108742936757033809800 q^{83} - 527989845644919754344 q^{85} - 24695689277789874000 q^{87} - 184207999274965368972 q^{89} - 126427419539014666912 q^{91} - 139841964206001442560 q^{93} + 1576185911066925315504 q^{95} + 739497785188476467140 q^{97} + 261627767970577922520 q^{99} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
4.22.a.a 4.a 1.a $2$ $11.179$ \(\Q(\sqrt{2161}) \) None 4.22.a.a \(0\) \(65640\) \(13689324\) \(-260508080\) $-$ $\mathrm{SU}(2)$ \(q+(32820-\beta )q^{3}+(6844662-204\beta )q^{5}+\cdots\)

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

\( S_{22}^{\mathrm{old}}(\Gamma_0(4)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)