Properties

Label 4.9.b
Level 4
Weight 9
Character orbit b
Rep. character \(\chi_{4}(3,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(4, [\chi])\).

Total New Old
Modular forms 5 5 0
Cusp forms 3 3 0
Eisenstein series 2 2 0

Trace form

\( 3q - 4q^{2} + 144q^{4} + 166q^{5} - 2496q^{6} + 11456q^{8} - 285q^{9} + O(q^{10}) \) \( 3q - 4q^{2} + 144q^{4} + 166q^{5} - 2496q^{6} + 11456q^{8} - 285q^{9} - 29064q^{10} + 49920q^{12} - 11418q^{13} - 34944q^{14} - 52992q^{16} + 82822q^{17} + 173436q^{18} - 338144q^{20} - 279552q^{21} + 461760q^{22} - 359424q^{24} + 683241q^{25} + 101752q^{26} + 698880q^{28} - 1664282q^{29} - 1522560q^{30} + 1534976q^{32} + 3694080q^{33} - 2475528q^{34} + 2062992q^{36} - 6018138q^{37} + 486720q^{38} + 172416q^{40} + 7871686q^{41} + 2795520q^{42} - 9235200q^{44} - 11091354q^{45} + 5925504q^{46} - 5591040q^{48} + 13380675q^{49} + 11895156q^{50} + 490272q^{52} - 7972058q^{53} - 7832448q^{54} - 5031936q^{56} + 3893760q^{57} - 19960968q^{58} + 30451200q^{60} - 8770074q^{61} + 1697280q^{62} + 37392384q^{64} - 6169588q^{65} - 36940800q^{66} - 24405728q^{68} + 47404032q^{69} - 21315840q^{70} + 1680576q^{72} - 66168378q^{73} + 84255352q^{74} - 9734400q^{76} + 51717120q^{77} + 13653120q^{78} - 141377024q^{80} - 64529469q^{81} + 14309112q^{82} + 15654912q^{84} + 155949132q^{85} + 148085184q^{86} + 66493440q^{88} - 196914362q^{89} - 68884104q^{90} - 118510080q^{92} + 13578240q^{93} - 190504704q^{94} + 203833344q^{96} + 67858182q^{97} + 16078076q^{98} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(4, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4.9.b.a \(1\) \(1.630\) \(\Q\) \(\Q(\sqrt{-1}) \) \(16\) \(0\) \(-1054\) \(0\) \(q+2^{4}q^{2}+2^{8}q^{4}-1054q^{5}+2^{12}q^{8}+\cdots\)
4.9.b.b \(2\) \(1.630\) \(\Q(\sqrt{-39}) \) None \(-20\) \(0\) \(1220\) \(0\) \(q+(-10-\beta )q^{2}-8\beta q^{3}+(-56+20\beta )q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 16 T \))(\( 1 + 20 T + 256 T^{2} \))
$3$ (\( ( 1 - 81 T )( 1 + 81 T ) \))(\( 1 - 3138 T^{2} + 43046721 T^{4} \))
$5$ (\( 1 + 1054 T + 390625 T^{2} \))(\( ( 1 - 610 T + 390625 T^{2} )^{2} \))
$7$ (\( ( 1 - 2401 T )( 1 + 2401 T ) \))(\( 1 - 9572738 T^{2} + 33232930569601 T^{4} \))
$11$ (\( ( 1 - 14641 T )( 1 + 14641 T ) \))(\( 1 - 87015362 T^{2} + 45949729863572161 T^{4} \))
$13$ (\( 1 + 478 T + 815730721 T^{2} \))(\( ( 1 + 5470 T + 815730721 T^{2} )^{2} \))
$17$ (\( 1 + 63358 T + 6975757441 T^{2} \))(\( ( 1 - 73090 T + 6975757441 T^{2} )^{2} \))
$19$ (\( ( 1 - 130321 T )( 1 + 130321 T ) \))(\( 1 - 33587484482 T^{2} + \)\(28\!\cdots\!81\)\( T^{4} \))
$23$ (\( ( 1 - 279841 T )( 1 + 279841 T ) \))(\( 1 - 100353384578 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \))
$29$ (\( 1 + 1407838 T + 500246412961 T^{2} \))(\( ( 1 + 128222 T + 500246412961 T^{2} )^{2} \))
$31$ (\( ( 1 - 923521 T )( 1 + 923521 T ) \))(\( 1 - 1701165473282 T^{2} + \)\(72\!\cdots\!81\)\( T^{4} \))
$37$ (\( 1 - 925922 T + 3512479453921 T^{2} \))(\( ( 1 + 3472030 T + 3512479453921 T^{2} )^{2} \))
$41$ (\( 1 - 3577922 T + 7984925229121 T^{2} \))(\( ( 1 - 2146882 T + 7984925229121 T^{2} )^{2} \))
$43$ (\( ( 1 - 3418801 T )( 1 + 3418801 T ) \))(\( 1 + 11766582970942 T^{2} + \)\(13\!\cdots\!01\)\( T^{4} \))
$47$ (\( ( 1 - 4879681 T )( 1 + 4879681 T ) \))(\( 1 + 10537750788862 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} \))
$53$ (\( 1 + 9620638 T + 62259690411361 T^{2} \))(\( ( 1 - 824290 T + 62259690411361 T^{2} )^{2} \))
$59$ (\( ( 1 - 12117361 T )( 1 + 12117361 T ) \))(\( 1 - 279781405698242 T^{2} + \)\(21\!\cdots\!41\)\( T^{4} \))
$61$ (\( 1 - 20722082 T + 191707312997281 T^{2} \))(\( ( 1 + 14746078 T + 191707312997281 T^{2} )^{2} \))
$67$ (\( ( 1 - 20151121 T )( 1 + 20151121 T ) \))(\( 1 - 579369070794818 T^{2} + \)\(16\!\cdots\!81\)\( T^{4} \))
$71$ (\( ( 1 - 25411681 T )( 1 + 25411681 T ) \))(\( 1 - 1290076545985922 T^{2} + \)\(41\!\cdots\!21\)\( T^{4} \))
$73$ (\( 1 + 54717118 T + 806460091894081 T^{2} \))(\( ( 1 + 5725630 T + 806460091894081 T^{2} )^{2} \))
$79$ (\( ( 1 - 38950081 T )( 1 + 38950081 T ) \))(\( 1 - 1744457179595522 T^{2} + \)\(23\!\cdots\!21\)\( T^{4} \))
$83$ (\( ( 1 - 47458321 T )( 1 + 47458321 T ) \))(\( 1 - 1804713576833858 T^{2} + \)\(50\!\cdots\!81\)\( T^{4} \))
$89$ (\( 1 + 30265918 T + 3936588805702081 T^{2} \))(\( ( 1 + 83324222 T + 3936588805702081 T^{2} )^{2} \))
$97$ (\( 1 + 173379838 T + 7837433594376961 T^{2} \))(\( ( 1 - 120619010 T + 7837433594376961 T^{2} )^{2} \))
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