Properties

Label 128.5.d
Level 128
Weight 5
Character orbit d
Rep. character \(\chi_{128}(63,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 4
Sturm bound 80
Trace bound 9

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(80\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(128, [\chi])\).

Total New Old
Modular forms 72 16 56
Cusp forms 56 16 40
Eisenstein series 16 0 16

Trace form

\( 16q + 432q^{9} + O(q^{10}) \) \( 16q + 432q^{9} - 480q^{17} - 2672q^{25} + 1984q^{33} + 2976q^{41} + 528q^{49} + 12736q^{57} - 13440q^{65} - 30560q^{73} + 9168q^{81} - 8544q^{89} + 31776q^{97} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(128, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
128.5.d.a \(2\) \(13.231\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-iq^{5}-3^{4}q^{9}+5iq^{13}-322q^{17}+\cdots\)
128.5.d.b \(2\) \(13.231\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{3}+47q^{9}+21\beta q^{11}+574q^{17}+\cdots\)
128.5.d.c \(4\) \(13.231\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+15q^{9}+\cdots\)
128.5.d.d \(8\) \(13.231\) 8.0.1871773696.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+(55-\beta _{6}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(128, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(128, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 81 T^{2} )^{2} \))(\( 1 + 34 T^{2} + 6561 T^{4} \))(\( ( 1 + 66 T^{2} + 6561 T^{4} )^{2} \))(\( ( 1 + 52 T^{2} + 486 T^{4} + 341172 T^{6} + 43046721 T^{8} )^{2} \))
$5$ (\( ( 1 - 14 T + 625 T^{2} )( 1 + 14 T + 625 T^{2} ) \))(\( ( 1 - 25 T )^{2}( 1 + 25 T )^{2} \))(\( ( 1 - 1186 T^{2} + 390625 T^{4} )^{2} \))(\( ( 1 - 548 T^{2} + 377094 T^{4} - 214062500 T^{6} + 152587890625 T^{8} )^{2} \))
$7$ (\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( ( 1 + 1342 T^{2} + 5764801 T^{4} )^{2} \))(\( ( 1 - 3138 T^{2} + 5764801 T^{4} )^{4} \))
$11$ (\( ( 1 + 14641 T^{2} )^{2} \))(\( 1 - 27166 T^{2} + 214358881 T^{4} \))(\( ( 1 + 17666 T^{2} + 214358881 T^{4} )^{2} \))(\( ( 1 + 45876 T^{2} + 934622054 T^{4} + 9833928024756 T^{6} + 45949729863572161 T^{8} )^{2} \))
$13$ (\( ( 1 - 238 T + 28561 T^{2} )( 1 + 238 T + 28561 T^{2} ) \))(\( ( 1 - 169 T )^{2}( 1 + 169 T )^{2} \))(\( ( 1 - 10466 T^{2} + 815730721 T^{4} )^{2} \))(\( ( 1 - 79268 T^{2} + 2902795398 T^{4} - 64661342792228 T^{6} + 665416609183179841 T^{8} )^{2} \))
$17$ (\( ( 1 + 322 T + 83521 T^{2} )^{2} \))(\( ( 1 - 574 T + 83521 T^{2} )^{2} \))(\( ( 1 + 162 T + 83521 T^{2} )^{4} \))(\( ( 1 + 84 T + 155494 T^{2} + 7015764 T^{3} + 6975757441 T^{4} )^{4} \))
$19$ (\( ( 1 + 130321 T^{2} )^{2} \))(\( 1 - 72286 T^{2} + 16983563041 T^{4} \))(\( ( 1 + 66242 T^{2} + 16983563041 T^{4} )^{2} \))(\( ( 1 + 191412 T^{2} + 35457974246 T^{4} + 3250857768803892 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))
$23$ (\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( ( 1 - 62018 T^{2} + 78310985281 T^{4} )^{2} \))(\( ( 1 - 108420 T^{2} - 36732732538 T^{4} - 8490477024166020 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} )^{2} \))
$29$ (\( ( 1 - 82 T + 707281 T^{2} )( 1 + 82 T + 707281 T^{2} ) \))(\( ( 1 - 841 T )^{2}( 1 + 841 T )^{2} \))(\( ( 1 + 285854 T^{2} + 500246412961 T^{4} )^{2} \))(\( ( 1 - 1076900 T^{2} + 796704441222 T^{4} - 538715362117700900 T^{6} + \)\(25\!\cdots\!21\)\( T^{8} )^{2} \))
$31$ (\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( ( 1 - 1453826 T^{2} + 852891037441 T^{4} )^{2} \))(\( ( 1 - 667394 T^{2} + 852891037441 T^{4} )^{4} \))
$37$ (\( ( 1 - 2162 T + 1874161 T^{2} )( 1 + 2162 T + 1874161 T^{2} ) \))(\( ( 1 - 1369 T )^{2}( 1 + 1369 T )^{2} \))(\( ( 1 - 1462178 T^{2} + 3512479453921 T^{4} )^{2} \))(\( ( 1 - 3128612 T^{2} + 6805143118086 T^{4} - 10989185369290687652 T^{6} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))
$41$ (\( ( 1 + 3038 T + 2825761 T^{2} )^{2} \))(\( ( 1 + 1246 T + 2825761 T^{2} )^{2} \))(\( ( 1 - 1890 T + 2825761 T^{2} )^{4} \))(\( ( 1 - 996 T + 5420294 T^{2} - 2814457956 T^{3} + 7984925229121 T^{4} )^{4} \))
$43$ (\( ( 1 + 3418801 T^{2} )^{2} \))(\( 1 + 5426402 T^{2} + 11688200277601 T^{4} \))(\( ( 1 - 1630462 T^{2} + 11688200277601 T^{4} )^{2} \))(\( ( 1 + 5340852 T^{2} + 29705752269926 T^{4} + 62424947829025856052 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \))
$47$ (\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( ( 1 - 7768706 T^{2} + 23811286661761 T^{4} )^{2} \))(\( ( 1 - 11288836 T^{2} + 65631563659014 T^{4} - \)\(26\!\cdots\!96\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} )^{2} \))
$53$ (\( ( 1 - 2482 T + 7890481 T^{2} )( 1 + 2482 T + 7890481 T^{2} ) \))(\( ( 1 - 2809 T )^{2}( 1 + 2809 T )^{2} \))(\( ( 1 - 11876386 T^{2} + 62259690411361 T^{4} )^{2} \))(\( ( 1 - 7465252 T^{2} + 129570259041798 T^{4} - \)\(46\!\cdots\!72\)\( T^{6} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))
$59$ (\( ( 1 + 12117361 T^{2} )^{2} \))(\( 1 - 24178078 T^{2} + 146830437604321 T^{4} \))(\( ( 1 + 19112066 T^{2} + 146830437604321 T^{4} )^{2} \))(\( ( 1 - 11608652 T^{2} + 321512309200230 T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} )^{2} \))
$61$ (\( ( 1 - 6958 T + 13845841 T^{2} )( 1 + 6958 T + 13845841 T^{2} ) \))(\( ( 1 - 3721 T )^{2}( 1 + 3721 T )^{2} \))(\( ( 1 - 22046306 T^{2} + 191707312997281 T^{4} )^{2} \))(\( ( 1 - 16591268 T^{2} + 144825482457990 T^{4} - \)\(31\!\cdots\!08\)\( T^{6} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \))
$67$ (\( ( 1 + 20151121 T^{2} )^{2} \))(\( 1 - 13944286 T^{2} + 406067677556641 T^{4} \))(\( ( 1 + 37495106 T^{2} + 406067677556641 T^{4} )^{2} \))(\( ( 1 + 68122548 T^{2} + 1934597002567910 T^{4} + \)\(27\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} )^{2} \))
$71$ (\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( ( 1 + 9393982 T^{2} + 645753531245761 T^{4} )^{2} \))(\( ( 1 - 10042500 T^{2} + 116338596303494 T^{4} - \)\(64\!\cdots\!00\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \))
$73$ (\( ( 1 + 1442 T + 28398241 T^{2} )^{2} \))(\( ( 1 + 9506 T + 28398241 T^{2} )^{2} \))(\( ( 1 - 2750 T + 28398241 T^{2} )^{4} \))(\( ( 1 + 4916 T + 10002918 T^{2} + 139605752756 T^{3} + 806460091894081 T^{4} )^{4} \))
$79$ (\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( ( 1 - 13977986 T^{2} + 1517108809906561 T^{4} )^{2} \))(\( ( 1 - 48021378 T^{2} + 1517108809906561 T^{4} )^{4} \))
$83$ (\( ( 1 + 47458321 T^{2} )^{2} \))(\( 1 + 30209954 T^{2} + 2252292232139041 T^{4} \))(\( ( 1 + 7728578 T^{2} + 2252292232139041 T^{4} )^{2} \))(\( ( 1 + 61584436 T^{2} + 3095005172414694 T^{4} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} )^{2} \))
$89$ (\( ( 1 - 9758 T + 62742241 T^{2} )^{2} \))(\( ( 1 + 5474 T + 62742241 T^{2} )^{2} \))(\( ( 1 - 2430 T + 62742241 T^{2} )^{4} \))(\( ( 1 + 6708 T + 129691750 T^{2} + 420874952628 T^{3} + 3936588805702081 T^{4} )^{4} \))
$97$ (\( ( 1 - 1918 T + 88529281 T^{2} )^{2} \))(\( ( 1 - 9982 T + 88529281 T^{2} )^{2} \))(\( ( 1 - 7454 T + 88529281 T^{2} )^{4} \))(\( ( 1 + 5460 T + 83752934 T^{2} + 483369874260 T^{3} + 7837433594376961 T^{4} )^{4} \))
show more
show less