Properties

Label 10.5.c
Level 10
Weight 5
Character orbit c
Rep. character \(\chi_{10}(3,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 2
Sturm bound 7
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(7\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(10, [\chi])\).

Total New Old
Modular forms 16 4 12
Cusp forms 8 4 4
Eisenstein series 8 0 8

Trace form

\( 4q + 20q^{3} - 60q^{5} - 64q^{6} + 20q^{7} + O(q^{10}) \) \( 4q + 20q^{3} - 60q^{5} - 64q^{6} + 20q^{7} + 160q^{10} + 168q^{11} + 160q^{12} - 60q^{13} + 20q^{15} - 256q^{16} - 1020q^{17} - 640q^{18} + 968q^{21} + 1280q^{22} + 1620q^{23} - 700q^{25} - 1344q^{26} + 320q^{27} - 160q^{28} + 960q^{30} - 1112q^{31} - 1720q^{33} - 2220q^{35} + 32q^{36} - 1020q^{37} + 960q^{38} + 1280q^{40} + 5448q^{41} - 2240q^{42} + 660q^{43} + 6400q^{45} + 2176q^{46} + 1620q^{47} - 1280q^{48} - 4800q^{50} - 10712q^{51} - 480q^{52} - 4860q^{53} - 2520q^{55} + 3072q^{56} + 5120q^{57} + 3520q^{58} - 4960q^{60} - 7032q^{61} - 3840q^{62} + 7700q^{63} + 7620q^{65} + 10112q^{66} + 8660q^{67} + 8160q^{68} + 1600q^{70} + 5928q^{71} - 5120q^{72} - 12860q^{73} - 13100q^{75} - 5120q^{76} - 14520q^{77} - 5760q^{78} + 3840q^{80} + 964q^{81} - 2560q^{82} - 300q^{83} + 16580q^{85} - 14784q^{86} + 11840q^{87} - 10240q^{88} + 9440q^{90} + 15528q^{91} + 12960q^{92} + 2120q^{93} - 9600q^{95} + 4096q^{96} - 13500q^{97} + 3840q^{98} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(10, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
10.5.c.a \(2\) \(1.034\) \(\Q(\sqrt{-1}) \) None \(-4\) \(18\) \(-30\) \(58\) \(q+(-2-2i)q^{2}+(9-9i)q^{3}+8iq^{4}+\cdots\)
10.5.c.b \(2\) \(1.034\) \(\Q(\sqrt{-1}) \) None \(4\) \(2\) \(-30\) \(-38\) \(q+(2+2i)q^{2}+(1-i)q^{3}+8iq^{4}+(-15+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(10, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 8 T^{2} \))(\( 1 - 4 T + 8 T^{2} \))
$3$ (\( ( 1 - 9 T )^{2}( 1 + 81 T^{2} ) \))(\( 1 - 2 T + 2 T^{2} - 162 T^{3} + 6561 T^{4} \))
$5$ (\( 1 + 30 T + 625 T^{2} \))(\( 1 + 30 T + 625 T^{2} \))
$7$ (\( 1 - 58 T + 1682 T^{2} - 139258 T^{3} + 5764801 T^{4} \))(\( 1 + 38 T + 722 T^{2} + 91238 T^{3} + 5764801 T^{4} \))
$11$ (\( ( 1 + 118 T + 14641 T^{2} )^{2} \))(\( ( 1 - 202 T + 14641 T^{2} )^{2} \))
$13$ (\( 1 - 138 T + 9522 T^{2} - 3941418 T^{3} + 815730721 T^{4} \))(\( 1 + 198 T + 19602 T^{2} + 5655078 T^{3} + 815730721 T^{4} \))
$17$ (\( 1 + 542 T + 146882 T^{2} + 45268382 T^{3} + 6975757441 T^{4} \))(\( 1 + 478 T + 114242 T^{2} + 39923038 T^{3} + 6975757441 T^{4} \))
$19$ (\( 1 - 182242 T^{2} + 16983563041 T^{4} \))(\( 1 - 259042 T^{2} + 16983563041 T^{4} \))
$23$ (\( 1 - 538 T + 144722 T^{2} - 150554458 T^{3} + 78310985281 T^{4} \))(\( 1 - 1082 T + 585362 T^{2} - 302787962 T^{3} + 78310985281 T^{4} \))
$29$ (\( 1 - 952162 T^{2} + 500246412961 T^{4} \))(\( 1 - 1374562 T^{2} + 500246412961 T^{4} \))
$31$ (\( ( 1 - 202 T + 923521 T^{2} )^{2} \))(\( ( 1 + 758 T + 923521 T^{2} )^{2} \))
$37$ (\( 1 + 1302 T + 847602 T^{2} + 2440157622 T^{3} + 3512479453921 T^{4} \))(\( 1 - 282 T + 39762 T^{2} - 528513402 T^{3} + 3512479453921 T^{4} \))
$41$ (\( ( 1 - 1682 T + 2825761 T^{2} )^{2} \))(\( ( 1 - 1042 T + 2825761 T^{2} )^{2} \))
$43$ (\( 1 - 2178 T + 2371842 T^{2} - 7446148578 T^{3} + 11688200277601 T^{4} \))(\( 1 + 1518 T + 1152162 T^{2} + 5189739918 T^{3} + 11688200277601 T^{4} \))
$47$ (\( 1 - 2538 T + 3220722 T^{2} - 12384630378 T^{3} + 23811286661761 T^{4} \))(\( 1 + 918 T + 421362 T^{2} + 4479547158 T^{3} + 23811286661761 T^{4} \))
$53$ (\( 1 + 1222 T + 746642 T^{2} + 9642167782 T^{3} + 62259690411361 T^{4} \))(\( 1 + 3638 T + 6617522 T^{2} + 28705569878 T^{3} + 62259690411361 T^{4} \))
$59$ (\( 1 - 22889122 T^{2} + 146830437604321 T^{4} \))(\( 1 - 3074722 T^{2} + 146830437604321 T^{4} \))
$61$ (\( ( 1 + 5598 T + 13845841 T^{2} )^{2} \))(\( ( 1 - 2082 T + 13845841 T^{2} )^{2} \))
$67$ (\( 1 + 1502 T + 1128002 T^{2} + 30266983742 T^{3} + 406067677556641 T^{4} \))(\( 1 - 10162 T + 51633122 T^{2} - 204775691602 T^{3} + 406067677556641 T^{4} \))
$71$ (\( ( 1 - 6442 T + 25411681 T^{2} )^{2} \))(\( ( 1 + 3478 T + 25411681 T^{2} )^{2} \))
$73$ (\( 1 + 5902 T + 17416802 T^{2} + 167606418382 T^{3} + 806460091894081 T^{4} \))(\( 1 + 6958 T + 24206882 T^{2} + 197594960878 T^{3} + 806460091894081 T^{4} \))
$79$ (\( 1 + 33613438 T^{2} + 1517108809906561 T^{4} \))(\( 1 - 18917762 T^{2} + 1517108809906561 T^{4} \))
$83$ (\( 1 + 12462 T + 77650722 T^{2} + 591425596302 T^{3} + 2252292232139041 T^{4} \))(\( 1 - 12162 T + 73957122 T^{2} - 577188100002 T^{3} + 2252292232139041 T^{4} \))
$89$ (\( 1 + 84185918 T^{2} + 3936588805702081 T^{4} \))(\( 1 - 93222082 T^{2} + 3936588805702081 T^{4} \))
$97$ (\( 1 + 14622 T + 106901442 T^{2} + 1294475146782 T^{3} + 7837433594376961 T^{4} \))(\( 1 - 1122 T + 629442 T^{2} - 99329853282 T^{3} + 7837433594376961 T^{4} \))
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