Properties

Label 10.6
Level 10
Weight 6
Dimension 5
Nonzero newspaces 2
Newform subspaces 4
Sturm bound 36
Trace bound 1

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Defining parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(10))\).

Total New Old
Modular forms 19 5 14
Cusp forms 11 5 6
Eisenstein series 8 0 8

Trace form

\( 5q - 4q^{2} + 4q^{3} + 16q^{4} + 85q^{5} - 80q^{6} - 312q^{7} - 64q^{8} + 653q^{9} + O(q^{10}) \) \( 5q - 4q^{2} + 4q^{3} + 16q^{4} + 85q^{5} - 80q^{6} - 312q^{7} - 64q^{8} + 653q^{9} - 180q^{10} - 740q^{11} + 64q^{12} + 114q^{13} + 1568q^{14} + 820q^{15} + 1280q^{16} + 918q^{17} - 3892q^{18} - 5580q^{19} - 2160q^{20} + 160q^{21} + 3312q^{22} + 3144q^{23} + 2304q^{24} + 7725q^{25} + 2920q^{26} - 5480q^{27} - 4992q^{28} + 3030q^{29} - 11760q^{30} - 12640q^{31} - 1024q^{32} + 24288q^{33} + 18808q^{34} + 2360q^{35} + 7440q^{36} - 16902q^{37} - 17360q^{38} - 34024q^{39} - 320q^{40} - 2390q^{41} + 11392q^{42} + 13164q^{43} - 2368q^{44} + 7845q^{45} + 9920q^{46} + 44448q^{47} + 1024q^{48} - 22743q^{49} - 11300q^{50} + 46760q^{51} + 1824q^{52} - 44406q^{53} - 32160q^{54} + 1420q^{55} - 15360q^{56} - 33040q^{57} + 37320q^{58} + 80860q^{59} + 22080q^{60} - 72290q^{61} - 20768q^{62} - 42376q^{63} + 4096q^{64} - 63830q^{65} - 71360q^{66} + 10788q^{67} + 14688q^{68} + 181456q^{69} + 96320q^{70} + 79560q^{71} - 62272q^{72} - 9666q^{73} - 11512q^{74} - 28300q^{75} + 52800q^{76} - 28464q^{77} + 112576q^{78} - 151920q^{79} + 21760q^{80} - 62595q^{81} - 12648q^{82} - 220476q^{83} - 139008q^{84} + 6910q^{85} - 90480q^{86} + 233880q^{87} + 52992q^{88} + 156290q^{89} + 26940q^{90} + 249360q^{91} + 50304q^{92} + 117968q^{93} - 158272q^{94} - 190700q^{95} - 20480q^{96} - 224682q^{97} + 2652q^{98} - 342244q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
10.6.a \(\chi_{10}(1, \cdot)\) 10.6.a.a 1 1
10.6.a.b 1
10.6.a.c 1
10.6.b \(\chi_{10}(9, \cdot)\) 10.6.b.a 2 1

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(10))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(10)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 + 16 T^{2} \))
$3$ (\( 1 + 26 T + 243 T^{2} \))(\( 1 - 24 T + 243 T^{2} \))(\( 1 - 6 T + 243 T^{2} \))(\( 1 - 290 T^{2} + 59049 T^{4} \))
$5$ (\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))(\( 1 - 110 T + 3125 T^{2} \))
$7$ (\( 1 + 22 T + 16807 T^{2} \))(\( 1 + 172 T + 16807 T^{2} \))(\( 1 + 118 T + 16807 T^{2} \))(\( 1 - 8650 T^{2} + 282475249 T^{4} \))
$11$ (\( 1 + 768 T + 161051 T^{2} \))(\( 1 - 132 T + 161051 T^{2} \))(\( 1 - 192 T + 161051 T^{2} \))(\( ( 1 + 148 T + 161051 T^{2} )^{2} \))
$13$ (\( 1 + 46 T + 371293 T^{2} \))(\( 1 + 946 T + 371293 T^{2} \))(\( 1 - 1106 T + 371293 T^{2} \))(\( 1 - 274730 T^{2} + 137858491849 T^{4} \))
$17$ (\( 1 - 378 T + 1419857 T^{2} \))(\( 1 + 222 T + 1419857 T^{2} \))(\( 1 - 762 T + 1419857 T^{2} \))(\( 1 + 1354590 T^{2} + 2015993900449 T^{4} \))
$19$ (\( 1 - 1100 T + 2476099 T^{2} \))(\( 1 - 500 T + 2476099 T^{2} \))(\( 1 + 2740 T + 2476099 T^{2} \))(\( ( 1 + 2220 T + 2476099 T^{2} )^{2} \))
$23$ (\( 1 + 1986 T + 6436343 T^{2} \))(\( 1 - 3564 T + 6436343 T^{2} \))(\( 1 - 1566 T + 6436343 T^{2} \))(\( 1 - 11320170 T^{2} + 41426511213649 T^{4} \))
$29$ (\( 1 + 5610 T + 20511149 T^{2} \))(\( 1 - 2190 T + 20511149 T^{2} \))(\( 1 - 5910 T + 20511149 T^{2} \))(\( ( 1 - 270 T + 20511149 T^{2} )^{2} \))
$31$ (\( 1 + 3988 T + 28629151 T^{2} \))(\( 1 - 2312 T + 28629151 T^{2} \))(\( 1 + 6868 T + 28629151 T^{2} \))(\( ( 1 + 2048 T + 28629151 T^{2} )^{2} \))
$37$ (\( 1 + 142 T + 69343957 T^{2} \))(\( 1 + 11242 T + 69343957 T^{2} \))(\( 1 + 5518 T + 69343957 T^{2} \))(\( 1 - 119573530 T^{2} + 4808584372417849 T^{4} \))
$41$ (\( 1 - 1542 T + 115856201 T^{2} \))(\( 1 - 1242 T + 115856201 T^{2} \))(\( 1 + 378 T + 115856201 T^{2} \))(\( ( 1 + 2398 T + 115856201 T^{2} )^{2} \))
$43$ (\( 1 + 5026 T + 147008443 T^{2} \))(\( 1 - 20624 T + 147008443 T^{2} \))(\( 1 + 2434 T + 147008443 T^{2} \))(\( 1 - 288754450 T^{2} + 21611482313284249 T^{4} \))
$47$ (\( 1 - 24738 T + 229345007 T^{2} \))(\( 1 - 6588 T + 229345007 T^{2} \))(\( 1 - 13122 T + 229345007 T^{2} \))(\( 1 - 344584890 T^{2} + 52599132235830049 T^{4} \))
$53$ (\( 1 + 14166 T + 418195493 T^{2} \))(\( 1 + 21066 T + 418195493 T^{2} \))(\( 1 + 9174 T + 418195493 T^{2} \))(\( 1 - 827605690 T^{2} + 174887470365513049 T^{4} \))
$59$ (\( 1 - 28380 T + 714924299 T^{2} \))(\( 1 - 7980 T + 714924299 T^{2} \))(\( 1 + 34980 T + 714924299 T^{2} \))(\( ( 1 - 39740 T + 714924299 T^{2} )^{2} \))
$61$ (\( 1 - 5522 T + 844596301 T^{2} \))(\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))(\( ( 1 + 42298 T + 844596301 T^{2} )^{2} \))
$67$ (\( 1 + 24742 T + 1350125107 T^{2} \))(\( 1 - 1808 T + 1350125107 T^{2} \))(\( 1 - 33722 T + 1350125107 T^{2} \))(\( 1 - 1669968610 T^{2} + 1822837804551761449 T^{4} \))
$71$ (\( 1 - 42372 T + 1804229351 T^{2} \))(\( 1 + 24528 T + 1804229351 T^{2} \))(\( 1 - 70212 T + 1804229351 T^{2} \))(\( ( 1 + 4248 T + 1804229351 T^{2} )^{2} \))
$73$ (\( 1 + 52126 T + 2073071593 T^{2} \))(\( 1 - 20474 T + 2073071593 T^{2} \))(\( 1 - 21986 T + 2073071593 T^{2} \))(\( 1 - 3239892370 T^{2} + 4297625829703557649 T^{4} \))
$79$ (\( 1 + 39640 T + 3077056399 T^{2} \))(\( 1 + 46240 T + 3077056399 T^{2} \))(\( 1 - 4520 T + 3077056399 T^{2} \))(\( ( 1 + 35280 T + 3077056399 T^{2} )^{2} \))
$83$ (\( 1 + 59826 T + 3939040643 T^{2} \))(\( 1 + 51576 T + 3939040643 T^{2} \))(\( 1 + 109074 T + 3939040643 T^{2} \))(\( 1 - 7103795010 T^{2} + 15516041187205853449 T^{4} \))
$89$ (\( 1 - 57690 T + 5584059449 T^{2} \))(\( 1 + 110310 T + 5584059449 T^{2} \))(\( 1 - 38490 T + 5584059449 T^{2} \))(\( ( 1 - 85210 T + 5584059449 T^{2} )^{2} \))
$97$ (\( 1 + 144382 T + 8587340257 T^{2} \))(\( 1 + 78382 T + 8587340257 T^{2} \))(\( 1 + 1918 T + 8587340257 T^{2} \))(\( 1 - 7720618690 T^{2} + 73742412689492826049 T^{4} \))
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