Properties

Label 10.6.a
Level 10
Weight 6
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 9
Trace bound 3

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(9\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(10))\).

Total New Old
Modular forms 9 3 6
Cusp forms 5 3 2
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3q - 4q^{2} + 4q^{3} + 48q^{4} - 25q^{5} + 32q^{6} - 312q^{7} - 64q^{8} + 559q^{9} + O(q^{10}) \) \( 3q - 4q^{2} + 4q^{3} + 48q^{4} - 25q^{5} + 32q^{6} - 312q^{7} - 64q^{8} + 559q^{9} - 100q^{10} - 444q^{11} + 64q^{12} + 114q^{13} + 304q^{14} + 1100q^{15} + 768q^{16} + 918q^{17} - 3892q^{18} - 1140q^{19} - 400q^{20} - 4264q^{21} + 3312q^{22} + 3144q^{23} + 512q^{24} + 1875q^{25} + 8392q^{26} - 5480q^{27} - 4992q^{28} + 2490q^{29} - 5600q^{30} - 8544q^{31} - 1024q^{32} + 24288q^{33} + 2424q^{34} - 800q^{35} + 8944q^{36} - 16902q^{37} - 17360q^{38} - 14872q^{39} - 1600q^{40} + 2406q^{41} + 11392q^{42} + 13164q^{43} - 7104q^{44} + 2675q^{45} - 48q^{46} + 44448q^{47} + 1024q^{48} - 6429q^{49} - 2500q^{50} - 10584q^{51} + 1824q^{52} - 44406q^{53} + 320q^{54} + 17700q^{55} + 4864q^{56} - 33040q^{57} + 37320q^{58} + 1380q^{59} + 17600q^{60} + 12306q^{61} - 20768q^{62} - 42376q^{63} + 12288q^{64} - 50150q^{65} - 87936q^{66} + 10788q^{67} + 14688q^{68} + 146568q^{69} + 26800q^{70} + 88056q^{71} - 62272q^{72} - 9666q^{73} + 23464q^{74} + 2500q^{75} - 18240q^{76} - 28464q^{77} + 112576q^{78} - 81360q^{79} - 6400q^{80} + 28243q^{81} - 12648q^{82} - 220476q^{83} - 68224q^{84} - 34050q^{85} - 72128q^{86} + 233880q^{87} + 52992q^{88} - 14130q^{89} + 30700q^{90} + 33216q^{91} + 50304q^{92} + 117968q^{93} - 72816q^{94} + 53500q^{95} + 8192q^{96} - 224682q^{97} + 2652q^{98} - 328332q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.6.a.a \(1\) \(1.604\) \(\Q\) None \(-4\) \(-26\) \(-25\) \(-22\) \(+\) \(+\) \(q-4q^{2}-26q^{3}+2^{4}q^{4}-5^{2}q^{5}+104q^{6}+\cdots\)
10.6.a.b \(1\) \(1.604\) \(\Q\) None \(-4\) \(24\) \(25\) \(-172\) \(+\) \(-\) \(q-4q^{2}+24q^{3}+2^{4}q^{4}+5^{2}q^{5}-96q^{6}+\cdots\)
10.6.a.c \(1\) \(1.604\) \(\Q\) None \(4\) \(6\) \(-25\) \(-118\) \(-\) \(+\) \(q+4q^{2}+6q^{3}+2^{4}q^{4}-5^{2}q^{5}+24q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))
$3$ (\( 1 + 26 T + 243 T^{2} \))(\( 1 - 24 T + 243 T^{2} \))(\( 1 - 6 T + 243 T^{2} \))
$5$ (\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))
$7$ (\( 1 + 22 T + 16807 T^{2} \))(\( 1 + 172 T + 16807 T^{2} \))(\( 1 + 118 T + 16807 T^{2} \))
$11$ (\( 1 + 768 T + 161051 T^{2} \))(\( 1 - 132 T + 161051 T^{2} \))(\( 1 - 192 T + 161051 T^{2} \))
$13$ (\( 1 + 46 T + 371293 T^{2} \))(\( 1 + 946 T + 371293 T^{2} \))(\( 1 - 1106 T + 371293 T^{2} \))
$17$ (\( 1 - 378 T + 1419857 T^{2} \))(\( 1 + 222 T + 1419857 T^{2} \))(\( 1 - 762 T + 1419857 T^{2} \))
$19$ (\( 1 - 1100 T + 2476099 T^{2} \))(\( 1 - 500 T + 2476099 T^{2} \))(\( 1 + 2740 T + 2476099 T^{2} \))
$23$ (\( 1 + 1986 T + 6436343 T^{2} \))(\( 1 - 3564 T + 6436343 T^{2} \))(\( 1 - 1566 T + 6436343 T^{2} \))
$29$ (\( 1 + 5610 T + 20511149 T^{2} \))(\( 1 - 2190 T + 20511149 T^{2} \))(\( 1 - 5910 T + 20511149 T^{2} \))
$31$ (\( 1 + 3988 T + 28629151 T^{2} \))(\( 1 - 2312 T + 28629151 T^{2} \))(\( 1 + 6868 T + 28629151 T^{2} \))
$37$ (\( 1 + 142 T + 69343957 T^{2} \))(\( 1 + 11242 T + 69343957 T^{2} \))(\( 1 + 5518 T + 69343957 T^{2} \))
$41$ (\( 1 - 1542 T + 115856201 T^{2} \))(\( 1 - 1242 T + 115856201 T^{2} \))(\( 1 + 378 T + 115856201 T^{2} \))
$43$ (\( 1 + 5026 T + 147008443 T^{2} \))(\( 1 - 20624 T + 147008443 T^{2} \))(\( 1 + 2434 T + 147008443 T^{2} \))
$47$ (\( 1 - 24738 T + 229345007 T^{2} \))(\( 1 - 6588 T + 229345007 T^{2} \))(\( 1 - 13122 T + 229345007 T^{2} \))
$53$ (\( 1 + 14166 T + 418195493 T^{2} \))(\( 1 + 21066 T + 418195493 T^{2} \))(\( 1 + 9174 T + 418195493 T^{2} \))
$59$ (\( 1 - 28380 T + 714924299 T^{2} \))(\( 1 - 7980 T + 714924299 T^{2} \))(\( 1 + 34980 T + 714924299 T^{2} \))
$61$ (\( 1 - 5522 T + 844596301 T^{2} \))(\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))
$67$ (\( 1 + 24742 T + 1350125107 T^{2} \))(\( 1 - 1808 T + 1350125107 T^{2} \))(\( 1 - 33722 T + 1350125107 T^{2} \))
$71$ (\( 1 - 42372 T + 1804229351 T^{2} \))(\( 1 + 24528 T + 1804229351 T^{2} \))(\( 1 - 70212 T + 1804229351 T^{2} \))
$73$ (\( 1 + 52126 T + 2073071593 T^{2} \))(\( 1 - 20474 T + 2073071593 T^{2} \))(\( 1 - 21986 T + 2073071593 T^{2} \))
$79$ (\( 1 + 39640 T + 3077056399 T^{2} \))(\( 1 + 46240 T + 3077056399 T^{2} \))(\( 1 - 4520 T + 3077056399 T^{2} \))
$83$ (\( 1 + 59826 T + 3939040643 T^{2} \))(\( 1 + 51576 T + 3939040643 T^{2} \))(\( 1 + 109074 T + 3939040643 T^{2} \))
$89$ (\( 1 - 57690 T + 5584059449 T^{2} \))(\( 1 + 110310 T + 5584059449 T^{2} \))(\( 1 - 38490 T + 5584059449 T^{2} \))
$97$ (\( 1 + 144382 T + 8587340257 T^{2} \))(\( 1 + 78382 T + 8587340257 T^{2} \))(\( 1 + 1918 T + 8587340257 T^{2} \))
show more
show less