Properties

Label 9.10.a
Level 9
Weight 10
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 10
Trace bound 2

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Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(9))\).

Total New Old
Modular forms 11 4 7
Cusp forms 7 3 4
Eisenstein series 4 1 3

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

\(3\)Dim.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3q + 18q^{2} + 84q^{4} + 2844q^{5} - 7932q^{7} + 22392q^{8} + O(q^{10}) \) \( 3q + 18q^{2} + 84q^{4} + 2844q^{5} - 7932q^{7} + 22392q^{8} + 19764q^{10} - 22608q^{11} - 1938q^{13} - 325584q^{14} + 82704q^{16} + 171180q^{17} - 239592q^{19} + 742536q^{20} + 327240q^{22} + 2732976q^{23} - 1791879q^{25} - 6016644q^{26} + 1212576q^{28} + 1950012q^{29} + 8392740q^{31} - 10875168q^{32} - 8921988q^{34} + 8079120q^{35} - 13509822q^{37} + 18758664q^{38} + 32144688q^{40} + 11576988q^{41} - 44083272q^{43} + 2815632q^{44} + 3158352q^{46} - 51281136q^{47} + 140586363q^{49} - 15134994q^{50} - 185266920q^{52} - 94588884q^{53} - 34271424q^{55} + 71144640q^{56} + 323595972q^{58} + 106386624q^{59} - 265029078q^{61} + 261185184q^{62} - 212374464q^{64} - 151342488q^{65} + 486602160q^{67} - 137315304q^{68} - 463307040q^{70} + 80513136q^{71} + 25718154q^{73} - 437254164q^{74} + 937970448q^{76} - 186280416q^{77} - 826612284q^{79} - 263981664q^{80} + 26522316q^{82} + 1020687984q^{83} + 285268392q^{85} + 259740792q^{86} - 280716192q^{88} - 90873684q^{89} - 525354648q^{91} + 428535072q^{92} - 605638944q^{94} + 999662976q^{95} + 1628401650q^{97} - 1503597438q^{98} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.10.a.a \(1\) \(4.635\) \(\Q\) None \(-18\) \(0\) \(1530\) \(9128\) \(-\) \(q-18q^{2}-188q^{4}+1530q^{5}+9128q^{7}+\cdots\)
9.10.a.b \(1\) \(4.635\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-12580\) \(+\) \(q-2^{9}q^{4}-12580q^{7}+118370q^{13}+\cdots\)
9.10.a.c \(1\) \(4.635\) \(\Q\) None \(36\) \(0\) \(1314\) \(-4480\) \(-\) \(q+6^{2}q^{2}+28^{2}q^{4}+1314q^{5}-4480q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 18 T + 512 T^{2} \))(\( 1 + 512 T^{2} \))(\( 1 - 36 T + 512 T^{2} \))
$3$ 1
$5$ (\( 1 - 1530 T + 1953125 T^{2} \))(\( 1 + 1953125 T^{2} \))(\( 1 - 1314 T + 1953125 T^{2} \))
$7$ (\( 1 - 9128 T + 40353607 T^{2} \))(\( 1 + 12580 T + 40353607 T^{2} \))(\( 1 + 4480 T + 40353607 T^{2} \))
$11$ (\( 1 + 21132 T + 2357947691 T^{2} \))(\( 1 + 2357947691 T^{2} \))(\( 1 + 1476 T + 2357947691 T^{2} \))
$13$ (\( 1 - 31214 T + 10604499373 T^{2} \))(\( 1 - 118370 T + 10604499373 T^{2} \))(\( 1 + 151522 T + 10604499373 T^{2} \))
$17$ (\( 1 - 279342 T + 118587876497 T^{2} \))(\( 1 + 118587876497 T^{2} \))(\( 1 + 108162 T + 118587876497 T^{2} \))
$19$ (\( 1 - 144020 T + 322687697779 T^{2} \))(\( 1 + 976696 T + 322687697779 T^{2} \))(\( 1 - 593084 T + 322687697779 T^{2} \))
$23$ (\( 1 - 1763496 T + 1801152661463 T^{2} \))(\( 1 + 1801152661463 T^{2} \))(\( 1 - 969480 T + 1801152661463 T^{2} \))
$29$ (\( 1 + 4692510 T + 14507145975869 T^{2} \))(\( 1 + 14507145975869 T^{2} \))(\( 1 - 6642522 T + 14507145975869 T^{2} \))
$31$ (\( 1 + 369088 T + 26439622160671 T^{2} \))(\( 1 - 1691228 T + 26439622160671 T^{2} \))(\( 1 - 7070600 T + 26439622160671 T^{2} \))
$37$ (\( 1 - 9347078 T + 129961739795077 T^{2} \))(\( 1 + 15384490 T + 129961739795077 T^{2} \))(\( 1 + 7472410 T + 129961739795077 T^{2} \))
$41$ (\( 1 - 7226838 T + 327381934393961 T^{2} \))(\( 1 + 327381934393961 T^{2} \))(\( 1 - 4350150 T + 327381934393961 T^{2} \))
$43$ (\( 1 + 23147476 T + 502592611936843 T^{2} \))(\( 1 + 16577080 T + 502592611936843 T^{2} \))(\( 1 + 4358716 T + 502592611936843 T^{2} \))
$47$ (\( 1 + 22971888 T + 1119130473102767 T^{2} \))(\( 1 + 1119130473102767 T^{2} \))(\( 1 + 28309248 T + 1119130473102767 T^{2} \))
$53$ (\( 1 + 78477174 T + 3299763591802133 T^{2} \))(\( 1 + 3299763591802133 T^{2} \))(\( 1 + 16111710 T + 3299763591802133 T^{2} \))
$59$ (\( 1 - 20310660 T + 8662995818654939 T^{2} \))(\( 1 + 8662995818654939 T^{2} \))(\( 1 - 86075964 T + 8662995818654939 T^{2} \))
$61$ (\( 1 + 179339938 T + 11694146092834141 T^{2} \))(\( 1 + 117903058 T + 11694146092834141 T^{2} \))(\( 1 - 32213918 T + 11694146092834141 T^{2} \))
$67$ (\( 1 - 274528388 T + 27206534396294947 T^{2} \))(\( 1 - 112542320 T + 27206534396294947 T^{2} \))(\( 1 - 99531452 T + 27206534396294947 T^{2} \))
$71$ (\( 1 - 36342648 T + 45848500718449031 T^{2} \))(\( 1 + 45848500718449031 T^{2} \))(\( 1 - 44170488 T + 45848500718449031 T^{2} \))
$73$ (\( 1 + 247089526 T + 58871586708267913 T^{2} \))(\( 1 - 296368310 T + 58871586708267913 T^{2} \))(\( 1 + 23560630 T + 58871586708267913 T^{2} \))
$79$ (\( 1 - 191874800 T + 119851595982618319 T^{2} \))(\( 1 + 616732324 T + 119851595982618319 T^{2} \))(\( 1 + 401754760 T + 119851595982618319 T^{2} \))
$83$ (\( 1 - 276159276 T + 186940255267540403 T^{2} \))(\( 1 + 186940255267540403 T^{2} \))(\( 1 - 744528708 T + 186940255267540403 T^{2} \))
$89$ (\( 1 - 678997350 T + 350356403707485209 T^{2} \))(\( 1 + 350356403707485209 T^{2} \))(\( 1 + 769871034 T + 350356403707485209 T^{2} \))
$97$ (\( 1 + 567657502 T + 760231058654565217 T^{2} \))(\( 1 - 1288928270 T + 760231058654565217 T^{2} \))(\( 1 - 907130882 T + 760231058654565217 T^{2} \))
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