Properties

Label 3.10.a
Level 3
Weight 10
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 3
Trace bound 2

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Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(3))\).

Total New Old
Modular forms 4 2 2
Cusp forms 2 2 0
Eisenstein series 2 0 2

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.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\( 2q - 18q^{2} + 596q^{4} - 2844q^{5} + 4374q^{6} + 4648q^{7} - 22392q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q - 18q^{2} + 596q^{4} - 2844q^{5} + 4374q^{6} + 4648q^{7} - 22392q^{8} + 13122q^{9} + 19764q^{10} + 22608q^{11} - 78732q^{12} - 120308q^{13} + 325584q^{14} - 17496q^{15} - 179440q^{16} - 171180q^{17} - 118098q^{18} + 737104q^{19} - 742536q^{20} + 1102248q^{21} + 327240q^{22} - 2732976q^{23} - 227448q^{24} + 161246q^{25} + 6016644q^{26} - 5228384q^{28} - 1950012q^{29} - 6062364q^{30} + 6701512q^{31} + 10875168q^{32} + 1592136q^{33} - 8921988q^{34} - 8079120q^{35} + 3910356q^{36} + 1874668q^{37} - 18758664q^{38} + 14801616q^{39} + 32144688q^{40} - 11576988q^{41} + 244944q^{42} - 27506192q^{43} - 2815632q^{44} - 18659484q^{45} + 3158352q^{46} + 51281136q^{47} - 6613488q^{48} + 22683570q^{49} + 15134994q^{50} - 31387824q^{51} - 124661480q^{52} + 94588884q^{53} + 28697814q^{54} - 34271424q^{55} - 71144640q^{56} - 36374184q^{57} + 323595972q^{58} - 106386624q^{59} + 106743096q^{60} - 147126020q^{61} - 261185184q^{62} + 30495528q^{63} - 78156736q^{64} + 151342488q^{65} + 35114472q^{66} + 374059840q^{67} + 137315304q^{68} - 64315296q^{69} - 463307040q^{70} - 80513136q^{71} - 146913912q^{72} - 270650156q^{73} + 437254164q^{74} + 49758624q^{75} + 437902096q^{76} + 186280416q^{77} - 396328140q^{78} - 209879960q^{79} + 263981664q^{80} + 86093442q^{81} + 26522316q^{82} - 1020687984q^{83} + 145496736q^{84} + 285268392q^{85} - 259740792q^{86} + 918137592q^{87} - 280716192q^{88} + 90873684q^{89} + 129671604q^{90} + 963739952q^{91} - 428535072q^{92} - 602614728q^{93} - 605638944q^{94} - 999662976q^{95} - 216460512q^{96} + 339473380q^{97} + 1503597438q^{98} + 148331088q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.10.a.a \(1\) \(1.545\) \(\Q\) None \(-36\) \(-81\) \(-1314\) \(-4480\) \(+\) \(q-6^{2}q^{2}-3^{4}q^{3}+28^{2}q^{4}-1314q^{5}+\cdots\)
3.10.a.b \(1\) \(1.545\) \(\Q\) None \(18\) \(81\) \(-1530\) \(9128\) \(-\) \(q+18q^{2}+3^{4}q^{3}-188q^{4}-1530q^{5}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 36 T + 512 T^{2} \))(\( 1 - 18 T + 512 T^{2} \))
$3$ (\( 1 + 81 T \))(\( 1 - 81 T \))
$5$ (\( 1 + 1314 T + 1953125 T^{2} \))(\( 1 + 1530 T + 1953125 T^{2} \))
$7$ (\( 1 + 4480 T + 40353607 T^{2} \))(\( 1 - 9128 T + 40353607 T^{2} \))
$11$ (\( 1 - 1476 T + 2357947691 T^{2} \))(\( 1 - 21132 T + 2357947691 T^{2} \))
$13$ (\( 1 + 151522 T + 10604499373 T^{2} \))(\( 1 - 31214 T + 10604499373 T^{2} \))
$17$ (\( 1 - 108162 T + 118587876497 T^{2} \))(\( 1 + 279342 T + 118587876497 T^{2} \))
$19$ (\( 1 - 593084 T + 322687697779 T^{2} \))(\( 1 - 144020 T + 322687697779 T^{2} \))
$23$ (\( 1 + 969480 T + 1801152661463 T^{2} \))(\( 1 + 1763496 T + 1801152661463 T^{2} \))
$29$ (\( 1 + 6642522 T + 14507145975869 T^{2} \))(\( 1 - 4692510 T + 14507145975869 T^{2} \))
$31$ (\( 1 - 7070600 T + 26439622160671 T^{2} \))(\( 1 + 369088 T + 26439622160671 T^{2} \))
$37$ (\( 1 + 7472410 T + 129961739795077 T^{2} \))(\( 1 - 9347078 T + 129961739795077 T^{2} \))
$41$ (\( 1 + 4350150 T + 327381934393961 T^{2} \))(\( 1 + 7226838 T + 327381934393961 T^{2} \))
$43$ (\( 1 + 4358716 T + 502592611936843 T^{2} \))(\( 1 + 23147476 T + 502592611936843 T^{2} \))
$47$ (\( 1 - 28309248 T + 1119130473102767 T^{2} \))(\( 1 - 22971888 T + 1119130473102767 T^{2} \))
$53$ (\( 1 - 16111710 T + 3299763591802133 T^{2} \))(\( 1 - 78477174 T + 3299763591802133 T^{2} \))
$59$ (\( 1 + 86075964 T + 8662995818654939 T^{2} \))(\( 1 + 20310660 T + 8662995818654939 T^{2} \))
$61$ (\( 1 - 32213918 T + 11694146092834141 T^{2} \))(\( 1 + 179339938 T + 11694146092834141 T^{2} \))
$67$ (\( 1 - 99531452 T + 27206534396294947 T^{2} \))(\( 1 - 274528388 T + 27206534396294947 T^{2} \))
$71$ (\( 1 + 44170488 T + 45848500718449031 T^{2} \))(\( 1 + 36342648 T + 45848500718449031 T^{2} \))
$73$ (\( 1 + 23560630 T + 58871586708267913 T^{2} \))(\( 1 + 247089526 T + 58871586708267913 T^{2} \))
$79$ (\( 1 + 401754760 T + 119851595982618319 T^{2} \))(\( 1 - 191874800 T + 119851595982618319 T^{2} \))
$83$ (\( 1 + 744528708 T + 186940255267540403 T^{2} \))(\( 1 + 276159276 T + 186940255267540403 T^{2} \))
$89$ (\( 1 - 769871034 T + 350356403707485209 T^{2} \))(\( 1 + 678997350 T + 350356403707485209 T^{2} \))
$97$ (\( 1 - 907130882 T + 760231058654565217 T^{2} \))(\( 1 + 567657502 T + 760231058654565217 T^{2} \))
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