Properties

Label 45.2.e
Level 45
Weight 2
Character orbit e
Rep. character \(\chi_{45}(16,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 45.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(45, [\chi])\).

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

Trace form

\( 8q - 2q^{2} - 2q^{3} - 4q^{4} - 2q^{5} - 4q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - 2q^{3} - 4q^{4} - 2q^{5} - 4q^{6} - 2q^{7} - 4q^{9} + 4q^{11} + 14q^{12} - 2q^{13} + 12q^{14} - 2q^{15} - 4q^{16} + 4q^{17} - 20q^{18} - 8q^{19} - 6q^{20} + 6q^{22} - 6q^{23} + 24q^{24} - 4q^{25} - 8q^{26} - 2q^{27} + 16q^{28} + 8q^{29} + 14q^{30} - 8q^{31} - 22q^{32} + 20q^{33} + 16q^{35} + 16q^{36} + 4q^{37} + 10q^{38} - 20q^{39} - 6q^{40} + 8q^{41} - 42q^{42} - 2q^{43} - 40q^{44} + 8q^{45} - 20q^{47} - 10q^{48} - 2q^{50} - 16q^{51} + 10q^{52} - 8q^{53} - 4q^{54} + 22q^{57} + 18q^{58} + 16q^{59} - 22q^{60} - 8q^{61} + 84q^{62} + 24q^{63} - 16q^{64} - 6q^{65} - 8q^{66} - 8q^{67} + 26q^{68} + 48q^{69} + 6q^{70} - 16q^{71} + 6q^{72} - 8q^{73} + 20q^{74} - 2q^{75} + 4q^{76} - 6q^{77} + 2q^{78} + 4q^{79} + 12q^{80} - 28q^{81} - 48q^{82} + 6q^{83} - 36q^{84} + 6q^{85} - 20q^{86} - 26q^{87} + 18q^{88} - 48q^{89} - 2q^{90} + 32q^{91} - 36q^{92} - 42q^{93} + 24q^{94} - 12q^{95} + 28q^{96} + 16q^{97} - 76q^{98} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(45, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
45.2.e.a \(2\) \(0.359\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(1\) \(3\) \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
45.2.e.b \(6\) \(0.359\) 6.0.954288.1 None \(-1\) \(1\) \(-3\) \(-5\) \(q+(-\beta _{1}+\beta _{2}-\beta _{4}-\beta _{5})q^{2}+\beta _{4}q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T - T^{3} - 2 T^{4} + T^{5} + 11 T^{6} + 2 T^{7} - 8 T^{8} - 8 T^{9} + 32 T^{11} + 64 T^{12} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 9 T^{5} + 27 T^{6} \))
$5$ (\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))
$7$ (\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 5 T + T^{2} - 14 T^{3} + 73 T^{4} + 173 T^{5} - 82 T^{6} + 1211 T^{7} + 3577 T^{8} - 4802 T^{9} + 2401 T^{10} + 84035 T^{11} + 117649 T^{12} \))
$11$ (\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 21 T^{2} + 14 T^{3} + 286 T^{4} + 58 T^{5} - 3673 T^{6} + 638 T^{7} + 34606 T^{8} + 18634 T^{9} - 307461 T^{10} - 322102 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 + 4 T - 19 T^{2} - 60 T^{3} + 370 T^{4} + 536 T^{5} - 4235 T^{6} + 6968 T^{7} + 62530 T^{8} - 131820 T^{9} - 542659 T^{10} + 1485172 T^{11} + 4826809 T^{12} \))
$17$ (\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} + 56 T^{3} + 731 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \))
$19$ (\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 1007 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2} \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 27 T^{2} + 66 T^{3} + 405 T^{4} - 2625 T^{5} - 9794 T^{6} - 60375 T^{7} + 214245 T^{8} + 803022 T^{9} - 7555707 T^{10} + 19309029 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 9 T^{2} + 304 T^{3} - 803 T^{4} - 2929 T^{5} + 36038 T^{6} - 84941 T^{7} - 675323 T^{8} + 7414256 T^{9} - 6365529 T^{10} - 143578043 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 31 T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 240436 T^{7} - 151838 T^{8} + 6196528 T^{9} + 28629151 T^{10} + 229033208 T^{11} + 887503681 T^{12} \))
$37$ (\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 99 T^{2} - 440 T^{3} + 3663 T^{4} - 8214 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 913439 T^{7} + 748045 T^{8} + 20124932 T^{9} + 76295547 T^{10} - 1506130613 T^{11} + 4750104241 T^{12} \))
$43$ (\( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))(\( 1 + 10 T - 25 T^{2} - 462 T^{3} + 1690 T^{4} + 17990 T^{5} + 34975 T^{6} + 773570 T^{7} + 3124810 T^{8} - 36732234 T^{9} - 85470025 T^{10} + 1470084430 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 + 13 T + 39 T^{2} - 16 T^{3} - 299 T^{4} - 19253 T^{5} - 232366 T^{6} - 904891 T^{7} - 660491 T^{8} - 1661168 T^{9} + 190307559 T^{10} + 2981485091 T^{11} + 10779215329 T^{12} \))
$53$ (\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 139 T^{2} + 188 T^{3} + 7367 T^{4} + 5618 T^{5} + 148877 T^{6} )^{2} \))
$59$ (\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 153 T^{2} + 110 T^{3} + 14962 T^{4} - 3194 T^{5} - 1012513 T^{6} - 188446 T^{7} + 52082722 T^{8} + 22591690 T^{9} - 1853956233 T^{10} - 1429848598 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 859307 T^{7} + 45459457 T^{8} - 54475440 T^{9} - 2007646945 T^{10} + 844596301 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 + 11 T - 41 T^{2} - 152 T^{3} + 4753 T^{4} - 32755 T^{5} - 764902 T^{6} - 2194585 T^{7} + 21336217 T^{8} - 45715976 T^{9} - 826195961 T^{10} + 14851376177 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 8591 T^{4} + 50410 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 11315 T^{4} + 42632 T^{5} + 389017 T^{6} )^{2} \))
$79$ (\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 1276166 T^{7} + 66416722 T^{8} - 184396586 T^{9} - 5803562069 T^{10} + 6154112798 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 - 15 T - 51 T^{2} + 678 T^{3} + 17133 T^{4} - 80979 T^{5} - 925778 T^{6} - 6721257 T^{7} + 118029237 T^{8} + 387671586 T^{9} - 2420374371 T^{10} - 59085609645 T^{11} + 326940373369 T^{12} \))
$89$ (\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 3 T + 89 T^{2} )^{6} \))
$97$ (\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 69 T^{2} - 214 T^{3} + 906 T^{4} + 163158 T^{5} - 2743251 T^{6} + 15826326 T^{7} + 8524554 T^{8} - 195312022 T^{9} + 6108520389 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \))
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