Properties

Label 45.2.j
Level 45
Weight 2
Character orbit j
Rep. character \(\chi_{45}(4,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 8
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 45.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

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

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

Trace form

\( 8q - 12q^{6} + O(q^{10}) \) \( 8q - 12q^{6} - 8q^{10} - 12q^{11} + 12q^{15} + 4q^{16} - 8q^{19} + 12q^{20} + 12q^{21} + 12q^{24} - 4q^{25} + 24q^{26} + 12q^{29} - 12q^{30} + 4q^{31} - 4q^{34} - 24q^{35} - 12q^{39} - 4q^{40} - 24q^{41} - 24q^{44} - 16q^{46} - 4q^{49} + 24q^{50} - 36q^{51} + 24q^{55} - 12q^{56} + 24q^{59} + 24q^{60} - 8q^{61} + 32q^{64} + 60q^{66} + 12q^{69} + 12q^{70} + 72q^{71} - 12q^{74} - 48q^{75} - 12q^{76} + 16q^{79} - 48q^{80} + 36q^{81} - 36q^{84} + 16q^{85} - 36q^{86} - 24q^{89} - 36q^{90} - 24q^{91} + 20q^{94} + 12q^{95} - 60q^{96} - 36q^{99} + O(q^{100}) \)

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.j.a \(8\) \(0.359\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{6}q^{2}+(\zeta_{24}^{5}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{3}q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + 7 T^{4} + 4 T^{6} + T^{8} + 16 T^{10} + 112 T^{12} + 256 T^{14} + 256 T^{16} \)
$3$ \( 1 - 9 T^{4} + 81 T^{8} \)
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( 1 + 16 T^{2} + 121 T^{4} + 592 T^{6} + 3280 T^{8} + 29008 T^{10} + 290521 T^{12} + 1882384 T^{14} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 396 T^{5} + 968 T^{6} + 7986 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 2 T + 36 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \)
$23$ \( 1 + 88 T^{2} + 4753 T^{4} + 170104 T^{6} + 4569664 T^{8} + 89985016 T^{10} + 1330084273 T^{12} + 13027158232 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 522 T^{5} - 15979 T^{6} - 146334 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 124 T^{5} - 53816 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 56 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 12 T + 29 T^{2} + 396 T^{3} + 5640 T^{4} + 16236 T^{5} + 48749 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( 1 + 88 T^{2} + 3838 T^{4} + 18304 T^{6} - 2480621 T^{8} + 33844096 T^{10} + 13121358238 T^{12} + 556279948312 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 + 160 T^{2} + 14809 T^{4} + 1019680 T^{6} + 55015600 T^{8} + 2252473120 T^{10} + 72263195929 T^{12} + 1724674452640 T^{14} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 196 T^{2} + 15174 T^{4} - 550564 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 12 T + 2 T^{2} - 288 T^{3} + 8187 T^{4} - 16992 T^{5} + 6962 T^{6} - 2464548 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 17324 T^{5} - 130235 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 184 T^{2} + 16657 T^{4} + 1512664 T^{6} + 123674896 T^{8} + 6790348696 T^{10} + 335657222497 T^{12} + 16644342319096 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 18 T + 196 T^{2} - 1278 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 74 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 8 T - 98 T^{2} - 32 T^{3} + 14947 T^{4} - 2528 T^{5} - 611618 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 + 160 T^{2} + 5929 T^{4} + 942880 T^{6} + 163131520 T^{8} + 6495500320 T^{10} + 281380385209 T^{12} + 52310459739040 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 6 T + 79 T^{2} + 534 T^{3} + 7921 T^{4} )^{4} \)
$97$ \( 1 + 52 T^{2} - 12902 T^{4} - 167024 T^{6} + 129576019 T^{8} - 1571528816 T^{10} - 1142204783462 T^{12} + 43314544256308 T^{14} + 7837433594376961 T^{16} \)
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