Properties

Label 15.3.f
Level 15
Weight 3
Character orbit f
Rep. character \(\chi_{15}(7,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 15.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(15, [\chi])\).

Total New Old
Modular forms 12 4 8
Cusp forms 4 4 0
Eisenstein series 8 0 8

Trace form

\( 4q - 4q^{2} - 4q^{5} - 12q^{6} + 4q^{7} + 12q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{5} - 12q^{6} + 4q^{7} + 12q^{8} + 4q^{10} + 16q^{11} + 24q^{12} - 32q^{13} + 24q^{15} - 20q^{16} - 40q^{17} - 12q^{18} - 36q^{20} - 24q^{21} + 20q^{22} + 56q^{23} + 16q^{25} + 88q^{26} + 44q^{28} - 24q^{30} - 16q^{31} - 76q^{32} - 36q^{33} - 40q^{35} + 12q^{36} + 64q^{37} - 96q^{38} + 48q^{40} - 56q^{41} + 12q^{42} - 8q^{43} + 36q^{45} - 136q^{46} + 128q^{47} + 48q^{48} + 164q^{50} + 72q^{51} - 80q^{52} + 56q^{53} - 124q^{55} - 72q^{57} - 12q^{58} - 84q^{60} + 200q^{61} + 88q^{62} + 12q^{63} - 112q^{65} + 24q^{66} - 200q^{67} - 104q^{68} - 60q^{70} - 272q^{71} - 36q^{72} + 76q^{73} + 24q^{75} + 312q^{76} + 88q^{77} + 120q^{78} + 164q^{80} - 36q^{81} + 128q^{82} - 16q^{83} + 232q^{85} - 224q^{86} - 84q^{87} + 12q^{88} - 96q^{90} - 16q^{91} + 104q^{92} - 72q^{93} + 144q^{95} - 84q^{96} - 20q^{97} - 188q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(15, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
15.3.f.a \(4\) \(0.409\) \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(-4\) \(4\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{3}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 17 T^{4} + 48 T^{5} + 128 T^{6} + 256 T^{7} + 256 T^{8} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( 1 + 4 T + 100 T^{3} + 625 T^{4} \)
$7$ \( 1 - 4 T + 8 T^{2} - 156 T^{3} + 2942 T^{4} - 7644 T^{5} + 19208 T^{6} - 470596 T^{7} + 5764801 T^{8} \)
$11$ \( ( 1 - 8 T + 204 T^{2} - 968 T^{3} + 14641 T^{4} )^{2} \)
$13$ \( 1 + 32 T + 512 T^{2} + 9120 T^{3} + 148994 T^{4} + 1541280 T^{5} + 14623232 T^{6} + 154457888 T^{7} + 815730721 T^{8} \)
$17$ \( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 4404360 T^{5} + 66816800 T^{6} + 965502760 T^{7} + 6975757441 T^{8} \)
$19$ \( 1 - 940 T^{2} + 450438 T^{4} - 122501740 T^{6} + 16983563041 T^{8} \)
$23$ \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 26928216 T^{5} + 438790688 T^{6} - 8290009784 T^{7} + 78310985281 T^{8} \)
$29$ \( 1 - 2128 T^{2} + 2165634 T^{4} - 1505093968 T^{6} + 500246412961 T^{8} \)
$31$ \( ( 1 + 8 T + 1722 T^{2} + 7688 T^{3} + 923521 T^{4} )^{2} \)
$37$ \( 1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 79642944 T^{5} + 3838281728 T^{6} - 164206490176 T^{7} + 3512479453921 T^{8} \)
$41$ \( ( 1 + 28 T + 3342 T^{2} + 47068 T^{3} + 2825761 T^{4} )^{2} \)
$43$ \( 1 + 8 T + 32 T^{2} + 5256 T^{3} - 557566 T^{4} + 9718344 T^{5} + 109401632 T^{6} + 50570904392 T^{7} + 11688200277601 T^{8} \)
$47$ \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 1118849664 T^{5} + 39974346752 T^{6} - 1379739562112 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 435574776 T^{5} + 12372274208 T^{6} - 1241204223224 T^{7} + 62259690411361 T^{8} \)
$59$ \( 1 + 200 T^{2} - 5646222 T^{4} + 2423472200 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 - 100 T + 7998 T^{2} - 372100 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( 1 + 200 T + 20000 T^{2} + 1888200 T^{3} + 153742658 T^{4} + 8476129800 T^{5} + 403022420000 T^{6} + 18091676433800 T^{7} + 406067677556641 T^{8} \)
$71$ \( ( 1 + 68 T + 5041 T^{2} )^{4} \)
$73$ \( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 348708444 T^{5} + 82014120008 T^{6} - 11501401197964 T^{7} + 806460091894081 T^{8} \)
$79$ \( ( 1 - 11882 T^{2} + 38950081 T^{4} )^{2} \)
$83$ \( 1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 698048592 T^{5} + 6074665088 T^{6} + 5231045973904 T^{7} + 2252292232139041 T^{8} \)
$89$ \( 1 - 16060 T^{2} + 188845638 T^{4} - 1007640390460 T^{6} + 3936588805702081 T^{8} \)
$97$ \( 1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 1635472380 T^{5} + 17705856200 T^{6} + 16659440098580 T^{7} + 7837433594376961 T^{8} \)
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