Properties

Label 15.4.e
Level 15
Weight 4
Character orbit e
Rep. character \(\chi_{15}(2,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 15.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(15, [\chi])\).

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

Trace form

\( 8q - 6q^{3} - 12q^{6} - 16q^{7} + O(q^{10}) \) \( 8q - 6q^{3} - 12q^{6} - 16q^{7} - 100q^{10} + 132q^{12} + 68q^{13} + 90q^{15} + 284q^{16} - 240q^{18} - 492q^{21} - 500q^{22} - 220q^{25} + 702q^{27} + 508q^{28} + 660q^{30} + 616q^{31} - 240q^{33} - 804q^{36} - 1156q^{37} - 600q^{40} + 540q^{42} + 548q^{43} + 180q^{45} + 736q^{46} - 1116q^{48} - 852q^{51} + 224q^{52} + 460q^{55} + 684q^{57} + 60q^{58} + 540q^{60} + 16q^{61} + 1428q^{63} + 2040q^{66} + 404q^{67} - 2220q^{70} - 1800q^{72} - 2512q^{73} - 2910q^{75} - 1488q^{76} - 360q^{78} + 288q^{81} + 2800q^{82} + 4940q^{85} - 1680q^{87} + 2460q^{88} + 600q^{90} - 1304q^{91} + 3408q^{93} + 4164q^{96} + 1904q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(15, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
15.4.e.a \(8\) \(0.885\) 8.0.\(\cdots\).8 None \(0\) \(-6\) \(0\) \(-16\) \(q-\beta _{3}q^{2}+(-1-\beta _{2}+\beta _{5})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 79 T^{4} + 2496 T^{8} - 323584 T^{12} + 16777216 T^{16} \)
$3$ \( 1 + 6 T + 18 T^{2} - 198 T^{3} - 1422 T^{4} - 5346 T^{5} + 13122 T^{6} + 118098 T^{7} + 531441 T^{8} \)
$5$ \( 1 + 110 T^{2} + 5250 T^{4} + 1718750 T^{6} + 244140625 T^{8} \)
$7$ \( ( 1 + 8 T + 32 T^{2} + 744 T^{3} - 45202 T^{4} + 255192 T^{5} + 3764768 T^{6} + 322828856 T^{7} + 13841287201 T^{8} )^{2} \)
$11$ \( ( 1 - 3334 T^{2} + 6292986 T^{4} - 5906384374 T^{6} + 3138428376721 T^{8} )^{2} \)
$13$ \( ( 1 - 34 T + 578 T^{2} - 77418 T^{3} + 10363058 T^{4} - 170087346 T^{5} + 2789895602 T^{6} - 360552978682 T^{7} + 23298085122481 T^{8} )^{2} \)
$17$ \( 1 - 10588864 T^{4} - 447530075229954 T^{8} - \)\(61\!\cdots\!04\)\( T^{12} + \)\(33\!\cdots\!21\)\( T^{16} \)
$19$ \( ( 1 - 24664 T^{2} + 245125086 T^{4} - 1160339608984 T^{6} + 2213314919066161 T^{8} )^{2} \)
$23$ \( 1 + 232150736 T^{4} + 54876630002867166 T^{8} + \)\(50\!\cdots\!56\)\( T^{12} + \)\(48\!\cdots\!41\)\( T^{16} \)
$29$ \( ( 1 + 79166 T^{2} + 2712313506 T^{4} + 47089783030286 T^{6} + 353814783205469041 T^{8} )^{2} \)
$31$ \( ( 1 - 154 T + 36486 T^{2} - 4587814 T^{3} + 887503681 T^{4} )^{4} \)
$37$ \( ( 1 + 578 T + 167042 T^{2} + 53079474 T^{3} + 15170807378 T^{4} + 2688634596522 T^{5} + 428584070812178 T^{6} + 75117885601554506 T^{7} + 6582952005840035281 T^{8} )^{2} \)
$41$ \( ( 1 - 115444 T^{2} + 7614039366 T^{4} - 548371033998004 T^{6} + 22563490300366186081 T^{8} )^{2} \)
$43$ \( ( 1 - 274 T + 37538 T^{2} - 17976318 T^{3} + 8415346898 T^{4} - 1429243115226 T^{5} + 237291326133362 T^{6} - 137710375670694982 T^{7} + 39959630797262576401 T^{8} )^{2} \)
$47$ \( 1 + 29984206736 T^{4} + \)\(42\!\cdots\!06\)\( T^{8} + \)\(34\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!81\)\( T^{16} \)
$53$ \( 1 - 24371904064 T^{4} + \)\(31\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!24\)\( T^{12} + \)\(24\!\cdots\!81\)\( T^{16} \)
$59$ \( ( 1 + 112106 T^{2} + 56049165066 T^{4} + 4728690904357946 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \)
$61$ \( ( 1 - 2 T + 226981 T^{2} )^{8} \)
$67$ \( ( 1 - 202 T + 20402 T^{2} - 3297246 T^{3} - 80373232942 T^{4} - 991689598698 T^{5} + 1845531913011938 T^{6} - 5495719948051579294 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \)
$71$ \( ( 1 - 863584 T^{2} + 377860054206 T^{4} - 110625355589632864 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \)
$73$ \( ( 1 + 1256 T + 788768 T^{2} + 733362072 T^{3} + 643873740638 T^{4} + 285290313163224 T^{5} + 119367595001521952 T^{6} + 73942712905584498728 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} )^{2} \)
$79$ \( ( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 394794447112367764 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \)
$83$ \( 1 + 1005705244496 T^{4} + \)\(46\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{12} + \)\(11\!\cdots\!21\)\( T^{16} \)
$89$ \( ( 1 - 806584 T^{2} + 1118488022286 T^{4} - 400857157588487224 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \)
$97$ \( ( 1 - 952 T + 453152 T^{2} - 121725576 T^{3} - 583228842562 T^{4} - 111095646624648 T^{5} + 377462929977586208 T^{6} - \)\(72\!\cdots\!84\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \)
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