Properties

Label 13.4.e
Level 13
Weight 4
Character orbit e
Rep. character \(\chi_{13}(4,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 4
Newform subspaces 2
Sturm bound 4
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

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

Trace form

\( 4q - 3q^{2} + 5q^{3} - q^{4} - 48q^{6} + 15q^{7} + q^{9} + O(q^{10}) \) \( 4q - 3q^{2} + 5q^{3} - q^{4} - 48q^{6} + 15q^{7} + q^{9} + 51q^{10} - 15q^{11} + 76q^{12} + 65q^{13} - 204q^{14} + 174q^{15} + 79q^{16} - 144q^{17} - 351q^{19} - 111q^{20} + 102q^{22} + 21q^{23} + 246q^{24} + 110q^{25} + 351q^{26} - 130q^{27} + 276q^{28} + 210q^{29} - 330q^{30} - 603q^{32} - 321q^{33} + 336q^{35} + 203q^{36} - 318q^{37} + 216q^{38} + 325q^{39} - 462q^{40} - 210q^{41} - 498q^{42} - 19q^{43} + 597q^{45} + 576q^{46} - 562q^{48} + 13q^{49} + 768q^{50} + 90q^{51} - 494q^{52} + 1038q^{53} - 510q^{54} + 336q^{55} + 624q^{56} + 9q^{58} - 525q^{59} - 128q^{61} - 522q^{62} - 1410q^{63} - 742q^{64} - 1209q^{65} + 996q^{66} - 1077q^{67} - 477q^{68} + 555q^{69} + 2841q^{71} + 1425q^{72} - 111q^{74} - 713q^{75} + 378q^{76} - 1398q^{77} - 858q^{78} + 64q^{79} + 1923q^{80} + 418q^{81} + 1833q^{82} + 852q^{84} + 297q^{85} - 201q^{87} + 624q^{88} - 1161q^{89} - 1974q^{90} - 741q^{91} - 1236q^{92} - 1422q^{93} - 1710q^{94} - 1026q^{95} + 2487q^{97} - 1437q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.4.e.a \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(-6\) \(7\) \(0\) \(39\) \(q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots\)
13.4.e.b \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(3\) \(-2\) \(0\) \(-24\) \(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T + 20 T^{2} + 48 T^{3} + 64 T^{4} \))(\( 1 - 3 T + 11 T^{2} - 24 T^{3} + 64 T^{4} \))
$3$ (\( 1 - 7 T + 22 T^{2} - 189 T^{3} + 729 T^{4} \))(\( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} \))
$5$ (\( 1 - 58 T^{2} + 15625 T^{4} \))(\( 1 - 247 T^{2} + 15625 T^{4} \))
$7$ (\( 1 - 39 T + 850 T^{2} - 13377 T^{3} + 117649 T^{4} \))(\( 1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4} \))
$11$ (\( 1 + 39 T + 1838 T^{2} + 51909 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T + 1523 T^{2} - 31944 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 + 26 T + 2197 T^{2} \))(\( 1 - 91 T + 2197 T^{2} \))
$17$ (\( 1 + 27 T - 4184 T^{2} + 132651 T^{3} + 24137569 T^{4} \))(\( 1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 153 T + 14662 T^{2} + 1049427 T^{3} + 47045881 T^{4} \))(\( 1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 + 57 T - 8918 T^{2} + 693519 T^{3} + 148035889 T^{4} \))(\( 1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 - 69 T - 19628 T^{2} - 1682841 T^{3} + 594823321 T^{4} \))(\( 1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 - 54290 T^{2} + 887503681 T^{4} \))(\( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \))
$37$ (\( 1 + 69 T + 52240 T^{2} + 3495057 T^{3} + 2565726409 T^{4} \))(\( 1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 + 681 T + 223508 T^{2} + 46935201 T^{3} + 4750104241 T^{4} \))(\( 1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 - 85 T - 72282 T^{2} - 6758095 T^{3} + 6321363049 T^{4} \))(\( 1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 - 90034 T^{2} + 10779215329 T^{4} \))(\( 1 - 116818 T^{2} + 10779215329 T^{4} \))
$53$ (\( ( 1 - 426 T + 148877 T^{2} )^{2} \))(\( ( 1 - 93 T + 148877 T^{2} )^{2} \))
$59$ (\( 1 + 33 T + 205742 T^{2} + 6777507 T^{3} + 42180533641 T^{4} \))(\( 1 + 492 T + 286067 T^{2} + 101046468 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 - 17 T - 226692 T^{2} - 3858677 T^{3} + 51520374361 T^{4} \))(\( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 285 T + 327838 T^{2} - 85717455 T^{3} + 90458382169 T^{4} \))(\( 1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 1011 T + 698618 T^{2} - 361848021 T^{3} + 128100283921 T^{4} \))(\( 1 - 1830 T + 1474211 T^{2} - 654977130 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 + 231166 T^{2} + 151334226289 T^{4} \))(\( 1 - 567359 T^{2} + 151334226289 T^{4} \))
$79$ (\( ( 1 + 1244 T + 493039 T^{2} )^{2} \))(\( ( 1 - 1276 T + 493039 T^{2} )^{2} \))
$83$ (\( 1 - 962026 T^{2} + 326940373369 T^{4} \))(\( 1 - 519766 T^{2} + 326940373369 T^{4} \))
$89$ (\( 1 - 531 T + 798956 T^{2} - 374338539 T^{3} + 496981290961 T^{4} \))(\( 1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 2139 T + 2437780 T^{2} - 1952207547 T^{3} + 832972004929 T^{4} \))(\( 1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4} \))
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