# Properties

 Label 13.3.f Level 13 Weight 3 Character orbit f Rep. character $$\chi_{13}(2,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 4 Newform subspaces 1 Sturm bound 3 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 13.f (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$1$$ Sturm bound: $$3$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(13, [\chi])$$.

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

## Trace form

 $$4q - 2q^{2} - 2q^{3} - 6q^{4} - 14q^{5} + 10q^{6} + 16q^{7} - 6q^{8} + 10q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{3} - 6q^{4} - 14q^{5} + 10q^{6} + 16q^{7} - 6q^{8} + 10q^{9} + 24q^{10} + 4q^{11} - 26q^{13} - 40q^{14} - 14q^{15} - 2q^{16} - 12q^{17} - 2q^{18} + 10q^{19} + 26q^{20} + 40q^{21} - 4q^{22} + 18q^{23} + 42q^{24} + 52q^{26} - 32q^{27} - 44q^{28} + 2q^{29} - 54q^{30} - 20q^{31} - 20q^{32} - 32q^{33} - 18q^{34} + 32q^{35} - 54q^{36} - 68q^{37} - 26q^{39} + 72q^{40} + 100q^{41} + 44q^{42} + 180q^{43} + 88q^{44} - 2q^{45} - 30q^{46} - 68q^{47} - 50q^{48} - 72q^{49} - 46q^{50} + 128q^{53} + 16q^{54} - 100q^{55} - 84q^{56} - 20q^{57} - 40q^{58} - 164q^{59} + 8q^{60} - 124q^{61} + 6q^{62} + 52q^{63} + 52q^{65} + 80q^{66} + 118q^{67} + 72q^{68} + 72q^{69} + 164q^{70} - 86q^{71} + 72q^{72} + 58q^{73} + 68q^{74} + 48q^{75} + 14q^{76} - 104q^{78} - 40q^{79} - 140q^{80} - 2q^{81} + 24q^{82} - 188q^{83} + 4q^{84} + 96q^{85} - 180q^{86} + 38q^{87} - 204q^{88} - 110q^{89} + 52q^{91} - 156q^{92} - 20q^{93} + 26q^{94} - 78q^{95} + 40q^{96} + 178q^{97} - 14q^{98} + 208q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
13.3.f.a $$4$$ $$0.354$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$-2$$ $$-14$$ $$16$$ $$q+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} + 12 T^{3} + 17 T^{4} + 48 T^{5} + 80 T^{6} + 128 T^{7} + 256 T^{8}$$
$3$ $$1 + 2 T - 12 T^{2} - 4 T^{3} + 139 T^{4} - 36 T^{5} - 972 T^{6} + 1458 T^{7} + 6561 T^{8}$$
$5$ $$( 1 + 6 T + 11 T^{2} + 150 T^{3} + 625 T^{4} )( 1 + 8 T + 39 T^{2} + 200 T^{3} + 625 T^{4} )$$
$7$ $$1 - 16 T + 164 T^{2} - 1236 T^{3} + 8927 T^{4} - 60564 T^{5} + 393764 T^{6} - 1882384 T^{7} + 5764801 T^{8}$$
$11$ $$1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 116160 T^{5} + 2928200 T^{6} - 7086244 T^{7} + 214358881 T^{8}$$
$13$ $$( 1 + 13 T + 169 T^{2} )^{2}$$
$17$ $$1 + 12 T + 413 T^{2} + 4380 T^{3} + 63576 T^{4} + 1265820 T^{5} + 34494173 T^{6} + 289650828 T^{7} + 6975757441 T^{8}$$
$19$ $$1 - 10 T + 74 T^{2} - 1752 T^{3} - 96625 T^{4} - 632472 T^{5} + 9643754 T^{6} - 470458810 T^{7} + 16983563041 T^{8}$$
$23$ $$1 - 18 T + 968 T^{2} - 15480 T^{3} + 516891 T^{4} - 8188920 T^{5} + 270886088 T^{6} - 2664646002 T^{7} + 78310985281 T^{8}$$
$29$ $$1 - 2 T - 1571 T^{2} + 214 T^{3} + 1769980 T^{4} + 179974 T^{5} - 1111138451 T^{6} - 1189646642 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 17586300 T^{5} + 184704200 T^{6} + 17750073620 T^{7} + 852891037441 T^{8}$$
$37$ $$1 + 68 T + 1517 T^{2} - 93168 T^{3} - 5925028 T^{4} - 127546992 T^{5} + 2843102237 T^{6} + 174469395812 T^{7} + 3512479453921 T^{8}$$
$41$ $$1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 12224232 T^{5} + 9779958821 T^{6} - 475010424100 T^{7} + 7984925229121 T^{8}$$
$43$ $$( 1 - 90 T + 4549 T^{2} - 166410 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 + 68 T + 2312 T^{2} + 148716 T^{3} + 9565454 T^{4} + 328513644 T^{5} + 11281822472 T^{6} + 732986642372 T^{7} + 23811286661761 T^{8}$$
$53$ $$( 1 - 64 T + 4455 T^{2} - 179776 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 + 164 T + 6980 T^{2} - 551844 T^{3} - 68910913 T^{4} - 1920968964 T^{5} + 84579179780 T^{6} + 6917607517124 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 1163199484 T^{5} + 74947537333 T^{6} + 6388526420764 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 - 118 T + 10706 T^{2} - 763488 T^{3} + 51177839 T^{4} - 3427297632 T^{5} + 215737901426 T^{6} - 10674089095942 T^{7} + 406067677556641 T^{8}$$
$71$ $$1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 1302271776 T^{5} + 118367610098 T^{6} + 11016624417206 T^{7} + 645753531245761 T^{8}$$
$73$ $$1 - 58 T + 1682 T^{2} - 201144 T^{3} + 20590727 T^{4} - 1071896376 T^{5} + 47765841362 T^{6} - 8777385124762 T^{7} + 806460091894081 T^{8}$$
$79$ $$( 1 + 20 T + 7290 T^{2} + 124820 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 + 188 T + 17672 T^{2} + 1935084 T^{3} + 200304482 T^{4} + 13330793676 T^{5} + 838683448712 T^{6} + 61464790193372 T^{7} + 2252292232139041 T^{8}$$
$89$ $$1 + 110 T + 12050 T^{2} + 1194180 T^{3} + 87746159 T^{4} + 9459099780 T^{5} + 756044004050 T^{6} + 54667942005710 T^{7} + 3936588805702081 T^{8}$$
$97$ $$1 - 178 T + 13250 T^{2} - 318828 T^{3} - 39375793 T^{4} - 2999852652 T^{5} + 1173012973250 T^{6} - 148269016877362 T^{7} + 7837433594376961 T^{8}$$