Properties

Label 14.3.d
Level 14
Weight 3
Character orbit d
Rep. character \(\chi_{14}(3,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 4
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 14.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(14, [\chi])\).

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

Trace form

\( 4q - 6q^{3} - 4q^{4} - 6q^{5} + 8q^{7} + O(q^{10}) \) \( 4q - 6q^{3} - 4q^{4} - 6q^{5} + 8q^{7} + 24q^{10} + 18q^{11} + 12q^{12} - 36q^{14} - 36q^{15} - 8q^{16} - 30q^{17} - 24q^{18} + 6q^{19} + 54q^{21} + 24q^{22} + 30q^{23} + 24q^{24} + 4q^{25} + 24q^{26} - 20q^{28} + 48q^{29} - 12q^{30} - 42q^{31} - 90q^{33} - 42q^{35} - 62q^{37} - 12q^{38} + 12q^{39} - 48q^{40} + 72q^{42} - 8q^{43} + 36q^{44} + 144q^{45} + 36q^{46} + 174q^{47} - 20q^{49} - 96q^{50} + 54q^{51} - 72q^{52} - 78q^{53} - 36q^{54} + 48q^{56} + 12q^{57} + 24q^{58} - 78q^{59} + 36q^{60} - 42q^{61} - 216q^{63} + 32q^{64} - 84q^{65} - 144q^{66} - 58q^{67} + 60q^{68} + 84q^{70} - 24q^{71} - 48q^{72} + 318q^{73} + 96q^{74} + 132q^{75} + 126q^{77} + 96q^{78} + 110q^{79} + 24q^{80} + 18q^{81} - 120q^{82} + 12q^{84} - 36q^{85} + 24q^{86} - 144q^{87} - 24q^{88} - 378q^{89} + 24q^{91} - 120q^{92} - 138q^{93} - 12q^{94} - 30q^{95} - 48q^{96} - 120q^{98} + 144q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(14, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
14.3.d.a \(4\) \(0.381\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-6\) \(-6\) \(8\) \(q+\beta _{1}q^{2}+(-2-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( 1 + 6 T + 27 T^{2} + 90 T^{3} + 252 T^{4} + 810 T^{5} + 2187 T^{6} + 4374 T^{7} + 6561 T^{8} \)
$5$ \( ( 1 + 2 T + 25 T^{2} )^{2}( 1 + 2 T - 21 T^{2} + 50 T^{3} + 625 T^{4} ) \)
$7$ \( 1 - 8 T + 42 T^{2} - 392 T^{3} + 2401 T^{4} \)
$11$ \( 1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 137214 T^{5} + 278179 T^{6} - 31888098 T^{7} + 214358881 T^{8} \)
$13$ \( 1 - 412 T^{2} + 89190 T^{4} - 11767132 T^{6} + 815730721 T^{8} \)
$17$ \( 1 + 30 T + 929 T^{2} + 18870 T^{3} + 398820 T^{4} + 5453430 T^{5} + 77591009 T^{6} + 724127070 T^{7} + 6975757441 T^{8} \)
$19$ \( 1 - 6 T + 731 T^{2} - 4314 T^{3} + 390972 T^{4} - 1557354 T^{5} + 95264651 T^{6} - 282275286 T^{7} + 16983563041 T^{8} \)
$23$ \( 1 - 30 T - 221 T^{2} - 1890 T^{3} + 500700 T^{4} - 999810 T^{5} - 61844861 T^{6} - 4441076670 T^{7} + 78310985281 T^{8} \)
$29$ \( ( 1 - 24 T + 1754 T^{2} - 20184 T^{3} + 707281 T^{4} )^{2} \)
$31$ \( 1 + 42 T + 1307 T^{2} + 30198 T^{3} + 158508 T^{4} + 29020278 T^{5} + 1207041947 T^{6} + 37275154602 T^{7} + 852891037441 T^{8} \)
$37$ \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 16211698 T^{5} + 2430786817 T^{6} + 159075037358 T^{7} + 3512479453921 T^{8} \)
$41$ \( 1 - 5500 T^{2} + 13185222 T^{4} - 15541685500 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 + 4 T + 3630 T^{2} + 7396 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 - 174 T + 17027 T^{2} - 1206690 T^{3} + 65507772 T^{4} - 2665578210 T^{5} + 83086328387 T^{6} - 1875583467246 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 + 78 T - 767 T^{2} + 96174 T^{3} + 21955764 T^{4} + 270152766 T^{5} - 6051998927 T^{6} + 1728820168062 T^{7} + 62259690411361 T^{8} \)
$59$ \( 1 + 78 T + 5747 T^{2} + 290082 T^{3} + 8773068 T^{4} + 1009775442 T^{5} + 69638473667 T^{6} + 3290081623998 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 225827490 T^{5} + 28148594753 T^{6} + 2163855723162 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 + 58 T - 2405 T^{2} - 186122 T^{3} - 1970756 T^{4} - 835501658 T^{5} - 48463446005 T^{6} + 5246586165802 T^{7} + 406067677556641 T^{8} \)
$71$ \( ( 1 + 12 T + 8318 T^{2} + 60492 T^{3} + 25411681 T^{4} )^{2} \)
$73$ \( 1 - 318 T + 51257 T^{2} - 5580582 T^{3} + 459199092 T^{4} - 29738921478 T^{5} + 1455608638937 T^{6} - 48124283959902 T^{7} + 806460091894081 T^{8} \)
$79$ \( 1 - 110 T - 2957 T^{2} - 283250 T^{3} + 112247068 T^{4} - 1767763250 T^{5} - 115175389517 T^{6} - 26739620107310 T^{7} + 1517108809906561 T^{8} \)
$83$ \( 1 + 380 T^{2} + 89625894 T^{4} + 18034161980 T^{6} + 2252292232139041 T^{8} \)
$89$ \( 1 + 378 T + 71921 T^{2} + 9182754 T^{3} + 904668996 T^{4} + 72736594434 T^{5} + 4512484714961 T^{6} + 187858927983258 T^{7} + 3936588805702081 T^{8} \)
$97$ \( 1 - 26620 T^{2} + 330657414 T^{4} - 2356649460220 T^{6} + 7837433594376961 T^{8} \)
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