Properties

Label 7.5.d
Level 7
Weight 5
Character orbit d
Rep. character \(\chi_{7}(3,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 4
Newform subspaces 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(7, [\chi])\).

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

Trace form

\( 4q - 4q^{2} + 6q^{3} - 20q^{4} - 30q^{5} + 304q^{8} - 24q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 6q^{3} - 20q^{4} - 30q^{5} + 304q^{8} - 24q^{9} - 204q^{10} - 58q^{11} - 588q^{12} + 560q^{14} + 468q^{15} - 72q^{16} - 246q^{17} + 216q^{18} + 642q^{19} - 1050q^{21} - 1264q^{22} + 290q^{23} + 720q^{24} - 572q^{25} + 1008q^{26} - 28q^{28} - 2176q^{29} - 72q^{30} + 3618q^{31} + 1584q^{32} + 2070q^{33} - 2478q^{35} - 1632q^{36} - 270q^{37} - 6168q^{38} - 1428q^{39} - 1752q^{40} + 6048q^{42} + 2472q^{43} + 2412q^{44} + 1944q^{45} + 2384q^{46} - 1542q^{47} - 980q^{49} + 7568q^{50} - 4734q^{51} - 3192q^{52} - 4510q^{53} - 11016q^{54} - 1232q^{56} + 11052q^{57} - 904q^{58} + 2526q^{59} - 756q^{60} - 282q^{61} - 336q^{63} - 5472q^{64} + 5796q^{65} - 2556q^{66} - 1318q^{67} + 20412q^{68} + 3360q^{70} - 10408q^{71} - 1296q^{72} + 5214q^{73} + 4036q^{74} - 9636q^{75} - 8890q^{77} - 9072q^{78} - 8110q^{79} - 3144q^{80} + 9306q^{81} - 4032q^{82} + 588q^{84} - 15492q^{85} + 12928q^{86} + 5976q^{87} - 2912q^{88} + 33990q^{89} + 17640q^{91} - 20232q^{92} + 7446q^{93} + 27768q^{94} + 6558q^{95} - 37828q^{98} + 10368q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(7, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
7.5.d.a \(4\) \(0.724\) \(\Q(\sqrt{-3}, \sqrt{22})\) None \(-4\) \(6\) \(-30\) \(0\) \(q+(-2+\beta _{1}-2\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 2 T^{2} - 72 T^{3} - 316 T^{4} - 1152 T^{5} + 512 T^{6} + 16384 T^{7} + 65536 T^{8} \)
$3$ \( 1 - 6 T + 111 T^{2} - 594 T^{3} + 4212 T^{4} - 48114 T^{5} + 728271 T^{6} - 3188646 T^{7} + 43046721 T^{8} \)
$5$ \( 1 + 30 T + 1361 T^{2} + 31830 T^{3} + 922596 T^{4} + 19893750 T^{5} + 531640625 T^{6} + 7324218750 T^{7} + 152587890625 T^{8} \)
$7$ \( 1 + 490 T^{2} + 5764801 T^{4} \)
$11$ \( 1 + 58 T - 20401 T^{2} - 319986 T^{3} + 301164020 T^{4} - 4684915026 T^{5} - 4373135531281 T^{6} + 182028845849818 T^{7} + 45949729863572161 T^{8} \)
$13$ \( 1 - 58972 T^{2} + 1740248838 T^{4} - 48105272078812 T^{6} + 665416609183179841 T^{8} \)
$17$ \( 1 + 246 T + 115961 T^{2} + 23564094 T^{3} + 3884560692 T^{4} + 1968096694974 T^{5} + 808915808615801 T^{6} + 143325070358521206 T^{7} + 48661191875666868481 T^{8} \)
$19$ \( 1 - 642 T + 342023 T^{2} - 131375670 T^{3} + 42796461732 T^{4} - 17121008690070 T^{5} + 5808769181971943 T^{6} - 1420948178040475362 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$23$ \( 1 - 290 T - 459625 T^{2} + 4627530 T^{3} + 193791262244 T^{4} + 1294972622730 T^{5} - 35993686609779625 T^{6} - 6355241085285893090 T^{7} + \)\(61\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 + 1088 T + 1602698 T^{2} + 769521728 T^{3} + 500246412961 T^{4} )^{2} \)
$31$ \( 1 - 3618 T + 7245671 T^{2} - 10428389334 T^{3} + 11484731993796 T^{4} - 9630836546125014 T^{5} + 6179767856146167911 T^{6} - \)\(28\!\cdots\!98\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} \)
$37$ \( 1 + 270 T - 3455695 T^{2} - 59326290 T^{3} + 8801876883204 T^{4} - 111187018992690 T^{5} - 12138057686517530095 T^{6} + \)\(17\!\cdots\!70\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$41$ \( 1 - 7249372 T^{2} + 28218527830086 T^{4} - 57885693378083362012 T^{6} + \)\(63\!\cdots\!41\)\( T^{8} \)
$43$ \( ( 1 - 1236 T + 4524526 T^{2} - 4225638036 T^{3} + 11688200277601 T^{4} )^{2} \)
$47$ \( 1 + 1542 T + 8442143 T^{2} + 11795613810 T^{3} + 38571981640692 T^{4} + 57558832591994610 T^{5} + \)\(20\!\cdots\!23\)\( T^{6} + \)\(17\!\cdots\!22\)\( T^{7} + \)\(56\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 4510 T + 2017313 T^{2} + 11463630750 T^{3} + 112971660447908 T^{4} + 90453560623890750 T^{5} + \)\(12\!\cdots\!93\)\( T^{6} + \)\(22\!\cdots\!10\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \)
$59$ \( 1 - 2526 T + 13587671 T^{2} - 28949927754 T^{3} + 10291335854532 T^{4} - 350796725519137194 T^{5} + \)\(19\!\cdots\!91\)\( T^{6} - \)\(44\!\cdots\!06\)\( T^{7} + \)\(21\!\cdots\!41\)\( T^{8} \)
$61$ \( 1 + 282 T + 23923217 T^{2} + 6738871938 T^{3} + 379712413586628 T^{4} + 93305349372909858 T^{5} + \)\(45\!\cdots\!77\)\( T^{6} + \)\(74\!\cdots\!22\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} \)
$67$ \( 1 + 1318 T - 38954849 T^{2} + 513665458 T^{3} + 1214763993160084 T^{4} + 10350934797678418 T^{5} - \)\(15\!\cdots\!09\)\( T^{6} + \)\(10\!\cdots\!98\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 + 5204 T + 56347598 T^{2} + 132242387924 T^{3} + 645753531245761 T^{4} )^{2} \)
$73$ \( 1 - 5214 T + 63240953 T^{2} - 282489415494 T^{3} + 2386249153485972 T^{4} - 8022202501147746054 T^{5} + \)\(51\!\cdots\!93\)\( T^{6} - \)\(11\!\cdots\!94\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \)
$79$ \( 1 + 8110 T - 25860665 T^{2} + 111371410330 T^{3} + 4317626189098564 T^{4} + 4337925453437736730 T^{5} - \)\(39\!\cdots\!65\)\( T^{6} + \)\(47\!\cdots\!10\)\( T^{7} + \)\(23\!\cdots\!21\)\( T^{8} \)
$83$ \( 1 - 166506148 T^{2} + 11414799577931910 T^{4} - \)\(37\!\cdots\!68\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} \)
$89$ \( 1 - 33990 T + 597537041 T^{2} - 7220507290590 T^{3} + 65352518353788900 T^{4} - \)\(45\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} - \)\(83\!\cdots\!90\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \)
$97$ \( 1 - 137047516 T^{2} + 19765432645146054 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!21\)\( T^{8} \)
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