Properties

Label 80.6.c
Level 80
Weight 6
Character orbit c
Rep. character \(\chi_{80}(49,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 4
Sturm bound 72
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(80, [\chi])\).

Total New Old
Modular forms 66 16 50
Cusp forms 54 14 40
Eisenstein series 12 2 10

Trace form

\( 14q + 18q^{5} - 974q^{9} + O(q^{10}) \) \( 14q + 18q^{5} - 974q^{9} + 728q^{11} + 1192q^{15} - 984q^{19} + 1304q^{21} - 586q^{25} + 8260q^{29} - 9344q^{31} - 12712q^{35} + 10544q^{39} + 3580q^{41} - 10578q^{45} - 31222q^{49} + 73088q^{51} + 22952q^{55} - 69896q^{59} + 21748q^{61} + 4016q^{65} - 21096q^{69} - 47792q^{71} - 85584q^{75} + 82912q^{79} + 141254q^{81} - 97920q^{85} + 110188q^{89} + 105680q^{91} - 195752q^{95} - 152024q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(80, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
80.6.c.a \(2\) \(12.831\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(-90\) \(0\) \(q-3\beta q^{3}+(-45-5\beta )q^{5}-9\beta q^{7}+\cdots\)
80.6.c.b \(2\) \(12.831\) \(\Q(\sqrt{-31}) \) None \(0\) \(0\) \(-10\) \(0\) \(q-\beta q^{3}+(-5+5\beta )q^{5}-11\beta q^{7}+119q^{9}+\cdots\)
80.6.c.c \(2\) \(12.831\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(110\) \(0\) \(q+7iq^{3}+(55-5i)q^{5}-79iq^{7}+47q^{9}+\cdots\)
80.6.c.d \(8\) \(12.831\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(8\) \(0\) \(q+\beta _{1}q^{3}+(1-\beta _{2})q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(80, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(80, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} ) \))(\( 1 - 362 T^{2} + 59049 T^{4} \))(\( 1 - 290 T^{2} + 59049 T^{4} \))(\( 1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16} \))
$5$ (\( 1 + 90 T + 3125 T^{2} \))(\( 1 + 10 T + 3125 T^{2} \))(\( 1 - 110 T + 3125 T^{2} \))(\( 1 - 8 T + 1100 T^{2} + 113000 T^{3} - 438250 T^{4} + 353125000 T^{5} + 10742187500 T^{6} - 244140625000 T^{7} + 95367431640625 T^{8} \))
$7$ (\( 1 - 30050 T^{2} + 282475249 T^{4} \))(\( 1 - 18610 T^{2} + 282475249 T^{4} \))(\( 1 - 8650 T^{2} + 282475249 T^{4} \))(\( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} \))
$11$ (\( ( 1 + 252 T + 161051 T^{2} )^{2} \))(\( ( 1 - 100 T + 161051 T^{2} )^{2} \))(\( ( 1 - 148 T + 161051 T^{2} )^{2} \))(\( ( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))
$13$ (\( 1 - 728330 T^{2} + 137858491849 T^{4} \))(\( 1 - 202442 T^{2} + 137858491849 T^{4} \))(\( 1 - 274730 T^{2} + 137858491849 T^{4} \))(\( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(41\!\cdots\!28\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \))
$17$ (\( 1 - 2363810 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1879458 T^{2} + 2015993900449 T^{4} \))(\( 1 + 1354590 T^{2} + 2015993900449 T^{4} \))(\( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!32\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \))
$19$ (\( ( 1 - 220 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2244 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 2220 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))
$23$ (\( 1 - 6946370 T^{2} + 41426511213649 T^{4} \))(\( 1 - 1185810 T^{2} + 41426511213649 T^{4} \))(\( 1 - 11320170 T^{2} + 41426511213649 T^{4} \))(\( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!28\)\( T^{12} - \)\(24\!\cdots\!04\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))
$29$ (\( ( 1 + 6930 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 7854 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 270 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))
$31$ (\( ( 1 + 6752 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2144 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2048 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))
$37$ (\( 1 + 56462470 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 30515770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 119573530 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} - \)\(27\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \))
$41$ (\( ( 1 + 198 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 7414 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 2398 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))
$43$ (\( 1 - 293842250 T^{2} + 21611482313284249 T^{4} \))(\( 1 + 21442214 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288754450 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 347593490 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 369731298 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 344584890 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(83\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} \))
$53$ (\( 1 - 802472090 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 248174170 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 827605690 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \))
$59$ (\( ( 1 - 24660 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 25972 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 39740 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} + \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \))
$61$ (\( ( 1 + 5698 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 3058 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 42298 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \))
$67$ (\( 1 - 795787610 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 755362070 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1669968610 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))
$71$ (\( ( 1 + 53352 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 37608 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 4248 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} - \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))
$73$ (\( 1 + 883886830 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3569749522 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3239892370 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!40\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \))
$79$ (\( ( 1 + 51920 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 79728 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 35280 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} + \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))
$83$ (\( 1 - 4053674810 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7612675530 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7103795010 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 6759897816 T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(98\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \))
$89$ (\( ( 1 + 9990 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 826 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 85210 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))
$97$ (\( 1 - 6923133890 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 15761405890 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 7720618690 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(55\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))
show more
show less