Properties

Label 50.6
Level 50
Weight 6
Dimension 115
Nonzero newspaces 4
Newform subspaces 14
Sturm bound 900
Trace bound 1

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 14 \)
Sturm bound: \(900\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(50))\).

Total New Old
Modular forms 403 115 288
Cusp forms 347 115 232
Eisenstein series 56 0 56

Trace form

\( 115q + 8q^{2} - 8q^{3} - 32q^{4} - 85q^{5} + 160q^{6} + 624q^{7} + 128q^{8} - 1306q^{9} + O(q^{10}) \) \( 115q + 8q^{2} - 8q^{3} - 32q^{4} - 85q^{5} + 160q^{6} + 624q^{7} + 128q^{8} - 1306q^{9} + 180q^{10} + 1480q^{11} - 128q^{12} - 228q^{13} - 3136q^{14} - 820q^{15} - 2560q^{16} + 1984q^{17} - 316q^{18} - 6400q^{19} + 640q^{20} - 2120q^{21} + 2816q^{22} + 12152q^{23} + 6912q^{24} + 25335q^{25} + 1760q^{26} + 15820q^{27} + 3904q^{28} - 34420q^{29} - 30000q^{30} - 22520q^{31} - 3072q^{32} - 63336q^{33} - 13716q^{34} + 60860q^{35} - 14880q^{36} + 64409q^{37} + 34720q^{38} + 60888q^{39} + 320q^{40} - 37120q^{41} - 22784q^{42} - 86288q^{43} + 4736q^{44} + 1835q^{45} - 19840q^{46} - 22456q^{47} - 2048q^{48} + 153531q^{49} + 11300q^{50} + 26780q^{51} - 3648q^{52} + 112257q^{53} + 64320q^{54} - 66180q^{55} + 30720q^{56} - 221840q^{57} - 74640q^{58} - 44860q^{59} + 112640q^{60} + 160880q^{61} - 90704q^{62} - 127508q^{63} - 8192q^{64} - 337865q^{65} - 110480q^{66} - 258296q^{67} - 114816q^{68} - 333372q^{69} + 47360q^{70} + 97280q^{71} + 124544q^{72} + 428212q^{73} + 570624q^{74} + 892460q^{75} + 89600q^{76} + 685888q^{77} + 34448q^{78} + 379120q^{79} - 21760q^{80} - 110410q^{81} - 368624q^{82} - 326088q^{83} - 210624q^{84} - 834505q^{85} - 360240q^{86} - 1636820q^{87} - 105984q^{88} - 509375q^{89} + 244020q^{90} - 186320q^{91} + 379712q^{92} + 525124q^{93} + 316544q^{94} + 379540q^{95} + 40960q^{96} + 614584q^{97} - 5304q^{98} + 1272548q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(50))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
50.6.a \(\chi_{50}(1, \cdot)\) 50.6.a.a 1 1
50.6.a.b 1
50.6.a.c 1
50.6.a.d 1
50.6.a.e 1
50.6.a.f 1
50.6.a.g 1
50.6.b \(\chi_{50}(49, \cdot)\) 50.6.b.a 2 1
50.6.b.b 2
50.6.b.c 2
50.6.b.d 2
50.6.d \(\chi_{50}(11, \cdot)\) 50.6.d.a 24 4
50.6.d.b 28
50.6.e \(\chi_{50}(9, \cdot)\) 50.6.e.a 48 4

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(50))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(50)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))
$3$ (\( 1 + 11 T + 243 T^{2} \))(\( 1 + 6 T + 243 T^{2} \))(\( 1 - 14 T + 243 T^{2} \))(\( 1 + 24 T + 243 T^{2} \))(\( 1 + 14 T + 243 T^{2} \))(\( 1 - 11 T + 243 T^{2} \))(\( 1 - 26 T + 243 T^{2} \))(\( 1 + 90 T^{2} + 59049 T^{4} \))(\( 1 - 450 T^{2} + 59049 T^{4} \))(\( 1 - 365 T^{2} + 59049 T^{4} \))(\( 1 + 190 T^{2} + 59049 T^{4} \))
$5$ 1
$7$ (\( 1 + 142 T + 16807 T^{2} \))(\( 1 - 118 T + 16807 T^{2} \))(\( 1 - 158 T + 16807 T^{2} \))(\( 1 - 172 T + 16807 T^{2} \))(\( 1 + 158 T + 16807 T^{2} \))(\( 1 - 142 T + 16807 T^{2} \))(\( 1 - 22 T + 16807 T^{2} \))(\( 1 - 4030 T^{2} + 282475249 T^{4} \))(\( 1 - 19690 T^{2} + 282475249 T^{4} \))(\( 1 - 13450 T^{2} + 282475249 T^{4} \))(\( 1 - 33130 T^{2} + 282475249 T^{4} \))
$11$ (\( 1 - 777 T + 161051 T^{2} \))(\( 1 - 192 T + 161051 T^{2} \))(\( 1 + 148 T + 161051 T^{2} \))(\( 1 - 132 T + 161051 T^{2} \))(\( 1 + 148 T + 161051 T^{2} \))(\( 1 - 777 T + 161051 T^{2} \))(\( 1 + 768 T + 161051 T^{2} \))(\( ( 1 - 132 T + 161051 T^{2} )^{2} \))(\( ( 1 - 192 T + 161051 T^{2} )^{2} \))(\( ( 1 - 777 T + 161051 T^{2} )^{2} \))(\( ( 1 + 768 T + 161051 T^{2} )^{2} \))
$13$ (\( 1 - 884 T + 371293 T^{2} \))(\( 1 + 1106 T + 371293 T^{2} \))(\( 1 - 684 T + 371293 T^{2} \))(\( 1 - 946 T + 371293 T^{2} \))(\( 1 + 684 T + 371293 T^{2} \))(\( 1 + 884 T + 371293 T^{2} \))(\( 1 - 46 T + 371293 T^{2} \))(\( 1 + 152330 T^{2} + 137858491849 T^{4} \))(\( 1 + 480650 T^{2} + 137858491849 T^{4} \))(\( 1 + 38870 T^{2} + 137858491849 T^{4} \))(\( 1 - 740470 T^{2} + 137858491849 T^{4} \))
$17$ (\( 1 + 27 T + 1419857 T^{2} \))(\( 1 + 762 T + 1419857 T^{2} \))(\( 1 - 2048 T + 1419857 T^{2} \))(\( 1 - 222 T + 1419857 T^{2} \))(\( 1 + 2048 T + 1419857 T^{2} \))(\( 1 - 27 T + 1419857 T^{2} \))(\( 1 + 378 T + 1419857 T^{2} \))(\( 1 - 2790430 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2259070 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2838985 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2696830 T^{2} + 2015993900449 T^{4} \))
$19$ (\( 1 - 1145 T + 2476099 T^{2} \))(\( 1 + 2740 T + 2476099 T^{2} \))(\( 1 - 2220 T + 2476099 T^{2} \))(\( 1 - 500 T + 2476099 T^{2} \))(\( 1 - 2220 T + 2476099 T^{2} \))(\( 1 - 1145 T + 2476099 T^{2} \))(\( 1 - 1100 T + 2476099 T^{2} \))(\( ( 1 + 500 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 2740 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1145 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1100 T + 2476099 T^{2} )^{2} \))
$23$ (\( 1 - 1854 T + 6436343 T^{2} \))(\( 1 + 1566 T + 6436343 T^{2} \))(\( 1 + 1246 T + 6436343 T^{2} \))(\( 1 + 3564 T + 6436343 T^{2} \))(\( 1 - 1246 T + 6436343 T^{2} \))(\( 1 + 1854 T + 6436343 T^{2} \))(\( 1 - 1986 T + 6436343 T^{2} \))(\( 1 - 170590 T^{2} + 41426511213649 T^{4} \))(\( 1 - 10420330 T^{2} + 41426511213649 T^{4} \))(\( 1 - 9435370 T^{2} + 41426511213649 T^{4} \))(\( 1 - 8928490 T^{2} + 41426511213649 T^{4} \))
$29$ (\( 1 + 4920 T + 20511149 T^{2} \))(\( 1 - 5910 T + 20511149 T^{2} \))(\( 1 + 270 T + 20511149 T^{2} \))(\( 1 - 2190 T + 20511149 T^{2} \))(\( 1 + 270 T + 20511149 T^{2} \))(\( 1 + 4920 T + 20511149 T^{2} \))(\( 1 + 5610 T + 20511149 T^{2} \))(\( ( 1 + 2190 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 5910 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 4920 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 5610 T + 20511149 T^{2} )^{2} \))
$31$ (\( 1 - 1802 T + 28629151 T^{2} \))(\( 1 + 6868 T + 28629151 T^{2} \))(\( 1 + 2048 T + 28629151 T^{2} \))(\( 1 - 2312 T + 28629151 T^{2} \))(\( 1 + 2048 T + 28629151 T^{2} \))(\( 1 - 1802 T + 28629151 T^{2} \))(\( 1 + 3988 T + 28629151 T^{2} \))(\( ( 1 - 2312 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 6868 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 1802 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 3988 T + 28629151 T^{2} )^{2} \))
$37$ (\( 1 - 13178 T + 69343957 T^{2} \))(\( 1 - 5518 T + 69343957 T^{2} \))(\( 1 + 4372 T + 69343957 T^{2} \))(\( 1 - 11242 T + 69343957 T^{2} \))(\( 1 - 4372 T + 69343957 T^{2} \))(\( 1 + 13178 T + 69343957 T^{2} \))(\( 1 - 142 T + 69343957 T^{2} \))(\( 1 - 12305350 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 108239590 T^{2} + 4808584372417849 T^{4} \))(\( 1 + 34971770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138667750 T^{2} + 4808584372417849 T^{4} \))
$41$ (\( 1 + 15123 T + 115856201 T^{2} \))(\( 1 + 378 T + 115856201 T^{2} \))(\( 1 + 2398 T + 115856201 T^{2} \))(\( 1 - 1242 T + 115856201 T^{2} \))(\( 1 + 2398 T + 115856201 T^{2} \))(\( 1 + 15123 T + 115856201 T^{2} \))(\( 1 - 1542 T + 115856201 T^{2} \))(\( ( 1 - 1242 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 378 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 15123 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 1542 T + 115856201 T^{2} )^{2} \))
$43$ (\( 1 - 7844 T + 147008443 T^{2} \))(\( 1 - 2434 T + 147008443 T^{2} \))(\( 1 - 2294 T + 147008443 T^{2} \))(\( 1 + 20624 T + 147008443 T^{2} \))(\( 1 + 2294 T + 147008443 T^{2} \))(\( 1 + 7844 T + 147008443 T^{2} \))(\( 1 - 5026 T + 147008443 T^{2} \))(\( 1 + 131332490 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288092530 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 232488550 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 268756210 T^{2} + 21611482313284249 T^{4} \))
$47$ (\( 1 + 6732 T + 229345007 T^{2} \))(\( 1 + 13122 T + 229345007 T^{2} \))(\( 1 + 10682 T + 229345007 T^{2} \))(\( 1 + 6588 T + 229345007 T^{2} \))(\( 1 - 10682 T + 229345007 T^{2} \))(\( 1 - 6732 T + 229345007 T^{2} \))(\( 1 + 24738 T + 229345007 T^{2} \))(\( 1 - 415288270 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 286503130 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 413370190 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 153278630 T^{2} + 52599132235830049 T^{4} \))
$53$ (\( 1 - 3414 T + 418195493 T^{2} \))(\( 1 - 9174 T + 418195493 T^{2} \))(\( 1 - 2964 T + 418195493 T^{2} \))(\( 1 - 21066 T + 418195493 T^{2} \))(\( 1 + 2964 T + 418195493 T^{2} \))(\( 1 + 3414 T + 418195493 T^{2} \))(\( 1 - 14166 T + 418195493 T^{2} \))(\( 1 - 392614630 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 752228710 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 824735590 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 635715430 T^{2} + 174887470365513049 T^{4} \))
$59$ (\( 1 - 33960 T + 714924299 T^{2} \))(\( 1 + 34980 T + 714924299 T^{2} \))(\( 1 + 39740 T + 714924299 T^{2} \))(\( 1 - 7980 T + 714924299 T^{2} \))(\( 1 + 39740 T + 714924299 T^{2} \))(\( 1 - 33960 T + 714924299 T^{2} \))(\( 1 - 28380 T + 714924299 T^{2} \))(\( ( 1 + 7980 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 34980 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 33960 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 28380 T + 714924299 T^{2} )^{2} \))
$61$ (\( 1 - 47402 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))(\( 1 + 42298 T + 844596301 T^{2} \))(\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 + 42298 T + 844596301 T^{2} \))(\( 1 - 47402 T + 844596301 T^{2} \))(\( 1 - 5522 T + 844596301 T^{2} \))(\( ( 1 - 16622 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 9838 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 47402 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 5522 T + 844596301 T^{2} )^{2} \))
$67$ (\( 1 + 13177 T + 1350125107 T^{2} \))(\( 1 + 33722 T + 1350125107 T^{2} \))(\( 1 - 32098 T + 1350125107 T^{2} \))(\( 1 + 1808 T + 1350125107 T^{2} \))(\( 1 + 32098 T + 1350125107 T^{2} \))(\( 1 - 13177 T + 1350125107 T^{2} \))(\( 1 - 24742 T + 1350125107 T^{2} \))(\( 1 - 2696981350 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1563076930 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2526616885 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2088083650 T^{2} + 1822837804551761449 T^{4} \))
$71$ (\( 1 + 7548 T + 1804229351 T^{2} \))(\( 1 - 70212 T + 1804229351 T^{2} \))(\( 1 + 4248 T + 1804229351 T^{2} \))(\( 1 + 24528 T + 1804229351 T^{2} \))(\( 1 + 4248 T + 1804229351 T^{2} \))(\( 1 + 7548 T + 1804229351 T^{2} \))(\( 1 - 42372 T + 1804229351 T^{2} \))(\( ( 1 + 24528 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 70212 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 7548 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 42372 T + 1804229351 T^{2} )^{2} \))
$73$ (\( 1 + 59821 T + 2073071593 T^{2} \))(\( 1 + 21986 T + 2073071593 T^{2} \))(\( 1 - 30104 T + 2073071593 T^{2} \))(\( 1 + 20474 T + 2073071593 T^{2} \))(\( 1 + 30104 T + 2073071593 T^{2} \))(\( 1 - 59821 T + 2073071593 T^{2} \))(\( 1 - 52126 T + 2073071593 T^{2} \))(\( 1 - 3726958510 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3662758990 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 567591145 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 1429023310 T^{2} + 4297625829703557649 T^{4} \))
$79$ (\( 1 - 75830 T + 3077056399 T^{2} \))(\( 1 - 4520 T + 3077056399 T^{2} \))(\( 1 - 35280 T + 3077056399 T^{2} \))(\( 1 + 46240 T + 3077056399 T^{2} \))(\( 1 - 35280 T + 3077056399 T^{2} \))(\( 1 - 75830 T + 3077056399 T^{2} \))(\( 1 + 39640 T + 3077056399 T^{2} \))(\( ( 1 - 46240 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 4520 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 75830 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 39640 T + 3077056399 T^{2} )^{2} \))
$83$ (\( 1 - 46299 T + 3939040643 T^{2} \))(\( 1 - 109074 T + 3939040643 T^{2} \))(\( 1 + 27826 T + 3939040643 T^{2} \))(\( 1 - 51576 T + 3939040643 T^{2} \))(\( 1 - 27826 T + 3939040643 T^{2} \))(\( 1 + 46299 T + 3939040643 T^{2} \))(\( 1 - 59826 T + 3939040643 T^{2} \))(\( 1 - 5217997510 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 4019056190 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 5734483885 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 4298931010 T^{2} + 15516041187205853449 T^{4} \))
$89$ (\( 1 + 30585 T + 5584059449 T^{2} \))(\( 1 - 38490 T + 5584059449 T^{2} \))(\( 1 + 85210 T + 5584059449 T^{2} \))(\( 1 + 110310 T + 5584059449 T^{2} \))(\( 1 + 85210 T + 5584059449 T^{2} \))(\( 1 + 30585 T + 5584059449 T^{2} \))(\( 1 - 57690 T + 5584059449 T^{2} \))(\( ( 1 - 110310 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 38490 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 30585 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 57690 T + 5584059449 T^{2} )^{2} \))
$97$ (\( 1 - 104018 T + 8587340257 T^{2} \))(\( 1 - 1918 T + 8587340257 T^{2} \))(\( 1 + 97232 T + 8587340257 T^{2} \))(\( 1 - 78382 T + 8587340257 T^{2} \))(\( 1 - 97232 T + 8587340257 T^{2} \))(\( 1 + 104018 T + 8587340257 T^{2} \))(\( 1 - 144382 T + 8587340257 T^{2} \))(\( 1 - 11030942590 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 17171001790 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 6354936190 T^{2} + 73742412689492826049 T^{4} \))(\( 1 + 3671481410 T^{2} + 73742412689492826049 T^{4} \))
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