Properties

Label 50.6.a
Level 50
Weight 6
Character orbit a
Rep. character \(\chi_{50}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 7
Sturm bound 45
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(45\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(50))\).

Total New Old
Modular forms 43 7 36
Cusp forms 31 7 24
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7q + 4q^{2} - 4q^{3} + 112q^{4} + 8q^{6} + 312q^{7} + 64q^{8} + 221q^{9} + O(q^{10}) \) \( 7q + 4q^{2} - 4q^{3} + 112q^{4} + 8q^{6} + 312q^{7} + 64q^{8} + 221q^{9} + 814q^{11} - 64q^{12} - 114q^{13} + 176q^{14} + 1792q^{16} - 918q^{17} + 3892q^{18} + 5590q^{19} + 3284q^{21} - 3312q^{22} - 3144q^{23} + 128q^{24} - 4152q^{26} + 5480q^{27} + 4992q^{28} - 7890q^{29} - 9036q^{31} + 1024q^{32} - 24288q^{33} - 13744q^{34} + 3536q^{36} + 16902q^{37} + 17360q^{38} - 15168q^{39} - 32636q^{41} - 11392q^{42} - 13164q^{43} + 13024q^{44} - 4912q^{46} - 44448q^{47} - 1024q^{48} + 16599q^{49} + 47354q^{51} - 1824q^{52} + 44406q^{53} + 680q^{54} + 2816q^{56} + 33040q^{57} - 37320q^{58} - 10180q^{59} + 22514q^{61} + 20768q^{62} + 42376q^{63} + 28672q^{64} - 2984q^{66} - 10788q^{67} - 14688q^{68} + 70892q^{69} + 64464q^{71} + 62272q^{72} + 9666q^{73} - 46984q^{74} + 89440q^{76} + 28464q^{77} - 112576q^{78} + 140860q^{79} - 91633q^{81} + 12648q^{82} + 220476q^{83} + 52544q^{84} - 153232q^{86} - 233880q^{87} - 52992q^{88} - 245720q^{89} - 1696q^{91} - 50304q^{92} - 117968q^{93} + 66496q^{94} + 2048q^{96} + 224682q^{97} - 2652q^{98} - 504008q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(50))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
50.6.a.a \(1\) \(8.019\) \(\Q\) None \(-4\) \(-11\) \(0\) \(-142\) \(+\) \(-\) \(q-4q^{2}-11q^{3}+2^{4}q^{4}+44q^{6}-142q^{7}+\cdots\)
50.6.a.b \(1\) \(8.019\) \(\Q\) None \(-4\) \(-6\) \(0\) \(118\) \(+\) \(+\) \(q-4q^{2}-6q^{3}+2^{4}q^{4}+24q^{6}+118q^{7}+\cdots\)
50.6.a.c \(1\) \(8.019\) \(\Q\) None \(-4\) \(14\) \(0\) \(158\) \(+\) \(-\) \(q-4q^{2}+14q^{3}+2^{4}q^{4}-56q^{6}+158q^{7}+\cdots\)
50.6.a.d \(1\) \(8.019\) \(\Q\) None \(4\) \(-24\) \(0\) \(172\) \(-\) \(+\) \(q+4q^{2}-24q^{3}+2^{4}q^{4}-96q^{6}+172q^{7}+\cdots\)
50.6.a.e \(1\) \(8.019\) \(\Q\) None \(4\) \(-14\) \(0\) \(-158\) \(-\) \(-\) \(q+4q^{2}-14q^{3}+2^{4}q^{4}-56q^{6}-158q^{7}+\cdots\)
50.6.a.f \(1\) \(8.019\) \(\Q\) None \(4\) \(11\) \(0\) \(142\) \(-\) \(+\) \(q+4q^{2}+11q^{3}+2^{4}q^{4}+44q^{6}+142q^{7}+\cdots\)
50.6.a.g \(1\) \(8.019\) \(\Q\) None \(4\) \(26\) \(0\) \(22\) \(-\) \(+\) \(q+4q^{2}+26q^{3}+2^{4}q^{4}+104q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(50)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))
$3$ (\( 1 + 11 T + 243 T^{2} \))(\( 1 + 6 T + 243 T^{2} \))(\( 1 - 14 T + 243 T^{2} \))(\( 1 + 24 T + 243 T^{2} \))(\( 1 + 14 T + 243 T^{2} \))(\( 1 - 11 T + 243 T^{2} \))(\( 1 - 26 T + 243 T^{2} \))
$5$ 1
$7$ (\( 1 + 142 T + 16807 T^{2} \))(\( 1 - 118 T + 16807 T^{2} \))(\( 1 - 158 T + 16807 T^{2} \))(\( 1 - 172 T + 16807 T^{2} \))(\( 1 + 158 T + 16807 T^{2} \))(\( 1 - 142 T + 16807 T^{2} \))(\( 1 - 22 T + 16807 T^{2} \))
$11$ (\( 1 - 777 T + 161051 T^{2} \))(\( 1 - 192 T + 161051 T^{2} \))(\( 1 + 148 T + 161051 T^{2} \))(\( 1 - 132 T + 161051 T^{2} \))(\( 1 + 148 T + 161051 T^{2} \))(\( 1 - 777 T + 161051 T^{2} \))(\( 1 + 768 T + 161051 T^{2} \))
$13$ (\( 1 - 884 T + 371293 T^{2} \))(\( 1 + 1106 T + 371293 T^{2} \))(\( 1 - 684 T + 371293 T^{2} \))(\( 1 - 946 T + 371293 T^{2} \))(\( 1 + 684 T + 371293 T^{2} \))(\( 1 + 884 T + 371293 T^{2} \))(\( 1 - 46 T + 371293 T^{2} \))
$17$ (\( 1 + 27 T + 1419857 T^{2} \))(\( 1 + 762 T + 1419857 T^{2} \))(\( 1 - 2048 T + 1419857 T^{2} \))(\( 1 - 222 T + 1419857 T^{2} \))(\( 1 + 2048 T + 1419857 T^{2} \))(\( 1 - 27 T + 1419857 T^{2} \))(\( 1 + 378 T + 1419857 T^{2} \))
$19$ (\( 1 - 1145 T + 2476099 T^{2} \))(\( 1 + 2740 T + 2476099 T^{2} \))(\( 1 - 2220 T + 2476099 T^{2} \))(\( 1 - 500 T + 2476099 T^{2} \))(\( 1 - 2220 T + 2476099 T^{2} \))(\( 1 - 1145 T + 2476099 T^{2} \))(\( 1 - 1100 T + 2476099 T^{2} \))
$23$ (\( 1 - 1854 T + 6436343 T^{2} \))(\( 1 + 1566 T + 6436343 T^{2} \))(\( 1 + 1246 T + 6436343 T^{2} \))(\( 1 + 3564 T + 6436343 T^{2} \))(\( 1 - 1246 T + 6436343 T^{2} \))(\( 1 + 1854 T + 6436343 T^{2} \))(\( 1 - 1986 T + 6436343 T^{2} \))
$29$ (\( 1 + 4920 T + 20511149 T^{2} \))(\( 1 - 5910 T + 20511149 T^{2} \))(\( 1 + 270 T + 20511149 T^{2} \))(\( 1 - 2190 T + 20511149 T^{2} \))(\( 1 + 270 T + 20511149 T^{2} \))(\( 1 + 4920 T + 20511149 T^{2} \))(\( 1 + 5610 T + 20511149 T^{2} \))
$31$ (\( 1 - 1802 T + 28629151 T^{2} \))(\( 1 + 6868 T + 28629151 T^{2} \))(\( 1 + 2048 T + 28629151 T^{2} \))(\( 1 - 2312 T + 28629151 T^{2} \))(\( 1 + 2048 T + 28629151 T^{2} \))(\( 1 - 1802 T + 28629151 T^{2} \))(\( 1 + 3988 T + 28629151 T^{2} \))
$37$ (\( 1 - 13178 T + 69343957 T^{2} \))(\( 1 - 5518 T + 69343957 T^{2} \))(\( 1 + 4372 T + 69343957 T^{2} \))(\( 1 - 11242 T + 69343957 T^{2} \))(\( 1 - 4372 T + 69343957 T^{2} \))(\( 1 + 13178 T + 69343957 T^{2} \))(\( 1 - 142 T + 69343957 T^{2} \))
$41$ (\( 1 + 15123 T + 115856201 T^{2} \))(\( 1 + 378 T + 115856201 T^{2} \))(\( 1 + 2398 T + 115856201 T^{2} \))(\( 1 - 1242 T + 115856201 T^{2} \))(\( 1 + 2398 T + 115856201 T^{2} \))(\( 1 + 15123 T + 115856201 T^{2} \))(\( 1 - 1542 T + 115856201 T^{2} \))
$43$ (\( 1 - 7844 T + 147008443 T^{2} \))(\( 1 - 2434 T + 147008443 T^{2} \))(\( 1 - 2294 T + 147008443 T^{2} \))(\( 1 + 20624 T + 147008443 T^{2} \))(\( 1 + 2294 T + 147008443 T^{2} \))(\( 1 + 7844 T + 147008443 T^{2} \))(\( 1 - 5026 T + 147008443 T^{2} \))
$47$ (\( 1 + 6732 T + 229345007 T^{2} \))(\( 1 + 13122 T + 229345007 T^{2} \))(\( 1 + 10682 T + 229345007 T^{2} \))(\( 1 + 6588 T + 229345007 T^{2} \))(\( 1 - 10682 T + 229345007 T^{2} \))(\( 1 - 6732 T + 229345007 T^{2} \))(\( 1 + 24738 T + 229345007 T^{2} \))
$53$ (\( 1 - 3414 T + 418195493 T^{2} \))(\( 1 - 9174 T + 418195493 T^{2} \))(\( 1 - 2964 T + 418195493 T^{2} \))(\( 1 - 21066 T + 418195493 T^{2} \))(\( 1 + 2964 T + 418195493 T^{2} \))(\( 1 + 3414 T + 418195493 T^{2} \))(\( 1 - 14166 T + 418195493 T^{2} \))
$59$ (\( 1 - 33960 T + 714924299 T^{2} \))(\( 1 + 34980 T + 714924299 T^{2} \))(\( 1 + 39740 T + 714924299 T^{2} \))(\( 1 - 7980 T + 714924299 T^{2} \))(\( 1 + 39740 T + 714924299 T^{2} \))(\( 1 - 33960 T + 714924299 T^{2} \))(\( 1 - 28380 T + 714924299 T^{2} \))
$61$ (\( 1 - 47402 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))(\( 1 + 42298 T + 844596301 T^{2} \))(\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 + 42298 T + 844596301 T^{2} \))(\( 1 - 47402 T + 844596301 T^{2} \))(\( 1 - 5522 T + 844596301 T^{2} \))
$67$ (\( 1 + 13177 T + 1350125107 T^{2} \))(\( 1 + 33722 T + 1350125107 T^{2} \))(\( 1 - 32098 T + 1350125107 T^{2} \))(\( 1 + 1808 T + 1350125107 T^{2} \))(\( 1 + 32098 T + 1350125107 T^{2} \))(\( 1 - 13177 T + 1350125107 T^{2} \))(\( 1 - 24742 T + 1350125107 T^{2} \))
$71$ (\( 1 + 7548 T + 1804229351 T^{2} \))(\( 1 - 70212 T + 1804229351 T^{2} \))(\( 1 + 4248 T + 1804229351 T^{2} \))(\( 1 + 24528 T + 1804229351 T^{2} \))(\( 1 + 4248 T + 1804229351 T^{2} \))(\( 1 + 7548 T + 1804229351 T^{2} \))(\( 1 - 42372 T + 1804229351 T^{2} \))
$73$ (\( 1 + 59821 T + 2073071593 T^{2} \))(\( 1 + 21986 T + 2073071593 T^{2} \))(\( 1 - 30104 T + 2073071593 T^{2} \))(\( 1 + 20474 T + 2073071593 T^{2} \))(\( 1 + 30104 T + 2073071593 T^{2} \))(\( 1 - 59821 T + 2073071593 T^{2} \))(\( 1 - 52126 T + 2073071593 T^{2} \))
$79$ (\( 1 - 75830 T + 3077056399 T^{2} \))(\( 1 - 4520 T + 3077056399 T^{2} \))(\( 1 - 35280 T + 3077056399 T^{2} \))(\( 1 + 46240 T + 3077056399 T^{2} \))(\( 1 - 35280 T + 3077056399 T^{2} \))(\( 1 - 75830 T + 3077056399 T^{2} \))(\( 1 + 39640 T + 3077056399 T^{2} \))
$83$ (\( 1 - 46299 T + 3939040643 T^{2} \))(\( 1 - 109074 T + 3939040643 T^{2} \))(\( 1 + 27826 T + 3939040643 T^{2} \))(\( 1 - 51576 T + 3939040643 T^{2} \))(\( 1 - 27826 T + 3939040643 T^{2} \))(\( 1 + 46299 T + 3939040643 T^{2} \))(\( 1 - 59826 T + 3939040643 T^{2} \))
$89$ (\( 1 + 30585 T + 5584059449 T^{2} \))(\( 1 - 38490 T + 5584059449 T^{2} \))(\( 1 + 85210 T + 5584059449 T^{2} \))(\( 1 + 110310 T + 5584059449 T^{2} \))(\( 1 + 85210 T + 5584059449 T^{2} \))(\( 1 + 30585 T + 5584059449 T^{2} \))(\( 1 - 57690 T + 5584059449 T^{2} \))
$97$ (\( 1 - 104018 T + 8587340257 T^{2} \))(\( 1 - 1918 T + 8587340257 T^{2} \))(\( 1 + 97232 T + 8587340257 T^{2} \))(\( 1 - 78382 T + 8587340257 T^{2} \))(\( 1 - 97232 T + 8587340257 T^{2} \))(\( 1 + 104018 T + 8587340257 T^{2} \))(\( 1 - 144382 T + 8587340257 T^{2} \))
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