Properties

Label 50.6.b
Level $50$
Weight $6$
Character orbit 50.b
Rep. character $\chi_{50}(49,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
Sturm bound $45$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(45\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(50, [\chi])\).

Total New Old
Modular forms 44 8 36
Cusp forms 32 8 24
Eisenstein series 12 0 12

Trace form

\( 8q - 128q^{4} + 152q^{6} - 874q^{9} + O(q^{10}) \) \( 8q - 128q^{4} + 152q^{6} - 874q^{9} + 666q^{11} - 1744q^{14} + 2048q^{16} - 10q^{19} - 5404q^{21} - 2432q^{24} + 9712q^{26} + 4860q^{29} - 13484q^{31} - 5064q^{34} + 13984q^{36} + 49192q^{39} - 25434q^{41} - 10656q^{44} - 14928q^{46} + 6144q^{49} - 20574q^{51} + 31480q^{54} + 27904q^{56} - 70680q^{59} + 119416q^{61} - 32768q^{64} - 107496q^{66} - 252348q^{69} + 161016q^{71} + 58496q^{74} + 160q^{76} + 11060q^{79} + 27448q^{81} + 86464q^{84} - 207008q^{86} + 89430q^{89} - 184624q^{91} + 91776q^{94} + 38912q^{96} + 846252q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
50.6.b.a \(2\) \(8.019\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}+12iq^{3}-2^{4}q^{4}-96q^{6}+\cdots\)
50.6.b.b \(2\) \(8.019\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}-3iq^{3}-2^{4}q^{4}+24q^{6}+\cdots\)
50.6.b.c \(2\) \(8.019\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{2}-11iq^{3}-2^{4}q^{4}+44q^{6}+\cdots\)
50.6.b.d \(2\) \(8.019\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}-13iq^{3}-2^{4}q^{4}+104q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))
$3$ (\( 1 + 90 T^{2} + 59049 T^{4} \))(\( 1 - 450 T^{2} + 59049 T^{4} \))(\( 1 - 365 T^{2} + 59049 T^{4} \))(\( 1 + 190 T^{2} + 59049 T^{4} \))
$5$ 1
$7$ (\( 1 - 4030 T^{2} + 282475249 T^{4} \))(\( 1 - 19690 T^{2} + 282475249 T^{4} \))(\( 1 - 13450 T^{2} + 282475249 T^{4} \))(\( 1 - 33130 T^{2} + 282475249 T^{4} \))
$11$ (\( ( 1 - 132 T + 161051 T^{2} )^{2} \))(\( ( 1 - 192 T + 161051 T^{2} )^{2} \))(\( ( 1 - 777 T + 161051 T^{2} )^{2} \))(\( ( 1 + 768 T + 161051 T^{2} )^{2} \))
$13$ (\( 1 + 152330 T^{2} + 137858491849 T^{4} \))(\( 1 + 480650 T^{2} + 137858491849 T^{4} \))(\( 1 + 38870 T^{2} + 137858491849 T^{4} \))(\( 1 - 740470 T^{2} + 137858491849 T^{4} \))
$17$ (\( 1 - 2790430 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2259070 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2838985 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2696830 T^{2} + 2015993900449 T^{4} \))
$19$ (\( ( 1 + 500 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 2740 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1145 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1100 T + 2476099 T^{2} )^{2} \))
$23$ (\( 1 - 170590 T^{2} + 41426511213649 T^{4} \))(\( 1 - 10420330 T^{2} + 41426511213649 T^{4} \))(\( 1 - 9435370 T^{2} + 41426511213649 T^{4} \))(\( 1 - 8928490 T^{2} + 41426511213649 T^{4} \))
$29$ (\( ( 1 + 2190 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 5910 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 4920 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 5610 T + 20511149 T^{2} )^{2} \))
$31$ (\( ( 1 - 2312 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 6868 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 1802 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 3988 T + 28629151 T^{2} )^{2} \))
$37$ (\( 1 - 12305350 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 108239590 T^{2} + 4808584372417849 T^{4} \))(\( 1 + 34971770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138667750 T^{2} + 4808584372417849 T^{4} \))
$41$ (\( ( 1 - 1242 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 378 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 15123 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 1542 T + 115856201 T^{2} )^{2} \))
$43$ (\( 1 + 131332490 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288092530 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 232488550 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 268756210 T^{2} + 21611482313284249 T^{4} \))
$47$ (\( 1 - 415288270 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 286503130 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 413370190 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 153278630 T^{2} + 52599132235830049 T^{4} \))
$53$ (\( 1 - 392614630 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 752228710 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 824735590 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 635715430 T^{2} + 174887470365513049 T^{4} \))
$59$ (\( ( 1 + 7980 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 34980 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 33960 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 28380 T + 714924299 T^{2} )^{2} \))
$61$ (\( ( 1 - 16622 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 9838 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 47402 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 5522 T + 844596301 T^{2} )^{2} \))
$67$ (\( 1 - 2696981350 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1563076930 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2526616885 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2088083650 T^{2} + 1822837804551761449 T^{4} \))
$71$ (\( ( 1 + 24528 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 70212 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 7548 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 42372 T + 1804229351 T^{2} )^{2} \))
$73$ (\( 1 - 3726958510 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3662758990 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 567591145 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 1429023310 T^{2} + 4297625829703557649 T^{4} \))
$79$ (\( ( 1 - 46240 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 4520 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 75830 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 39640 T + 3077056399 T^{2} )^{2} \))
$83$ (\( 1 - 5217997510 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 4019056190 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 5734483885 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 4298931010 T^{2} + 15516041187205853449 T^{4} \))
$89$ (\( ( 1 - 110310 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 38490 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 30585 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 57690 T + 5584059449 T^{2} )^{2} \))
$97$ (\( 1 - 11030942590 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 17171001790 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 6354936190 T^{2} + 73742412689492826049 T^{4} \))(\( 1 + 3671481410 T^{2} + 73742412689492826049 T^{4} \))
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