# 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 $6$

# Related objects

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