Properties

Label 50.2.e
Level 50
Weight 2
Character orbit e
Rep. character \(\chi_{50}(9,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newform subspaces 1
Sturm bound 15
Trace bound 0

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 50.e (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(15\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(50, [\chi])\).

Total New Old
Modular forms 40 8 32
Cusp forms 24 8 16
Eisenstein series 16 0 16

Trace form

\( 8q + 2q^{4} - 10q^{5} + 2q^{6} - 4q^{9} + O(q^{10}) \) \( 8q + 2q^{4} - 10q^{5} + 2q^{6} - 4q^{9} - 4q^{11} - 10q^{12} - 4q^{14} - 20q^{15} - 2q^{16} + 10q^{17} + 10q^{19} + 16q^{21} + 20q^{22} + 10q^{23} + 8q^{24} + 10q^{25} - 28q^{26} + 10q^{28} + 10q^{29} + 20q^{30} + 6q^{31} - 30q^{33} - 4q^{34} + 10q^{35} + 4q^{36} - 10q^{37} - 8q^{39} - 14q^{41} - 10q^{42} - 6q^{44} + 10q^{45} - 8q^{46} - 30q^{47} - 10q^{48} - 16q^{49} + 16q^{51} - 20q^{54} - 10q^{55} + 4q^{56} - 14q^{61} + 20q^{63} + 2q^{64} + 50q^{65} + 24q^{66} + 10q^{67} - 8q^{69} - 34q^{71} + 36q^{74} + 10q^{75} + 40q^{77} + 20q^{78} - 12q^{81} + 50q^{83} + 14q^{84} - 20q^{85} + 22q^{86} + 20q^{87} - 10q^{88} + 20q^{89} - 30q^{90} - 4q^{91} - 10q^{92} - 24q^{94} + 2q^{96} - 20q^{97} - 40q^{98} - 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
50.2.e.a \(8\) \(0.399\) \(\Q(\zeta_{20})\) None \(0\) \(0\) \(-10\) \(0\) \(q+\zeta_{20}q^{2}+(-1+\zeta_{20}^{2}-\zeta_{20}^{4}+2\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( ( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} )^{2}( 1 - 4 T + 13 T^{2} - 30 T^{3} + 61 T^{4} - 90 T^{5} + 117 T^{6} - 108 T^{7} + 81 T^{8} ) \)
$5$ \( 1 + 10 T + 45 T^{2} + 130 T^{3} + 305 T^{4} + 650 T^{5} + 1125 T^{6} + 1250 T^{7} + 625 T^{8} \)
$7$ \( 1 - 20 T^{2} + 146 T^{4} - 160 T^{6} - 2789 T^{8} - 7840 T^{10} + 350546 T^{12} - 2352980 T^{14} + 5764801 T^{16} \)
$11$ \( 1 + 4 T - 25 T^{2} - 80 T^{3} + 260 T^{4} + 752 T^{5} + 1353 T^{6} - 2310 T^{7} - 44765 T^{8} - 25410 T^{9} + 163713 T^{10} + 1000912 T^{11} + 3806660 T^{12} - 12884080 T^{13} - 44289025 T^{14} + 77948684 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 5 T^{2} + 20 T^{3} + 96 T^{4} - 100 T^{5} + 25 T^{6} + 6470 T^{7} - 6509 T^{8} + 84110 T^{9} + 4225 T^{10} - 219700 T^{11} + 2741856 T^{12} + 7425860 T^{13} - 24134045 T^{14} + 815730721 T^{16} \)
$17$ \( 1 - 10 T + 75 T^{2} - 420 T^{3} + 2171 T^{4} - 9960 T^{5} + 43585 T^{6} - 177290 T^{7} + 711536 T^{8} - 3013930 T^{9} + 12596065 T^{10} - 48933480 T^{11} + 181324091 T^{12} - 596339940 T^{13} + 1810317675 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 10 T + 22 T^{2} - 30 T^{3} + 1023 T^{4} - 4210 T^{5} - 4736 T^{6} - 17200 T^{7} + 424105 T^{8} - 326800 T^{9} - 1709696 T^{10} - 28876390 T^{11} + 133318383 T^{12} - 74282970 T^{13} + 1035009382 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 10 T + 105 T^{2} - 790 T^{3} + 5156 T^{4} - 30260 T^{5} + 156735 T^{6} - 798320 T^{7} + 3801671 T^{8} - 18361360 T^{9} + 82912815 T^{10} - 368173420 T^{11} + 1442860196 T^{12} - 5084710970 T^{13} + 15543768345 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 10 T + 47 T^{2} - 470 T^{3} + 4888 T^{4} - 28460 T^{5} + 139689 T^{6} - 999650 T^{7} + 6593475 T^{8} - 28989850 T^{9} + 117478449 T^{10} - 694110940 T^{11} + 3457189528 T^{12} - 9640240030 T^{13} + 27956696087 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 3519120 T^{9} - 9049737 T^{10} - 120593968 T^{11} + 1699278640 T^{12} - 3149206610 T^{13} + 4437518405 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 10 T + 65 T^{2} + 230 T^{3} + 1471 T^{4} + 9410 T^{5} + 48855 T^{6} - 117890 T^{7} - 967744 T^{8} - 4361930 T^{9} + 66882495 T^{10} + 476644730 T^{11} + 2756890831 T^{12} + 15949110110 T^{13} + 166772216585 T^{14} + 949318771330 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 14 T + 45 T^{2} + 130 T^{3} + 2680 T^{4} + 7652 T^{5} - 46977 T^{6} - 402900 T^{7} - 2944785 T^{8} - 16518900 T^{9} - 78968337 T^{10} + 527383492 T^{11} + 7573039480 T^{12} + 15061306130 T^{13} + 213754690845 T^{14} + 2726559834334 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 120 T^{2} + 10666 T^{4} - 636400 T^{6} + 31214171 T^{8} - 1176703600 T^{10} + 36464931466 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 + 30 T + 425 T^{2} + 3450 T^{3} + 13416 T^{4} - 40020 T^{5} - 992525 T^{6} - 8396100 T^{7} - 57415609 T^{8} - 394616700 T^{9} - 2192487725 T^{10} - 4154996460 T^{11} + 65465800296 T^{12} + 791240274150 T^{13} + 4581166514825 T^{14} + 15198693613890 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 90 T^{2} + 900 T^{3} + 5291 T^{4} + 81000 T^{5} + 628380 T^{6} + 3853800 T^{7} + 44701381 T^{8} + 204251400 T^{9} + 1765119420 T^{10} + 12059037000 T^{11} + 41748534971 T^{12} + 376375943700 T^{13} + 1994792501610 T^{14} + 62259690411361 T^{16} \)
$59$ \( 1 - 118 T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 1127857924 T^{10} + 43174157243 T^{12} - 4977302969638 T^{14} + 146830437604321 T^{16} \)
$61$ \( 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 91989220 T^{9} - 692820432 T^{10} + 5733994022 T^{11} + 72206060815 T^{12} - 25337889030 T^{13} + 1545611230830 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 10 T + 195 T^{2} - 2790 T^{3} + 29716 T^{4} - 356960 T^{5} + 3452465 T^{6} - 30345750 T^{7} + 284471411 T^{8} - 2033165250 T^{9} + 15498115385 T^{10} - 107360360480 T^{11} + 598810711636 T^{12} - 3766849048530 T^{13} + 17639384522955 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 34 T + 425 T^{2} + 2570 T^{3} + 16960 T^{4} + 219512 T^{5} + 2212323 T^{6} + 19311440 T^{7} + 171308535 T^{8} + 1371112240 T^{9} + 11152320243 T^{10} + 78565759432 T^{11} + 430982109760 T^{12} + 4636869432070 T^{13} + 54442620666425 T^{14} + 309234085385294 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 110 T^{2} + 840 T^{3} + 6291 T^{4} + 92400 T^{5} + 907660 T^{6} + 5561520 T^{7} + 91719461 T^{8} + 405990960 T^{9} + 4836920140 T^{10} + 35945170800 T^{11} + 178653334131 T^{12} + 1741380138120 T^{13} + 16646764891790 T^{14} + 806460091894081 T^{16} \)
$79$ \( 1 - 138 T^{2} + 1500 T^{3} + 4403 T^{4} - 205200 T^{5} + 1430844 T^{6} + 7779600 T^{7} - 206126795 T^{8} + 614588400 T^{9} + 8929897404 T^{10} - 101171602800 T^{11} + 171497206643 T^{12} + 4615584598500 T^{13} - 33546068861898 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 - 50 T + 1335 T^{2} - 23750 T^{3} + 305836 T^{4} - 2886800 T^{5} + 19447445 T^{6} - 89310250 T^{7} + 442867371 T^{8} - 7412750750 T^{9} + 133973448605 T^{10} - 1650634711600 T^{11} + 14514463061356 T^{12} - 93552215271250 T^{13} + 436465398447615 T^{14} - 1356802549481350 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 10 T + 71 T^{2} - 1290 T^{3} + 19001 T^{4} - 114810 T^{5} + 562391 T^{6} - 7049690 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 + 20 T + 270 T^{2} + 3780 T^{3} + 51031 T^{4} + 591220 T^{5} + 6636980 T^{6} + 69655800 T^{7} + 673071701 T^{8} + 6756612600 T^{9} + 62447344820 T^{10} + 539590531060 T^{11} + 4517737738711 T^{12} + 32460146171460 T^{13} + 224902441330830 T^{14} + 1615965689562260 T^{15} + 7837433594376961 T^{16} \)
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