Properties

Label 57.2.e
Level 57
Weight 2
Character orbit e
Rep. character \(\chi_{57}(7,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 2
Sturm bound 13
Trace bound 1

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Defining parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(57, [\chi])\).

Total New Old
Modular forms 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

Trace form

\( 8q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{6} - 12q^{8} - 4q^{9} + O(q^{10}) \) \( 8q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{6} - 12q^{8} - 4q^{9} + 4q^{10} - 4q^{11} + 12q^{12} - 4q^{13} + 2q^{14} - 2q^{15} - 4q^{16} + 4q^{17} - 4q^{19} + 44q^{20} + 2q^{21} - 16q^{22} - 10q^{23} - 32q^{26} + 4q^{27} - 16q^{28} + 4q^{29} - 8q^{30} + 24q^{31} + 12q^{32} - 2q^{33} + 4q^{34} - 18q^{35} - 4q^{36} + 42q^{38} - 12q^{39} + 24q^{40} + 16q^{41} + 4q^{42} + 4q^{43} - 14q^{44} + 4q^{45} + 32q^{46} + 24q^{47} - 6q^{48} - 16q^{49} - 56q^{50} - 4q^{51} + 2q^{53} + 2q^{54} - 12q^{55} - 48q^{56} - 6q^{57} - 32q^{58} - 10q^{59} - 22q^{60} + 26q^{62} - 16q^{64} + 12q^{65} - 20q^{66} - 20q^{67} + 8q^{68} + 36q^{69} + 24q^{70} + 12q^{71} + 6q^{72} - 8q^{73} - 4q^{74} + 20q^{75} + 20q^{76} + 20q^{77} + 26q^{78} - 12q^{79} - 10q^{80} - 4q^{81} + 4q^{82} - 8q^{83} + 36q^{84} + 18q^{86} + 24q^{87} + 24q^{88} + 22q^{89} + 4q^{90} - 12q^{91} - 2q^{92} - 18q^{93} + 2q^{95} - 44q^{96} + 20q^{98} + 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(57, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
57.2.e.a \(2\) \(0.455\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(2\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
57.2.e.b \(6\) \(0.455\) 6.0.954288.1 None \(1\) \(-3\) \(-2\) \(-2\) \(q+(-\beta _{4}+\beta _{5})q^{2}-\beta _{3}q^{3}+(-2-\beta _{1}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + T^{3} - 2 T^{4} - T^{5} + 11 T^{6} - 2 T^{7} - 8 T^{8} + 8 T^{9} - 32 T^{11} + 64 T^{12} \))
$3$ (\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 170 T^{7} - 50 T^{8} - 250 T^{9} - 1875 T^{10} + 6250 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 + T + 12 T^{2} + 17 T^{3} + 84 T^{4} + 49 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} - 36 T^{3} + 99 T^{4} + 1331 T^{6} )^{2} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 2899 T^{7} + 10309 T^{8} + 87880 T^{9} - 485537 T^{10} - 371293 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{3} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 136 T^{3} + 323 T^{4} - 1444 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 14 T + 99 T^{2} + 382 T^{3} + 346 T^{4} - 7726 T^{5} - 57613 T^{6} - 177698 T^{7} + 183034 T^{8} + 4647794 T^{9} + 27704259 T^{10} + 90108802 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 - 8 T + 35 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 4 T - 39 T^{2} - 52 T^{3} + 886 T^{4} - 1916 T^{5} - 33907 T^{6} - 55564 T^{7} + 745126 T^{8} - 1268228 T^{9} - 27583959 T^{10} + 82044596 T^{11} + 594823321 T^{12} \))
$31$ (\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 - 15 T + 144 T^{2} - 899 T^{3} + 4464 T^{4} - 14415 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{6} \))
$41$ (\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 - 4 T - 75 T^{2} + 100 T^{3} + 3622 T^{4} + 2012 T^{5} - 174751 T^{6} + 82492 T^{7} + 6088582 T^{8} + 6892100 T^{9} - 211932075 T^{10} - 463424804 T^{11} + 4750104241 T^{12} \))
$43$ (\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 99 T^{2} + 218 T^{3} + 6207 T^{4} - 6951 T^{5} - 285738 T^{6} - 298893 T^{7} + 11476743 T^{8} + 17332526 T^{9} - 338461299 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{3} \))
$53$ (\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 51 T^{2} + 102 T^{3} + 1086 T^{4} + 11334 T^{5} - 107183 T^{6} + 600702 T^{7} + 3050574 T^{8} + 15185454 T^{9} - 402414531 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 153 T^{2} + 72 T^{3} + 14382 T^{4} - 5508 T^{5} - 969077 T^{6} - 324972 T^{7} + 50063742 T^{8} + 14787288 T^{9} - 1853956233 T^{10} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 2086871 T^{7} + 22820893 T^{8} - 114398424 T^{9} - 346146025 T^{10} + 10979751913 T^{11} + 51520374361 T^{12} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 9 T - 39 T^{2} - 250 T^{3} + 375 T^{4} - 24519 T^{5} - 284658 T^{6} - 1642773 T^{7} + 1683375 T^{8} - 75190750 T^{9} - 785893719 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \))
$71$ (\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 - 18 T + 99 T^{2} - 234 T^{3} + 306 T^{4} + 55062 T^{5} - 871373 T^{6} + 3909402 T^{7} + 1542546 T^{8} - 83751174 T^{9} + 2515756419 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \))
$73$ (\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 6707897 T^{7} + 90811489 T^{8} + 98032284 T^{9} + 1675496219 T^{10} + 39388360267 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 11 T - 119 T^{2} - 494 T^{3} + 21403 T^{4} + 27611 T^{5} - 1863994 T^{6} + 2181269 T^{7} + 133576123 T^{8} - 243561266 T^{9} - 4635059639 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 169 T^{2} + 472 T^{3} + 14027 T^{4} + 27556 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 16 T - 87 T^{2} + 424 T^{3} + 39826 T^{4} - 163780 T^{5} - 2350663 T^{6} - 14576420 T^{7} + 315461746 T^{8} + 298906856 T^{9} - 5458574967 T^{10} - 89344951184 T^{11} + 496981290961 T^{12} \))
$97$ (\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 1405142 T^{7} + 3932962 T^{8} - 1976849718 T^{9} - 1682056339 T^{10} - 17174680514 T^{11} + 832972004929 T^{12} \))
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