Properties

Label 28.3.c
Level 28
Weight 3
Character orbit c
Rep. character \(\chi_{28}(15,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 28.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(28, [\chi])\).

Total New Old
Modular forms 10 6 4
Cusp forms 6 6 0
Eisenstein series 4 0 4

Trace form

\( 6q - q^{2} + q^{4} - 4q^{5} + 6q^{6} - 13q^{8} - 10q^{9} + O(q^{10}) \) \( 6q - q^{2} + q^{4} - 4q^{5} + 6q^{6} - 13q^{8} - 10q^{9} - 28q^{10} + 6q^{12} + 12q^{13} + 7q^{14} + 17q^{16} - 4q^{17} + 43q^{18} - 32q^{20} + 52q^{22} + 122q^{24} - 30q^{25} - 56q^{26} - 35q^{28} - 36q^{29} - 64q^{30} - 101q^{32} + 80q^{33} + 58q^{34} - 131q^{36} + 28q^{37} - 190q^{38} + 40q^{40} - 20q^{41} + 70q^{42} + 164q^{44} + 12q^{45} + 120q^{46} - 98q^{48} - 42q^{49} + 161q^{50} + 292q^{52} + 92q^{53} - 44q^{54} - 49q^{56} + 160q^{57} - 166q^{58} - 176q^{60} - 164q^{61} + 148q^{62} - 215q^{64} - 136q^{65} - 408q^{66} + 62q^{68} - 48q^{69} + 84q^{70} + 151q^{72} - 132q^{73} + 250q^{74} - 78q^{76} + 112q^{77} + 248q^{78} + 312q^{80} - 218q^{81} - 86q^{82} - 98q^{84} - 232q^{85} - 164q^{86} - 100q^{88} + 348q^{89} + 52q^{90} - 104q^{92} + 288q^{93} - 276q^{94} + 170q^{96} + 252q^{97} + 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(28, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
28.3.c.a \(6\) \(0.763\) 6.0.1539727.2 None \(-1\) \(0\) \(-4\) \(0\) \(q-\beta _{2}q^{2}-\beta _{3}q^{3}+(-\beta _{1}-\beta _{4})q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 4 T^{3} + 16 T^{5} + 64 T^{6} \)
$3$ \( 1 - 22 T^{2} + 319 T^{4} - 3188 T^{6} + 25839 T^{8} - 144342 T^{10} + 531441 T^{12} \)
$5$ \( ( 1 + 2 T + 47 T^{2} + 132 T^{3} + 1175 T^{4} + 1250 T^{5} + 15625 T^{6} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{3} \)
$11$ \( 1 - 214 T^{2} + 51679 T^{4} - 6313780 T^{6} + 756632239 T^{8} - 45872800534 T^{10} + 3138428376721 T^{12} \)
$13$ \( ( 1 - 6 T + 143 T^{2} - 316 T^{3} + 24167 T^{4} - 171366 T^{5} + 4826809 T^{6} )^{2} \)
$17$ \( ( 1 + 2 T + 751 T^{2} + 668 T^{3} + 217039 T^{4} + 167042 T^{5} + 24137569 T^{6} )^{2} \)
$19$ \( 1 - 566 T^{2} + 358783 T^{4} - 123926324 T^{6} + 46756959343 T^{8} - 9612696681206 T^{10} + 2213314919066161 T^{12} \)
$23$ \( 1 - 2246 T^{2} + 2391919 T^{4} - 1569561620 T^{6} + 669357004879 T^{8} - 175886472941126 T^{10} + 21914624432020321 T^{12} \)
$29$ \( ( 1 + 18 T + 1575 T^{2} + 35228 T^{3} + 1324575 T^{4} + 12731058 T^{5} + 594823321 T^{6} )^{2} \)
$31$ \( 1 - 3718 T^{2} + 6570127 T^{4} - 7534084372 T^{6} + 6067650257167 T^{8} - 3171048877205638 T^{10} + 787662783788549761 T^{12} \)
$37$ \( ( 1 - 14 T + 2807 T^{2} - 42660 T^{3} + 3842783 T^{4} - 26238254 T^{5} + 2565726409 T^{6} )^{2} \)
$41$ \( ( 1 + 10 T + 3647 T^{2} + 18252 T^{3} + 6130607 T^{4} + 28257610 T^{5} + 4750104241 T^{6} )^{2} \)
$43$ \( 1 - 6998 T^{2} + 25469023 T^{4} - 57500194868 T^{6} + 87073521301423 T^{8} - 81794025542651798 T^{10} + 39959630797262576401 T^{12} \)
$47$ \( 1 - 7814 T^{2} + 34231759 T^{4} - 91924200980 T^{6} + 167040063988879 T^{8} - 186061393975000454 T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \)
$53$ \( ( 1 - 46 T + 8343 T^{2} - 249892 T^{3} + 23435487 T^{4} - 362962126 T^{5} + 22164361129 T^{6} )^{2} \)
$59$ \( 1 - 15670 T^{2} + 117060767 T^{4} - 516019884468 T^{6} + 1418467572675887 T^{8} - 2300832957259710070 T^{10} + \)\(17\!\cdots\!81\)\( T^{12} \)
$61$ \( ( 1 + 82 T + 11743 T^{2} + 602692 T^{3} + 43695703 T^{4} + 1135358962 T^{5} + 51520374361 T^{6} )^{2} \)
$67$ \( 1 - 6870 T^{2} + 51039551 T^{4} - 270200089268 T^{6} + 1028504167986671 T^{8} - 2789684944814123670 T^{10} + \)\(81\!\cdots\!61\)\( T^{12} \)
$71$ \( 1 - 25318 T^{2} + 288430255 T^{4} - 1875969873364 T^{6} + 7329497630808655 T^{8} - 16349187904080176998 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} \)
$73$ \( ( 1 + 66 T + 15647 T^{2} + 699740 T^{3} + 83382863 T^{4} + 1874283906 T^{5} + 151334226289 T^{6} )^{2} \)
$79$ \( 1 - 31494 T^{2} + 446220879 T^{4} - 3596761635604 T^{6} + 17380339380941199 T^{8} - 47779824859197232134 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \)
$83$ \( 1 - 15734 T^{2} + 12104959 T^{4} + 672850782796 T^{6} + 574481029913839 T^{8} - 35437565980475671094 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \)
$89$ \( ( 1 - 174 T + 24959 T^{2} - 2135620 T^{3} + 197700239 T^{4} - 10917149934 T^{5} + 496981290961 T^{6} )^{2} \)
$97$ \( ( 1 - 126 T + 29295 T^{2} - 2174884 T^{3} + 275636655 T^{4} - 11154689406 T^{5} + 832972004929 T^{6} )^{2} \)
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