Properties

Label 224.2.q
Level 224
Weight 2
Character orbit q
Rep. character \(\chi_{224}(47,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 1
Sturm bound 64
Trace bound 0

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Defining parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 56 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(64\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(224, [\chi])\).

Total New Old
Modular forms 80 20 60
Cusp forms 48 12 36
Eisenstein series 32 8 24

Trace form

\( 12q + 6q^{3} + O(q^{10}) \) \( 12q + 6q^{3} + 6q^{11} - 6q^{17} + 6q^{19} - 6q^{33} - 18q^{35} - 12q^{49} - 6q^{51} - 36q^{57} - 42q^{59} - 12q^{65} - 30q^{67} + 18q^{73} - 24q^{75} + 6q^{81} + 18q^{89} + 72q^{91} + 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.2.q.a \(12\) \(1.789\) 12.0.\(\cdots\).2 None \(0\) \(6\) \(0\) \(0\) \(q+\beta _{8}q^{3}+(-\beta _{4}-\beta _{7})q^{5}+(\beta _{5}-\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 9 T^{2} - 18 T^{3} + 33 T^{4} - 63 T^{5} + 102 T^{6} - 189 T^{7} + 297 T^{8} - 486 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} )^{2} \)
$5$ \( 1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24} \)
$7$ \( 1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12} \)
$11$ \( ( 1 - 3 T - 21 T^{2} + 28 T^{3} + 393 T^{4} - 153 T^{5} - 4846 T^{6} - 1683 T^{7} + 47553 T^{8} + 37268 T^{9} - 307461 T^{10} - 483153 T^{11} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 729 T^{4} + 753 T^{5} + 12014 T^{6} + 12801 T^{7} + 210681 T^{8} + 530604 T^{9} + 3257319 T^{10} + 4259571 T^{11} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 3 T + 39 T^{2} - 108 T^{3} + 705 T^{4} - 2265 T^{5} + 12706 T^{6} - 43035 T^{7} + 254505 T^{8} - 740772 T^{9} + 5082519 T^{10} - 7428297 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( 1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24} \)
$29$ \( ( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( 1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24} \)
$37$ \( 1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24} \)
$41$ \( ( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 + 43 T^{2} )^{12} \)
$47$ \( 1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 + 21 T + 309 T^{2} + 3402 T^{3} + 29601 T^{4} + 225789 T^{5} + 1760606 T^{6} + 13321551 T^{7} + 103041081 T^{8} + 698699358 T^{9} + 3744264549 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} )^{2} \)
$61$ \( 1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} - \)\(11\!\cdots\!59\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( ( 1 + 15 T - 3 T^{2} - 142 T^{3} + 9993 T^{4} + 15123 T^{5} - 719466 T^{6} + 1013241 T^{7} + 44858577 T^{8} - 42708346 T^{9} - 60453363 T^{10} + 20251876605 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6} \)
$73$ \( ( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2} \)
$79$ \( 1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} + \)\(24\!\cdots\!61\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( ( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{6} \)
$97$ \( ( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2} \)
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