Properties

Label 224.3.h
Level 224
Weight 3
Character orbit h
Rep. character \(\chi_{224}(209,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 4
Sturm bound 96
Trace bound 9

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 56 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(224, [\chi])\).

Total New Old
Modular forms 72 18 54
Cusp forms 56 14 42
Eisenstein series 16 4 12

Trace form

\( 14q + 2q^{7} + 26q^{9} + O(q^{10}) \) \( 14q + 2q^{7} + 26q^{9} + 40q^{15} + 36q^{23} + 26q^{25} + 40q^{39} + 14q^{49} - 136q^{57} - 42q^{63} - 104q^{65} - 316q^{71} - 156q^{79} - 90q^{81} - 280q^{95} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.3.h.a \(2\) \(6.104\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(-14\) \(q-7q^{7}-9q^{9}-\beta q^{11}-18q^{23}-5^{2}q^{25}+\cdots\)
224.3.h.b \(2\) \(6.104\) \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-14}) \) \(0\) \(0\) \(0\) \(-14\) \(q+\beta q^{3}+\beta q^{5}-7q^{7}+19q^{9}+\beta q^{13}+\cdots\)
224.3.h.c \(2\) \(6.104\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-14}) \) \(0\) \(0\) \(0\) \(14\) \(q+\beta q^{3}+3\beta q^{5}+7q^{7}-q^{9}-9\beta q^{13}+\cdots\)
224.3.h.d \(8\) \(6.104\) 8.0.\(\cdots\).51 None \(0\) \(0\) \(0\) \(16\) \(q-\beta _{3}q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 9 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 81 T^{4} \))(\( 1 + 10 T^{2} + 81 T^{4} \))(\( ( 1 + 16 T^{2} + 178 T^{4} + 1296 T^{6} + 6561 T^{8} )^{2} \))
$5$ (\( ( 1 + 25 T^{2} )^{2} \))(\( 1 + 22 T^{2} + 625 T^{4} \))(\( 1 - 22 T^{2} + 625 T^{4} \))(\( ( 1 + 56 T^{2} + 1602 T^{4} + 35000 T^{6} + 390625 T^{8} )^{2} \))
$7$ (\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{2} \))(\( ( 1 - 8 T + 102 T^{2} - 392 T^{3} + 2401 T^{4} )^{2} \))
$11$ (\( ( 1 - 6 T + 121 T^{2} )( 1 + 6 T + 121 T^{2} ) \))(\( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \))(\( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \))(\( ( 1 - 364 T^{2} + 59334 T^{4} - 5329324 T^{6} + 214358881 T^{8} )^{2} \))
$13$ (\( ( 1 + 169 T^{2} )^{2} \))(\( 1 + 310 T^{2} + 28561 T^{4} \))(\( 1 - 310 T^{2} + 28561 T^{4} \))(\( ( 1 + 344 T^{2} + 86658 T^{4} + 9824984 T^{6} + 815730721 T^{8} )^{2} \))
$17$ (\( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \))(\( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \))(\( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \))(\( ( 1 - 148 T^{2} + 165606 T^{4} - 12361108 T^{6} + 6975757441 T^{8} )^{2} \))
$19$ (\( ( 1 + 361 T^{2} )^{2} \))(\( 1 - 650 T^{2} + 130321 T^{4} \))(\( 1 + 650 T^{2} + 130321 T^{4} \))(\( ( 1 + 656 T^{2} + 327858 T^{4} + 85490576 T^{6} + 16983563041 T^{8} )^{2} \))
$23$ (\( ( 1 + 18 T + 529 T^{2} )^{2} \))(\( ( 1 - 10 T + 529 T^{2} )^{2} \))(\( ( 1 - 10 T + 529 T^{2} )^{2} \))(\( ( 1 - 8 T + 642 T^{2} - 4232 T^{3} + 279841 T^{4} )^{4} \))
$29$ (\( ( 1 - 54 T + 841 T^{2} )( 1 + 54 T + 841 T^{2} ) \))(\( ( 1 - 29 T )^{2}( 1 + 29 T )^{2} \))(\( ( 1 - 29 T )^{2}( 1 + 29 T )^{2} \))(\( ( 1 - 1444 T^{2} + 1048038 T^{4} - 1021313764 T^{6} + 500246412961 T^{8} )^{2} \))
$31$ (\( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \))(\( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \))(\( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \))(\( ( 1 + 140 T^{2} + 1206054 T^{4} + 129292940 T^{6} + 852891037441 T^{8} )^{2} \))
$37$ (\( ( 1 - 38 T + 1369 T^{2} )( 1 + 38 T + 1369 T^{2} ) \))(\( ( 1 - 37 T )^{2}( 1 + 37 T )^{2} \))(\( ( 1 - 37 T )^{2}( 1 + 37 T )^{2} \))(\( ( 1 - 4612 T^{2} + 8955366 T^{4} - 8643630532 T^{6} + 3512479453921 T^{8} )^{2} \))
$41$ (\( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \))(\( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \))(\( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \))(\( ( 1 - 5380 T^{2} + 12875334 T^{4} - 15202594180 T^{6} + 7984925229121 T^{8} )^{2} \))
$43$ (\( ( 1 - 58 T + 1849 T^{2} )( 1 + 58 T + 1849 T^{2} ) \))(\( ( 1 - 43 T )^{2}( 1 + 43 T )^{2} \))(\( ( 1 - 43 T )^{2}( 1 + 43 T )^{2} \))(\( ( 1 - 1324 T^{2} + 5788998 T^{4} - 4526492524 T^{6} + 11688200277601 T^{8} )^{2} \))
$47$ (\( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \))(\( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \))(\( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \))(\( ( 1 - 4852 T^{2} + 14998950 T^{4} - 23676212212 T^{6} + 23811286661761 T^{8} )^{2} \))
$53$ (\( ( 1 - 6 T + 2809 T^{2} )( 1 + 6 T + 2809 T^{2} ) \))(\( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \))(\( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \))(\( ( 1 - 7780 T^{2} + 30664230 T^{4} - 61387942180 T^{6} + 62259690411361 T^{8} )^{2} \))
$59$ (\( ( 1 + 3481 T^{2} )^{2} \))(\( 1 - 1130 T^{2} + 12117361 T^{4} \))(\( 1 + 1130 T^{2} + 12117361 T^{4} \))(\( ( 1 + 9200 T^{2} + 44845170 T^{4} + 111479721200 T^{6} + 146830437604321 T^{8} )^{2} \))
$61$ (\( ( 1 + 3721 T^{2} )^{2} \))(\( 1 - 7370 T^{2} + 13845841 T^{4} \))(\( 1 + 7370 T^{2} + 13845841 T^{4} \))(\( ( 1 + 14840 T^{2} + 82747650 T^{4} + 205472280440 T^{6} + 191707312997281 T^{8} )^{2} \))
$67$ (\( ( 1 - 118 T + 4489 T^{2} )( 1 + 118 T + 4489 T^{2} ) \))(\( ( 1 - 67 T )^{2}( 1 + 67 T )^{2} \))(\( ( 1 - 67 T )^{2}( 1 + 67 T )^{2} \))(\( ( 1 - 11884 T^{2} + 74122758 T^{4} - 239475921964 T^{6} + 406067677556641 T^{8} )^{2} \))
$71$ (\( ( 1 - 114 T + 5041 T^{2} )^{2} \))(\( ( 1 + 110 T + 5041 T^{2} )^{2} \))(\( ( 1 - 110 T + 5041 T^{2} )^{2} \))(\( ( 1 + 136 T + 12678 T^{2} + 685576 T^{3} + 25411681 T^{4} )^{4} \))
$73$ (\( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \))(\( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \))(\( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \))(\( ( 1 - 9364 T^{2} + 72904614 T^{4} - 265921128724 T^{6} + 806460091894081 T^{8} )^{2} \))
$79$ (\( ( 1 + 94 T + 6241 T^{2} )^{2} \))(\( ( 1 - 130 T + 6241 T^{2} )^{2} \))(\( ( 1 + 130 T + 6241 T^{2} )^{2} \))(\( ( 1 - 8 T + 11526 T^{2} - 49928 T^{3} + 38950081 T^{4} )^{4} \))
$83$ (\( ( 1 + 6889 T^{2} )^{2} \))(\( 1 - 13130 T^{2} + 47458321 T^{4} \))(\( 1 + 13130 T^{2} + 47458321 T^{4} \))(\( ( 1 + 6608 T^{2} - 3756750 T^{4} + 313604585168 T^{6} + 2252292232139041 T^{8} )^{2} \))
$89$ (\( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \))(\( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \))(\( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \))(\( ( 1 - 17428 T^{2} + 180509478 T^{4} - 1093471776148 T^{6} + 3936588805702081 T^{8} )^{2} \))
$97$ (\( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \))(\( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \))(\( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \))(\( ( 1 - 13588 T^{2} + 180804198 T^{4} - 1202935870228 T^{6} + 7837433594376961 T^{8} )^{2} \))
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