Properties

Label 672.2.k
Level 672
Weight 2
Character orbit k
Rep. character \(\chi_{672}(545,\cdot)\)
Character field \(\Q\)
Dimension 32
Newform subspaces 4
Sturm bound 256
Trace bound 9

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.k (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(256\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(5\), \(43\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(672, [\chi])\).

Total New Old
Modular forms 144 32 112
Cusp forms 112 32 80
Eisenstein series 32 0 32

Trace form

\( 32q + O(q^{10}) \) \( 32q + 8q^{21} + 48q^{25} - 16q^{37} + 16q^{57} + 64q^{81} - 16q^{85} - 32q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(672, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
672.2.k.a \(8\) \(5.366\) 8.0.342102016.5 None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(-1-\beta _{4}+\cdots)q^{7}+\cdots\)
672.2.k.b \(8\) \(5.366\) 8.0.49787136.1 \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+\beta _{5}q^{5}-\beta _{2}q^{7}-3q^{9}+\beta _{3}q^{11}+\cdots\)
672.2.k.c \(8\) \(5.366\) 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{4}+\beta _{5})q^{3}+\beta _{7}q^{5}+(-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
672.2.k.d \(8\) \(5.366\) 8.0.342102016.5 None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{2}q^{3}+(\beta _{2}+\beta _{3})q^{5}+(1+\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 + 3 T^{2} )^{4} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))
$5$ (\( ( 1 + 6 T^{2} + 42 T^{4} + 150 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 34 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 + 8 T^{2} + 25 T^{4} )^{4} \))(\( ( 1 + 6 T^{2} + 42 T^{4} + 150 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} )^{4} \))(\( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 6 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{8} \))(\( ( 1 - 6 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 6 T^{2} + 330 T^{4} - 1014 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} )^{8} \))(\( ( 1 - 16 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 6 T^{2} + 330 T^{4} - 1014 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 28 T^{2} + 502 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 178 T^{4} + 83521 T^{8} )^{2} \))(\( ( 1 + 26 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 + 28 T^{2} + 502 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 40 T^{2} + 846 T^{4} - 21160 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 10 T + 50 T^{2} - 230 T^{3} + 529 T^{4} )^{2}( 1 + 10 T + 50 T^{2} + 230 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 40 T^{2} + 846 T^{4} - 21160 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 64 T^{2} + 2094 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 29 T^{2} )^{8} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 64 T^{2} + 2094 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 30 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 + 6 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{8} \))(\( ( 1 + 6 T^{2} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 + 124 T^{2} + 6934 T^{4} + 208444 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 1262 T^{4} + 2825761 T^{8} )^{2} \))(\( ( 1 + 10 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 124 T^{2} + 6934 T^{4} + 208444 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{8} \))(\( ( 1 + 43 T^{2} )^{8} \))(\( ( 1 + 6 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 + 8 T + 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 132 T^{2} + 8502 T^{4} + 291588 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 47 T^{2} )^{8} \))(\( ( 1 + 54 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 + 132 T^{2} + 8502 T^{4} + 291588 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 53 T^{2} )^{8} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{4} \))(\( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 78 T^{2} + 7650 T^{4} + 271518 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 108 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 78 T^{2} + 7650 T^{4} + 271518 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 54 T^{2} + 7338 T^{4} - 200934 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 112 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 54 T^{2} + 7338 T^{4} - 200934 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 + 12 T + 102 T^{2} + 804 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 67 T^{2} )^{8} \))(\( ( 1 + 67 T^{2} )^{8} \))(\( ( 1 - 12 T + 102 T^{2} - 804 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 76 T^{2} + 1734 T^{4} - 383116 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 22 T + 242 T^{2} - 1562 T^{3} + 5041 T^{4} )^{2}( 1 + 22 T + 242 T^{2} + 1562 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 78 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 76 T^{2} + 1734 T^{4} - 383116 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 + 14 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 73 T^{2} )^{8} \))
$79$ (\( ( 1 + 2 T + 142 T^{2} + 158 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 79 T^{2} )^{8} \))(\( ( 1 - 22 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 2 T + 142 T^{2} - 158 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 - 2 T^{2} - 2558 T^{4} - 13778 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 83 T^{2} )^{8} \))(\( ( 1 + 156 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 2 T^{2} - 2558 T^{4} - 13778 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 244 T^{2} + 29638 T^{4} + 1932724 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 13742 T^{4} + 62742241 T^{8} )^{2} \))(\( ( 1 + 50 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 244 T^{2} + 29638 T^{4} + 1932724 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 140 T^{2} + 10390 T^{4} - 1317260 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 97 T^{2} )^{8} \))(\( ( 1 + 166 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 140 T^{2} + 10390 T^{4} - 1317260 T^{6} + 88529281 T^{8} )^{2} \))
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