Properties

Label 672.2.h
Level $672$
Weight $2$
Character orbit 672.h
Rep. character $\chi_{672}(575,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $5$
Sturm bound $256$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(256\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

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

Total New Old
Modular forms 144 24 120
Cusp forms 112 24 88
Eisenstein series 32 0 32

Trace form

\( 24q - 8q^{9} + O(q^{10}) \) \( 24q - 8q^{9} - 24q^{25} + 32q^{33} - 32q^{37} - 48q^{45} - 24q^{49} - 32q^{57} + 32q^{61} + 48q^{69} + 48q^{73} - 8q^{81} + 16q^{85} + 48q^{93} - 48q^{97} + O(q^{100}) \)

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.h.a \(4\) \(5.366\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(0\) \(q+(-1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(\zeta_{8}+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
672.2.h.b \(4\) \(5.366\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}-3\zeta_{8}^{2}q^{5}-\zeta_{8}q^{7}+\cdots\)
672.2.h.c \(4\) \(5.366\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{5}-\zeta_{8}q^{7}+\cdots\)
672.2.h.d \(4\) \(5.366\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(0\) \(q+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(-\zeta_{8}+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
672.2.h.e \(8\) \(5.366\) 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}-\beta _{4}q^{5}+\beta _{1}q^{7}+(-2+\beta _{6}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} \))(\( ( 1 + 4 T^{2} + 9 T^{4} )^{2} \))
$5$ (\( 1 - 8 T^{2} + 34 T^{4} - 200 T^{6} + 625 T^{8} \))(\( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 8 T^{2} + 34 T^{4} - 200 T^{6} + 625 T^{8} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \))
$7$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$11$ (\( ( 1 + 2 T + 11 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 4 T + 28 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 6 T + 13 T^{2} )^{4} \))(\( ( 1 - 4 T + 28 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )^{4} \))
$17$ (\( 1 - 44 T^{2} + 934 T^{4} - 12716 T^{6} + 83521 T^{8} \))(\( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 44 T^{2} + 934 T^{4} - 12716 T^{6} + 83521 T^{8} \))(\( ( 1 - 12 T^{2} - 26 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 32 T^{2} + 690 T^{4} - 11552 T^{6} + 130321 T^{8} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( 1 - 32 T^{2} + 690 T^{4} - 11552 T^{6} + 130321 T^{8} \))(\( ( 1 - 24 T^{2} + 226 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} \))(\( ( 1 - 60 T^{2} + 1942 T^{4} - 50460 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - 76 T^{2} + 2854 T^{4} - 73036 T^{6} + 923521 T^{8} \))(\( ( 1 - 58 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 961 T^{4} )^{2} \))(\( 1 - 76 T^{2} + 2854 T^{4} - 73036 T^{6} + 923521 T^{8} \))(\( ( 1 - 22 T^{2} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 - 4 T + 70 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( ( 1 - 4 T + 70 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 38 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 60 T^{2} + 1702 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 20 T^{2} + 2646 T^{4} - 36980 T^{6} + 3418801 T^{8} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( 1 - 20 T^{2} + 2646 T^{4} - 36980 T^{6} + 3418801 T^{8} \))(\( ( 1 - 84 T^{2} + 4822 T^{4} - 155316 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 8 T + 78 T^{2} + 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 2209 T^{4} )^{4} \))
$53$ (\( 1 + 52 T^{2} + 4246 T^{4} + 146068 T^{6} + 7890481 T^{8} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( 1 + 52 T^{2} + 4246 T^{4} + 146068 T^{6} + 7890481 T^{8} \))(\( ( 1 - 108 T^{2} + 5974 T^{4} - 303372 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 - 4 T + 104 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 4 T + 104 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 116 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 - 20 T + 204 T^{2} - 1220 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{4} \))(\( ( 1 - 20 T + 204 T^{2} - 1220 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \))
$67$ (\( 1 - 4 T^{2} + 6934 T^{4} - 17956 T^{6} + 20151121 T^{8} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 122 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 4 T^{2} + 6934 T^{4} - 17956 T^{6} + 20151121 T^{8} \))(\( ( 1 - 130 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 12 T + 146 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 124 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 12 T + 146 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 180 T^{2} + 15622 T^{4} + 907380 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{4} \))(\( ( 1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 110 T^{2} + 292 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 + 20 T + 264 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 94 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 20 T + 264 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 168 T^{2} + 20194 T^{4} + 1157352 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 284 T^{2} + 35494 T^{4} - 2249564 T^{6} + 62742241 T^{8} \))(\( ( 1 - 80 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 128 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 284 T^{2} + 35494 T^{4} - 2249564 T^{6} + 62742241 T^{8} \))(\( ( 1 - 252 T^{2} + 29158 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{4} \))(\( ( 1 - 6 T + 97 T^{2} )^{4} \))(\( ( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 4 T + 38 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \))
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