Properties

Label 672.2.bc
Level 672
Weight 2
Character orbit bc
Rep. character \(\chi_{672}(257,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 64
Newform subspaces 5
Sturm bound 256
Trace bound 5

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.bc (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 288 64 224
Cusp forms 224 64 160
Eisenstein series 64 0 64

Trace form

\( 64q + O(q^{10}) \) \( 64q - 8q^{21} - 24q^{25} - 8q^{37} + 24q^{45} + 32q^{57} + 24q^{73} + 8q^{81} - 32q^{85} - 16q^{93} + O(q^{100}) \)

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.bc.a \(4\) \(5.366\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+(-2\zeta_{12}+\zeta_{12}^{3})q^{3}+(-1+\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
672.2.bc.b \(4\) \(5.366\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{5}+(2\zeta_{12}+\cdots)q^{7}+\cdots\)
672.2.bc.c \(8\) \(5.366\) 8.0.49787136.1 None \(0\) \(0\) \(0\) \(0\) \(q+(2\beta _{2}+\beta _{4})q^{3}+\beta _{7}q^{5}+\beta _{1}q^{7}+(-3+\cdots)q^{9}+\cdots\)
672.2.bc.d \(16\) \(5.366\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{10}q^{3}+\beta _{8}q^{5}+\beta _{9}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
672.2.bc.e \(32\) \(5.366\) None \(0\) \(0\) \(0\) \(0\)

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

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( 1 - T^{2} - 3 T^{4} + 14 T^{6} - 74 T^{8} + 126 T^{10} - 243 T^{12} - 729 T^{14} + 6561 T^{16} \))
$5$ (\( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T^{2} - 16 T^{4} - 75 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 7 T^{2} + T^{4} + 14 T^{6} + 454 T^{8} + 350 T^{10} + 625 T^{12} - 109375 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 3 T^{2} - 28 T^{4} + 147 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - 3 T^{2} - 112 T^{4} - 363 T^{6} + 14641 T^{8} \))(\( 1 - 3 T^{2} - 112 T^{4} - 363 T^{6} + 14641 T^{8} \))(\( ( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 11 T^{2} - 23 T^{4} - 1078 T^{6} - 8546 T^{8} - 130438 T^{10} - 336743 T^{12} + 19487171 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 5 T^{2} - 12 T^{4} - 845 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} - 253 T^{4} - 1734 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 55 T^{2} + 1705 T^{4} - 40810 T^{6} + 784654 T^{8} - 11794090 T^{10} + 142403305 T^{12} - 1327566295 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + 11 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \))(\( ( 1 + 11 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \))(\( ( 1 + 10 T^{2} - 261 T^{4} + 3610 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 379 T^{4} - 8876 T^{6} - 249320 T^{8} - 3204236 T^{10} + 49391659 T^{12} + 1317284668 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))(\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))(\( ( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 23 T^{2} + 493 T^{4} - 23506 T^{6} - 558386 T^{8} - 12434674 T^{10} + 137961613 T^{12} + 3404825447 T^{14} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 + 56 T^{2} + 2238 T^{4} + 47096 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( 1 - 19 T^{2} - 600 T^{4} - 18259 T^{6} + 923521 T^{8} \))(\( 1 - 19 T^{2} - 600 T^{4} - 18259 T^{6} + 923521 T^{8} \))(\( ( 1 - T^{2} - 960 T^{4} - 961 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 3427 T^{4} + 143388 T^{6} + 5465304 T^{8} + 137795868 T^{10} + 3164906467 T^{12} + 74550309204 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 4 T - 5 T^{2} + 212 T^{3} - 1520 T^{4} + 7844 T^{5} - 6845 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 + 8 T + 41 T^{2} )^{4} \))(\( ( 1 - 8 T + 41 T^{2} )^{4} \))(\( ( 1 - 30 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 41 T^{2} )^{16} \))
$43$ (\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{8} \))(\( ( 1 + 87 T^{2} + 5576 T^{4} + 160863 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8} \))(\( 1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8} \))(\( ( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 165 T^{2} + 16129 T^{4} - 1101870 T^{6} + 58101510 T^{8} - 2434030830 T^{10} + 78704374849 T^{12} - 1778570529285 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 + 21 T + 200 T^{2} + 1113 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 21 T + 200 T^{2} - 1113 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 85 T^{2} + 4416 T^{4} + 238765 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 173 T^{2} + 16957 T^{4} + 1272242 T^{6} + 76311742 T^{8} + 3573727778 T^{10} + 133798886317 T^{12} + 3834434475317 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \))(\( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \))(\( ( 1 + 29 T^{2} - 2640 T^{4} + 100949 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 189 T^{2} + 20185 T^{4} - 1620486 T^{6} + 104905134 T^{8} - 5640911766 T^{10} + 244588931785 T^{12} - 7972120858149 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 24 T + 253 T^{2} - 1464 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 21 T + 301 T^{2} + 3234 T^{3} + 28962 T^{4} + 197274 T^{5} + 1120021 T^{6} + 4766601 T^{7} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 50 T^{2} - 1989 T^{4} - 224450 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 124 T^{2} + 7171 T^{4} + 95852 T^{6} - 21851960 T^{8} + 430279628 T^{10} + 144503688691 T^{12} - 11216839388956 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 126 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 42 T^{2} + 5041 T^{4} )^{8} \))
$73$ (\( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 12 T + 187 T^{2} + 1668 T^{3} + 17496 T^{4} + 121764 T^{5} + 996523 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{4} \))
$79$ (\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( ( 1 - 137 T^{2} + 12528 T^{4} - 855017 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 196 T^{2} + 16843 T^{4} - 1781836 T^{6} + 186689992 T^{8} - 11120438476 T^{10} + 656036214283 T^{12} - 47645141282116 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 163 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 + 288 T^{2} + 34286 T^{4} + 1984032 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 150 T^{2} + 14579 T^{4} - 1188150 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 91 T^{2} + 757 T^{4} + 756938 T^{6} - 71725466 T^{8} + 5995705898 T^{10} + 47495876437 T^{12} - 45225297477451 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 119 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 40 T^{2} + 17166 T^{4} - 376360 T^{6} + 88529281 T^{8} )^{4} \))
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