# 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

# Related objects

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