# 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 $13$

# Related objects

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