# Properties

 Label 672.2.i Level 672 Weight 2 Character orbit i Rep. character $$\chi_{672}(209,\cdot)$$ Character field $$\Q$$ Dimension 28 Newform subspaces 5 Sturm bound 256 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 672.i (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$168$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$256$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

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

Total New Old
Modular forms 144 36 108
Cusp forms 112 28 84
Eisenstein series 32 8 24

## Trace form

 $$28q + 4q^{7} - 4q^{9} + O(q^{10})$$ $$28q + 4q^{7} - 4q^{9} - 8q^{15} - 20q^{25} + 16q^{39} + 4q^{49} - 16q^{57} + 36q^{63} - 24q^{79} + 12q^{81} + O(q^{100})$$

## 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.i.a $$4$$ $$5.366$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$-4$$ $$0$$ $$8$$ $$q+(-1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots$$
672.2.i.b $$4$$ $$5.366$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{2}q^{3}+2\beta _{2}q^{5}+(-1+\beta _{3})q^{7}+\cdots$$
672.2.i.c $$4$$ $$5.366$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots$$
672.2.i.d $$8$$ $$5.366$$ 8.0.3317760000.1 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(-\beta _{2}-\beta _{3})q^{3}-\beta _{3}q^{5}+(-1-\beta _{6}+\cdots)q^{7}+\cdots$$
672.2.i.e $$8$$ $$5.366$$ 8.0.$$\cdots$$.11 $$\Q(\sqrt{-14})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{6}-\beta _{7})q^{5}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(672, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 3}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 + 2 T + 3 T^{2} )^{2}$$)($$( 1 + 3 T^{2} )^{2}$$)($$( 1 - 2 T + 3 T^{2} )^{2}$$)($$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$)($$1 - 10 T^{4} + 81 T^{8}$$)
$5$ ($$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 25 T^{4} )^{4}$$)($$( 1 + 22 T^{4} + 625 T^{8} )^{2}$$)
$7$ ($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{4}$$)($$( 1 - 7 T^{2} )^{4}$$)
$11$ ($$( 1 + 16 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 11 T^{2} )^{8}$$)
$13$ ($$( 1 + 2 T + 13 T^{2} )^{4}$$)($$( 1 + 13 T^{2} )^{4}$$)($$( 1 - 2 T + 13 T^{2} )^{4}$$)($$( 1 + 16 T^{2} + 169 T^{4} )^{4}$$)($$( 1 + 310 T^{4} + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{8}$$)($$( 1 + 17 T^{2} )^{8}$$)
$19$ ($$( 1 + 4 T + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 - 4 T + 19 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 650 T^{4} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 23 T^{2} )^{4}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4}$$)
$29$ ($$( 1 + 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 50 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 841 T^{4} )^{4}$$)($$( 1 + 29 T^{2} )^{8}$$)
$31$ ($$( 1 - 14 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{8}$$)
$37$ ($$( 1 + 34 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 + 34 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 37 T^{2} )^{8}$$)($$( 1 - 37 T^{2} )^{8}$$)
$41$ ($$( 1 + 76 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 76 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 + 41 T^{2} )^{8}$$)
$43$ ($$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 43 T^{2} )^{8}$$)
$47$ ($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 47 T^{2} )^{8}$$)
$53$ ($$( 1 + 53 T^{2} )^{4}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 + 53 T^{2} )^{8}$$)($$( 1 + 53 T^{2} )^{8}$$)
$59$ ($$( 1 - 86 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 1130 T^{4} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 10 T + 61 T^{2} )^{4}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 + 112 T^{2} + 3721 T^{4} )^{4}$$)($$( 1 - 7370 T^{4} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$)($$( 1 - 74 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 - 67 T^{2} )^{8}$$)
$71$ ($$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 71 T^{2} )^{4}$$)($$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 62 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 73 T^{2} )^{8}$$)
$79$ ($$( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 + 10 T + 79 T^{2} )^{4}$$)($$( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 - 10 T + 79 T^{2} )^{8}$$)($$( 1 + 130 T^{2} + 6241 T^{4} )^{4}$$)
$83$ ($$( 1 - 38 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 134 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 116 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 - 13130 T^{4} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 124 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 124 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)($$( 1 + 89 T^{2} )^{8}$$)
$97$ ($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{4}$$)($$( 1 - 97 T^{2} )^{8}$$)