# Properties

 Label 84.2.k Level 84 Weight 2 Character orbit k Rep. character $$\chi_{84}(5,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 6 Newform subspaces 3 Sturm bound 32 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 84.k (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$32$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(84, [\chi])$$.

Total New Old
Modular forms 44 6 38
Cusp forms 20 6 14
Eisenstein series 24 0 24

## Trace form

 $$6q + 3q^{7} + O(q^{10})$$ $$6q + 3q^{7} - 18q^{15} - 9q^{19} - 9q^{21} - 3q^{25} - 9q^{31} + 27q^{33} - 3q^{37} + 9q^{39} + 42q^{43} + 27q^{45} + 15q^{49} + 9q^{51} - 36q^{57} - 54q^{61} - 15q^{67} - 45q^{73} + 27q^{75} - 15q^{79} + 36q^{85} - 9q^{91} - 54q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(84, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
84.2.k.a $$2$$ $$0.671$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$3$$ $$4$$ $$q+(-2+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots$$
84.2.k.b $$2$$ $$0.671$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$4$$ $$q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots$$
84.2.k.c $$2$$ $$0.671$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$-5$$ $$q+(1+\zeta_{6})q^{3}+(-2-\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(84, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(84, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)
$5$ ($$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$)($$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 - 5 T^{2} + 25 T^{4}$$)
$7$ ($$1 - 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)
$11$ ($$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$)($$1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}$$)($$1 + 11 T^{2} + 121 T^{4}$$)
$13$ ($$( 1 - 13 T^{2} )^{2}$$)($$( 1 - 13 T^{2} )^{2}$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)
$17$ ($$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - 17 T^{2} + 289 T^{4}$$)
$19$ ($$1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4}$$)($$1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4}$$)($$( 1 + 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)
$23$ ($$1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4}$$)($$1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 + 23 T^{2} + 529 T^{4}$$)
$29$ ($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)
$37$ ($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$)
$41$ ($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)
$43$ ($$( 1 - 4 T + 43 T^{2} )^{2}$$)($$( 1 - 4 T + 43 T^{2} )^{2}$$)($$( 1 - 13 T + 43 T^{2} )^{2}$$)
$47$ ($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)
$53$ ($$1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 + 53 T^{2} + 2809 T^{4}$$)
$59$ ($$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$)($$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$)($$( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)
$67$ ($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)
$71$ ($$1 - 34 T^{2} + 5041 T^{4}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)
$73$ ($$1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4}$$)($$1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4}$$)($$( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)
$79$ ($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$)
$83$ ($$( 1 - 12 T + 83 T^{2} )^{2}$$)($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)
$89$ ($$1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4}$$)($$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 - 89 T^{2} + 7921 T^{4}$$)
$97$ ($$1 - 146 T^{2} + 9409 T^{4}$$)($$1 - 146 T^{2} + 9409 T^{4}$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)