# Properties

 Label 336.2.bj Level 336 Weight 2 Character orbit bj Rep. character $$\chi_{336}(95,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 32 Newform subspaces 7 Sturm bound 128 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bj (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$84$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$128$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$13$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(336, [\chi])$$.

Total New Old
Modular forms 152 32 120
Cusp forms 104 32 72
Eisenstein series 48 0 48

## Trace form

 $$32q + O(q^{10})$$ $$32q + 8q^{13} - 12q^{21} + 28q^{25} + 4q^{37} - 12q^{45} + 8q^{49} - 48q^{57} - 8q^{61} - 96q^{69} + 20q^{73} - 36q^{81} - 48q^{85} - 24q^{93} - 40q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
336.2.bj.a $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$-5$$ $$q+(-2+\zeta_{6})q^{3}+(-3+\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots$$
336.2.bj.b $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$-1$$ $$q+(-2+\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots$$
336.2.bj.c $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$1$$ $$q+(2-\zeta_{6})q^{3}+(-1+3\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots$$
336.2.bj.d $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$5$$ $$q+(2-\zeta_{6})q^{3}+(3-\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots$$
336.2.bj.e $$8$$ $$2.683$$ 8.0.8275904784.2 None $$0$$ $$-3$$ $$0$$ $$-6$$ $$q+(1-\beta _{1}-\beta _{5}+\beta _{6})q^{3}+(1-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$$
336.2.bj.f $$8$$ $$2.683$$ 8.0.303595776.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(1-2\beta _{2}+\beta _{4}+2\beta _{6})q^{5}+\cdots$$
336.2.bj.g $$8$$ $$2.683$$ 8.0.8275904784.2 None $$0$$ $$3$$ $$0$$ $$6$$ $$q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{4}-\beta _{7})q^{5}+(3+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()()()
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} + 18 T^{5} + 45 T^{6} + 81 T^{7} + 81 T^{8}$$)($$1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8}$$)($$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} - 18 T^{5} + 45 T^{6} - 81 T^{7} + 81 T^{8}$$)
$5$ ($$1 + 5 T^{2} + 25 T^{4}$$)($$1 + 5 T^{2} + 25 T^{4}$$)($$1 + 5 T^{2} + 25 T^{4}$$)($$1 + 5 T^{2} + 25 T^{4}$$)($$1 + 4 T^{2} + 19 T^{4} - 212 T^{6} - 1016 T^{8} - 5300 T^{10} + 11875 T^{12} + 62500 T^{14} + 390625 T^{16}$$)($$( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$)($$1 + 4 T^{2} + 19 T^{4} - 212 T^{6} - 1016 T^{8} - 5300 T^{10} + 11875 T^{12} + 62500 T^{14} + 390625 T^{16}$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} )^{2}$$)
$11$ ($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 4 T^{2} - 173 T^{4} + 212 T^{6} + 19144 T^{8} + 25652 T^{10} - 2532893 T^{12} - 7086244 T^{14} + 214358881 T^{16}$$)($$( 1 - 11 T^{2} )^{4}( 1 + 11 T^{2} + 121 T^{4} )^{2}$$)($$1 - 4 T^{2} - 173 T^{4} + 212 T^{6} + 19144 T^{8} + 25652 T^{10} - 2532893 T^{12} - 7086244 T^{14} + 214358881 T^{16}$$)
$13$ ($$( 1 + 5 T + 13 T^{2} )^{2}$$)($$( 1 - 7 T + 13 T^{2} )^{2}$$)($$( 1 - 7 T + 13 T^{2} )^{2}$$)($$( 1 + 5 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{8}$$)($$( 1 + 4 T + 13 T^{2} )^{8}$$)($$( 1 - 2 T + 13 T^{2} )^{8}$$)
$17$ ($$1 + 17 T^{2} + 289 T^{4}$$)($$1 + 17 T^{2} + 289 T^{4}$$)($$1 + 17 T^{2} + 289 T^{4}$$)($$1 + 17 T^{2} + 289 T^{4}$$)($$1 + 25 T^{2} - 95 T^{4} + 3550 T^{6} + 254254 T^{8} + 1025950 T^{10} - 7934495 T^{12} + 603439225 T^{14} + 6975757441 T^{16}$$)($$( 1 + 23 T^{2} + 240 T^{4} + 6647 T^{6} + 83521 T^{8} )^{2}$$)($$1 + 25 T^{2} - 95 T^{4} + 3550 T^{6} + 254254 T^{8} + 1025950 T^{10} - 7934495 T^{12} + 603439225 T^{14} + 6975757441 T^{16}$$)
$19$ ($$( 1 + 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$)($$( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} )$$)($$( 1 - 3 T + 37 T^{2} - 102 T^{3} + 852 T^{4} - 1938 T^{5} + 13357 T^{6} - 20577 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 3 T + 37 T^{2} + 102 T^{3} + 852 T^{4} + 1938 T^{5} + 13357 T^{6} + 20577 T^{7} + 130321 T^{8} )^{2}$$)
$23$ ($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 79 T^{2} + 3637 T^{4} - 122134 T^{6} + 3199486 T^{8} - 64608886 T^{10} + 1017781717 T^{12} - 11694835231 T^{14} + 78310985281 T^{16}$$)($$( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} )^{2}$$)($$1 - 79 T^{2} + 3637 T^{4} - 122134 T^{6} + 3199486 T^{8} - 64608886 T^{10} + 1017781717 T^{12} - 11694835231 T^{14} + 78310985281 T^{16}$$)
$29$ ($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 31 T^{2} + 1908 T^{4} - 26071 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 14 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 31 T^{2} + 1908 T^{4} - 26071 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} )$$)($$( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4}$$)($$( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{4}( 1 + 4 T + 31 T^{2} )^{4}$$)
$37$ ($$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$( 1 - 5 T - 41 T^{2} + 40 T^{3} + 2062 T^{4} + 1480 T^{5} - 56129 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 11 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$)($$( 1 - 5 T - 41 T^{2} + 40 T^{3} + 2062 T^{4} + 1480 T^{5} - 56129 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{2}$$)($$( 1 - 64 T^{2} + 2334 T^{4} - 107584 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 38 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 64 T^{2} + 2334 T^{4} - 107584 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$( 1 - 80 T^{2} + 3246 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 82 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 80 T^{2} + 3246 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 71 T^{2} + 517 T^{4} - 7526 T^{6} + 4451326 T^{8} - 16624934 T^{10} + 2522795077 T^{12} - 765324288359 T^{14} + 23811286661761 T^{16}$$)($$( 1 + 5 T^{2} - 2184 T^{4} + 11045 T^{6} + 4879681 T^{8} )^{2}$$)($$1 - 71 T^{2} + 517 T^{4} - 7526 T^{6} + 4451326 T^{8} - 16624934 T^{10} + 2522795077 T^{12} - 765324288359 T^{14} + 23811286661761 T^{16}$$)
$53$ ($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 - 20 T^{2} - 4805 T^{4} + 8260 T^{6} + 18077944 T^{8} + 23202340 T^{10} - 37913761205 T^{12} - 443287222580 T^{14} + 62259690411361 T^{16}$$)($$( 1 + 95 T^{2} + 6216 T^{4} + 266855 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 20 T^{2} - 4805 T^{4} + 8260 T^{6} + 18077944 T^{8} + 23202340 T^{10} - 37913761205 T^{12} - 443287222580 T^{14} + 62259690411361 T^{16}$$)
$59$ ($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 76 T^{2} - 2117 T^{4} - 70756 T^{6} + 30075832 T^{8} - 246301636 T^{10} - 25652453237 T^{12} - 3205720556716 T^{14} + 146830437604321 T^{16}$$)($$( 1 - 107 T^{2} + 7968 T^{4} - 372467 T^{6} + 12117361 T^{8} )^{2}$$)($$1 - 76 T^{2} - 2117 T^{4} - 70756 T^{6} + 30075832 T^{8} - 246301636 T^{10} - 25652453237 T^{12} - 3205720556716 T^{14} + 146830437604321 T^{16}$$)
$61$ ($$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 - 16 T + 67 T^{2} )( 1 - 11 T + 67 T^{2} )$$)($$( 1 + 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} )$$)($$( 1 + 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 33 T + 545 T^{2} - 6006 T^{3} + 52956 T^{4} - 402402 T^{5} + 2446505 T^{6} - 9925179 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 + 33 T + 545 T^{2} + 6006 T^{3} + 52956 T^{4} + 402402 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 160 T^{2} + 14430 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 34 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 + 160 T^{2} + 14430 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} )$$)($$( 1 - 7 T - 95 T^{2} + 14 T^{3} + 11830 T^{4} + 1022 T^{5} - 506255 T^{6} - 2723119 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{4}( 1 + 17 T + 73 T^{2} )^{4}$$)($$( 1 - 7 T - 95 T^{2} + 14 T^{3} + 11830 T^{4} + 1022 T^{5} - 506255 T^{6} - 2723119 T^{7} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} )$$)($$( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} )$$)($$( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$)($$( 1 - 4 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4}$$)($$( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 13 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4}$$)
$83$ ($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 301 T^{2} + 36300 T^{4} + 2073589 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 10 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 301 T^{2} + 36300 T^{4} + 2073589 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$1 + 89 T^{2} + 7921 T^{4}$$)($$1 + 89 T^{2} + 7921 T^{4}$$)($$1 + 89 T^{2} + 7921 T^{4}$$)($$1 + 89 T^{2} + 7921 T^{4}$$)($$1 + 109 T^{2} - 1787 T^{4} - 236966 T^{6} + 29962582 T^{8} - 1877007686 T^{10} - 112120384667 T^{12} + 54170960714749 T^{14} + 3936588805702081 T^{16}$$)($$( 1 + 167 T^{2} + 19968 T^{4} + 1322807 T^{6} + 62742241 T^{8} )^{2}$$)($$1 + 109 T^{2} - 1787 T^{4} - 236966 T^{6} + 29962582 T^{8} - 1877007686 T^{10} - 112120384667 T^{12} + 54170960714749 T^{14} + 3936588805702081 T^{16}$$)
$97$ ($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - T + 180 T^{2} - 97 T^{3} + 9409 T^{4} )^{4}$$)($$( 1 - 8 T + 97 T^{2} )^{8}$$)($$( 1 - T + 180 T^{2} - 97 T^{3} + 9409 T^{4} )^{4}$$)