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

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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

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