Properties

Label 336.2.bl
Level 336
Weight 2
Character orbit bl
Rep. character \(\chi_{336}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 16
Newform subspaces 8
Sturm bound 128
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bl (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

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

Total New Old
Modular forms 152 16 136
Cusp forms 104 16 88
Eisenstein series 48 0 48

Trace form

\( 16q - 8q^{9} + O(q^{10}) \) \( 16q - 8q^{9} + 8q^{21} + 20q^{25} + 36q^{33} + 4q^{37} - 56q^{49} - 24q^{53} - 8q^{57} - 72q^{61} - 24q^{65} + 12q^{73} - 8q^{81} + 96q^{85} - 28q^{93} + O(q^{100}) \)

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.bl.a \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-6\) \(1\) \(q+(-1+\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(2+\cdots)q^{7}+\cdots\)
336.2.bl.b \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-3\) \(-5\) \(q+(-1+\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+(-3+\cdots)q^{7}+\cdots\)
336.2.bl.c \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(3\) \(1\) \(q+(-1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-1+\cdots)q^{7}+\cdots\)
336.2.bl.d \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(6\) \(1\) \(q+(-1+\zeta_{6})q^{3}+(4-2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
336.2.bl.e \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-6\) \(-1\) \(q+(1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(-2+\cdots)q^{7}+\cdots\)
336.2.bl.f \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-3\) \(5\) \(q+(1-\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+\cdots\)
336.2.bl.g \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(3\) \(-1\) \(q+(1-\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(1-3\zeta_{6})q^{7}+\cdots\)
336.2.bl.h \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(6\) \(-1\) \(q+(1-\zeta_{6})q^{3}+(4-2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))
$5$ (\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))
$11$ (\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))
$17$ (\( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))
$41$ (\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))
$43$ (\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 11 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 11 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 61 T^{2} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 3 T + 70 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 + 67 T^{2} + 4489 T^{4} \))(\( 1 + 24 T + 259 T^{2} + 1608 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 3 T + 70 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 + 67 T^{2} + 4489 T^{4} \))(\( 1 - 24 T + 259 T^{2} - 1608 T^{3} + 4489 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} \))(\( 1 - 3 T + 82 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( 1 - 27 T + 322 T^{2} - 2133 T^{3} + 6241 T^{4} \))(\( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))
$83$ (\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 18 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} )^{2} \))(\( ( 1 - 18 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 101 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 101 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))
show more
show less