Properties

Label 336.2.b
Level 336
Weight 2
Character orbit b
Rep. character \(\chi_{336}(223,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 4
Sturm bound 128
Trace bound 7

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(128\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(19\)

Dimensions

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

Total New Old
Modular forms 76 8 68
Cusp forms 52 8 44
Eisenstein series 24 0 24

Trace form

\( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 4q^{21} + 16q^{25} + 8q^{37} - 16q^{49} - 48q^{53} + 8q^{57} - 48q^{65} + 8q^{81} - 24q^{85} + 16q^{93} + O(q^{100}) \)

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.b.a \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(-4\) \(q-q^{3}+(-2-\zeta_{6})q^{7}+q^{9}-2\zeta_{6}q^{11}+\cdots\)
336.2.b.b \(2\) \(2.683\) \(\Q(\sqrt{-6}) \) None \(0\) \(-2\) \(0\) \(2\) \(q-q^{3}+\beta q^{5}+(1-\beta )q^{7}+q^{9}+\beta q^{11}+\cdots\)
336.2.b.c \(2\) \(2.683\) \(\Q(\sqrt{-6}) \) None \(0\) \(2\) \(0\) \(-2\) \(q+q^{3}+\beta q^{5}+(-1+\beta )q^{7}+q^{9}-\beta q^{11}+\cdots\)
336.2.b.d \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(4\) \(q+q^{3}+(2+\zeta_{6})q^{7}+q^{9}+2\zeta_{6}q^{11}+\cdots\)

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
$3$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 4 T^{2} + 25 T^{4} \))(\( 1 - 4 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))
$11$ (\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 16 T^{2} + 121 T^{4} \))(\( 1 - 16 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 169 T^{4} \))(\( 1 - 2 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))
$17$ (\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 - 28 T^{2} + 289 T^{4} \))(\( 1 - 28 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 + 8 T^{2} + 529 T^{4} \))(\( 1 + 8 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 28 T^{2} + 1681 T^{4} \))(\( 1 - 28 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 62 T^{2} + 1849 T^{4} \))(\( 1 - 62 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))
$53$ (\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))
$59$ (\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))
$61$ (\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))
$71$ (\( 1 - 34 T^{2} + 5041 T^{4} \))(\( 1 + 8 T^{2} + 5041 T^{4} \))(\( 1 + 8 T^{2} + 5041 T^{4} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 70 T^{2} + 5329 T^{4} \))(\( 1 + 70 T^{2} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))
$79$ (\( 1 - 50 T^{2} + 6241 T^{4} \))(\( 1 - 62 T^{2} + 6241 T^{4} \))(\( 1 - 62 T^{2} + 6241 T^{4} \))(\( 1 - 50 T^{2} + 6241 T^{4} \))
$83$ (\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 28 T^{2} + 7921 T^{4} \))(\( 1 - 28 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))
$97$ (\( ( 1 - 97 T^{2} )^{2} \))(\( 1 - 170 T^{2} + 9409 T^{4} \))(\( 1 - 170 T^{2} + 9409 T^{4} \))(\( ( 1 - 97 T^{2} )^{2} \))
show more
show less