Properties

Label 336.2.h
Level 336
Weight 2
Character orbit h
Rep. character \(\chi_{336}(239,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 2
Sturm bound 128
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(128\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 76 12 64
Cusp forms 52 12 40
Eisenstein series 24 0 24

Trace form

\( 12q + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{9} - 12q^{25} + 48q^{37} - 24q^{45} - 12q^{49} - 48q^{61} + 24q^{69} - 72q^{73} - 36q^{81} - 24q^{85} + 24q^{93} + 72q^{97} + O(q^{100}) \)

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.h.a \(4\) \(2.683\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{5}-\zeta_{8}q^{7}+\cdots\)
336.2.h.b \(8\) \(2.683\) 8.0.56070144.2 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{3}+(\beta _{3}+\beta _{5})q^{5}+\beta _{4}q^{7}+(2+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))
$3$ (\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} \))
$5$ (\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 42 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$11$ (\( ( 1 + 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 54 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 26 T^{3} + 169 T^{4} )^{4} \))
$17$ (\( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 28 T^{2} + 582 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 68 T^{2} + 1866 T^{4} - 24548 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 1062 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 76 T^{2} + 2934 T^{4} - 63916 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 26 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 28 T^{2} + 390 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( ( 1 - 16 T + 126 T^{2} - 592 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 76 T^{2} + 3078 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 + 22 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{8} \))
$53$ (\( ( 1 + 22 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 124 T^{2} + 7734 T^{4} - 348316 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 76 T^{2} + 8298 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 + 6 T + 61 T^{2} )^{4} \))(\( ( 1 + 6 T + 128 T^{2} + 366 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{4}( 1 + 16 T + 67 T^{2} )^{4} \))
$71$ (\( ( 1 + 92 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 196 T^{2} + 17958 T^{4} + 988036 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 2 T + 73 T^{2} )^{4} \))(\( ( 1 + 16 T + 198 T^{2} + 1168 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 - 34 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 172 T^{2} + 21066 T^{4} + 1184908 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 80 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 15750 T^{4} + 158420 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{4} \))(\( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \))
show more
show less