Properties

Label 336.2.k
Level 336
Weight 2
Character orbit k
Rep. character \(\chi_{336}(209,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 3
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.k (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
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 18 58
Cusp forms 52 14 38
Eisenstein series 24 4 20

Trace form

\( 14q + 4q^{7} - 2q^{9} + O(q^{10}) \) \( 14q + 4q^{7} - 2q^{9} + 8q^{15} - 2q^{21} - 6q^{25} - 4q^{37} - 16q^{39} + 32q^{43} - 2q^{49} - 4q^{57} - 20q^{63} - 16q^{67} - 8q^{79} - 18q^{81} - 16q^{85} - 24q^{91} + 20q^{93} - 64q^{99} + O(q^{100}) \)

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.k.a \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{3}+(-2-\zeta_{6})q^{7}-3q^{9}-4\zeta_{6}q^{13}+\cdots\)
336.2.k.b \(4\) \(2.683\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+\cdots\)
336.2.k.c \(8\) \(2.683\) 8.0.342102016.5 None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(1+\beta _{5})q^{7}+(-\beta _{5}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 9 T^{4} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 58 T^{4} + 250 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 8 T^{2} + 190 T^{4} - 968 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 48 T^{2} + 1086 T^{4} + 13872 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 706 T^{4} - 722 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 42 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 238 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 32 T^{2} + 2110 T^{4} - 30752 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{8} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 48 T^{2} + 606 T^{4} + 80688 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 - 4 T + 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 4150 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 8610 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 118 T^{2} + 9546 T^{4} - 439078 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )^{2} \))(\( ( 1 + 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 88 T^{2} + 6510 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 + 14 T + 190 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 318 T^{2} + 39042 T^{4} + 2190702 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 - 24 T^{2} + 12654 T^{4} - 190104 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} )^{2} \))
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