Properties

Label 1344.2.s
Level 1344
Weight 2
Character orbit s
Rep. character \(\chi_{1344}(239,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 96
Newform subspaces 4
Sturm bound 512
Trace bound 3

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1344.s (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 48 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(512\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1344, [\chi])\).

Total New Old
Modular forms 544 96 448
Cusp forms 480 96 384
Eisenstein series 64 0 64

Trace form

\( 96q + O(q^{10}) \) \( 96q - 16q^{19} + 24q^{27} + 48q^{39} + 32q^{43} + 96q^{49} + 64q^{55} - 32q^{61} + 32q^{67} - 56q^{75} - 32q^{85} - 112q^{87} - 48q^{93} - 64q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1344, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1344.2.s.a \(4\) \(10.732\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(4\) \(4\) \(q+(-\zeta_{8}-\zeta_{8}^{3})q^{3}+(1+\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
1344.2.s.b \(4\) \(10.732\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(-4\) \(4\) \(q+(1-\zeta_{8}^{2})q^{3}+(-1-\zeta_{8}+\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+\cdots\)
1344.2.s.c \(40\) \(10.732\) None \(0\) \(-4\) \(0\) \(40\)
1344.2.s.d \(48\) \(10.732\) None \(0\) \(0\) \(0\) \(-48\)

Decomposition of \(S_{2}^{\mathrm{old}}(1344, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1344, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 3}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))
$3$ (\( 1 - 2 T^{2} + 9 T^{4} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))
$5$ (\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 60 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 60 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} \))
$7$ (\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))
$11$ (\( 1 + 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 132 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 132 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} \))
$17$ (\( 1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8} \))(\( 1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8} \))
$19$ (\( 1 - 4 T + 8 T^{2} - 68 T^{3} + 574 T^{4} - 1292 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 68 T^{3} + 574 T^{4} - 1292 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} \))
$23$ (\( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} \))(\( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))
$31$ (\( 1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8} \))(\( 1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8} \))
$37$ (\( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} \))
$41$ (\( ( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 4 T + 8 T^{2} - 116 T^{3} + 1486 T^{4} - 4988 T^{5} + 14792 T^{6} - 318028 T^{7} + 3418801 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 116 T^{3} + 1486 T^{4} - 4988 T^{5} + 14792 T^{6} - 318028 T^{7} + 3418801 T^{8} \))
$47$ (\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 34980 T^{5} + 202248 T^{6} - 1786524 T^{7} + 7890481 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 34980 T^{5} + 202248 T^{6} + 1786524 T^{7} + 7890481 T^{8} \))
$59$ (\( ( 1 + 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} ) \))(\( ( 1 - 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} ) \))
$61$ (\( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8} \))
$67$ (\( ( 1 - 14 T + 98 T^{2} - 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 14 T + 98 T^{2} - 938 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8} \))(\( 1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 4 T + 8 T^{2} - 444 T^{3} - 12994 T^{4} - 36852 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 36852 T^{5} + 55112 T^{6} - 2287148 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 20 T + 246 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 20 T + 246 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 2 T + 97 T^{2} )^{4} \))(\( ( 1 + 2 T + 97 T^{2} )^{4} \))
show more
show less