Properties

Label 16.8.a
Level 16
Weight 8
Character orbit a
Rep. character \(\chi_{16}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 16
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(16))\).

Total New Old
Modular forms 17 4 13
Cusp forms 11 3 8
Eisenstein series 6 1 5

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3q + 28q^{3} + 138q^{5} + 664q^{7} + 2575q^{9} + O(q^{10}) \) \( 3q + 28q^{3} + 138q^{5} + 664q^{7} + 2575q^{9} + 4596q^{11} - 3278q^{13} - 23288q^{15} - 12714q^{17} + 41292q^{19} - 3360q^{21} - 151992q^{23} + 1349q^{25} + 383320q^{27} + 120066q^{29} - 438432q^{31} + 85904q^{33} + 702288q^{35} - 311254q^{37} - 1191128q^{39} - 167154q^{41} + 610388q^{43} - 78158q^{45} - 74736q^{47} + 267739q^{49} - 397192q^{51} + 615162q^{53} + 1382872q^{55} - 1066640q^{57} - 777084q^{59} + 1830562q^{61} + 3988728q^{63} + 3224316q^{65} - 7499556q^{67} - 5969248q^{69} + 6673752q^{71} - 6370898q^{73} - 10287484q^{75} + 6133152q^{77} + 4652912q^{79} + 7963579q^{81} + 1276428q^{83} - 9170060q^{85} + 4209000q^{87} - 6818562q^{89} + 1169552q^{91} + 17190784q^{93} - 5694552q^{95} + 13548006q^{97} + 13726148q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
16.8.a.a \(1\) \(4.998\) \(\Q\) None \(0\) \(-44\) \(430\) \(1224\) \(+\) \(q-44q^{3}+430q^{5}+1224q^{7}-251q^{9}+\cdots\)
16.8.a.b \(1\) \(4.998\) \(\Q\) None \(0\) \(-12\) \(-210\) \(-1016\) \(-\) \(q-12q^{3}-210q^{5}-1016q^{7}-2043q^{9}+\cdots\)
16.8.a.c \(1\) \(4.998\) \(\Q\) None \(0\) \(84\) \(-82\) \(456\) \(+\) \(q+84q^{3}-82q^{5}+456q^{7}+4869q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 44 T + 2187 T^{2} \))(\( 1 + 12 T + 2187 T^{2} \))(\( 1 - 84 T + 2187 T^{2} \))
$5$ (\( 1 - 430 T + 78125 T^{2} \))(\( 1 + 210 T + 78125 T^{2} \))(\( 1 + 82 T + 78125 T^{2} \))
$7$ (\( 1 - 1224 T + 823543 T^{2} \))(\( 1 + 1016 T + 823543 T^{2} \))(\( 1 - 456 T + 823543 T^{2} \))
$11$ (\( 1 - 3164 T + 19487171 T^{2} \))(\( 1 + 1092 T + 19487171 T^{2} \))(\( 1 - 2524 T + 19487171 T^{2} \))
$13$ (\( 1 - 6118 T + 62748517 T^{2} \))(\( 1 - 1382 T + 62748517 T^{2} \))(\( 1 + 10778 T + 62748517 T^{2} \))
$17$ (\( 1 + 16270 T + 410338673 T^{2} \))(\( 1 - 14706 T + 410338673 T^{2} \))(\( 1 + 11150 T + 410338673 T^{2} \))
$19$ (\( 1 - 5476 T + 893871739 T^{2} \))(\( 1 - 39940 T + 893871739 T^{2} \))(\( 1 + 4124 T + 893871739 T^{2} \))
$23$ (\( 1 + 1576 T + 3404825447 T^{2} \))(\( 1 + 68712 T + 3404825447 T^{2} \))(\( 1 + 81704 T + 3404825447 T^{2} \))
$29$ (\( 1 - 122838 T + 17249876309 T^{2} \))(\( 1 + 102570 T + 17249876309 T^{2} \))(\( 1 - 99798 T + 17249876309 T^{2} \))
$31$ (\( 1 + 251360 T + 27512614111 T^{2} \))(\( 1 + 227552 T + 27512614111 T^{2} \))(\( 1 - 40480 T + 27512614111 T^{2} \))
$37$ (\( 1 + 52338 T + 94931877133 T^{2} \))(\( 1 - 160526 T + 94931877133 T^{2} \))(\( 1 + 419442 T + 94931877133 T^{2} \))
$41$ (\( 1 + 319398 T + 194754273881 T^{2} \))(\( 1 - 10842 T + 194754273881 T^{2} \))(\( 1 - 141402 T + 194754273881 T^{2} \))
$43$ (\( 1 + 710788 T + 271818611107 T^{2} \))(\( 1 - 630748 T + 271818611107 T^{2} \))(\( 1 - 690428 T + 271818611107 T^{2} \))
$47$ (\( 1 + 284112 T + 506623120463 T^{2} \))(\( 1 + 472656 T + 506623120463 T^{2} \))(\( 1 - 682032 T + 506623120463 T^{2} \))
$53$ (\( 1 - 296062 T + 1174711139837 T^{2} \))(\( 1 + 1494018 T + 1174711139837 T^{2} \))(\( 1 - 1813118 T + 1174711139837 T^{2} \))
$59$ (\( 1 - 897548 T + 2488651484819 T^{2} \))(\( 1 + 2640660 T + 2488651484819 T^{2} \))(\( 1 - 966028 T + 2488651484819 T^{2} \))
$61$ (\( 1 + 884810 T + 3142742836021 T^{2} \))(\( 1 - 827702 T + 3142742836021 T^{2} \))(\( 1 - 1887670 T + 3142742836021 T^{2} \))
$67$ (\( 1 + 4659692 T + 6060711605323 T^{2} \))(\( 1 - 126004 T + 6060711605323 T^{2} \))(\( 1 + 2965868 T + 6060711605323 T^{2} \))
$71$ (\( 1 - 2710792 T + 9095120158391 T^{2} \))(\( 1 - 1414728 T + 9095120158391 T^{2} \))(\( 1 - 2548232 T + 9095120158391 T^{2} \))
$73$ (\( 1 + 5670854 T + 11047398519097 T^{2} \))(\( 1 - 980282 T + 11047398519097 T^{2} \))(\( 1 + 1680326 T + 11047398519097 T^{2} \))
$79$ (\( 1 - 5124176 T + 19203908986159 T^{2} \))(\( 1 - 3566800 T + 19203908986159 T^{2} \))(\( 1 + 4038064 T + 19203908986159 T^{2} \))
$83$ (\( 1 - 1563556 T + 27136050989627 T^{2} \))(\( 1 + 5672892 T + 27136050989627 T^{2} \))(\( 1 - 5385764 T + 27136050989627 T^{2} \))
$89$ (\( 1 - 11605674 T + 44231334895529 T^{2} \))(\( 1 + 11951190 T + 44231334895529 T^{2} \))(\( 1 + 6473046 T + 44231334895529 T^{2} \))
$97$ (\( 1 - 10931618 T + 80798284478113 T^{2} \))(\( 1 - 8682146 T + 80798284478113 T^{2} \))(\( 1 + 6065758 T + 80798284478113 T^{2} \))
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