Properties

Label 8.16.a
Level $8$
Weight $16$
Character orbit 8.a
Rep. character $\chi_{8}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
Sturm bound $16$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 8.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 \(M_{16}(\Gamma_0(8))\).

Total New Old
Modular forms 17 4 13
Cusp forms 13 4 9
Eisenstein series 4 0 4

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
\(+\)\(2\)
\(-\)\(2\)

Trace form

\( 4 q - 4816 q^{3} - 78792 q^{5} - 824352 q^{7} + 17559700 q^{9} + O(q^{10}) \) \( 4 q - 4816 q^{3} - 78792 q^{5} - 824352 q^{7} + 17559700 q^{9} - 77853168 q^{11} + 66385432 q^{13} + 236751008 q^{15} + 346434504 q^{17} - 5038000912 q^{19} + 15735268992 q^{21} - 42325541472 q^{23} + 110985555164 q^{25} - 190050794272 q^{27} + 209796530136 q^{29} - 377063644288 q^{31} + 853611748544 q^{33} - 929457797568 q^{35} + 603580348344 q^{37} - 1399226132704 q^{39} + 1411163409000 q^{41} + 2916482940304 q^{43} - 7867578801128 q^{45} + 10055753636160 q^{47} - 11305510260700 q^{49} + 10407251548000 q^{51} - 14453600213640 q^{53} + 21743718954208 q^{55} - 9768419244224 q^{57} - 5476736925360 q^{59} + 11293162832728 q^{61} - 19618814290080 q^{63} + 15801538329936 q^{65} - 101238902133712 q^{67} + 215730284047232 q^{69} - 87771720342816 q^{71} + 111037285914088 q^{73} - 405206000663216 q^{75} + 144267440340864 q^{77} + 171350784855232 q^{79} + 135699750198436 q^{81} - 33861880122384 q^{83} - 686208914866960 q^{85} + 1340070554352672 q^{87} - 946249239678552 q^{89} - 146193789167808 q^{91} - 1147797513827840 q^{93} + 2775060805039392 q^{95} - 76978025186680 q^{97} - 2625158795769776 q^{99} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
8.16.a.a 8.a 1.a $1$ $11.415$ \(\Q\) None \(0\) \(-3444\) \(313358\) \(-2324616\) $-$ $\mathrm{SU}(2)$ \(q-3444q^{3}+313358q^{5}-2324616q^{7}+\cdots\)
8.16.a.b 8.a 1.a $1$ $11.415$ \(\Q\) None \(0\) \(2700\) \(-251890\) \(1374072\) $-$ $\mathrm{SU}(2)$ \(q+2700q^{3}-251890q^{5}+1374072q^{7}+\cdots\)
8.16.a.c 8.a 1.a $2$ $11.415$ \(\Q(\sqrt{58}) \) None \(0\) \(-4072\) \(-140260\) \(126192\) $+$ $\mathrm{SU}(2)$ \(q+(-2036+\beta )q^{3}+(-70130+6^{2}\beta )q^{5}+\cdots\)

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

\( S_{16}^{\mathrm{old}}(\Gamma_0(8)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)