Properties

Label 8.10.a
Level 8
Weight 10
Character orbit a
Rep. character \(\chi_{8}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 10
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(8))\).

Total New Old
Modular forms 11 2 9
Cusp forms 7 2 5
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.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\( 2q + 8q^{3} - 564q^{5} + 5904q^{7} - 31142q^{9} + O(q^{10}) \) \( 2q + 8q^{3} - 564q^{5} + 5904q^{7} - 31142q^{9} + 97560q^{11} - 188836q^{13} + 227120q^{15} - 171228q^{17} - 53720q^{19} + 957504q^{21} - 2602704q^{23} + 2675326q^{25} - 216496q^{27} + 1154940q^{29} - 5041088q^{31} - 5352352q^{33} + 24483936q^{35} - 13132788q^{37} - 11273872q^{39} - 15066924q^{41} + 45379928q^{43} + 10617052q^{45} - 79274400q^{47} + 43184626q^{49} + 29265040q^{51} + 39751980q^{53} - 188304496q^{55} + 67197088q^{57} + 173485944q^{59} - 12522052q^{61} - 84460080q^{63} - 241267032q^{65} + 195391624q^{67} - 85728064q^{69} + 335441808q^{71} - 7997164q^{73} - 118666760q^{75} - 366658368q^{77} - 34286944q^{79} + 323563378q^{81} - 291672408q^{83} + 886884952q^{85} - 187171344q^{87} - 6641100q^{89} - 1756556064q^{91} + 479023360q^{93} + 1902684144q^{95} + 204662980q^{97} - 1565047496q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(8))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
8.10.a.a \(1\) \(4.120\) \(\Q\) None \(0\) \(-60\) \(-2074\) \(-4344\) \(+\) \(q-60q^{3}-2074q^{5}-4344q^{7}-16083q^{9}+\cdots\)
8.10.a.b \(1\) \(4.120\) \(\Q\) None \(0\) \(68\) \(1510\) \(10248\) \(-\) \(q+68q^{3}+1510q^{5}+10248q^{7}-15059q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(8))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(8)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 60 T + 19683 T^{2} \))(\( 1 - 68 T + 19683 T^{2} \))
$5$ (\( 1 + 2074 T + 1953125 T^{2} \))(\( 1 - 1510 T + 1953125 T^{2} \))
$7$ (\( 1 + 4344 T + 40353607 T^{2} \))(\( 1 - 10248 T + 40353607 T^{2} \))
$11$ (\( 1 - 93644 T + 2357947691 T^{2} \))(\( 1 - 3916 T + 2357947691 T^{2} \))
$13$ (\( 1 + 12242 T + 10604499373 T^{2} \))(\( 1 + 176594 T + 10604499373 T^{2} \))
$17$ (\( 1 + 319598 T + 118587876497 T^{2} \))(\( 1 - 148370 T + 118587876497 T^{2} \))
$19$ (\( 1 + 553516 T + 322687697779 T^{2} \))(\( 1 - 499796 T + 322687697779 T^{2} \))
$23$ (\( 1 + 712936 T + 1801152661463 T^{2} \))(\( 1 + 1889768 T + 1801152661463 T^{2} \))
$29$ (\( 1 - 2075838 T + 14507145975869 T^{2} \))(\( 1 + 920898 T + 14507145975869 T^{2} \))
$31$ (\( 1 + 6420448 T + 26439622160671 T^{2} \))(\( 1 - 1379360 T + 26439622160671 T^{2} \))
$37$ (\( 1 + 18197754 T + 129961739795077 T^{2} \))(\( 1 - 5064966 T + 129961739795077 T^{2} \))
$41$ (\( 1 - 9033834 T + 327381934393961 T^{2} \))(\( 1 + 24100758 T + 327381934393961 T^{2} \))
$43$ (\( 1 - 19594732 T + 502592611936843 T^{2} \))(\( 1 - 25785196 T + 502592611936843 T^{2} \))
$47$ (\( 1 + 18484176 T + 1119130473102767 T^{2} \))(\( 1 + 60790224 T + 1119130473102767 T^{2} \))
$53$ (\( 1 - 10255766 T + 3299763591802133 T^{2} \))(\( 1 - 29496214 T + 3299763591802133 T^{2} \))
$59$ (\( 1 - 121666556 T + 8662995818654939 T^{2} \))(\( 1 - 51819388 T + 8662995818654939 T^{2} \))
$61$ (\( 1 + 45948962 T + 11694146092834141 T^{2} \))(\( 1 - 33426910 T + 11694146092834141 T^{2} \))
$67$ (\( 1 - 50535428 T + 27206534396294947 T^{2} \))(\( 1 - 144856196 T + 27206534396294947 T^{2} \))
$71$ (\( 1 - 267044680 T + 45848500718449031 T^{2} \))(\( 1 - 68397128 T + 45848500718449031 T^{2} \))
$73$ (\( 1 + 176213366 T + 58871586708267913 T^{2} \))(\( 1 - 168216202 T + 58871586708267913 T^{2} \))
$79$ (\( 1 + 269685680 T + 119851595982618319 T^{2} \))(\( 1 - 235398736 T + 119851595982618319 T^{2} \))
$83$ (\( 1 + 227032556 T + 186940255267540403 T^{2} \))(\( 1 + 64639852 T + 186940255267540403 T^{2} \))
$89$ (\( 1 - 72141594 T + 350356403707485209 T^{2} \))(\( 1 + 78782694 T + 350356403707485209 T^{2} \))
$97$ (\( 1 - 228776546 T + 760231058654565217 T^{2} \))(\( 1 + 24113566 T + 760231058654565217 T^{2} \))
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