Properties

Label 16.10.a
Level $16$
Weight $10$
Character orbit 16.a
Rep. character $\chi_{16}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $20$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(20\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(16))\).

Total New Old
Modular forms 21 5 16
Cusp forms 15 4 11
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.
\(+\)\(2\)
\(-\)\(2\)

Trace form

\( 4q - 80q^{3} - 360q^{5} + 1376q^{7} + 5812q^{9} + O(q^{10}) \) \( 4q - 80q^{3} - 360q^{5} + 1376q^{7} + 5812q^{9} - 10992q^{11} - 43080q^{13} + 60448q^{15} + 172104q^{17} - 296336q^{19} - 336768q^{21} + 1349664q^{23} - 30468q^{25} - 5005088q^{27} + 1723896q^{29} + 13751680q^{31} - 3529024q^{33} - 27870144q^{35} + 1408792q^{37} + 46429600q^{39} + 4797864q^{41} - 78798192q^{43} - 6847304q^{45} + 139372608q^{47} + 3427300q^{49} - 202646944q^{51} - 10028712q^{53} + 216893536q^{55} + 25903936q^{57} - 224197296q^{59} - 10080648q^{61} + 293290464q^{63} - 64788144q^{65} - 255761744q^{67} + 121435520q^{69} - 39174048q^{71} - 15735768q^{73} + 276237392q^{75} - 120707712q^{77} - 341312064q^{79} - 113638172q^{81} + 758656368q^{83} + 277817008q^{85} - 1417283424q^{87} + 573376104q^{89} + 1721466688q^{91} - 1004178944q^{93} - 2153970528q^{95} - 728181880q^{97} + 2808900560q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(16))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
16.10.a.a \(1\) \(8.241\) \(\Q\) None \(0\) \(-228\) \(-666\) \(6328\) \(-\) \(q-228q^{3}-666q^{5}+6328q^{7}+32301q^{9}+\cdots\)
16.10.a.b \(1\) \(8.241\) \(\Q\) None \(0\) \(-68\) \(1510\) \(-10248\) \(+\) \(q-68q^{3}+1510q^{5}-10248q^{7}-15059q^{9}+\cdots\)
16.10.a.c \(1\) \(8.241\) \(\Q\) None \(0\) \(60\) \(-2074\) \(4344\) \(+\) \(q+60q^{3}-2074q^{5}+4344q^{7}-16083q^{9}+\cdots\)
16.10.a.d \(1\) \(8.241\) \(\Q\) None \(0\) \(156\) \(870\) \(952\) \(-\) \(q+156q^{3}+870q^{5}+952q^{7}+4653q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 228 T + 19683 T^{2} \))(\( 1 + 68 T + 19683 T^{2} \))(\( 1 - 60 T + 19683 T^{2} \))(\( 1 - 156 T + 19683 T^{2} \))
$5$ (\( 1 + 666 T + 1953125 T^{2} \))(\( 1 - 1510 T + 1953125 T^{2} \))(\( 1 + 2074 T + 1953125 T^{2} \))(\( 1 - 870 T + 1953125 T^{2} \))
$7$ (\( 1 - 6328 T + 40353607 T^{2} \))(\( 1 + 10248 T + 40353607 T^{2} \))(\( 1 - 4344 T + 40353607 T^{2} \))(\( 1 - 952 T + 40353607 T^{2} \))
$11$ (\( 1 - 30420 T + 2357947691 T^{2} \))(\( 1 + 3916 T + 2357947691 T^{2} \))(\( 1 + 93644 T + 2357947691 T^{2} \))(\( 1 - 56148 T + 2357947691 T^{2} \))
$13$ (\( 1 + 32338 T + 10604499373 T^{2} \))(\( 1 + 176594 T + 10604499373 T^{2} \))(\( 1 + 12242 T + 10604499373 T^{2} \))(\( 1 - 178094 T + 10604499373 T^{2} \))
$17$ (\( 1 - 590994 T + 118587876497 T^{2} \))(\( 1 - 148370 T + 118587876497 T^{2} \))(\( 1 + 319598 T + 118587876497 T^{2} \))(\( 1 + 247662 T + 118587876497 T^{2} \))
$19$ (\( 1 + 34676 T + 322687697779 T^{2} \))(\( 1 + 499796 T + 322687697779 T^{2} \))(\( 1 - 553516 T + 322687697779 T^{2} \))(\( 1 + 315380 T + 322687697779 T^{2} \))
$23$ (\( 1 + 1048536 T + 1801152661463 T^{2} \))(\( 1 - 1889768 T + 1801152661463 T^{2} \))(\( 1 - 712936 T + 1801152661463 T^{2} \))(\( 1 + 204504 T + 1801152661463 T^{2} \))
$29$ (\( 1 - 4409406 T + 14507145975869 T^{2} \))(\( 1 + 920898 T + 14507145975869 T^{2} \))(\( 1 - 2075838 T + 14507145975869 T^{2} \))(\( 1 + 3840450 T + 14507145975869 T^{2} \))
$31$ (\( 1 - 7401184 T + 26439622160671 T^{2} \))(\( 1 + 1379360 T + 26439622160671 T^{2} \))(\( 1 - 6420448 T + 26439622160671 T^{2} \))(\( 1 - 1309408 T + 26439622160671 T^{2} \))
$37$ (\( 1 - 10234502 T + 129961739795077 T^{2} \))(\( 1 - 5064966 T + 129961739795077 T^{2} \))(\( 1 + 18197754 T + 129961739795077 T^{2} \))(\( 1 - 4307078 T + 129961739795077 T^{2} \))
$41$ (\( 1 - 18352746 T + 327381934393961 T^{2} \))(\( 1 + 24100758 T + 327381934393961 T^{2} \))(\( 1 - 9033834 T + 327381934393961 T^{2} \))(\( 1 - 1512042 T + 327381934393961 T^{2} \))
$43$ (\( 1 - 252340 T + 502592611936843 T^{2} \))(\( 1 + 25785196 T + 502592611936843 T^{2} \))(\( 1 + 19594732 T + 502592611936843 T^{2} \))(\( 1 + 33670604 T + 502592611936843 T^{2} \))
$47$ (\( 1 - 49517136 T + 1119130473102767 T^{2} \))(\( 1 - 60790224 T + 1119130473102767 T^{2} \))(\( 1 - 18484176 T + 1119130473102767 T^{2} \))(\( 1 - 10581072 T + 1119130473102767 T^{2} \))
$53$ (\( 1 + 66396906 T + 3299763591802133 T^{2} \))(\( 1 - 29496214 T + 3299763591802133 T^{2} \))(\( 1 - 10255766 T + 3299763591802133 T^{2} \))(\( 1 - 16616214 T + 3299763591802133 T^{2} \))
$59$ (\( 1 - 61523748 T + 8662995818654939 T^{2} \))(\( 1 + 51819388 T + 8662995818654939 T^{2} \))(\( 1 + 121666556 T + 8662995818654939 T^{2} \))(\( 1 + 112235100 T + 8662995818654939 T^{2} \))
$61$ (\( 1 - 35638622 T + 11694146092834141 T^{2} \))(\( 1 - 33426910 T + 11694146092834141 T^{2} \))(\( 1 + 45948962 T + 11694146092834141 T^{2} \))(\( 1 + 33197218 T + 11694146092834141 T^{2} \))
$67$ (\( 1 + 181742372 T + 27206534396294947 T^{2} \))(\( 1 + 144856196 T + 27206534396294947 T^{2} \))(\( 1 + 50535428 T + 27206534396294947 T^{2} \))(\( 1 - 121372252 T + 27206534396294947 T^{2} \))
$71$ (\( 1 + 90904968 T + 45848500718449031 T^{2} \))(\( 1 + 68397128 T + 45848500718449031 T^{2} \))(\( 1 + 267044680 T + 45848500718449031 T^{2} \))(\( 1 - 387172728 T + 45848500718449031 T^{2} \))
$73$ (\( 1 + 262978678 T + 58871586708267913 T^{2} \))(\( 1 - 168216202 T + 58871586708267913 T^{2} \))(\( 1 + 176213366 T + 58871586708267913 T^{2} \))(\( 1 - 255240074 T + 58871586708267913 T^{2} \))
$79$ (\( 1 - 116502832 T + 119851595982618319 T^{2} \))(\( 1 + 235398736 T + 119851595982618319 T^{2} \))(\( 1 - 269685680 T + 119851595982618319 T^{2} \))(\( 1 + 492101840 T + 119851595982618319 T^{2} \))
$83$ (\( 1 - 9563724 T + 186940255267540403 T^{2} \))(\( 1 - 64639852 T + 186940255267540403 T^{2} \))(\( 1 - 227032556 T + 186940255267540403 T^{2} \))(\( 1 - 457420236 T + 186940255267540403 T^{2} \))
$89$ (\( 1 - 611826714 T + 350356403707485209 T^{2} \))(\( 1 + 78782694 T + 350356403707485209 T^{2} \))(\( 1 - 72141594 T + 350356403707485209 T^{2} \))(\( 1 + 31809510 T + 350356403707485209 T^{2} \))
$97$ (\( 1 + 259312798 T + 760231058654565217 T^{2} \))(\( 1 + 24113566 T + 760231058654565217 T^{2} \))(\( 1 - 228776546 T + 760231058654565217 T^{2} \))(\( 1 + 673532062 T + 760231058654565217 T^{2} \))
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