Properties

Label 4.10
Level 4
Weight 10
Dimension 1
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4\( 4 = 2^{2} \) \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(4))\).

Total New Old
Modular forms 6 1 5
Cusp forms 3 1 2
Eisenstein series 3 0 3

Trace form

\( q + 228q^{3} - 666q^{5} - 6328q^{7} + 32301q^{9} + O(q^{10}) \) \( q + 228q^{3} - 666q^{5} - 6328q^{7} + 32301q^{9} - 30420q^{11} - 32338q^{13} - 151848q^{15} + 590994q^{17} + 34676q^{19} - 1442784q^{21} + 1048536q^{23} - 1509569q^{25} + 2876904q^{27} + 4409406q^{29} - 7401184q^{31} - 6935760q^{33} + 4214448q^{35} + 10234502q^{37} - 7373064q^{39} + 18352746q^{41} - 252340q^{43} - 21512466q^{45} - 49517136q^{47} - 310023q^{49} + 134746632q^{51} - 66396906q^{53} + 20259720q^{55} + 7906128q^{57} - 61523748q^{59} + 35638622q^{61} - 204400728q^{63} + 21537108q^{65} + 181742372q^{67} + 239066208q^{69} + 90904968q^{71} - 262978678q^{73} - 344181732q^{75} + 192497760q^{77} - 116502832q^{79} + 20153529q^{81} - 9563724q^{83} - 393602004q^{85} + 1005344568q^{87} + 611826714q^{89} + 204634864q^{91} - 1687469952q^{93} - 23094216q^{95} - 259312798q^{97} - 982596420q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(4))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
4.10.a \(\chi_{4}(1, \cdot)\) 4.10.a.a 1 1

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(4))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(4)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 228 T + 19683 T^{2} \)
$5$ \( 1 + 666 T + 1953125 T^{2} \)
$7$ \( 1 + 6328 T + 40353607 T^{2} \)
$11$ \( 1 + 30420 T + 2357947691 T^{2} \)
$13$ \( 1 + 32338 T + 10604499373 T^{2} \)
$17$ \( 1 - 590994 T + 118587876497 T^{2} \)
$19$ \( 1 - 34676 T + 322687697779 T^{2} \)
$23$ \( 1 - 1048536 T + 1801152661463 T^{2} \)
$29$ \( 1 - 4409406 T + 14507145975869 T^{2} \)
$31$ \( 1 + 7401184 T + 26439622160671 T^{2} \)
$37$ \( 1 - 10234502 T + 129961739795077 T^{2} \)
$41$ \( 1 - 18352746 T + 327381934393961 T^{2} \)
$43$ \( 1 + 252340 T + 502592611936843 T^{2} \)
$47$ \( 1 + 49517136 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 66396906 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 61523748 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 35638622 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 181742372 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 90904968 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 262978678 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 116502832 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 9563724 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 611826714 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 259312798 T + 760231058654565217 T^{2} \)
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