Properties

Label 100.9.d
Level $100$
Weight $9$
Character orbit 100.d
Rep. character $\chi_{100}(99,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $4$
Sturm bound $135$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(135\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(100, [\chi])\).

Total New Old
Modular forms 126 74 52
Cusp forms 114 70 44
Eisenstein series 12 4 8

Trace form

\( 70 q + 478 q^{4} + 2778 q^{6} + 144346 q^{9} + O(q^{10}) \) \( 70 q + 478 q^{4} + 2778 q^{6} + 144346 q^{9} - 107772 q^{14} - 14110 q^{16} + 367936 q^{21} + 1859618 q^{24} + 40352 q^{26} - 1333964 q^{29} - 7090802 q^{34} - 3068292 q^{36} + 2071340 q^{41} - 14424090 q^{44} + 6929908 q^{46} + 49848826 q^{49} + 30719506 q^{54} + 47382228 q^{56} + 64259660 q^{61} + 8910478 q^{64} - 10514590 q^{66} - 16422144 q^{69} + 77566828 q^{74} + 91340010 q^{76} + 16585926 q^{81} + 97196996 q^{84} - 100634352 q^{86} + 19314196 q^{89} - 141340392 q^{94} - 19366962 q^{96} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.9.d.a 100.d 20.d $2$ $40.738$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 4.9.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-8iq^{2}-2^{8}q^{4}+2^{11}iq^{8}-3^{8}q^{9}+\cdots\)
100.9.d.b 100.d 20.d $4$ $40.738$ \(\Q(i, \sqrt{39})\) None 4.9.b.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{2}-8\beta _{1}q^{3}+(56-2\beta _{3})q^{4}+\cdots\)
100.9.d.c 100.d 20.d $32$ $40.738$ None 20.9.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
100.9.d.d 100.d 20.d $32$ $40.738$ None 100.9.b.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{9}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(100, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)