Properties

Label 100.9.f
Level $100$
Weight $9$
Character orbit 100.f
Rep. character $\chi_{100}(57,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $24$
Newform subspaces $3$
Sturm bound $135$
Trace bound $1$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 258 24 234
Cusp forms 222 24 198
Eisenstein series 36 0 36

Trace form

\( 24 q + 70 q^{3} + 2030 q^{7} + O(q^{10}) \) \( 24 q + 70 q^{3} + 2030 q^{7} + 840 q^{11} - 33180 q^{13} - 43620 q^{17} - 627196 q^{21} + 663270 q^{23} - 1576040 q^{27} + 3166524 q^{31} + 944020 q^{33} - 5344080 q^{37} + 7246344 q^{41} + 10342710 q^{43} - 19232250 q^{47} + 42485848 q^{51} + 24320640 q^{53} - 88218320 q^{57} + 45976748 q^{61} + 77441350 q^{63} - 100675930 q^{67} + 16468572 q^{71} + 93528520 q^{73} - 134199660 q^{77} + 115768396 q^{81} + 10450350 q^{83} - 164801600 q^{87} + 347762196 q^{91} + 50183620 q^{93} + 179570760 q^{97} + O(q^{100}) \)

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.f.a 100.f 5.c $4$ $40.738$ \(\Q(i, \sqrt{3309})\) None 100.9.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+7\beta _{2}q^{7}-57\beta _{1}q^{9}+420q^{11}+\cdots\)
100.9.f.b 100.f 5.c $8$ $40.738$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 20.9.f.a \(0\) \(70\) \(0\) \(2030\) $\mathrm{SU}(2)[C_{4}]$ \(q+(9+9\beta _{1}+\beta _{2})q^{3}+(254-254\beta _{1}+\cdots)q^{7}+\cdots\)
100.9.f.c 100.f 5.c $12$ $40.738$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 100.9.f.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{3}+(7\beta _{7}-20\beta _{8}-\beta _{11})q^{7}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(100, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)