Properties

Label 100.9.b
Level $100$
Weight $9$
Character orbit 100.b
Rep. character $\chi_{100}(51,\cdot)$
Character field $\Q$
Dimension $73$
Newform subspaces $7$
Sturm bound $135$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(135\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

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

Total New Old
Modular forms 126 79 47
Cusp forms 114 73 41
Eisenstein series 12 6 6

Trace form

\( 73 q - 2 q^{2} - 330 q^{4} - 742 q^{6} + 2728 q^{8} - 144623 q^{9} + O(q^{10}) \) \( 73 q - 2 q^{2} - 330 q^{4} - 742 q^{6} + 2728 q^{8} - 144623 q^{9} + 14120 q^{12} - 39974 q^{13} + 3892 q^{14} + 192178 q^{16} - 110374 q^{17} + 443558 q^{18} + 35896 q^{21} - 72640 q^{22} - 212978 q^{24} - 1538972 q^{26} - 1987400 q^{28} - 468070 q^{29} + 2844928 q^{32} + 1827520 q^{33} + 1429838 q^{34} + 2568656 q^{36} - 2991334 q^{37} + 2600640 q^{38} - 993774 q^{41} + 1271880 q^{42} + 4071210 q^{44} + 16298188 q^{46} + 8287680 q^{48} - 38003503 q^{49} - 7169624 q^{52} + 5520026 q^{53} - 9023426 q^{54} - 1019932 q^{56} - 15851520 q^{57} - 32195604 q^{58} + 24874386 q^{61} - 2987280 q^{62} - 52515150 q^{64} + 38970930 q^{66} + 8310536 q^{68} - 83516536 q^{69} + 40562088 q^{72} + 4261146 q^{73} - 42725672 q^{74} + 43505610 q^{76} + 105280320 q^{77} + 90379280 q^{78} + 136079489 q^{81} - 98230124 q^{82} - 128420836 q^{84} - 133189792 q^{86} - 111221920 q^{88} + 179555930 q^{89} + 132386280 q^{92} - 119142080 q^{93} + 161554592 q^{94} - 72309682 q^{96} - 239709414 q^{97} + 269309638 q^{98} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.9.b.a 100.b 4.b $1$ $40.738$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-16\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{4}q^{2}+2^{8}q^{4}-2^{12}q^{8}+3^{8}q^{9}+\cdots\)
100.9.b.b 100.b 4.b $2$ $40.738$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8iq^{2}+79iq^{3}-2^{8}q^{4}-2528q^{6}+\cdots\)
100.9.b.c 100.b 4.b $2$ $40.738$ \(\Q(\sqrt{-39}) \) None \(20\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(10-\beta )q^{2}-8\beta q^{3}+(-56-20\beta )q^{4}+\cdots\)
100.9.b.d 100.b 4.b $16$ $40.738$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(-3-\beta _{3}+\cdots)q^{4}+\cdots\)
100.9.b.e 100.b 4.b $16$ $40.738$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-21-\beta _{4})q^{4}+\cdots\)
100.9.b.f 100.b 4.b $16$ $40.738$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-21+\beta _{4})q^{4}+\cdots\)
100.9.b.g 100.b 4.b $20$ $40.738$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{8})q^{3}+(38+\beta _{2}+\cdots)q^{4}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(100, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)