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Properties

Label 100.4.e
Level $100$
Weight $4$
Character orbit 100.e
Rep. character $\chi_{100}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $50$
Newform subspaces $6$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(100, [\chi])\).

Total New Old
Modular forms 102 58 44
Cusp forms 78 50 28
Eisenstein series 24 8 16

Trace form

\( 50 q + 2 q^{2} - 28 q^{6} + 44 q^{8} + O(q^{10}) \) \( 50 q + 2 q^{2} - 28 q^{6} + 44 q^{8} + 80 q^{12} - 42 q^{13} - 380 q^{16} + 134 q^{17} - 306 q^{18} + 264 q^{21} - 360 q^{22} + 908 q^{26} + 880 q^{28} + 632 q^{32} - 80 q^{33} - 1028 q^{36} - 326 q^{37} - 1600 q^{38} - 600 q^{41} - 1160 q^{42} + 632 q^{46} + 2720 q^{48} + 1524 q^{52} + 698 q^{53} - 4528 q^{56} + 960 q^{57} - 2712 q^{58} + 3640 q^{61} - 2440 q^{62} + 5940 q^{66} + 2428 q^{68} + 2172 q^{72} - 1942 q^{73} - 2380 q^{76} - 3120 q^{77} - 3720 q^{78} - 7626 q^{81} - 536 q^{82} + 112 q^{86} + 2400 q^{88} + 1840 q^{92} + 3280 q^{93} + 8252 q^{96} + 5994 q^{97} - 326 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.4.e.a 100.e 20.e $2$ $5.900$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2-2i)q^{2}+8iq^{4}+(2^{4}-2^{4}i)q^{8}+\cdots\)
100.4.e.b 100.e 20.e $2$ $5.900$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(-4\) \(14\) \(0\) \(-18\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2+2i)q^{2}+(7+7i)q^{3}-8iq^{4}+\cdots\)
100.4.e.c 100.e 20.e $2$ $5.900$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(4\) \(-14\) \(0\) \(18\) $\mathrm{U}(1)[D_{4}]$ \(q+(2-2i)q^{2}+(-7-7i)q^{3}-8iq^{4}+\cdots\)
100.4.e.d 100.e 20.e $8$ $5.900$ 8.0.2342560000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{6})q^{3}+(-\beta _{1}-3\beta _{4}+\cdots)q^{4}+\cdots\)
100.4.e.e 100.e 20.e $12$ $5.900$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{3}+\beta _{5})q^{2}+\beta _{9}q^{3}+(-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
100.4.e.f 100.e 20.e $24$ $5.900$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(100, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)