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Properties

Label 100.6.e
Level $100$
Weight $6$
Character orbit 100.e
Rep. character $\chi_{100}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $86$
Newform subspaces $6$
Sturm bound $90$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(90\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(100, [\chi])\).

Total New Old
Modular forms 162 94 68
Cusp forms 138 86 52
Eisenstein series 24 8 16

Trace form

\( 86 q + 2 q^{2} + 356 q^{6} - 244 q^{8} + O(q^{10}) \) \( 86 q + 2 q^{2} + 356 q^{6} - 244 q^{8} + 1280 q^{12} - 118 q^{13} - 3964 q^{16} - 1006 q^{17} + 3246 q^{18} - 3288 q^{21} + 2440 q^{22} - 13684 q^{26} - 5920 q^{28} - 17608 q^{32} - 10400 q^{33} + 68092 q^{36} + 9414 q^{37} + 22160 q^{38} + 19272 q^{41} + 39400 q^{42} - 1544 q^{46} - 108160 q^{48} - 38476 q^{52} + 58982 q^{53} + 10256 q^{56} - 40320 q^{57} + 183672 q^{58} - 192328 q^{61} + 109400 q^{62} - 278700 q^{66} - 313988 q^{68} - 309828 q^{72} - 39298 q^{73} + 582900 q^{76} + 53280 q^{77} + 586200 q^{78} + 94098 q^{81} + 458744 q^{82} - 632144 q^{86} - 690080 q^{88} - 576800 q^{92} - 182240 q^{93} + 879356 q^{96} - 129426 q^{97} + 1036906 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.6.e.a 100.e 20.e $2$ $16.038$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(-8\) \(-38\) \(0\) \(-366\) $\mathrm{U}(1)[D_{4}]$ \(q+(-4+4i)q^{2}+(-19-19i)q^{3}+\cdots\)
100.6.e.b 100.e 20.e $2$ $16.038$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-8\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-4-4i)q^{2}+2^{5}iq^{4}+(2^{7}-2^{7}i)q^{8}+\cdots\)
100.6.e.c 100.e 20.e $2$ $16.038$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(8\) \(38\) \(0\) \(366\) $\mathrm{U}(1)[D_{4}]$ \(q+(4-4i)q^{2}+(19+19i)q^{3}-2^{5}iq^{4}+\cdots\)
100.6.e.d 100.e 20.e $16$ $16.038$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{8}q^{2}-\beta _{10}q^{3}+(9\beta _{6}+\beta _{11})q^{4}+\cdots\)
100.6.e.e 100.e 20.e $24$ $16.038$ None \(10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
100.6.e.f 100.e 20.e $40$ $16.038$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{6}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(100, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)