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Properties

Label 68.3.g
Level $68$
Weight $3$
Character orbit 68.g
Rep. character $\chi_{68}(15,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $64$
Newform subspaces $5$
Sturm bound $27$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 68.g (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 68 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 5 \)
Sturm bound: \(27\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(68, [\chi])\).

Total New Old
Modular forms 80 80 0
Cusp forms 64 64 0
Eisenstein series 16 16 0

Trace form

\( 64 q - 4 q^{2} - 8 q^{5} + 20 q^{6} - 4 q^{8} + 8 q^{9} + O(q^{10}) \) \( 64 q - 4 q^{2} - 8 q^{5} + 20 q^{6} - 4 q^{8} + 8 q^{9} - 4 q^{10} - 40 q^{12} - 20 q^{14} - 8 q^{16} - 8 q^{17} - 8 q^{18} + 60 q^{20} - 28 q^{22} - 216 q^{24} - 128 q^{25} - 100 q^{26} - 188 q^{28} - 48 q^{29} + 76 q^{32} - 16 q^{33} + 28 q^{34} + 156 q^{36} - 8 q^{37} + 136 q^{40} + 192 q^{41} + 448 q^{42} + 108 q^{44} + 112 q^{45} + 324 q^{46} + 84 q^{48} - 8 q^{49} - 368 q^{50} + 120 q^{52} - 64 q^{53} + 4 q^{54} + 244 q^{56} - 520 q^{57} + 472 q^{58} + 176 q^{60} - 232 q^{61} + 356 q^{62} + 88 q^{65} - 268 q^{66} + 276 q^{68} + 48 q^{69} - 536 q^{70} + 304 q^{73} - 4 q^{74} - 672 q^{76} + 56 q^{77} - 72 q^{78} - 776 q^{80} - 352 q^{82} - 128 q^{84} + 64 q^{85} - 1152 q^{86} - 24 q^{88} - 876 q^{90} - 852 q^{92} + 184 q^{93} + 216 q^{94} - 956 q^{96} + 232 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(68, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
68.3.g.a 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q-2\zeta_{8}^{2}q^{2}+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{3}+\cdots\)
68.3.g.b 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-16\) \(0\) $\mathrm{U}(1)[D_{8}]$ \(q-2\zeta_{8}^{3}q^{2}-4\zeta_{8}^{2}q^{4}+(-4+4\zeta_{8}+\cdots)q^{5}+\cdots\)
68.3.g.c 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(16\) \(0\) $\mathrm{U}(1)[D_{8}]$ \(q+2\zeta_{8}^{3}q^{2}-4\zeta_{8}^{2}q^{4}+(4-4\zeta_{8}-3\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
68.3.g.d 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) None \(8\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+2q^{2}+(2\zeta_{8}^{2}+2\zeta_{8}^{3})q^{3}+4q^{4}+\cdots\)
68.3.g.e 68.g 68.g $48$ $1.853$ None \(-12\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{8}]$