Properties

Label 81.10.c
Level $81$
Weight $10$
Character orbit 81.c
Rep. character $\chi_{81}(28,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $70$
Newform subspaces $12$
Sturm bound $90$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(90\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(81, [\chi])\).

Total New Old
Modular forms 174 74 100
Cusp forms 150 70 80
Eisenstein series 24 4 20

Trace form

\( 70 q - 8702 q^{4} + 1712 q^{7} - 2052 q^{10} - 162178 q^{13} - 2096126 q^{16} - 329044 q^{19} - 1055646 q^{22} - 10647917 q^{25} - 4552708 q^{28} + 16432778 q^{31} - 19250784 q^{34} - 52213756 q^{37} + 77162814 q^{40}+ \cdots - 1560307930 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{10}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.10.c.a 81.c 9.c $2$ $41.718$ \(\Q(\sqrt{-3}) \) None 3.10.a.a \(-36\) \(0\) \(-1314\) \(4480\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-6^{2}+6^{2}\zeta_{6})q^{2}-28^{2}\zeta_{6}q^{4}-1314\zeta_{6}q^{5}+\cdots\)
81.10.c.b 81.c 9.c $2$ $41.718$ \(\Q(\sqrt{-3}) \) None 3.10.a.b \(-18\) \(0\) \(1530\) \(-9128\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-18+18\zeta_{6})q^{2}+188\zeta_{6}q^{4}+1530\zeta_{6}q^{5}+\cdots\)
81.10.c.c 81.c 9.c $2$ $41.718$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 9.10.a.b \(0\) \(0\) \(0\) \(12580\) $\mathrm{U}(1)[D_{3}]$ \(q+2^{9}\zeta_{6}q^{4}+(12580-12580\zeta_{6})q^{7}+\cdots\)
81.10.c.d 81.c 9.c $2$ $41.718$ \(\Q(\sqrt{-3}) \) None 3.10.a.b \(18\) \(0\) \(-1530\) \(-9128\) $\mathrm{SU}(2)[C_{3}]$ \(q+(18-18\zeta_{6})q^{2}+188\zeta_{6}q^{4}-1530\zeta_{6}q^{5}+\cdots\)
81.10.c.e 81.c 9.c $2$ $41.718$ \(\Q(\sqrt{-3}) \) None 3.10.a.a \(36\) \(0\) \(1314\) \(4480\) $\mathrm{SU}(2)[C_{3}]$ \(q+(6^{2}-6^{2}\zeta_{6})q^{2}-28^{2}\zeta_{6}q^{4}+1314\zeta_{6}q^{5}+\cdots\)
81.10.c.f 81.c 9.c $4$ $41.718$ \(\Q(\sqrt{-3}, \sqrt{14})\) None 27.10.a.a \(0\) \(0\) \(0\) \(1526\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}-8\beta _{2}q^{4}+(22\beta _{1}+22\beta _{3})q^{5}+\cdots\)
81.10.c.g 81.c 9.c $6$ $41.718$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 27.10.a.b \(-3\) \(0\) \(1983\) \(3693\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2}-\beta _{3})q^{2}+(2\beta _{1}+199\beta _{2}+\cdots)q^{4}+\cdots\)
81.10.c.h 81.c 9.c $6$ $41.718$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 27.10.a.b \(3\) \(0\) \(-1983\) \(3693\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2}+\beta _{3})q^{2}+(2\beta _{1}+199\beta _{2}+\cdots)q^{4}+\cdots\)
81.10.c.i 81.c 9.c $8$ $41.718$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 81.10.a.a \(-33\) \(0\) \(-570\) \(3238\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\beta _{3}+\beta _{4})q^{2}+(-6\beta _{2}-212\beta _{3}+\cdots)q^{4}+\cdots\)
81.10.c.j 81.c 9.c $8$ $41.718$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 27.10.a.d \(0\) \(0\) \(0\) \(-11852\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(-559+559\beta _{1}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
81.10.c.k 81.c 9.c $8$ $41.718$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 81.10.a.a \(33\) \(0\) \(570\) \(3238\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+8\beta _{3})q^{2}+(-212+\beta _{1}+212\beta _{3}+\cdots)q^{4}+\cdots\)
81.10.c.l 81.c 9.c $20$ $41.718$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 81.10.a.e \(0\) \(0\) \(0\) \(-5108\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{9}-\beta _{10})q^{2}+(-290-290\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)