Properties

Label 729.2.e
Level $729$
Weight $2$
Character orbit 729.e
Rep. character $\chi_{729}(82,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $198$
Newform subspaces $21$
Sturm bound $162$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 27 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 21 \)
Sturm bound: \(162\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(729, [\chi])\).

Total New Old
Modular forms 594 234 360
Cusp forms 378 198 180
Eisenstein series 216 36 180

Trace form

\( 198 q + O(q^{10}) \) \( 198 q + 18 q^{10} + 18 q^{19} - 36 q^{28} + 18 q^{37} + 18 q^{46} - 36 q^{55} - 18 q^{64} - 117 q^{73} - 36 q^{82} - 117 q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(729, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
729.2.e.a 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(-6\) \(0\) \(-3\) \(-9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1-\zeta_{18}+\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
729.2.e.b 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(-3\) \(0\) \(-6\) \(9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1+\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.c 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(-3\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4})q^{2}+\cdots\)
729.2.e.d 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) 243.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{9}]$ \(q+2\zeta_{18}^{5}q^{4}+5\zeta_{18}^{4}q^{7}+2\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.e 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) 243.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{9}]$ \(q+2\zeta_{18}^{5}q^{4}-4\zeta_{18}^{4}q^{7}-7\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.f 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) 27.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{9}]$ \(q+2\zeta_{18}^{5}q^{4}-\zeta_{18}^{4}q^{7}+5\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.g 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(3\) \(0\) \(6\) \(9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.h 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(3\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.i 729.e 27.e $6$ $5.821$ \(\Q(\zeta_{18})\) None 243.2.a.e \(6\) \(0\) \(3\) \(-9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(1+\cdots)q^{4}+\cdots\)
729.2.e.j 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(-6\) \(0\) \(6\) \(-3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1-\beta _{6}-\beta _{7}+\beta _{11})q^{2}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
729.2.e.k 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(-3\) \(0\) \(-6\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\beta _{5}-\beta _{7}+\beta _{10})q^{2}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)
729.2.e.l 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(-3\) \(0\) \(12\) \(-3\) $\mathrm{SU}(2)[C_{9}]$ \(q+\beta _{2}q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots\)
729.2.e.m 729.e 27.e $12$ $5.821$ \(\Q(\zeta_{36})\) None 729.2.a.c \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{36}^{2}+\zeta_{36}^{7}-\zeta_{36}^{9}+\zeta_{36}^{11})q^{2}+\cdots\)
729.2.e.n 729.e 27.e $12$ $5.821$ \(\Q(\zeta_{36})\) None 243.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q-\zeta_{36}^{8}q^{2}+(-\zeta_{36}^{2}+\zeta_{36}^{7})q^{4}+\cdots\)
729.2.e.o 729.e 27.e $12$ $5.821$ \(\Q(\zeta_{36})\) None 81.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q-\zeta_{36}^{8}q^{2}+(-\zeta_{36}^{2}+\zeta_{36}^{7})q^{4}+\cdots\)
729.2.e.p 729.e 27.e $12$ $5.821$ 12.0.\(\cdots\).1 None 243.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q+\beta _{4}q^{2}+4\beta _{5}q^{4}+\beta _{11}q^{5}-2\beta _{7}q^{7}+\cdots\)
729.2.e.q 729.e 27.e $12$ $5.821$ \(\Q(\zeta_{36})\) None 729.2.a.c \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{9}]$ \(q+\zeta_{36}^{9}q^{2}+(1+\zeta_{36}-\zeta_{36}^{4})q^{4}+\cdots\)
729.2.e.r 729.e 27.e $12$ $5.821$ \(\Q(\zeta_{36})\) None 729.2.a.c \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{36}^{9}+\zeta_{36}^{10})q^{2}+(1+\zeta_{36}^{3}+\cdots)q^{4}+\cdots\)
729.2.e.s 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(3\) \(0\) \(-12\) \(-3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1-\beta _{5}+\beta _{6}+\beta _{10}-\beta _{11})q^{2}+(\beta _{1}+\cdots)q^{4}+\cdots\)
729.2.e.t 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(3\) \(0\) \(6\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\beta _{1}+\beta _{2}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(1+\cdots)q^{4}+\cdots\)
729.2.e.u 729.e 27.e $12$ $5.821$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 729.2.a.b \(6\) \(0\) \(-6\) \(-3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\beta _{1}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(-1+2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(729, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(729, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(243, [\chi])\)\(^{\oplus 2}\)