Properties

Label 810.4.e
Level $810$
Weight $4$
Character orbit 810.e
Rep. character $\chi_{810}(271,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $34$
Sturm bound $648$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 34 \)
Sturm bound: \(648\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(810, [\chi])\).

Total New Old
Modular forms 1020 96 924
Cusp forms 924 96 828
Eisenstein series 96 0 96

Trace form

\( 96 q - 192 q^{4} + 120 q^{7} + O(q^{10}) \) \( 96 q - 192 q^{4} + 120 q^{7} - 240 q^{13} - 768 q^{16} + 1560 q^{19} - 72 q^{22} - 1200 q^{25} - 960 q^{28} + 300 q^{31} + 360 q^{34} + 3360 q^{37} + 264 q^{43} + 1008 q^{46} - 5148 q^{49} - 960 q^{52} - 204 q^{61} + 6144 q^{64} + 1380 q^{67} + 360 q^{70} + 1848 q^{73} - 3120 q^{76} - 6072 q^{79} - 4176 q^{82} - 1440 q^{85} - 288 q^{88} + 8376 q^{91} + 1224 q^{94} - 5244 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(810, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
810.4.e.a 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.b 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.c 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.d 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(34\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.e 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(-32\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.f 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.g 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.h 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.i 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.j 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.k 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.l 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.m 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-32\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.n 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.o 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.p 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.q 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.r 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.s 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.t 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.u 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(5\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.v 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(5\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.w 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.x 810.e 9.c $2$ $47.792$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(5\) \(34\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.y 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{-163})\) None \(-4\) \(0\) \(-10\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)
810.4.e.z 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{401})\) None \(-4\) \(0\) \(-10\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-5+5\beta _{1}+\cdots)q^{5}+\cdots\)
810.4.e.ba 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{-1027})\) None \(-4\) \(0\) \(-10\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)
810.4.e.bb 810.e 9.c $4$ $47.792$ \(\Q(\zeta_{12})\) None \(-4\) \(0\) \(-10\) \(26\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}-5\zeta_{12}q^{5}+\cdots\)
810.4.e.bc 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{-163})\) None \(4\) \(0\) \(10\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bd 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{401})\) None \(4\) \(0\) \(10\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(5-5\beta _{1}+\cdots)q^{5}+\cdots\)
810.4.e.be 810.e 9.c $4$ $47.792$ \(\Q(\sqrt{-3}, \sqrt{-1027})\) None \(4\) \(0\) \(10\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bf 810.e 9.c $4$ $47.792$ \(\Q(\zeta_{12})\) None \(4\) \(0\) \(10\) \(26\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+5\zeta_{12}q^{5}+\cdots\)
810.4.e.bg 810.e 9.c $8$ $47.792$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-8\) \(0\) \(20\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bh 810.e 9.c $8$ $47.792$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(8\) \(0\) \(-20\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(810, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(810, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 2}\)