Properties

Label 1620.4.i
Level $1620$
Weight $4$
Character orbit 1620.i
Rep. character $\chi_{1620}(541,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $24$
Sturm bound $1296$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 24 \)
Sturm bound: \(1296\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1620, [\chi])\).

Total New Old
Modular forms 2016 96 1920
Cusp forms 1872 96 1776
Eisenstein series 144 0 144

Trace form

\( 96 q - 60 q^{7} + O(q^{10}) \) \( 96 q - 60 q^{7} + 120 q^{13} - 240 q^{19} - 1200 q^{25} + 300 q^{31} - 1680 q^{37} + 264 q^{43} - 1548 q^{49} + 156 q^{61} + 1632 q^{67} + 1560 q^{73} + 3036 q^{79} + 720 q^{85} + 5712 q^{91} + 2676 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1620, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1620.4.i.a 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(-32\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(-2^{5}+2^{5}\zeta_{6})q^{7}+(-6^{2}+\cdots)q^{11}+\cdots\)
1620.4.i.b 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(-17+17\zeta_{6})q^{7}+(30+\cdots)q^{11}+\cdots\)
1620.4.i.c 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-30+\cdots)q^{11}+\cdots\)
1620.4.i.d 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(2^{4}-2^{4}\zeta_{6})q^{7}+(60-60\zeta_{6})q^{11}+\cdots\)
1620.4.i.e 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(22-22\zeta_{6})q^{7}+(-9+9\zeta_{6})q^{11}+\cdots\)
1620.4.i.f 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{5}+(28-28\zeta_{6})q^{7}+(-24+\cdots)q^{11}+\cdots\)
1620.4.i.g 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(-32\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(-2^{5}+2^{5}\zeta_{6})q^{7}+(6^{2}+\cdots)q^{11}+\cdots\)
1620.4.i.h 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(-17+17\zeta_{6})q^{7}+(-30+\cdots)q^{11}+\cdots\)
1620.4.i.i 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(30-30\zeta_{6})q^{11}+\cdots\)
1620.4.i.j 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(2^{4}-2^{4}\zeta_{6})q^{7}+(-60+\cdots)q^{11}+\cdots\)
1620.4.i.k 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(22-22\zeta_{6})q^{7}+(9-9\zeta_{6})q^{11}+\cdots\)
1620.4.i.l 1620.i 9.c $2$ $95.583$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{5}+(28-28\zeta_{6})q^{7}+(24-24\zeta_{6})q^{11}+\cdots\)
1620.4.i.m 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{41})\) None \(0\) \(0\) \(-10\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-5+5\beta _{1})q^{5}+(-7\beta _{1}+\beta _{2})q^{7}+\cdots\)
1620.4.i.n 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{-23})\) None \(0\) \(0\) \(-10\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\beta _{1}q^{5}+(-5-5\beta _{1}-\beta _{3})q^{7}+(-9+\cdots)q^{11}+\cdots\)
1620.4.i.o 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(0\) \(0\) \(-10\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+(21+21\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.p 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{41})\) None \(0\) \(0\) \(10\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5-5\beta _{1})q^{5}+(-7\beta _{1}+\beta _{2})q^{7}+(10\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.q 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{-23})\) None \(0\) \(0\) \(10\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\beta _{1}q^{5}+(-5-5\beta _{1}-\beta _{3})q^{7}+(9+\cdots)q^{11}+\cdots\)
1620.4.i.r 1620.i 9.c $4$ $95.583$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(0\) \(0\) \(10\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+(-21+\cdots)q^{11}+\cdots\)
1620.4.i.s 1620.i 9.c $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-15\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-5-5\beta _{2})q^{5}+(5\beta _{2}-\beta _{4})q^{7}+(8\beta _{2}+\cdots)q^{11}+\cdots\)
1620.4.i.t 1620.i 9.c $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-15\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-5-5\beta _{1})q^{5}+(-\beta _{1}-\beta _{4})q^{7}+\cdots\)
1620.4.i.u 1620.i 9.c $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(15\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5+5\beta _{2})q^{5}+(5\beta _{2}-\beta _{4})q^{7}+(-8\beta _{2}+\cdots)q^{11}+\cdots\)
1620.4.i.v 1620.i 9.c $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(15\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5+5\beta _{1})q^{5}+(-\beta _{1}-\beta _{4})q^{7}+(8\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.w 1620.i 9.c $12$ $95.583$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-30\) \(-12\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\beta _{6}q^{5}+(-2-\beta _{2}-2\beta _{6}+\beta _{8}+\cdots)q^{7}+\cdots\)
1620.4.i.x 1620.i 9.c $12$ $95.583$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(30\) \(-12\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\beta _{6}q^{5}+(-2-\beta _{2}-2\beta _{6}+\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1620, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1620, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(540, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(810, [\chi])\)\(^{\oplus 2}\)