Properties

Label 2160.4.a
Level $2160$
Weight $4$
Character orbit 2160.a
Rep. character $\chi_{2160}(1,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $48$
Sturm bound $1728$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 48 \)
Sturm bound: \(1728\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2160))\).

Total New Old
Modular forms 1332 96 1236
Cusp forms 1260 96 1164
Eisenstein series 72 0 72

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(11\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(13\)
\(-\)\(+\)\(+\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(+\)\(13\)
\(-\)\(-\)\(+\)\(+\)\(13\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(50\)
Minus space\(-\)\(46\)

Trace form

\( 96 q + 36 q^{7} + O(q^{10}) \) \( 96 q + 36 q^{7} + 192 q^{19} + 2400 q^{25} + 348 q^{31} - 672 q^{43} + 4344 q^{49} - 456 q^{61} - 2508 q^{67} + 216 q^{73} - 1920 q^{79} + 120 q^{85} - 3588 q^{91} + 792 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
2160.4.a.a 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(-17\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-17q^{7}-30q^{11}-61q^{13}+\cdots\)
2160.4.a.b 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-14q^{7}+3q^{11}+47q^{13}+\cdots\)
2160.4.a.c 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-8q^{7}+18q^{11}+8q^{13}-15q^{17}+\cdots\)
2160.4.a.d 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+10q^{11}-80q^{13}-7q^{17}+\cdots\)
2160.4.a.e 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+4q^{7}-42q^{11}+20q^{13}+\cdots\)
2160.4.a.f 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+6q^{7}-47q^{11}-5q^{13}+131q^{17}+\cdots\)
2160.4.a.g 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+13q^{7}+30q^{11}-61q^{13}+\cdots\)
2160.4.a.h 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+22q^{7}+9q^{11}+17q^{13}+\cdots\)
2160.4.a.i 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(22\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+22q^{7}+12q^{11}+38q^{13}+\cdots\)
2160.4.a.j 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(-5\) \(34\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+34q^{7}-48q^{11}-70q^{13}+\cdots\)
2160.4.a.k 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(-17\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-17q^{7}+30q^{11}-61q^{13}+\cdots\)
2160.4.a.l 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-14q^{7}-3q^{11}+47q^{13}+\cdots\)
2160.4.a.m 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(-8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-8q^{7}-18q^{11}+8q^{13}+15q^{17}+\cdots\)
2160.4.a.n 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-10q^{11}-80q^{13}+7q^{17}+\cdots\)
2160.4.a.o 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+4q^{7}+42q^{11}+20q^{13}+\cdots\)
2160.4.a.p 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+6q^{7}+47q^{11}-5q^{13}-131q^{17}+\cdots\)
2160.4.a.q 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(13\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+13q^{7}-30q^{11}-61q^{13}+\cdots\)
2160.4.a.r 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+22q^{7}-12q^{11}+38q^{13}+\cdots\)
2160.4.a.s 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+22q^{7}-9q^{11}+17q^{13}+\cdots\)
2160.4.a.t 2160.a 1.a $1$ $127.444$ \(\Q\) None \(0\) \(0\) \(5\) \(34\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+34q^{7}+48q^{11}-70q^{13}+\cdots\)
2160.4.a.u 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(-10\) \(-13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-7-\beta )q^{7}+(10+5\beta )q^{11}+\cdots\)
2160.4.a.v 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{69}) \) None \(0\) \(0\) \(-10\) \(-10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-5-\beta )q^{7}+(9-\beta )q^{11}+\cdots\)
2160.4.a.w 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{401}) \) None \(0\) \(0\) \(-10\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-1-\beta )q^{7}+(2^{4}-\beta )q^{11}+\cdots\)
2160.4.a.x 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{21}) \) None \(0\) \(0\) \(-10\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(1+\beta )q^{7}+(-21+\beta )q^{11}+\cdots\)
2160.4.a.y 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(-10\) \(5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(3-\beta )q^{7}+(-2-\beta )q^{11}+\cdots\)
2160.4.a.z 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(10\) \(-13\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-7-\beta )q^{7}+(-10-5\beta )q^{11}+\cdots\)
2160.4.a.ba 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{69}) \) None \(0\) \(0\) \(10\) \(-10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-5-\beta )q^{7}+(-9+\beta )q^{11}+\cdots\)
2160.4.a.bb 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{401}) \) None \(0\) \(0\) \(10\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-1-\beta )q^{7}+(-2^{4}+\beta )q^{11}+\cdots\)
2160.4.a.bc 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{21}) \) None \(0\) \(0\) \(10\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(1+\beta )q^{7}+(21-\beta )q^{11}+(-7+\cdots)q^{13}+\cdots\)
2160.4.a.bd 2160.a 1.a $2$ $127.444$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(10\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(3-\beta )q^{7}+(2+\beta )q^{11}+(-9+\cdots)q^{13}+\cdots\)
2160.4.a.be 2160.a 1.a $3$ $127.444$ 3.3.5637.1 None \(0\) \(0\) \(-15\) \(-44\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-15-\beta _{1})q^{7}+(12-\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bf 2160.a 1.a $3$ $127.444$ 3.3.47977.1 None \(0\) \(0\) \(-15\) \(-9\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-3-\beta _{2})q^{7}+(6+\beta _{1}-\beta _{2})q^{11}+\cdots\)
2160.4.a.bg 2160.a 1.a $3$ $127.444$ 3.3.985.1 None \(0\) \(0\) \(-15\) \(-6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-2+\beta _{2})q^{7}+(4-\beta _{1}-\beta _{2})q^{11}+\cdots\)
2160.4.a.bh 2160.a 1.a $3$ $127.444$ 3.3.1765.1 None \(0\) \(0\) \(-15\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+\beta _{1}q^{7}+(-9-\beta _{1}+2\beta _{2})q^{11}+\cdots\)
2160.4.a.bi 2160.a 1.a $3$ $127.444$ 3.3.1772.1 None \(0\) \(0\) \(-15\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(3\beta _{1}+\beta _{2})q^{7}+(3-5\beta _{1}+\beta _{2})q^{11}+\cdots\)
2160.4.a.bj 2160.a 1.a $3$ $127.444$ 3.3.4281.1 None \(0\) \(0\) \(-15\) \(8\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(3-\beta _{1})q^{7}+(-4+\beta _{1}-\beta _{2})q^{11}+\cdots\)
2160.4.a.bk 2160.a 1.a $3$ $127.444$ 3.3.1257.1 None \(0\) \(0\) \(-15\) \(10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(4-\beta _{1}-\beta _{2})q^{7}+(11-5\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bl 2160.a 1.a $3$ $127.444$ 3.3.697.1 None \(0\) \(0\) \(-15\) \(24\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(8+\beta _{2})q^{7}+(2-\beta _{1}+3\beta _{2})q^{11}+\cdots\)
2160.4.a.bm 2160.a 1.a $3$ $127.444$ 3.3.5637.1 None \(0\) \(0\) \(15\) \(-44\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-15-\beta _{1})q^{7}+(-12+\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bn 2160.a 1.a $3$ $127.444$ 3.3.47977.1 None \(0\) \(0\) \(15\) \(-9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-3-\beta _{2})q^{7}+(-6-\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bo 2160.a 1.a $3$ $127.444$ 3.3.985.1 None \(0\) \(0\) \(15\) \(-6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-2+\beta _{2})q^{7}+(-4+\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bp 2160.a 1.a $3$ $127.444$ 3.3.1765.1 None \(0\) \(0\) \(15\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+\beta _{1}q^{7}+(9+\beta _{1}-2\beta _{2})q^{11}+\cdots\)
2160.4.a.bq 2160.a 1.a $3$ $127.444$ 3.3.1772.1 None \(0\) \(0\) \(15\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(3\beta _{1}+\beta _{2})q^{7}+(-3+5\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.br 2160.a 1.a $3$ $127.444$ 3.3.4281.1 None \(0\) \(0\) \(15\) \(8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(3-\beta _{1})q^{7}+(4-\beta _{1}+\beta _{2})q^{11}+\cdots\)
2160.4.a.bs 2160.a 1.a $3$ $127.444$ 3.3.1257.1 None \(0\) \(0\) \(15\) \(10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(4-\beta _{1}-\beta _{2})q^{7}+(-11+5\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bt 2160.a 1.a $3$ $127.444$ 3.3.697.1 None \(0\) \(0\) \(15\) \(24\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(8+\beta _{2})q^{7}+(-2+\beta _{1}-3\beta _{2})q^{11}+\cdots\)
2160.4.a.bu 2160.a 1.a $4$ $127.444$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(-20\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-3+\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
2160.4.a.bv 2160.a 1.a $4$ $127.444$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(20\) \(-14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-3+\beta _{2})q^{7}+(1-\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2160))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2160)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1080))\)\(^{\oplus 2}\)