Properties

Label 1440.4.a
Level $1440$
Weight $4$
Character orbit 1440.a
Rep. character $\chi_{1440}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $37$
Sturm bound $1152$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 37 \)
Sturm bound: \(1152\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1440))\).

Total New Old
Modular forms 896 60 836
Cusp forms 832 60 772
Eisenstein series 64 0 64

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.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(10\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(33\)
Minus space\(-\)\(27\)

Trace form

\( 60 q + O(q^{10}) \) \( 60 q - 144 q^{13} - 152 q^{17} + 1500 q^{25} - 56 q^{29} + 1008 q^{37} - 416 q^{41} + 2116 q^{49} - 784 q^{53} + 1760 q^{61} + 280 q^{65} - 1224 q^{73} + 1680 q^{77} - 3656 q^{89} + 680 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\) 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
1440.4.a.a 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(-32\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{5}q^{7}+2^{6}q^{11}-6q^{13}+\cdots\)
1440.4.a.b 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(-30\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-30q^{7}-50q^{11}-88q^{13}+\cdots\)
1440.4.a.c 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(-16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{4}q^{7}+24q^{11}-14q^{13}+\cdots\)
1440.4.a.d 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-12q^{7}+24q^{11}+38q^{13}+\cdots\)
1440.4.a.e 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-4q^{7}-40q^{11}-90q^{13}+\cdots\)
1440.4.a.f 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+4q^{7}+40q^{11}-90q^{13}+\cdots\)
1440.4.a.g 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+12q^{7}-24q^{11}+38q^{13}+\cdots\)
1440.4.a.h 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2^{4}q^{7}-24q^{11}-14q^{13}+\cdots\)
1440.4.a.i 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(30\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+30q^{7}+50q^{11}-88q^{13}+\cdots\)
1440.4.a.j 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(-5\) \(32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2^{5}q^{7}-2^{6}q^{11}-6q^{13}+\cdots\)
1440.4.a.k 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(-30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-30q^{7}+50q^{11}-88q^{13}+\cdots\)
1440.4.a.l 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(-12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-12q^{7}-20q^{11}-58q^{13}+\cdots\)
1440.4.a.m 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(-8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-8q^{7}-4q^{11}-6q^{13}+2q^{17}+\cdots\)
1440.4.a.n 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-6q^{7}+60q^{11}+50q^{13}+\cdots\)
1440.4.a.o 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+6q^{7}-60q^{11}+50q^{13}+\cdots\)
1440.4.a.p 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+8q^{7}+4q^{11}-6q^{13}+2q^{17}+\cdots\)
1440.4.a.q 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+12q^{7}+20q^{11}-58q^{13}+\cdots\)
1440.4.a.r 1440.a 1.a $1$ $84.963$ \(\Q\) None \(0\) \(0\) \(5\) \(30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+30q^{7}-50q^{11}-88q^{13}+\cdots\)
1440.4.a.s 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{89}) \) None \(0\) \(0\) \(-10\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-6-\beta )q^{7}+(12-2\beta )q^{11}+\cdots\)
1440.4.a.t 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-10\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-4+5\beta )q^{7}+(2^{5}+6\beta )q^{11}+\cdots\)
1440.4.a.u 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{65}) \) None \(0\) \(0\) \(-10\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-\beta q^{7}+\beta q^{11}-8q^{13}+26q^{17}+\cdots\)
1440.4.a.v 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(-10\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+3\beta q^{7}+2\beta q^{11}+38q^{13}+\cdots\)
1440.4.a.w 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{85}) \) None \(0\) \(0\) \(-10\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-\beta q^{7}+3\beta q^{11}+52q^{13}+\cdots\)
1440.4.a.x 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-10\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(4+5\beta )q^{7}+(-2^{5}+6\beta )q^{11}+\cdots\)
1440.4.a.y 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{89}) \) None \(0\) \(0\) \(-10\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(6+\beta )q^{7}+(-12+2\beta )q^{11}+\cdots\)
1440.4.a.z 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(10\) \(-12\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-6-\beta )q^{7}+(-12-4\beta )q^{11}+\cdots\)
1440.4.a.ba 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(10\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-2-\beta )q^{7}+20q^{11}+(-12+\cdots)q^{13}+\cdots\)
1440.4.a.bb 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(10\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-7\beta q^{7}+2\beta q^{11}-62q^{13}+\cdots\)
1440.4.a.bc 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{65}) \) None \(0\) \(0\) \(10\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-\beta q^{7}-\beta q^{11}-8q^{13}-26q^{17}+\cdots\)
1440.4.a.bd 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(10\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-\beta q^{7}+6\beta q^{11}+34q^{13}+\cdots\)
1440.4.a.be 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{85}) \) None \(0\) \(0\) \(10\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-\beta q^{7}-3\beta q^{11}+52q^{13}+\cdots\)
1440.4.a.bf 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(10\) \(4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(2+\beta )q^{7}-20q^{11}+(-12+\cdots)q^{13}+\cdots\)
1440.4.a.bg 1440.a 1.a $2$ $84.963$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(10\) \(12\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(6+\beta )q^{7}+(12+4\beta )q^{11}+\cdots\)
1440.4.a.bh 1440.a 1.a $3$ $84.963$ 3.3.16773.1 None \(0\) \(0\) \(-15\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-5+\beta _{1})q^{7}+(7+2\beta _{1}+\beta _{2})q^{11}+\cdots\)
1440.4.a.bi 1440.a 1.a $3$ $84.963$ 3.3.16773.1 None \(0\) \(0\) \(-15\) \(14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(5-\beta _{1})q^{7}+(-7-2\beta _{1}-\beta _{2})q^{11}+\cdots\)
1440.4.a.bj 1440.a 1.a $3$ $84.963$ 3.3.16773.1 None \(0\) \(0\) \(15\) \(-14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-5+\beta _{1})q^{7}+(-7-2\beta _{1}+\cdots)q^{11}+\cdots\)
1440.4.a.bk 1440.a 1.a $3$ $84.963$ 3.3.16773.1 None \(0\) \(0\) \(15\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(5-\beta _{1})q^{7}+(7+2\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1440))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1440)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 2}\)