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Properties

Label 480.4.a
Level $480$
Weight $4$
Character orbit 480.a
Rep. character $\chi_{480}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $18$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(480))\).

Total New Old
Modular forms 304 24 280
Cusp forms 272 24 248
Eisenstein series 32 0 32

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
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(14\)
Minus space\(-\)\(10\)

Trace form

\( 24 q + 216 q^{9} + O(q^{10}) \) \( 24 q + 216 q^{9} - 144 q^{13} + 240 q^{21} + 600 q^{25} + 1008 q^{37} + 592 q^{41} + 664 q^{49} + 1568 q^{53} - 336 q^{57} + 2960 q^{61} - 560 q^{65} + 432 q^{73} - 736 q^{77} + 1944 q^{81} + 1008 q^{89} + 3696 q^{93} + 3536 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(480))\) 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
480.4.a.a 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(-5\) \(-12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}-12q^{7}+9q^{9}+20q^{11}+\cdots\)
480.4.a.b 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(-5\) \(8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+8q^{7}+9q^{9}-4q^{11}+\cdots\)
480.4.a.c 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(5\) \(-32\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}-2^{6}q^{11}+\cdots\)
480.4.a.d 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(5\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}-4q^{7}+9q^{9}+40q^{11}+\cdots\)
480.4.a.e 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(5\) \(12\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+12q^{7}+9q^{9}+24q^{11}+\cdots\)
480.4.a.f 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(-3\) \(5\) \(16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+2^{4}q^{7}+9q^{9}+24q^{11}+\cdots\)
480.4.a.g 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(-5\) \(-8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}-8q^{7}+9q^{9}+4q^{11}+\cdots\)
480.4.a.h 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(-5\) \(12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+12q^{7}+9q^{9}-20q^{11}+\cdots\)
480.4.a.i 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(5\) \(-16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}-2^{4}q^{7}+9q^{9}-24q^{11}+\cdots\)
480.4.a.j 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(5\) \(-12\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}-12q^{7}+9q^{9}-24q^{11}+\cdots\)
480.4.a.k 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(5\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+4q^{7}+9q^{9}-40q^{11}+\cdots\)
480.4.a.l 480.a 1.a $1$ $28.321$ \(\Q\) None \(0\) \(3\) \(5\) \(32\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+2^{5}q^{7}+9q^{9}+2^{6}q^{11}+\cdots\)
480.4.a.m 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{41}) \) None \(0\) \(-6\) \(-10\) \(-12\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+(-6-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.n 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{201}) \) None \(0\) \(-6\) \(-10\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+(-2-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.o 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{89}) \) None \(0\) \(-6\) \(10\) \(-12\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(-6-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.p 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{201}) \) None \(0\) \(6\) \(-10\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(2+\beta )q^{7}+9q^{9}+20q^{11}+\cdots\)
480.4.a.q 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{41}) \) None \(0\) \(6\) \(-10\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(6+\beta )q^{7}+9q^{9}+(-12+\cdots)q^{11}+\cdots\)
480.4.a.r 480.a 1.a $2$ $28.321$ \(\Q(\sqrt{89}) \) None \(0\) \(6\) \(10\) \(12\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+(6+\beta )q^{7}+9q^{9}+(12+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(480))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(480)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)