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Properties

Label 459.2.a
Level $459$
Weight $2$
Character orbit 459.a
Rep. character $\chi_{459}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $14$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 459 = 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 459.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(459))\).

Total New Old
Modular forms 60 22 38
Cusp forms 49 22 27
Eisenstein series 11 0 11

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

\(3\)\(17\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(15\)

Trace form

\( 22 q + 24 q^{4} + 2 q^{7} + O(q^{10}) \) \( 22 q + 24 q^{4} + 2 q^{7} + 12 q^{10} + 4 q^{13} + 4 q^{16} + 4 q^{19} - 20 q^{22} + 40 q^{25} - 32 q^{28} - 12 q^{31} + 6 q^{37} + 56 q^{40} + 2 q^{43} - 12 q^{46} + 52 q^{49} + 32 q^{52} - 2 q^{55} - 32 q^{58} + 14 q^{61} + 24 q^{64} - 50 q^{67} - 68 q^{70} + 34 q^{73} - 52 q^{76} - 66 q^{79} - 32 q^{82} + 2 q^{85} - 88 q^{88} - 58 q^{91} - 108 q^{94} + 50 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(459))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 17
459.2.a.a 459.a 1.a $1$ $3.665$ \(\Q\) None \(-2\) \(0\) \(-4\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-4q^{5}+q^{7}+8q^{10}+\cdots\)
459.2.a.b 459.a 1.a $1$ $3.665$ \(\Q\) None \(-2\) \(0\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+2q^{5}+4q^{7}-4q^{10}+\cdots\)
459.2.a.c 459.a 1.a $1$ $3.665$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}-2q^{7}+3q^{8}-q^{10}+\cdots\)
459.2.a.d 459.a 1.a $1$ $3.665$ \(\Q\) None \(0\) \(0\) \(-3\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{5}+2q^{7}+3q^{11}+2q^{13}+\cdots\)
459.2.a.e 459.a 1.a $1$ $3.665$ \(\Q\) None \(0\) \(0\) \(3\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+3q^{5}+2q^{7}-3q^{11}+2q^{13}+\cdots\)
459.2.a.f 459.a 1.a $1$ $3.665$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-2q^{7}-3q^{8}-q^{10}+\cdots\)
459.2.a.g 459.a 1.a $1$ $3.665$ \(\Q\) None \(2\) \(0\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-2q^{5}+4q^{7}-4q^{10}+\cdots\)
459.2.a.h 459.a 1.a $1$ $3.665$ \(\Q\) None \(2\) \(0\) \(4\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+4q^{5}+q^{7}+8q^{10}+\cdots\)
459.2.a.i 459.a 1.a $2$ $3.665$ \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(3\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(1+\beta )q^{5}-3\beta q^{7}+\cdots\)
459.2.a.j 459.a 1.a $2$ $3.665$ \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(-5\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+(-3+\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
459.2.a.k 459.a 1.a $2$ $3.665$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(-3\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
459.2.a.l 459.a 1.a $2$ $3.665$ \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(5\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+(3-\beta )q^{5}+(2-\beta )q^{7}+\cdots\)
459.2.a.m 459.a 1.a $3$ $3.665$ 3.3.404.1 None \(-1\) \(0\) \(-3\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(3+\beta _{1}-\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
459.2.a.n 459.a 1.a $3$ $3.665$ 3.3.404.1 None \(1\) \(0\) \(3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(3+\beta _{1}-\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(459))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(459)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 2}\)