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Properties

Label 513.2.a
Level $513$
Weight $2$
Character orbit 513.a
Rep. character $\chi_{513}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $9$
Sturm bound $120$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(120\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(513))\).

Total New Old
Modular forms 66 24 42
Cusp forms 55 24 31
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.

\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(14\)

Trace form

\( 24 q + 24 q^{4} + 4 q^{7} + O(q^{10}) \) \( 24 q + 24 q^{4} + 4 q^{7} - 4 q^{10} - 16 q^{13} - 8 q^{22} + 20 q^{25} + 8 q^{28} + 16 q^{31} + 20 q^{34} - 44 q^{37} + 24 q^{40} + 12 q^{43} + 4 q^{46} + 28 q^{49} - 8 q^{52} - 12 q^{55} - 52 q^{58} - 32 q^{61} - 12 q^{64} - 36 q^{67} + 24 q^{70} - 68 q^{79} + 56 q^{82} - 60 q^{85} - 84 q^{88} + 68 q^{91} - 12 q^{94} + 4 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(513))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 19
513.2.a.a 513.a 1.a $1$ $4.096$ \(\Q\) None 513.2.a.a \(-1\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-2q^{7}+3q^{8}+5q^{11}+\cdots\)
513.2.a.b 513.a 1.a $1$ $4.096$ \(\Q\) None 513.2.a.a \(1\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-2q^{7}-3q^{8}-5q^{11}+\cdots\)
513.2.a.c 513.a 1.a $2$ $4.096$ \(\Q(\sqrt{3}) \) None 513.2.a.c \(0\) \(0\) \(0\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-2\beta q^{5}-4q^{7}-\beta q^{8}+\cdots\)
513.2.a.d 513.a 1.a $3$ $4.096$ \(\Q(\zeta_{18})^+\) None 513.2.a.d \(-3\) \(0\) \(-3\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
513.2.a.e 513.a 1.a $3$ $4.096$ 3.3.321.1 None 513.2.a.e \(-1\) \(0\) \(-5\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
513.2.a.f 513.a 1.a $3$ $4.096$ 3.3.321.1 None 513.2.a.e \(1\) \(0\) \(5\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
513.2.a.g 513.a 1.a $3$ $4.096$ \(\Q(\zeta_{18})^+\) None 513.2.a.d \(3\) \(0\) \(3\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
513.2.a.h 513.a 1.a $4$ $4.096$ 4.4.27648.1 None 513.2.a.h \(0\) \(0\) \(0\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)
513.2.a.i 513.a 1.a $4$ $4.096$ 4.4.29952.1 None 513.2.a.i \(0\) \(0\) \(0\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{3}q^{5}+(3+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(513))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(513)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)