Properties

Label 405.2.a
Level $405$
Weight $2$
Character orbit 405.a
Rep. character $\chi_{405}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $10$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(405))\).

Total New Old
Modular forms 66 16 50
Cusp forms 43 16 27
Eisenstein series 23 0 23

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

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(10\)

Trace form

\( 16 q + 16 q^{4} - 4 q^{7} + O(q^{10}) \) \( 16 q + 16 q^{4} - 4 q^{7} - 4 q^{13} + 40 q^{16} - 16 q^{19} + 12 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{31} - 24 q^{34} - 4 q^{37} - 12 q^{40} - 28 q^{43} - 52 q^{52} - 12 q^{58} + 8 q^{61} + 16 q^{64} + 56 q^{67} - 12 q^{70} - 16 q^{73} - 64 q^{76} + 32 q^{79} - 48 q^{82} - 12 q^{85} + 36 q^{88} + 16 q^{91} - 24 q^{94} - 16 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(405))\) 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 5
405.2.a.a 405.a 1.a $1$ $3.234$ \(\Q\) None \(-2\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}+2q^{10}-5q^{11}+\cdots\)
405.2.a.b 405.a 1.a $1$ $3.234$ \(\Q\) None \(-1\) \(0\) \(1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}-3q^{7}+3q^{8}-q^{10}+\cdots\)
405.2.a.c 405.a 1.a $1$ $3.234$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{5}+2q^{7}-3q^{11}-4q^{13}+\cdots\)
405.2.a.d 405.a 1.a $1$ $3.234$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}+2q^{7}+3q^{11}-4q^{13}+\cdots\)
405.2.a.e 405.a 1.a $1$ $3.234$ \(\Q\) None \(1\) \(0\) \(-1\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-3q^{7}-3q^{8}-q^{10}+\cdots\)
405.2.a.f 405.a 1.a $1$ $3.234$ \(\Q\) None \(2\) \(0\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+q^{5}+2q^{10}+5q^{11}+\cdots\)
405.2.a.g 405.a 1.a $2$ $3.234$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(2\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+q^{5}+(-3+\cdots)q^{7}+\cdots\)
405.2.a.h 405.a 1.a $2$ $3.234$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(2+2\beta )q^{4}-q^{5}+(-3+\cdots)q^{7}+\cdots\)
405.2.a.i 405.a 1.a $3$ $3.234$ 3.3.564.1 None \(-1\) \(0\) \(-3\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
405.2.a.j 405.a 1.a $3$ $3.234$ 3.3.564.1 None \(1\) \(0\) \(3\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(405))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(405)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)