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Properties

Label 405.4.a
Level $405$
Weight $4$
Character orbit 405.a
Rep. character $\chi_{405}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $14$
Sturm bound $216$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(216\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(405))\).

Total New Old
Modular forms 174 48 126
Cusp forms 150 48 102
Eisenstein series 24 0 24

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
\(+\)\(+\)$+$\(13\)
\(+\)\(-\)$-$\(11\)
\(-\)\(+\)$-$\(11\)
\(-\)\(-\)$+$\(13\)
Plus space\(+\)\(26\)
Minus space\(-\)\(22\)

Trace form

\( 48 q + 192 q^{4} + 24 q^{7} + O(q^{10}) \) \( 48 q + 192 q^{4} + 24 q^{7} - 48 q^{13} + 408 q^{16} + 60 q^{19} + 144 q^{22} + 1200 q^{25} + 888 q^{28} + 240 q^{31} + 36 q^{34} - 336 q^{37} + 180 q^{40} - 1020 q^{43} + 2268 q^{46} + 2268 q^{49} + 1068 q^{52} + 3852 q^{58} - 1236 q^{61} + 48 q^{64} - 4836 q^{67} - 360 q^{70} + 2004 q^{73} - 876 q^{76} - 624 q^{79} - 3492 q^{82} + 720 q^{85} + 6264 q^{88} - 3624 q^{91} - 5796 q^{94} - 3468 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\) 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
405.4.a.a 405.a 1.a $1$ $23.896$ \(\Q\) None \(-5\) \(0\) \(5\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}+5q^{5}+9q^{7}-45q^{8}+\cdots\)
405.4.a.b 405.a 1.a $1$ $23.896$ \(\Q\) None \(5\) \(0\) \(-5\) \(9\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}-5q^{5}+9q^{7}+45q^{8}+\cdots\)
405.4.a.c 405.a 1.a $2$ $23.896$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-10\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(-4-2\beta )q^{4}-5q^{5}+\cdots\)
405.4.a.d 405.a 1.a $2$ $23.896$ \(\Q(\sqrt{33}) \) None \(-1\) \(0\) \(10\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+\beta q^{4}+5q^{5}+(5-\beta )q^{7}+(-8+\cdots)q^{8}+\cdots\)
405.4.a.e 405.a 1.a $2$ $23.896$ \(\Q(\sqrt{33}) \) None \(1\) \(0\) \(-10\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+\beta q^{4}-5q^{5}+(5-\beta )q^{7}+(8+\cdots)q^{8}+\cdots\)
405.4.a.f 405.a 1.a $2$ $23.896$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(10\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+5q^{5}+\cdots\)
405.4.a.g 405.a 1.a $3$ $23.896$ 3.3.7032.1 None \(-1\) \(0\) \(15\) \(-25\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
405.4.a.h 405.a 1.a $3$ $23.896$ 3.3.2292.1 None \(-1\) \(0\) \(15\) \(-43\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}+5q^{5}+\cdots\)
405.4.a.i 405.a 1.a $3$ $23.896$ 3.3.7032.1 None \(1\) \(0\) \(-15\) \(-25\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)
405.4.a.j 405.a 1.a $3$ $23.896$ 3.3.2292.1 None \(1\) \(0\) \(-15\) \(-43\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}-5q^{5}+\cdots\)
405.4.a.k 405.a 1.a $6$ $23.896$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-4\) \(0\) \(-30\) \(40\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+\cdots\)
405.4.a.l 405.a 1.a $6$ $23.896$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(4\) \(0\) \(30\) \(40\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+5q^{5}+\cdots\)
405.4.a.m 405.a 1.a $7$ $23.896$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-2\) \(0\) \(-35\) \(22\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}-5q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)
405.4.a.n 405.a 1.a $7$ $23.896$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(0\) \(35\) \(22\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+5q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(405))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(405)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)