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Properties

Label 4020.2.a
Level $4020$
Weight $2$
Character orbit 4020.a
Rep. character $\chi_{4020}(1,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $9$
Sturm bound $1632$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(1632\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4020))\).

Total New Old
Modular forms 828 44 784
Cusp forms 805 44 761
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.

\(2\)\(3\)\(5\)\(67\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(5\)
\(-\)\(+\)\(-\)\(-\)$-$\(6\)
\(-\)\(-\)\(+\)\(+\)$+$\(5\)
\(-\)\(-\)\(+\)\(-\)$-$\(6\)
\(-\)\(-\)\(-\)\(+\)$-$\(7\)
\(-\)\(-\)\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(18\)
Minus space\(-\)\(26\)

Trace form

\( 44 q + 44 q^{9} + O(q^{10}) \) \( 44 q + 44 q^{9} - 8 q^{11} + 4 q^{17} - 4 q^{19} - 12 q^{23} + 44 q^{25} + 4 q^{29} - 8 q^{31} - 8 q^{35} - 12 q^{37} - 16 q^{41} + 16 q^{43} - 4 q^{47} + 20 q^{49} + 8 q^{51} - 8 q^{53} - 4 q^{59} + 16 q^{61} - 8 q^{65} - 4 q^{67} - 16 q^{69} + 8 q^{71} + 4 q^{73} + 16 q^{77} + 24 q^{79} + 44 q^{81} + 16 q^{83} + 8 q^{87} - 28 q^{89} - 24 q^{91} - 8 q^{93} - 16 q^{97} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\) 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 67
4020.2.a.a 4020.a 1.a $1$ $32.100$ \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
4020.2.a.b 4020.a 1.a $4$ $32.100$ 4.4.9301.1 None \(0\) \(-4\) \(-4\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(\beta _{1}+\beta _{2})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
4020.2.a.c 4020.a 1.a $4$ $32.100$ 4.4.98117.1 None \(0\) \(4\) \(-4\) \(1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
4020.2.a.d 4020.a 1.a $4$ $32.100$ \(\Q(\zeta_{15})^+\) None \(0\) \(4\) \(4\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+(-\beta _{2}+\beta _{3})q^{7}+q^{9}+\cdots\)
4020.2.a.e 4020.a 1.a $5$ $32.100$ 5.5.1257629.1 None \(0\) \(-5\) \(5\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(-1+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
4020.2.a.f 4020.a 1.a $6$ $32.100$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-6\) \(6\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{5})q^{11}+\cdots\)
4020.2.a.g 4020.a 1.a $6$ $32.100$ 6.6.195727752.1 None \(0\) \(6\) \(-6\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(\beta _{1}+\beta _{4})q^{7}+q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
4020.2.a.h 4020.a 1.a $7$ $32.100$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-7\) \(-7\) \(3\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+(-1+\beta _{2}+\cdots)q^{11}+\cdots\)
4020.2.a.i 4020.a 1.a $7$ $32.100$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(7\) \(7\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-\beta _{4}q^{7}+q^{9}+(\beta _{3}+\beta _{6})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4020))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4020)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(804))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1340))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2010))\)\(^{\oplus 2}\)