Properties

Label 1620.4.a
Level $1620$
Weight $4$
Character orbit 1620.a
Rep. character $\chi_{1620}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $12$
Sturm bound $1296$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(1296\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1620))\).

Total New Old
Modular forms 1008 48 960
Cusp forms 936 48 888
Eisenstein series 72 0 72

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

\(2\)\(3\)\(5\)FrickeDim
\(-\)\(+\)\(+\)$-$\(11\)
\(-\)\(+\)\(-\)$+$\(13\)
\(-\)\(-\)\(+\)$+$\(13\)
\(-\)\(-\)\(-\)$-$\(11\)
Plus space\(+\)\(26\)
Minus space\(-\)\(22\)

Trace form

\( 48 q + 24 q^{7} + O(q^{10}) \) \( 48 q + 24 q^{7} - 48 q^{13} - 300 q^{19} + 1200 q^{25} - 120 q^{31} - 336 q^{37} + 852 q^{43} + 1476 q^{49} - 1164 q^{61} + 708 q^{67} - 1308 q^{73} + 96 q^{79} + 720 q^{85} + 4512 q^{91} - 3324 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1620))\) 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
1620.4.a.a 1620.a 1.a $1$ $95.583$ \(\Q\) None \(0\) \(0\) \(-5\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-7q^{7}-30q^{11}-22q^{13}+\cdots\)
1620.4.a.b 1620.a 1.a $1$ $95.583$ \(\Q\) None \(0\) \(0\) \(5\) \(-7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-7q^{7}+30q^{11}-22q^{13}+\cdots\)
1620.4.a.c 1620.a 1.a $3$ $95.583$ 3.3.244785.1 None \(0\) \(0\) \(-15\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-1-\beta _{1})q^{7}+(8+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1620.4.a.d 1620.a 1.a $3$ $95.583$ 3.3.560145.1 None \(0\) \(0\) \(-15\) \(15\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(5-\beta _{1})q^{7}+(-8-\beta _{2})q^{11}+\cdots\)
1620.4.a.e 1620.a 1.a $3$ $95.583$ 3.3.244785.1 None \(0\) \(0\) \(15\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-1-\beta _{1})q^{7}+(-8-\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.a.f 1620.a 1.a $3$ $95.583$ 3.3.560145.1 None \(0\) \(0\) \(15\) \(15\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(5-\beta _{1})q^{7}+(8+\beta _{2})q^{11}+\cdots\)
1620.4.a.g 1620.a 1.a $4$ $95.583$ 4.4.438516.1 None \(0\) \(0\) \(-20\) \(-13\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(-3+2\beta _{1}-\beta _{3})q^{7}+(15+\cdots)q^{11}+\cdots\)
1620.4.a.h 1620.a 1.a $4$ $95.583$ 4.4.438516.1 None \(0\) \(0\) \(20\) \(-13\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(-3+2\beta _{1}-\beta _{3})q^{7}+(-15+\cdots)q^{11}+\cdots\)
1620.4.a.i 1620.a 1.a $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-30\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(2+\beta _{1})q^{7}+(\beta _{2}+\beta _{4}-\beta _{5})q^{11}+\cdots\)
1620.4.a.j 1620.a 1.a $6$ $95.583$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(30\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(2+\beta _{1})q^{7}+(-\beta _{2}-\beta _{4}+\beta _{5})q^{11}+\cdots\)
1620.4.a.k 1620.a 1.a $7$ $95.583$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(-35\) \(8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+(1-\beta _{2})q^{7}+(-4-\beta _{5})q^{11}+\cdots\)
1620.4.a.l 1620.a 1.a $7$ $95.583$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(35\) \(8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+(1-\beta _{2})q^{7}+(4+\beta _{5})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1620))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1620)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 2}\)