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Properties

Label 1860.2.a
Level $1860$
Weight $2$
Character orbit 1860.a
Rep. character $\chi_{1860}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $9$
Sturm bound $768$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(768\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1860))\).

Total New Old
Modular forms 396 20 376
Cusp forms 373 20 353
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.

\(2\)\(3\)\(5\)\(31\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(+\)\(+\)\(20\)\(0\)\(20\)\(19\)\(0\)\(19\)\(1\)\(0\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(28\)\(0\)\(28\)\(26\)\(0\)\(26\)\(2\)\(0\)\(2\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(30\)\(0\)\(30\)\(28\)\(0\)\(28\)\(2\)\(0\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(22\)\(0\)\(22\)\(20\)\(0\)\(20\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(24\)\(0\)\(24\)\(22\)\(0\)\(22\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(28\)\(0\)\(28\)\(26\)\(0\)\(26\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(26\)\(0\)\(26\)\(24\)\(0\)\(24\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(22\)\(0\)\(22\)\(20\)\(0\)\(20\)\(2\)\(0\)\(2\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(25\)\(3\)\(22\)\(24\)\(3\)\(21\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(23\)\(2\)\(21\)\(22\)\(2\)\(20\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(24\)\(1\)\(23\)\(23\)\(1\)\(22\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(26\)\(4\)\(22\)\(25\)\(4\)\(21\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(24\)\(2\)\(22\)\(23\)\(2\)\(21\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(26\)\(3\)\(23\)\(25\)\(3\)\(22\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(25\)\(4\)\(21\)\(24\)\(4\)\(20\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(23\)\(1\)\(22\)\(22\)\(1\)\(21\)\(1\)\(0\)\(1\)
Plus space\(+\)\(190\)\(6\)\(184\)\(179\)\(6\)\(173\)\(11\)\(0\)\(11\)
Minus space\(-\)\(206\)\(14\)\(192\)\(194\)\(14\)\(180\)\(12\)\(0\)\(12\)

Trace form

\( 20 q + 20 q^{9} - 8 q^{11} + 8 q^{17} + 8 q^{23} + 20 q^{25} + 8 q^{29} - 8 q^{33} - 8 q^{35} - 8 q^{37} + 8 q^{43} + 32 q^{47} + 20 q^{49} + 8 q^{51} + 24 q^{53} + 8 q^{55} + 24 q^{59} + 24 q^{61} + 16 q^{67}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1860))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 31
1860.2.a.a 1860.a 1.a $1$ $14.852$ \(\Q\) None 1860.2.a.a \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}-2q^{13}-q^{15}-4q^{17}+\cdots\)
1860.2.a.b 1860.a 1.a $1$ $14.852$ \(\Q\) None 1860.2.a.b \(0\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
1860.2.a.c 1860.a 1.a $1$ $14.852$ \(\Q\) None 1860.2.a.c \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
1860.2.a.d 1860.a 1.a $2$ $14.852$ \(\Q(\sqrt{6}) \) None 1860.2.a.d \(0\) \(-2\) \(-2\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+\beta q^{7}+q^{9}+(-2-\beta )q^{13}+\cdots\)
1860.2.a.e 1860.a 1.a $2$ $14.852$ \(\Q(\sqrt{3}) \) None 1860.2.a.e \(0\) \(2\) \(-2\) \(2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}-4q^{11}+\cdots\)
1860.2.a.f 1860.a 1.a $3$ $14.852$ 3.3.564.1 None 1860.2.a.f \(0\) \(-3\) \(-3\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
1860.2.a.g 1860.a 1.a $3$ $14.852$ 3.3.404.1 None 1860.2.a.g \(0\) \(3\) \(-3\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(-1+\beta _{2})q^{7}+q^{9}+2\beta _{2}q^{11}+\cdots\)
1860.2.a.h 1860.a 1.a $3$ $14.852$ 3.3.7636.1 None 1860.2.a.h \(0\) \(3\) \(3\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-\beta _{1}q^{7}+q^{9}+2q^{11}+\cdots\)
1860.2.a.i 1860.a 1.a $4$ $14.852$ 4.4.224148.1 None 1860.2.a.i \(0\) \(-4\) \(4\) \(-4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(-1+\beta _{1})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1860))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1860)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(372))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(930))\)\(^{\oplus 2}\)