Properties

Label 1728.2.a
Level $1728$
Weight $2$
Character orbit 1728.a
Rep. character $\chi_{1728}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $30$
Sturm bound $576$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(576\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1728))\).

Total New Old
Modular forms 324 32 292
Cusp forms 253 32 221
Eisenstein series 71 0 71

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(18\)

Trace form

\( 32 q + O(q^{10}) \) \( 32 q - 8 q^{13} + 32 q^{25} - 8 q^{37} + 32 q^{49} + 8 q^{61} + 16 q^{85} + 16 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\) 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
1728.2.a.a 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-4\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}-3q^{7}-4q^{11}-q^{13}-4q^{17}+\cdots\)
1728.2.a.b 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-4\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}+3q^{7}+4q^{11}-q^{13}-4q^{17}+\cdots\)
1728.2.a.c 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-3\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}-q^{7}+3q^{11}+4q^{13}-2q^{19}+\cdots\)
1728.2.a.d 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{5}+q^{7}-3q^{11}+4q^{13}+2q^{19}+\cdots\)
1728.2.a.e 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-2\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-3q^{7}+6q^{11}+3q^{13}-2q^{17}+\cdots\)
1728.2.a.f 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-q^{7}+2q^{11}-q^{13}+6q^{17}+\cdots\)
1728.2.a.g 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}-2q^{11}-q^{13}+6q^{17}+\cdots\)
1728.2.a.h 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+3q^{7}-6q^{11}+3q^{13}-2q^{17}+\cdots\)
1728.2.a.i 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}-5q^{11}-4q^{13}+8q^{17}+\cdots\)
1728.2.a.j 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots\)
1728.2.a.k 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-1\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots\)
1728.2.a.l 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(-1\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}+5q^{11}-4q^{13}+8q^{17}+\cdots\)
1728.2.a.m 1728.a 1.a $1$ $13.798$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-5q^{7}+7q^{13}-q^{19}-5q^{25}+4q^{31}+\cdots\)
1728.2.a.n 1728.a 1.a $1$ $13.798$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-1\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-q^{7}-5q^{13}+7q^{19}-5q^{25}-4q^{31}+\cdots\)
1728.2.a.o 1728.a 1.a $1$ $13.798$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+q^{7}-5q^{13}-7q^{19}-5q^{25}+4q^{31}+\cdots\)
1728.2.a.p 1728.a 1.a $1$ $13.798$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+5q^{7}+7q^{13}+q^{19}-5q^{25}-4q^{31}+\cdots\)
1728.2.a.q 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots\)
1728.2.a.r 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}+5q^{11}-4q^{13}-8q^{17}+\cdots\)
1728.2.a.s 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(1\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}-5q^{11}-4q^{13}-8q^{17}+\cdots\)
1728.2.a.t 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots\)
1728.2.a.u 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(2\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-3q^{7}-6q^{11}+3q^{13}+2q^{17}+\cdots\)
1728.2.a.v 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots\)
1728.2.a.w 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots\)
1728.2.a.x 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}+3q^{7}+6q^{11}+3q^{13}+2q^{17}+\cdots\)
1728.2.a.y 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(3\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}-q^{7}-3q^{11}+4q^{13}-2q^{19}+\cdots\)
1728.2.a.z 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+q^{7}+3q^{11}+4q^{13}+2q^{19}+\cdots\)
1728.2.a.ba 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(4\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{5}-3q^{7}+4q^{11}-q^{13}+4q^{17}+\cdots\)
1728.2.a.bb 1728.a 1.a $1$ $13.798$ \(\Q\) None \(0\) \(0\) \(4\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{5}+3q^{7}-4q^{11}-q^{13}+4q^{17}+\cdots\)
1728.2.a.bc 1728.a 1.a $2$ $13.798$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+\beta q^{7}-q^{11}-4q^{13}+2\beta q^{19}+\cdots\)
1728.2.a.bd 1728.a 1.a $2$ $13.798$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{5}-\beta q^{7}+q^{11}-4q^{13}-2\beta q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1728))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1728)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(864))\)\(^{\oplus 2}\)