Properties

Label 3072.2.a
Level $3072$
Weight $2$
Character orbit 3072.a
Rep. character $\chi_{3072}(1,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $20$
Sturm bound $1024$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(1024\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3072))\).

Total New Old
Modular forms 560 64 496
Cusp forms 465 64 401
Eisenstein series 95 0 95

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.
\(+\)\(+\)\(+\)\(16\)
\(+\)\(-\)\(-\)\(20\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(12\)
Plus space\(+\)\(28\)
Minus space\(-\)\(36\)

Trace form

\( 64 q + 64 q^{9} + O(q^{10}) \) \( 64 q + 64 q^{9} + 64 q^{25} + 64 q^{49} + 64 q^{81} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3072))\) 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
3072.2.a.a 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{7}+q^{9}+\beta q^{13}-6q^{17}+\cdots\)
3072.2.a.b 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+2\beta q^{7}+q^{9}-3\beta q^{13}+\cdots\)
3072.2.a.c 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+q^{9}+4q^{11}-\beta q^{13}+\cdots\)
3072.2.a.d 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2\beta q^{5}-3\beta q^{7}+q^{9}+4q^{11}+\cdots\)
3072.2.a.e 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{9}-4q^{11}-\beta q^{13}+\cdots\)
3072.2.a.f 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2\beta q^{5}+3\beta q^{7}+q^{9}-4q^{11}+\cdots\)
3072.2.a.g 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{7}+q^{9}+\beta q^{13}-6q^{17}+\cdots\)
3072.2.a.h 3072.a 1.a $2$ $24.530$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-2\beta q^{7}+q^{9}-3\beta q^{13}+\cdots\)
3072.2.a.i 3072.a 1.a $4$ $24.530$ 4.4.4352.1 None \(0\) \(-4\) \(-4\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{3})q^{5}+(1+\beta _{2})q^{7}+\cdots\)
3072.2.a.j 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{16})^+\) None \(0\) \(-4\) \(0\) \(-8\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(\beta _{2}-\beta _{3})q^{5}+(-2-\beta _{3})q^{7}+\cdots\)
3072.2.a.k 3072.a 1.a $4$ $24.530$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(-4\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
3072.2.a.l 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{24})^+\) None \(0\) \(-4\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{7}+q^{9}+\cdots\)
3072.2.a.m 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{16})^+\) None \(0\) \(-4\) \(0\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(2-\beta _{3})q^{7}+\cdots\)
3072.2.a.n 3072.a 1.a $4$ $24.530$ 4.4.4352.1 None \(0\) \(-4\) \(4\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta _{3})q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
3072.2.a.o 3072.a 1.a $4$ $24.530$ 4.4.4352.1 None \(0\) \(4\) \(-4\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta _{3})q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
3072.2.a.p 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{16})^+\) None \(0\) \(4\) \(0\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(-2+\beta _{3})q^{7}+\cdots\)
3072.2.a.q 3072.a 1.a $4$ $24.530$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(4\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
3072.2.a.r 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(\beta _{1}+\beta _{2})q^{5}-\beta _{2}q^{7}+q^{9}+\cdots\)
3072.2.a.s 3072.a 1.a $4$ $24.530$ \(\Q(\zeta_{16})^+\) None \(0\) \(4\) \(0\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(\beta _{2}-\beta _{3})q^{5}+(2+\beta _{3})q^{7}+q^{9}+\cdots\)
3072.2.a.t 3072.a 1.a $4$ $24.530$ 4.4.4352.1 None \(0\) \(4\) \(4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta _{3})q^{5}+(1+\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3072))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3072)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1024))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\)\(^{\oplus 2}\)