Properties

Label 3200.2.a
Level $3200$
Weight $2$
Character orbit 3200.a
Rep. character $\chi_{3200}(1,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $48$
Sturm bound $960$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 48 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3200))\).

Total New Old
Modular forms 528 76 452
Cusp forms 433 76 357
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.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(22\)
\(-\)\(+\)\(-\)\(21\)
\(-\)\(-\)\(+\)\(18\)
Plus space\(+\)\(33\)
Minus space\(-\)\(43\)

Trace form

\( 76 q + 76 q^{9} + O(q^{10}) \) \( 76 q + 76 q^{9} + 8 q^{17} - 16 q^{33} - 8 q^{41} + 108 q^{49} + 16 q^{57} + 8 q^{73} + 92 q^{81} + 8 q^{89} + 72 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\) 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 5
3200.2.a.a 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-3\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-4q^{7}+6q^{9}-3q^{11}-2q^{13}+\cdots\)
3200.2.a.b 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-3\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-4q^{7}+6q^{9}+3q^{11}+2q^{13}+\cdots\)
3200.2.a.c 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-3\) \(0\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{7}+6q^{9}-3q^{11}+2q^{13}+\cdots\)
3200.2.a.d 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-3\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{7}+6q^{9}+3q^{11}-2q^{13}+\cdots\)
3200.2.a.e 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
3200.2.a.f 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
3200.2.a.g 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+2q^{11}+2q^{13}-6q^{17}+\cdots\)
3200.2.a.h 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-2\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+4q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3200.2.a.i 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-q^{11}-2q^{13}-3q^{17}+\cdots\)
3200.2.a.j 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-q^{11}+2q^{13}-3q^{17}+\cdots\)
3200.2.a.k 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}+q^{11}-2q^{13}+3q^{17}+\cdots\)
3200.2.a.l 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}+q^{11}+2q^{13}+3q^{17}+\cdots\)
3200.2.a.m 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{9}-6q^{11}+2q^{13}+6q^{17}+\cdots\)
3200.2.a.n 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{9}+6q^{11}-2q^{13}+6q^{17}+\cdots\)
3200.2.a.o 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{7}-3q^{9}-6q^{11}-2q^{13}+6q^{17}+\cdots\)
3200.2.a.p 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{7}-3q^{9}+6q^{11}+2q^{13}+6q^{17}+\cdots\)
3200.2.a.q 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-q^{11}-2q^{13}+3q^{17}+\cdots\)
3200.2.a.r 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots\)
3200.2.a.s 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}+q^{11}-2q^{13}-3q^{17}+\cdots\)
3200.2.a.t 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}+q^{11}+2q^{13}-3q^{17}+\cdots\)
3200.2.a.u 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(2\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3200.2.a.v 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-2q^{11}-2q^{13}-6q^{17}+\cdots\)
3200.2.a.w 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-2q^{11}+2q^{13}-6q^{17}+\cdots\)
3200.2.a.x 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(2\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
3200.2.a.y 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(3\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{7}+6q^{9}-3q^{11}-2q^{13}+\cdots\)
3200.2.a.z 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(3\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{7}+6q^{9}+3q^{11}+2q^{13}+\cdots\)
3200.2.a.ba 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(3\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{7}+6q^{9}-3q^{11}+2q^{13}+\cdots\)
3200.2.a.bb 3200.a 1.a $1$ $25.552$ \(\Q\) None \(0\) \(3\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{7}+6q^{9}+3q^{11}-2q^{13}+\cdots\)
3200.2.a.bc 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(-2-2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bd 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(-2-2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.be 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bf 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bg 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(2+2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bh 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(2+2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bi 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-2+2\beta )q^{7}+2\beta q^{9}+\cdots\)
3200.2.a.bj 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-2+2\beta )q^{7}+2\beta q^{9}+\cdots\)
3200.2.a.bk 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bl 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bm 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2-2\beta )q^{7}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
3200.2.a.bn 3200.a 1.a $2$ $25.552$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2-2\beta )q^{7}+2\beta q^{9}+(1+\cdots)q^{11}+\cdots\)
3200.2.a.bo 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(-2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
3200.2.a.bp 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(-2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
3200.2.a.bq 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(-2\) \(0\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.br 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(-2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bs 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(2\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bt 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bu 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bv 3200.a 1.a $3$ $25.552$ 3.3.148.1 None \(0\) \(2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\)\(^{\oplus 2}\)