Properties

Label 3840.2.a
Level $3840$
Weight $2$
Character orbit 3840.a
Rep. character $\chi_{3840}(1,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $44$
Sturm bound $1536$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 44 \)
Sturm bound: \(1536\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3840))\).

Total New Old
Modular forms 816 64 752
Cusp forms 721 64 657
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.

\(2\)\(3\)\(5\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(8\)
\(+\)\(+\)\(-\)\(-\)\(10\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(10\)
Plus space\(+\)\(28\)
Minus space\(-\)\(36\)

Trace form

\( 64 q + 64 q^{9} + O(q^{10}) \) \( 64 q + 64 q^{9} + 64 q^{25} - 64 q^{57} - 64 q^{73} + 64 q^{81} - 64 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3840))\) 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
3840.2.a.a 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.a \(0\) \(-1\) \(-1\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3840.2.a.b 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.a \(0\) \(-1\) \(-1\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
3840.2.a.c 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.b \(0\) \(-1\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
3840.2.a.d 3840.a 1.a $1$ $30.663$ \(\Q\) None 120.2.k.a \(0\) \(-1\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}+4q^{11}+q^{15}+\cdots\)
3840.2.a.e 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.b \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{9}+6q^{13}+q^{15}+2q^{19}+\cdots\)
3840.2.a.f 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.c \(0\) \(-1\) \(-1\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3840.2.a.g 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.d \(0\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}+6q^{11}+6q^{13}+\cdots\)
3840.2.a.h 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.c \(0\) \(-1\) \(1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
3840.2.a.i 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.d \(0\) \(-1\) \(1\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}+6q^{11}-6q^{13}+\cdots\)
3840.2.a.j 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.b \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}-6q^{13}-q^{15}+2q^{19}+\cdots\)
3840.2.a.k 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.a \(0\) \(-1\) \(1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3840.2.a.l 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.b \(0\) \(-1\) \(1\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3840.2.a.m 3840.a 1.a $1$ $30.663$ \(\Q\) None 120.2.k.a \(0\) \(-1\) \(1\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}+q^{9}+4q^{11}-q^{15}+\cdots\)
3840.2.a.n 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.a \(0\) \(-1\) \(1\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3840.2.a.o 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.d \(0\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-6q^{11}+6q^{13}+\cdots\)
3840.2.a.p 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.c \(0\) \(1\) \(-1\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3840.2.a.q 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.b \(0\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{9}+6q^{13}-q^{15}-2q^{19}+\cdots\)
3840.2.a.r 3840.a 1.a $1$ $30.663$ \(\Q\) None 120.2.k.a \(0\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+2q^{7}+q^{9}-4q^{11}-q^{15}+\cdots\)
3840.2.a.s 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.a \(0\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
3840.2.a.t 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.b \(0\) \(1\) \(-1\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
3840.2.a.u 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.a \(0\) \(1\) \(-1\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3840.2.a.v 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.a \(0\) \(1\) \(1\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3840.2.a.w 3840.a 1.a $1$ $30.663$ \(\Q\) None 120.2.k.a \(0\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}+q^{15}+\cdots\)
3840.2.a.x 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.a \(0\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3840.2.a.y 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.b \(0\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3840.2.a.z 3840.a 1.a $1$ $30.663$ \(\Q\) None 960.2.k.b \(0\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{9}-6q^{13}+q^{15}-2q^{19}+\cdots\)
3840.2.a.ba 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.d \(0\) \(1\) \(1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}-6q^{11}-6q^{13}+\cdots\)
3840.2.a.bb 3840.a 1.a $1$ $30.663$ \(\Q\) None 1920.2.k.c \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
3840.2.a.bc 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.i \(0\) \(-2\) \(-2\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(-2+\beta )q^{7}+q^{9}+(2+\cdots)q^{11}+\cdots\)
3840.2.a.bd 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.j \(0\) \(-2\) \(-2\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
3840.2.a.be 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{3}) \) None 960.2.k.e \(0\) \(-2\) \(-2\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
3840.2.a.bf 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{3}) \) None 960.2.k.e \(0\) \(-2\) \(2\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
3840.2.a.bg 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.j \(0\) \(-2\) \(2\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
3840.2.a.bh 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.i \(0\) \(-2\) \(2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(2+\beta )q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
3840.2.a.bi 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{3}) \) None 960.2.k.e \(0\) \(2\) \(-2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
3840.2.a.bj 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.j \(0\) \(2\) \(-2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
3840.2.a.bk 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.i \(0\) \(2\) \(-2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(2+\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
3840.2.a.bl 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.i \(0\) \(2\) \(2\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+(-2+\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
3840.2.a.bm 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{2}) \) None 1920.2.k.j \(0\) \(2\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
3840.2.a.bn 3840.a 1.a $2$ $30.663$ \(\Q(\sqrt{3}) \) None 960.2.k.e \(0\) \(2\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
3840.2.a.bo 3840.a 1.a $3$ $30.663$ 3.3.316.1 None 120.2.k.b \(0\) \(-3\) \(-3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(1+\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3840.2.a.bp 3840.a 1.a $3$ $30.663$ 3.3.316.1 None 120.2.k.b \(0\) \(-3\) \(3\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(-1-\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3840.2.a.bq 3840.a 1.a $3$ $30.663$ 3.3.316.1 None 120.2.k.b \(0\) \(3\) \(-3\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(-1-\beta _{1})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
3840.2.a.br 3840.a 1.a $3$ $30.663$ 3.3.316.1 None 120.2.k.b \(0\) \(3\) \(3\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+(1+\beta _{1})q^{7}+q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3840)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(960))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1920))\)\(^{\oplus 2}\)