Properties

Label 4160.2.a
Level $4160$
Weight $2$
Character orbit 4160.a
Rep. character $\chi_{4160}(1,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $50$
Sturm bound $1344$
Trace bound $29$

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Defining parameters

Level: \( N \) \(=\) \( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4160.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 50 \)
Sturm bound: \(1344\)
Trace bound: \(29\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4160))\).

Total New Old
Modular forms 696 96 600
Cusp forms 649 96 553
Eisenstein series 47 0 47

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

\(2\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(14\)
\(+\)\(-\)\(+\)\(-\)\(14\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(16\)
Plus space\(+\)\(40\)
Minus space\(-\)\(56\)

Trace form

\( 96 q + 96 q^{9} + O(q^{10}) \) \( 96 q + 96 q^{9} + 96 q^{25} + 32 q^{29} + 32 q^{33} + 32 q^{37} + 32 q^{41} + 96 q^{49} + 32 q^{53} + 32 q^{57} + 32 q^{61} + 32 q^{69} + 32 q^{77} + 160 q^{81} + 32 q^{85} + 32 q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4160))\) 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 5 13
4160.2.a.a 4160.a 1.a $1$ $33.218$ \(\Q\) None 520.2.a.b \(0\) \(-2\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+q^{9}-2q^{11}-q^{13}+\cdots\)
4160.2.a.b 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.a \(0\) \(-2\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+4q^{7}+q^{9}-6q^{11}+\cdots\)
4160.2.a.c 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.c \(0\) \(-2\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}-4q^{7}+q^{9}+2q^{11}+\cdots\)
4160.2.a.d 4160.a 1.a $1$ $33.218$ \(\Q\) None 260.2.a.a \(0\) \(-2\) \(1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)
4160.2.a.e 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.a \(0\) \(-2\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+4q^{7}+q^{9}-6q^{11}+\cdots\)
4160.2.a.f 4160.a 1.a $1$ $33.218$ \(\Q\) None 65.2.a.a \(0\) \(-2\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
4160.2.a.g 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.d \(0\) \(0\) \(-1\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}-3q^{9}+4q^{11}-q^{13}+\cdots\)
4160.2.a.h 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.b \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}-q^{13}+2q^{17}-8q^{19}+\cdots\)
4160.2.a.i 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.b \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}-q^{13}+2q^{17}+8q^{19}+\cdots\)
4160.2.a.j 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.d \(0\) \(0\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}-3q^{9}-4q^{11}-q^{13}+\cdots\)
4160.2.a.k 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.b \(0\) \(0\) \(1\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-3q^{9}+2q^{11}+q^{13}+\cdots\)
4160.2.a.l 4160.a 1.a $1$ $33.218$ \(\Q\) None 520.2.a.a \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}-4q^{11}+q^{13}-6q^{17}+\cdots\)
4160.2.a.m 4160.a 1.a $1$ $33.218$ \(\Q\) None 520.2.a.a \(0\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}+4q^{11}+q^{13}-6q^{17}+\cdots\)
4160.2.a.n 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.b \(0\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}-3q^{9}-2q^{11}+q^{13}+\cdots\)
4160.2.a.o 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.a \(0\) \(2\) \(-1\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}-4q^{7}+q^{9}+6q^{11}+\cdots\)
4160.2.a.p 4160.a 1.a $1$ $33.218$ \(\Q\) None 520.2.a.b \(0\) \(2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}+2q^{11}-q^{13}+\cdots\)
4160.2.a.q 4160.a 1.a $1$ $33.218$ \(\Q\) None 65.2.a.a \(0\) \(2\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
4160.2.a.r 4160.a 1.a $1$ $33.218$ \(\Q\) None 2080.2.a.a \(0\) \(2\) \(1\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-4q^{7}+q^{9}+6q^{11}+\cdots\)
4160.2.a.s 4160.a 1.a $1$ $33.218$ \(\Q\) None 260.2.a.a \(0\) \(2\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
4160.2.a.t 4160.a 1.a $1$ $33.218$ \(\Q\) None 130.2.a.c \(0\) \(2\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+4q^{7}+q^{9}-2q^{11}+\cdots\)
4160.2.a.u 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 520.2.a.c \(0\) \(-4\) \(-2\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}-q^{5}+2q^{7}+(3-4\beta )q^{9}+\cdots\)
4160.2.a.v 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 520.2.a.d \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
4160.2.a.w 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 520.2.a.e \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
4160.2.a.x 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{5}) \) None 520.2.a.g \(0\) \(-2\) \(2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+q^{5}+(3+2\beta )q^{9}+(-3+\cdots)q^{11}+\cdots\)
4160.2.a.y 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 65.2.a.c \(0\) \(-2\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}+2q^{7}+(1-2\beta )q^{9}+\cdots\)
4160.2.a.z 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 65.2.a.b \(0\) \(0\) \(-2\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+(-2+2\beta )q^{7}-q^{9}+\cdots\)
4160.2.a.ba 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{6}) \) None 520.2.a.f \(0\) \(0\) \(-2\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}-2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
4160.2.a.bb 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.i \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta q^{7}-3q^{9}+\beta q^{11}-q^{13}+\cdots\)
4160.2.a.bc 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.j \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+2\beta q^{7}-q^{9}+\beta q^{11}+\cdots\)
4160.2.a.bd 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.k \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}-2\beta q^{7}-q^{9}+\beta q^{11}+\cdots\)
4160.2.a.be 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.l \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+5q^{9}+\beta q^{11}-q^{13}+\cdots\)
4160.2.a.bf 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 65.2.a.b \(0\) \(0\) \(-2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+(2+2\beta )q^{7}-q^{9}+(-2+\cdots)q^{11}+\cdots\)
4160.2.a.bg 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{6}) \) None 520.2.a.f \(0\) \(0\) \(-2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+2q^{7}+3q^{9}+(2+\beta )q^{11}+\cdots\)
4160.2.a.bh 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.h \(0\) \(0\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}-q^{9}+\beta q^{11}-q^{13}+\cdots\)
4160.2.a.bi 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 2080.2.a.g \(0\) \(0\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}-q^{9}-3\beta q^{11}-q^{13}+\cdots\)
4160.2.a.bj 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 65.2.a.c \(0\) \(2\) \(2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}-2q^{7}+(1+2\beta )q^{9}+\cdots\)
4160.2.a.bk 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 520.2.a.e \(0\) \(2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}-2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
4160.2.a.bl 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{3}) \) None 520.2.a.d \(0\) \(2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
4160.2.a.bm 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{5}) \) None 520.2.a.g \(0\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+(3+2\beta )q^{9}+(3-\beta )q^{11}+\cdots\)
4160.2.a.bn 4160.a 1.a $2$ $33.218$ \(\Q(\sqrt{2}) \) None 520.2.a.c \(0\) \(4\) \(-2\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}-q^{5}-2q^{7}+(3+4\beta )q^{9}+\cdots\)
4160.2.a.bo 4160.a 1.a $3$ $33.218$ 3.3.564.1 None 260.2.a.b \(0\) \(-2\) \(-3\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}-q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
4160.2.a.bp 4160.a 1.a $3$ $33.218$ 3.3.148.1 None 2080.2.a.m \(0\) \(-2\) \(-3\) \(6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}-q^{5}+2q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
4160.2.a.bq 4160.a 1.a $3$ $33.218$ 3.3.148.1 None 2080.2.a.m \(0\) \(2\) \(-3\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}-q^{5}-2q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
4160.2.a.br 4160.a 1.a $3$ $33.218$ 3.3.564.1 None 260.2.a.b \(0\) \(2\) \(-3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}-q^{5}+(1+\beta _{1})q^{7}+(4+\cdots)q^{9}+\cdots\)
4160.2.a.bs 4160.a 1.a $4$ $33.218$ 4.4.4752.1 None 2080.2.a.o \(0\) \(-4\) \(-4\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}-q^{5}+(1+\beta _{1})q^{7}+\cdots\)
4160.2.a.bt 4160.a 1.a $4$ $33.218$ 4.4.5744.1 None 2080.2.a.p \(0\) \(-2\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+q^{5}-\beta _{1}q^{7}+(1-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
4160.2.a.bu 4160.a 1.a $4$ $33.218$ \(\Q(\zeta_{16})^+\) None 2080.2.a.q \(0\) \(0\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+q^{5}+(\beta _{1}+\beta _{2})q^{7}+(1+\beta _{3})q^{9}+\cdots\)
4160.2.a.bv 4160.a 1.a $4$ $33.218$ 4.4.25088.1 None 2080.2.a.r \(0\) \(0\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(3+\cdots)q^{9}+\cdots\)
4160.2.a.bw 4160.a 1.a $4$ $33.218$ 4.4.5744.1 None 2080.2.a.p \(0\) \(2\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+q^{5}+\beta _{1}q^{7}+(1-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
4160.2.a.bx 4160.a 1.a $4$ $33.218$ 4.4.4752.1 None 2080.2.a.o \(0\) \(4\) \(-4\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4160))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4160)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(832))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2080))\)\(^{\oplus 2}\)