Properties

Label 1040.6.a
Level $1040$
Weight $6$
Character orbit 1040.a
Rep. character $\chi_{1040}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $27$
Sturm bound $1008$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1040.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(1008\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1040))\).

Total New Old
Modular forms 852 120 732
Cusp forms 828 120 708
Eisenstein series 24 0 24

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
\(+\)\(+\)\(+\)\(+\)\(15\)
\(+\)\(+\)\(-\)\(-\)\(16\)
\(+\)\(-\)\(+\)\(-\)\(16\)
\(+\)\(-\)\(-\)\(+\)\(13\)
\(-\)\(+\)\(+\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(+\)\(14\)
\(-\)\(-\)\(+\)\(+\)\(14\)
\(-\)\(-\)\(-\)\(-\)\(17\)
Plus space\(+\)\(56\)
Minus space\(-\)\(64\)

Trace form

\( 120 q + 196 q^{7} + 9632 q^{9} - 1208 q^{11} + 900 q^{15} - 2008 q^{17} - 2360 q^{19} + 1640 q^{21} + 1672 q^{23} + 75000 q^{25} - 16104 q^{29} - 7160 q^{31} + 3840 q^{33} + 21296 q^{37} + 11608 q^{41}+ \cdots - 121248 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\) 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
1040.6.a.a 1040.a 1.a $1$ $166.799$ \(\Q\) None 65.6.a.a \(0\) \(-6\) \(-25\) \(244\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-6q^{3}-5^{2}q^{5}+244q^{7}-207q^{9}+\cdots\)
1040.6.a.b 1040.a 1.a $1$ $166.799$ \(\Q\) None 260.6.a.a \(0\) \(14\) \(-25\) \(212\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+14q^{3}-5^{2}q^{5}+212q^{7}-47q^{9}+\cdots\)
1040.6.a.c 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{145}) \) None 130.6.a.c \(0\) \(-6\) \(-50\) \(-130\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{3}-5^{2}q^{5}+(-65+\beta )q^{7}+\cdots\)
1040.6.a.d 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{19}) \) None 130.6.a.b \(0\) \(-6\) \(-50\) \(208\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+3\beta )q^{3}-5^{2}q^{5}+(104+10\beta )q^{7}+\cdots\)
1040.6.a.e 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{10}) \) None 130.6.a.a \(0\) \(4\) \(50\) \(-300\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+5^{2}q^{5}+(-150+4\beta )q^{7}+\cdots\)
1040.6.a.f 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{51}) \) None 260.6.a.c \(0\) \(6\) \(-50\) \(140\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+3\beta )q^{3}-5^{2}q^{5}+(70+8\beta )q^{7}+\cdots\)
1040.6.a.g 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{67}) \) None 260.6.a.b \(0\) \(14\) \(-50\) \(-164\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(7+\beta )q^{3}-5^{2}q^{5}+(-82-4\beta )q^{7}+\cdots\)
1040.6.a.h 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{14}) \) None 130.6.a.e \(0\) \(16\) \(50\) \(252\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(8+3\beta )q^{3}+5^{2}q^{5}+(126-20\beta )q^{7}+\cdots\)
1040.6.a.i 1040.a 1.a $2$ $166.799$ \(\Q(\sqrt{235}) \) None 130.6.a.d \(0\) \(26\) \(-50\) \(-160\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(13+\beta )q^{3}-5^{2}q^{5}+(-80-6\beta )q^{7}+\cdots\)
1040.6.a.j 1040.a 1.a $3$ $166.799$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 130.6.a.g \(0\) \(8\) \(-75\) \(-42\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}-5^{2}q^{5}+(-12+6\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.k 1040.a 1.a $3$ $166.799$ 3.3.49857.1 None 65.6.a.b \(0\) \(16\) \(75\) \(208\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(5+\beta _{2})q^{3}+5^{2}q^{5}+(72+2\beta _{1}-6\beta _{2})q^{7}+\cdots\)
1040.6.a.l 1040.a 1.a $3$ $166.799$ 3.3.1458804.1 None 130.6.a.f \(0\) \(22\) \(75\) \(234\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(7+\beta _{1})q^{3}+5^{2}q^{5}+(76+7\beta _{1}+\beta _{2})q^{7}+\cdots\)
1040.6.a.m 1040.a 1.a $4$ $166.799$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 130.6.a.h \(0\) \(-20\) \(100\) \(-62\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-5+\beta _{1})q^{3}+5^{2}q^{5}+(-2^{4}-\beta _{3})q^{7}+\cdots\)
1040.6.a.n 1040.a 1.a $4$ $166.799$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 260.6.a.e \(0\) \(-12\) \(-100\) \(-126\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}-5^{2}q^{5}+(-2^{5}+6\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.o 1040.a 1.a $4$ $166.799$ 4.4.1878612.1 None 65.6.a.c \(0\) \(4\) \(-100\) \(136\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1}+\beta _{2})q^{3}-5^{2}q^{5}+(34+4\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.p 1040.a 1.a $4$ $166.799$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 260.6.a.d \(0\) \(20\) \(100\) \(24\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(5+\beta _{3})q^{3}+5^{2}q^{5}+(6-\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.q 1040.a 1.a $6$ $166.799$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.d \(0\) \(-38\) \(-150\) \(-220\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-6-\beta _{1})q^{3}-5^{2}q^{5}+(-37+\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.r 1040.a 1.a $6$ $166.799$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.e \(0\) \(-20\) \(150\) \(-172\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}+(-28+\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.s 1040.a 1.a $6$ $166.799$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 520.6.a.a \(0\) \(-12\) \(150\) \(146\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+5^{2}q^{5}+(5^{2}-3\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.t 1040.a 1.a $7$ $166.799$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 520.6.a.c \(0\) \(-26\) \(175\) \(-146\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4+\beta _{1})q^{3}+5^{2}q^{5}+(-21-\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.u 1040.a 1.a $7$ $166.799$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 520.6.a.b \(0\) \(-2\) \(-175\) \(-54\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-5^{2}q^{5}+(-8-\beta _{4})q^{7}+(74+\cdots)q^{9}+\cdots\)
1040.6.a.v 1040.a 1.a $7$ $166.799$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 260.6.a.f \(0\) \(2\) \(175\) \(-86\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+5^{2}q^{5}+(-13+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.w 1040.a 1.a $8$ $166.799$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 520.6.a.h \(0\) \(-36\) \(-200\) \(54\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{3}-5^{2}q^{5}+(7-\beta _{1}+\beta _{5}+\cdots)q^{7}+\cdots\)
1040.6.a.x 1040.a 1.a $8$ $166.799$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 520.6.a.g \(0\) \(-12\) \(200\) \(80\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+5^{2}q^{5}+(10+\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.y 1040.a 1.a $8$ $166.799$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 520.6.a.f \(0\) \(0\) \(-200\) \(184\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-5^{2}q^{5}+(23+2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.z 1040.a 1.a $8$ $166.799$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 520.6.a.e \(0\) \(16\) \(-200\) \(-184\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}-5^{2}q^{5}+(-23+\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.ba 1040.a 1.a $8$ $166.799$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 520.6.a.d \(0\) \(28\) \(200\) \(-80\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{3}+5^{2}q^{5}+(-11+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1040))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1040)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 2}\)