Properties

Label 560.6.a
Level $560$
Weight $6$
Character orbit 560.a
Rep. character $\chi_{560}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $26$
Sturm bound $576$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(576\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(560))\).

Total New Old
Modular forms 492 60 432
Cusp forms 468 60 408
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.

\(2\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(27\)
Minus space\(-\)\(33\)

Trace form

\( 60 q + 98 q^{7} + 4772 q^{9} + O(q^{10}) \) \( 60 q + 98 q^{7} + 4772 q^{9} - 604 q^{11} + 900 q^{15} - 2008 q^{17} - 2360 q^{19} + 7184 q^{23} + 37500 q^{25} - 7464 q^{27} - 4072 q^{29} - 7160 q^{31} + 3840 q^{33} - 7350 q^{35} + 10648 q^{37} - 43764 q^{39} + 11608 q^{41} - 6152 q^{43} - 5016 q^{47} + 144060 q^{49} + 47580 q^{51} + 24728 q^{53} + 96000 q^{57} + 59520 q^{59} - 48080 q^{61} + 39690 q^{63} - 23288 q^{67} + 22320 q^{69} + 77600 q^{71} + 107256 q^{73} + 7448 q^{77} + 163844 q^{79} + 393484 q^{81} + 118824 q^{83} + 132400 q^{85} + 150936 q^{87} + 321016 q^{89} - 362352 q^{93} + 144400 q^{95} - 228920 q^{97} - 66520 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\) 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 7
560.6.a.a 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.d \(0\) \(-11\) \(-25\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-11q^{3}-5^{2}q^{5}+7^{2}q^{7}-122q^{9}+\cdots\)
560.6.a.b 560.a 1.a $1$ $89.815$ \(\Q\) None 280.6.a.b \(0\) \(-4\) \(25\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{3}+5^{2}q^{5}+7^{2}q^{7}-227q^{9}+\cdots\)
560.6.a.c 560.a 1.a $1$ $89.815$ \(\Q\) None 35.6.a.a \(0\) \(-1\) \(25\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+5^{2}q^{5}-7^{2}q^{7}-242q^{9}+453q^{11}+\cdots\)
560.6.a.d 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.c \(0\) \(3\) \(-25\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5^{2}q^{5}-7^{2}q^{7}-234q^{9}+\cdots\)
560.6.a.e 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.b \(0\) \(9\) \(25\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+5^{2}q^{5}+7^{2}q^{7}-162q^{9}+\cdots\)
560.6.a.f 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.f \(0\) \(11\) \(-25\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+11q^{3}-5^{2}q^{5}-7^{2}q^{7}-122q^{9}+\cdots\)
560.6.a.g 560.a 1.a $1$ $89.815$ \(\Q\) None 280.6.a.a \(0\) \(12\) \(25\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+12q^{3}+5^{2}q^{5}+7^{2}q^{7}-99q^{9}+\cdots\)
560.6.a.h 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.e \(0\) \(17\) \(25\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+17q^{3}+5^{2}q^{5}+7^{2}q^{7}+46q^{9}+\cdots\)
560.6.a.i 560.a 1.a $1$ $89.815$ \(\Q\) None 70.6.a.a \(0\) \(23\) \(25\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+23q^{3}+5^{2}q^{5}-7^{2}q^{7}+286q^{9}+\cdots\)
560.6.a.j 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{37}) \) None 280.6.a.d \(0\) \(-26\) \(50\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-13-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(74+\cdots)q^{9}+\cdots\)
560.6.a.k 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{1129}) \) None 70.6.a.h \(0\) \(-5\) \(50\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{3}+5^{2}q^{5}-7^{2}q^{7}+(43+\cdots)q^{9}+\cdots\)
560.6.a.l 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{65}) \) None 35.6.a.b \(0\) \(-3\) \(-50\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3\beta q^{3}-5^{2}q^{5}+7^{2}q^{7}+(-99+9\beta )q^{9}+\cdots\)
560.6.a.m 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{3369}) \) None 70.6.a.g \(0\) \(-3\) \(-50\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}-5^{2}q^{5}+7^{2}q^{7}+(600+\cdots)q^{9}+\cdots\)
560.6.a.n 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{109}) \) None 280.6.a.c \(0\) \(-2\) \(50\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(194+\cdots)q^{9}+\cdots\)
560.6.a.o 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{1009}) \) None 140.6.a.b \(0\) \(17\) \(50\) \(98\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(9-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(90-17\beta )q^{9}+\cdots\)
560.6.a.p 560.a 1.a $2$ $89.815$ \(\Q(\sqrt{1009}) \) None 140.6.a.a \(0\) \(23\) \(-50\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(12-\beta )q^{3}-5^{2}q^{5}-7^{2}q^{7}+(153+\cdots)q^{9}+\cdots\)
560.6.a.q 560.a 1.a $3$ $89.815$ 3.3.577880.1 None 35.6.a.c \(0\) \(-26\) \(-75\) \(-147\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-9-\beta _{2})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(166+\cdots)q^{9}+\cdots\)
560.6.a.r 560.a 1.a $3$ $89.815$ 3.3.996509.1 None 280.6.a.f \(0\) \(-14\) \(-75\) \(-147\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta _{2})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(4+\cdots)q^{9}+\cdots\)
560.6.a.s 560.a 1.a $3$ $89.815$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 280.6.a.e \(0\) \(-6\) \(-75\) \(-147\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(70+\cdots)q^{9}+\cdots\)
560.6.a.t 560.a 1.a $3$ $89.815$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 140.6.a.d \(0\) \(-6\) \(75\) \(-147\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(94+\cdots)q^{9}+\cdots\)
560.6.a.u 560.a 1.a $3$ $89.815$ 3.3.3101016.1 None 140.6.a.c \(0\) \(10\) \(-75\) \(147\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(-23+\cdots)q^{9}+\cdots\)
560.6.a.v 560.a 1.a $4$ $89.815$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 35.6.a.d \(0\) \(-14\) \(100\) \(196\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}+7^{2}q^{7}+(14^{2}+\cdots)q^{9}+\cdots\)
560.6.a.w 560.a 1.a $4$ $89.815$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 280.6.a.h \(0\) \(-5\) \(-100\) \(196\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(142+\cdots)q^{9}+\cdots\)
560.6.a.x 560.a 1.a $4$ $89.815$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 280.6.a.g \(0\) \(13\) \(100\) \(-196\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(-7^{2}+\cdots)q^{9}+\cdots\)
560.6.a.y 560.a 1.a $5$ $89.815$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 280.6.a.j \(0\) \(-15\) \(125\) \(-245\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(226+\cdots)q^{9}+\cdots\)
560.6.a.z 560.a 1.a $5$ $89.815$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 280.6.a.i \(0\) \(3\) \(-125\) \(245\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(74+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(560))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(560)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)