Properties

Label 585.6.a
Level $585$
Weight $6$
Character orbit 585.a
Rep. character $\chi_{585}(1,\cdot)$
Character field $\Q$
Dimension $100$
Newform subspaces $18$
Sturm bound $504$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(504\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(585))\).

Total New Old
Modular forms 428 100 328
Cusp forms 412 100 312
Eisenstein series 16 0 16

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

\(3\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(11\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(+\)\(14\)
\(-\)\(-\)\(+\)\(+\)\(14\)
\(-\)\(-\)\(-\)\(-\)\(17\)
Plus space\(+\)\(48\)
Minus space\(-\)\(52\)

Trace form

\( 100 q + 12 q^{2} + 1664 q^{4} + 50 q^{5} - 196 q^{7} + 156 q^{8} - 100 q^{10} + 2124 q^{11} - 338 q^{13} - 4432 q^{14} + 28032 q^{16} - 2884 q^{17} - 3864 q^{19} + 2400 q^{20} + 1544 q^{22} + 480 q^{23}+ \cdots + 812052 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(585))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 13
585.6.a.a 585.a 1.a $1$ $93.825$ \(\Q\) None 65.6.a.a \(-5\) \(0\) \(25\) \(-244\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}-7q^{4}+5^{2}q^{5}-244q^{7}+195q^{8}+\cdots\)
585.6.a.b 585.a 1.a $1$ $93.825$ \(\Q\) None 195.6.a.a \(-2\) \(0\) \(-25\) \(-168\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-28q^{4}-5^{2}q^{5}-168q^{7}+\cdots\)
585.6.a.c 585.a 1.a $3$ $93.825$ 3.3.49857.1 None 65.6.a.b \(2\) \(0\) \(-75\) \(-208\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.d 585.a 1.a $4$ $93.825$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 195.6.a.e \(-2\) \(0\) \(100\) \(87\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(14-\beta _{1}+\beta _{3})q^{4}+\cdots\)
585.6.a.e 585.a 1.a $4$ $93.825$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 195.6.a.d \(3\) \(0\) \(-100\) \(-87\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
585.6.a.f 585.a 1.a $4$ $93.825$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 195.6.a.c \(7\) \(0\) \(100\) \(73\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(3-2\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
585.6.a.g 585.a 1.a $4$ $93.825$ 4.4.1878612.1 None 65.6.a.c \(9\) \(0\) \(100\) \(-136\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1}-\beta _{2})q^{2}+(22-4\beta _{2}+\beta _{3})q^{4}+\cdots\)
585.6.a.h 585.a 1.a $4$ $93.825$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 195.6.a.b \(12\) \(0\) \(-100\) \(-73\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+(13-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
585.6.a.i 585.a 1.a $5$ $93.825$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 195.6.a.g \(-5\) \(0\) \(-125\) \(164\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(8+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.j 585.a 1.a $5$ $93.825$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 195.6.a.f \(-1\) \(0\) \(125\) \(128\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+4\beta _{1}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
585.6.a.k 585.a 1.a $6$ $93.825$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.e \(-2\) \(0\) \(-150\) \(172\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(22+\beta _{3})q^{4}-5^{2}q^{5}+(3^{3}+\cdots)q^{7}+\cdots\)
585.6.a.l 585.a 1.a $6$ $93.825$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 195.6.a.h \(-2\) \(0\) \(-150\) \(236\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(35+\beta _{3})q^{4}-5^{2}q^{5}+(37+\cdots)q^{7}+\cdots\)
585.6.a.m 585.a 1.a $6$ $93.825$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.d \(0\) \(0\) \(150\) \(220\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(22+\beta _{2})q^{4}+5^{2}q^{5}+(35+\cdots)q^{7}+\cdots\)
585.6.a.n 585.a 1.a $7$ $93.825$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 195.6.a.i \(-2\) \(0\) \(175\) \(32\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(26-\beta _{1}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
585.6.a.o 585.a 1.a $9$ $93.825$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 585.6.a.o \(-5\) \(0\) \(225\) \(-153\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2^{4}-\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.p 585.a 1.a $9$ $93.825$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 585.6.a.o \(5\) \(0\) \(-225\) \(-153\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2^{4}-\beta _{1}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
585.6.a.q 585.a 1.a $11$ $93.825$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None 585.6.a.q \(-3\) \(0\) \(-275\) \(-43\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(21+\beta _{2})q^{4}-5^{2}q^{5}+(-4+\cdots)q^{7}+\cdots\)
585.6.a.r 585.a 1.a $11$ $93.825$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None 585.6.a.q \(3\) \(0\) \(275\) \(-43\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(21+\beta _{2})q^{4}+5^{2}q^{5}+(-4+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(585))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(585)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)