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Properties

Label 490.6.a
Level $490$
Weight $6$
Character orbit 490.a
Rep. character $\chi_{490}(1,\cdot)$
Character field $\Q$
Dimension $67$
Newform subspaces $31$
Sturm bound $504$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(504\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(490))\).

Total New Old
Modular forms 436 67 369
Cusp forms 404 67 337
Eisenstein series 32 0 32

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
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(31\)
Minus space\(-\)\(36\)

Trace form

\( 67 q - 4 q^{2} + 40 q^{3} + 1072 q^{4} + 25 q^{5} - 48 q^{6} - 64 q^{8} + 5155 q^{9} + O(q^{10}) \) \( 67 q - 4 q^{2} + 40 q^{3} + 1072 q^{4} + 25 q^{5} - 48 q^{6} - 64 q^{8} + 5155 q^{9} + 100 q^{10} - 1184 q^{11} + 640 q^{12} + 694 q^{13} - 1100 q^{15} + 17152 q^{16} + 1062 q^{17} - 1908 q^{18} - 80 q^{19} + 400 q^{20} - 1456 q^{22} + 2880 q^{23} - 768 q^{24} + 41875 q^{25} - 5864 q^{26} - 9920 q^{27} + 2342 q^{29} + 2000 q^{30} - 1384 q^{31} - 1024 q^{32} - 18080 q^{33} + 14648 q^{34} + 82480 q^{36} - 38094 q^{37} + 320 q^{38} + 16272 q^{39} + 1600 q^{40} - 30818 q^{41} + 61092 q^{43} - 18944 q^{44} + 8925 q^{45} + 33456 q^{46} + 39848 q^{47} + 10240 q^{48} - 2500 q^{50} + 50064 q^{51} + 11104 q^{52} + 21522 q^{53} - 61344 q^{54} + 10100 q^{55} + 264920 q^{57} - 86072 q^{58} + 37944 q^{59} - 17600 q^{60} + 50494 q^{61} + 87168 q^{62} + 274432 q^{64} + 16550 q^{65} + 21696 q^{66} + 83716 q^{67} + 16992 q^{68} + 14056 q^{69} - 44152 q^{71} - 30528 q^{72} + 62102 q^{73} + 33144 q^{74} + 25000 q^{75} - 1280 q^{76} + 14336 q^{78} + 460736 q^{79} + 6400 q^{80} + 423555 q^{81} - 21448 q^{82} - 82192 q^{83} + 5150 q^{85} + 40144 q^{86} - 450480 q^{87} - 23296 q^{88} - 202866 q^{89} + 111700 q^{90} + 46080 q^{92} + 173464 q^{93} + 102096 q^{94} + 18100 q^{95} - 12288 q^{96} + 116342 q^{97} - 953168 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\) 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
490.6.a.a 490.a 1.a $1$ $78.588$ \(\Q\) None 10.6.a.b \(-4\) \(-24\) \(-25\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-24q^{3}+2^{4}q^{4}-5^{2}q^{5}+96q^{6}+\cdots\)
490.6.a.b 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.e.b \(-4\) \(-18\) \(25\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-18q^{3}+2^{4}q^{4}+5^{2}q^{5}+72q^{6}+\cdots\)
490.6.a.c 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.d \(-4\) \(-11\) \(25\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-11q^{3}+2^{4}q^{4}+5^{2}q^{5}+44q^{6}+\cdots\)
490.6.a.d 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.e.a \(-4\) \(-5\) \(-25\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-5q^{3}+2^{4}q^{4}-5^{2}q^{5}+20q^{6}+\cdots\)
490.6.a.e 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.c \(-4\) \(3\) \(25\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+3q^{3}+2^{4}q^{4}+5^{2}q^{5}-12q^{6}+\cdots\)
490.6.a.f 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.e.a \(-4\) \(5\) \(25\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+5q^{3}+2^{4}q^{4}+5^{2}q^{5}-20q^{6}+\cdots\)
490.6.a.g 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.b \(-4\) \(9\) \(-25\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
490.6.a.h 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.e.b \(-4\) \(18\) \(-25\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+18q^{3}+2^{4}q^{4}-5^{2}q^{5}-72q^{6}+\cdots\)
490.6.a.i 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.a \(-4\) \(23\) \(-25\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+23q^{3}+2^{4}q^{4}-5^{2}q^{5}-92q^{6}+\cdots\)
490.6.a.j 490.a 1.a $1$ $78.588$ \(\Q\) None 10.6.a.a \(-4\) \(26\) \(25\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+26q^{3}+2^{4}q^{4}+5^{2}q^{5}-104q^{6}+\cdots\)
490.6.a.k 490.a 1.a $1$ $78.588$ \(\Q\) None 10.6.a.c \(4\) \(-6\) \(25\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-6q^{3}+2^{4}q^{4}+5^{2}q^{5}-24q^{6}+\cdots\)
490.6.a.l 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.f \(4\) \(11\) \(25\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+11q^{3}+2^{4}q^{4}+5^{2}q^{5}+44q^{6}+\cdots\)
490.6.a.m 490.a 1.a $1$ $78.588$ \(\Q\) None 70.6.a.e \(4\) \(17\) \(-25\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+17q^{3}+2^{4}q^{4}-5^{2}q^{5}+68q^{6}+\cdots\)
490.6.a.n 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{106}) \) None 70.6.e.d \(-8\) \(-22\) \(-50\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-11+\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.o 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{337}) \) None 490.6.a.o \(-8\) \(-20\) \(50\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-10-\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.p 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{79}) \) None 70.6.e.e \(-8\) \(-6\) \(50\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-3+\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.q 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{79}) \) None 70.6.e.e \(-8\) \(6\) \(-50\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(3+\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.r 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{337}) \) None 490.6.a.o \(-8\) \(20\) \(-50\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(10-\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.s 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{106}) \) None 70.6.e.d \(-8\) \(22\) \(50\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(11+\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.t 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{46}) \) None 70.6.e.c \(8\) \(-18\) \(-50\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-9+\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.u 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{1129}) \) None 70.6.a.h \(8\) \(-5\) \(-50\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-2-\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.v 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{3369}) \) None 70.6.a.g \(8\) \(-3\) \(50\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-1-\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.w 490.a 1.a $2$ $78.588$ \(\Q(\sqrt{46}) \) None 70.6.e.c \(8\) \(18\) \(50\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(9+\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.x 490.a 1.a $3$ $78.588$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 490.6.a.x \(12\) \(-2\) \(75\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-1+\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.y 490.a 1.a $3$ $78.588$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 490.6.a.x \(12\) \(2\) \(-75\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(1-\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.z 490.a 1.a $4$ $78.588$ \(\Q(\sqrt{2}, \sqrt{193})\) None 490.6.a.z \(16\) \(-40\) \(100\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-10-\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.ba 490.a 1.a $4$ $78.588$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 70.6.e.f \(16\) \(-23\) \(-100\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-6+\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.bb 490.a 1.a $4$ $78.588$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 70.6.e.f \(16\) \(23\) \(100\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(6-\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.bc 490.a 1.a $4$ $78.588$ \(\Q(\sqrt{2}, \sqrt{193})\) None 490.6.a.z \(16\) \(40\) \(-100\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(10-\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
490.6.a.bd 490.a 1.a $6$ $78.588$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 490.6.a.bd \(-24\) \(-4\) \(150\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-1-\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
490.6.a.be 490.a 1.a $6$ $78.588$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 490.6.a.bd \(-24\) \(4\) \(-150\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(1+\beta _{2})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(490))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(490)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)