Properties

Label 63.6.a
Level $63$
Weight $6$
Character orbit 63.a
Rep. character $\chi_{63}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $8$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(63))\).

Total New Old
Modular forms 44 13 31
Cusp forms 36 13 23
Eisenstein series 8 0 8

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

\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(8\)

Trace form

\( 13 q - 9 q^{2} + 193 q^{4} + 42 q^{5} + 49 q^{7} + 99 q^{8} + O(q^{10}) \) \( 13 q - 9 q^{2} + 193 q^{4} + 42 q^{5} + 49 q^{7} + 99 q^{8} - 156 q^{10} - 876 q^{11} - 742 q^{13} + 147 q^{14} + 4645 q^{16} + 3090 q^{17} + 4868 q^{19} + 7788 q^{20} + 1788 q^{22} - 3168 q^{23} + 11083 q^{25} - 9372 q^{26} + 637 q^{28} - 3066 q^{29} - 9976 q^{31} - 741 q^{32} - 13230 q^{34} - 7938 q^{35} + 3902 q^{37} + 29862 q^{38} - 27144 q^{40} + 6138 q^{41} + 5276 q^{43} - 3168 q^{44} - 43020 q^{46} - 9840 q^{47} + 31213 q^{49} - 70131 q^{50} - 67084 q^{52} + 31110 q^{53} + 17856 q^{55} - 20433 q^{56} - 39414 q^{58} + 40092 q^{59} - 63190 q^{61} - 157404 q^{62} + 7201 q^{64} + 146412 q^{65} + 158996 q^{67} + 108798 q^{68} + 37044 q^{70} - 42864 q^{71} + 39146 q^{73} - 194946 q^{74} + 210194 q^{76} - 39396 q^{77} - 103504 q^{79} + 491748 q^{80} - 178806 q^{82} - 90108 q^{83} + 206004 q^{85} + 216840 q^{86} + 460572 q^{88} - 17070 q^{89} + 44198 q^{91} - 314760 q^{92} - 758436 q^{94} - 467832 q^{95} + 98282 q^{97} - 21609 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\) 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 7
63.6.a.a 63.a 1.a $1$ $10.104$ \(\Q\) None 21.6.a.d \(-10\) \(0\) \(106\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-10q^{2}+68q^{4}+106q^{5}-7^{2}q^{7}+\cdots\)
63.6.a.b 63.a 1.a $1$ $10.104$ \(\Q\) None 21.6.a.c \(-5\) \(0\) \(-94\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}-7q^{4}-94q^{5}-7^{2}q^{7}+195q^{8}+\cdots\)
63.6.a.c 63.a 1.a $1$ $10.104$ \(\Q\) None 21.6.a.b \(-1\) \(0\) \(34\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-31q^{4}+34q^{5}-7^{2}q^{7}+63q^{8}+\cdots\)
63.6.a.d 63.a 1.a $1$ $10.104$ \(\Q\) None 21.6.a.a \(6\) \(0\) \(-78\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+6q^{2}+4q^{4}-78q^{5}+7^{2}q^{7}-168q^{8}+\cdots\)
63.6.a.e 63.a 1.a $1$ $10.104$ \(\Q\) None 7.6.a.a \(10\) \(0\) \(56\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{2}+68q^{4}+56q^{5}-7^{2}q^{7}+\cdots\)
63.6.a.f 63.a 1.a $2$ $10.104$ \(\Q(\sqrt{57}) \) None 7.6.a.b \(-9\) \(0\) \(18\) \(98\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}+(-2+9\beta )q^{4}+(14+\cdots)q^{5}+\cdots\)
63.6.a.g 63.a 1.a $2$ $10.104$ \(\Q(\sqrt{7}) \) None 63.6.a.g \(0\) \(0\) \(0\) \(-98\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-4q^{4}-7\beta q^{5}-7^{2}q^{7}-6^{2}\beta q^{8}+\cdots\)
63.6.a.h 63.a 1.a $4$ $10.104$ 4.4.358541904.1 None 63.6.a.h \(0\) \(0\) \(0\) \(196\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(24+\beta _{3})q^{4}+(3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(63)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)