↑

Properties

Label 126.8.a
Level $126$
Weight $8$
Character orbit 126.a
Rep. character $\chi_{126}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $13$
Sturm bound $192$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(126))\).

Total New Old
Modular forms 176 18 158
Cusp forms 160 18 142
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.

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(10\)
Minus space\(-\)\(8\)

Trace form

\( 18 q + 1152 q^{4} - 394 q^{5} + O(q^{10}) \) \( 18 q + 1152 q^{4} - 394 q^{5} - 3120 q^{10} + 6244 q^{11} - 17754 q^{13} - 5488 q^{14} + 73728 q^{16} - 23456 q^{17} + 3558 q^{19} - 25216 q^{20} - 52128 q^{22} + 144376 q^{23} + 394098 q^{25} - 143728 q^{26} - 391840 q^{29} - 565092 q^{31} + 297504 q^{34} - 144746 q^{35} + 162384 q^{37} + 1325808 q^{38} - 199680 q^{40} + 734328 q^{41} + 568452 q^{43} + 399616 q^{44} - 68544 q^{46} + 159084 q^{47} + 2117682 q^{49} - 2014432 q^{50} - 1136256 q^{52} + 615924 q^{53} - 2161272 q^{55} - 351232 q^{56} + 1034592 q^{58} + 8634890 q^{59} + 2943066 q^{61} + 907232 q^{62} + 4718592 q^{64} - 3431340 q^{65} + 13239600 q^{67} - 1501184 q^{68} - 1300656 q^{70} - 3554984 q^{71} - 6246084 q^{73} + 2636064 q^{74} + 227712 q^{76} - 5664988 q^{77} + 2286312 q^{79} - 1613824 q^{80} + 8628576 q^{82} + 356290 q^{83} + 9288564 q^{85} + 2107104 q^{86} - 3336192 q^{88} + 13108188 q^{89} - 10827138 q^{91} + 9240064 q^{92} - 12376992 q^{94} - 4505864 q^{95} + 2555472 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(126))\) 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 3 7
126.8.a.a 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.f \(-8\) \(0\) \(-470\) \(-343\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}-470q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.b 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.e \(-8\) \(0\) \(-30\) \(343\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}-30q^{5}+7^{3}q^{7}-2^{9}q^{8}+\cdots\)
126.8.a.c 126.a 1.a $1$ $39.361$ \(\Q\) None 14.8.a.b \(-8\) \(0\) \(400\) \(-343\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+20^{2}q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.d 126.a 1.a $1$ $39.361$ \(\Q\) None 14.8.a.a \(8\) \(0\) \(-448\) \(-343\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}-448q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.e 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.d \(8\) \(0\) \(-270\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}-270q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.f 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.b \(8\) \(0\) \(18\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+18q^{5}+7^{3}q^{7}+2^{9}q^{8}+\cdots\)
126.8.a.g 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.c \(8\) \(0\) \(122\) \(-343\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+122q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.h 126.a 1.a $1$ $39.361$ \(\Q\) None 42.8.a.a \(8\) \(0\) \(410\) \(-343\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+410q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.i 126.a 1.a $2$ $39.361$ \(\Q(\sqrt{1969}) \) None 14.8.a.c \(-16\) \(0\) \(-126\) \(686\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-63-\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.j 126.a 1.a $2$ $39.361$ \(\Q(\sqrt{499}) \) None 126.8.a.j \(-16\) \(0\) \(56\) \(-686\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(28+\beta )q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.k 126.a 1.a $2$ $39.361$ \(\Q(\sqrt{3691}) \) None 126.8.a.k \(-16\) \(0\) \(168\) \(686\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(84+\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.l 126.a 1.a $2$ $39.361$ \(\Q(\sqrt{3691}) \) None 126.8.a.k \(16\) \(0\) \(-168\) \(686\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-84+\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.m 126.a 1.a $2$ $39.361$ \(\Q(\sqrt{499}) \) None 126.8.a.j \(16\) \(0\) \(-56\) \(-686\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-28+\beta )q^{5}-7^{3}q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(126)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)