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Properties

Label 126.10.a
Level $126$
Weight $10$
Character orbit 126.a
Rep. character $\chi_{126}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $15$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(240\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(126))\).

Total New Old
Modular forms 224 22 202
Cusp forms 208 22 186
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
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(10\)
Minus space\(-\)\(12\)

Trace form

\( 22 q + 5632 q^{4} + 4742 q^{5} + O(q^{10}) \) \( 22 q + 5632 q^{4} + 4742 q^{5} + 2528 q^{10} - 64172 q^{11} + 191014 q^{13} + 76832 q^{14} + 1441792 q^{16} - 643352 q^{17} + 1848578 q^{19} + 1213952 q^{20} + 247744 q^{22} - 77744 q^{23} + 8192814 q^{25} - 47776 q^{26} - 5883688 q^{29} + 12769780 q^{31} - 19548224 q^{34} - 3202934 q^{35} - 42102376 q^{37} + 89184 q^{38} + 647168 q^{40} - 18702432 q^{41} - 48768100 q^{43} - 16428032 q^{44} + 22897792 q^{46} + 28040100 q^{47} + 126825622 q^{49} - 32535616 q^{50} + 48899584 q^{52} - 43802340 q^{53} + 45599560 q^{55} + 19668992 q^{56} + 72779840 q^{58} + 359458910 q^{59} - 238402262 q^{61} - 224753728 q^{62} + 369098752 q^{64} + 502575108 q^{65} + 523412496 q^{67} - 164698112 q^{68} - 201530336 q^{70} + 334697152 q^{71} + 78556164 q^{73} + 38415552 q^{74} + 473235968 q^{76} + 401917796 q^{77} - 1023944792 q^{79} + 310771712 q^{80} - 514093248 q^{82} - 2304938618 q^{83} - 438131228 q^{85} + 1383069888 q^{86} + 63422464 q^{88} + 2096357988 q^{89} + 893772250 q^{91} - 19902464 q^{92} - 764224320 q^{94} - 3274066520 q^{95} + 5138870712 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(126))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
126.10.a.a 126.a 1.a $1$ $64.895$ \(\Q\) None \(-16\) \(0\) \(-544\) \(-2401\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}-544q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.b 126.a 1.a $1$ $64.895$ \(\Q\) None \(-16\) \(0\) \(474\) \(2401\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+474q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.c 126.a 1.a $1$ $64.895$ \(\Q\) None \(-16\) \(0\) \(1634\) \(-2401\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+1634q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.d 126.a 1.a $1$ $64.895$ \(\Q\) None \(16\) \(0\) \(-1590\) \(2401\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}-1590q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.e 126.a 1.a $1$ $64.895$ \(\Q\) None \(16\) \(0\) \(-560\) \(-2401\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}-560q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.f 126.a 1.a $1$ $64.895$ \(\Q\) None \(16\) \(0\) \(76\) \(-2401\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+76q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.g 126.a 1.a $1$ $64.895$ \(\Q\) None \(16\) \(0\) \(624\) \(2401\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+624q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.h 126.a 1.a $1$ $64.895$ \(\Q\) None \(16\) \(0\) \(2290\) \(-2401\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+2290q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.i 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{66739}) \) None \(-32\) \(0\) \(-1064\) \(-4802\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+(-532+\beta )q^{5}+\cdots\)
126.10.a.j 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{243601}) \) None \(-32\) \(0\) \(-534\) \(4802\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+(-267-\beta )q^{5}+\cdots\)
126.10.a.k 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{474769}) \) None \(-32\) \(0\) \(142\) \(-4802\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+(71-\beta )q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.l 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{211}) \) None \(-32\) \(0\) \(2184\) \(4802\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+(1092+13\beta )q^{5}+\cdots\)
126.10.a.m 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{211}) \) None \(32\) \(0\) \(-2184\) \(4802\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+(-1092+13\beta )q^{5}+\cdots\)
126.10.a.n 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{66739}) \) None \(32\) \(0\) \(1064\) \(-4802\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+(532+\beta )q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.o 126.a 1.a $2$ $64.895$ \(\Q(\sqrt{2305}) \) None \(32\) \(0\) \(2730\) \(4802\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+(1365-7\beta )q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(126)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)