Properties

Label 112.12.a
Level $112$
Weight $12$
Character orbit 112.a
Rep. character $\chi_{112}(1,\cdot)$
Character field $\Q$
Dimension $33$
Newform subspaces $12$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(112))\).

Total New Old
Modular forms 182 33 149
Cusp forms 170 33 137
Eisenstein series 12 0 12

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

\(2\)\(7\)FrickeDim
\(+\)\(+\)$+$\(8\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(8\)
\(-\)\(-\)$+$\(9\)
Plus space\(+\)\(17\)
Minus space\(-\)\(16\)

Trace form

\( 33 q + 2642 q^{5} + 16807 q^{7} + 1741373 q^{9} + O(q^{10}) \) \( 33 q + 2642 q^{5} + 16807 q^{7} + 1741373 q^{9} - 695884 q^{11} - 246046 q^{13} - 7082808 q^{15} - 4772598 q^{17} + 11291288 q^{19} + 35248184 q^{23} + 305174527 q^{25} + 195277248 q^{27} + 134315902 q^{29} - 270260488 q^{31} + 239902080 q^{33} - 315131250 q^{35} - 432471498 q^{37} + 2043645048 q^{39} + 18833234 q^{41} - 2542721820 q^{43} + 780037290 q^{45} - 929140200 q^{47} + 9321683217 q^{49} - 9086805168 q^{51} + 806103502 q^{53} + 21985844680 q^{55} + 6551146984 q^{57} - 22168610880 q^{59} + 161020986 q^{61} + 4962182715 q^{63} - 9845180756 q^{65} - 49960234844 q^{67} + 16193415024 q^{69} + 72360801096 q^{71} + 22279340666 q^{73} - 154194104800 q^{75} + 13333262012 q^{77} + 114341663920 q^{79} + 131882443649 q^{81} - 76324012456 q^{83} - 166918358300 q^{85} + 38093428976 q^{87} - 32241031782 q^{89} - 37441928706 q^{91} - 124934984976 q^{93} + 174814540072 q^{95} - 2742758406 q^{97} - 153974683772 q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(112))\) 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 7
112.12.a.a 112.a 1.a $1$ $86.054$ \(\Q\) None \(0\) \(90\) \(-7480\) \(16807\) $-$ $-$ $\mathrm{SU}(2)$ \(q+90q^{3}-7480q^{5}+7^{5}q^{7}-169047q^{9}+\cdots\)
112.12.a.b 112.a 1.a $1$ $86.054$ \(\Q\) None \(0\) \(396\) \(7350\) \(-16807\) $-$ $+$ $\mathrm{SU}(2)$ \(q+396q^{3}+7350q^{5}-7^{5}q^{7}-20331q^{9}+\cdots\)
112.12.a.c 112.a 1.a $2$ $86.054$ \(\Q(\sqrt{153169}) \) None \(0\) \(-350\) \(266\) \(33614\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-175-\beta )q^{3}+(133+21\beta )q^{5}+\cdots\)
112.12.a.d 112.a 1.a $2$ $86.054$ \(\Q(\sqrt{3369}) \) None \(0\) \(-120\) \(-13500\) \(-33614\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-60-3\beta )q^{3}+(-6750-5\beta )q^{5}+\cdots\)
112.12.a.e 112.a 1.a $2$ $86.054$ \(\Q(\sqrt{352969}) \) None \(0\) \(350\) \(3738\) \(-33614\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(175-\beta )q^{3}+(1869+3\beta )q^{5}-7^{5}q^{7}+\cdots\)
112.12.a.f 112.a 1.a $3$ $86.054$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-1132\) \(4986\) \(-50421\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-377+\beta _{1})q^{3}+(1662+\beta _{2})q^{5}+\cdots\)
112.12.a.g 112.a 1.a $3$ $86.054$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(100\) \(4762\) \(50421\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(33+\beta _{1})q^{3}+(1585+7\beta _{1}-\beta _{2})q^{5}+\cdots\)
112.12.a.h 112.a 1.a $3$ $86.054$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(140\) \(5026\) \(50421\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(47+\beta _{2})q^{3}+(1679-7\beta _{1}+4\beta _{2})q^{5}+\cdots\)
112.12.a.i 112.a 1.a $3$ $86.054$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(276\) \(-870\) \(-50421\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(92+\beta _{1})q^{3}+(-290-15\beta _{1}-5\beta _{2})q^{5}+\cdots\)
112.12.a.j 112.a 1.a $4$ $86.054$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-210\) \(-6062\) \(67228\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-53-\beta _{1})q^{3}+(-1513+5\beta _{1}+\cdots)q^{5}+\cdots\)
112.12.a.k 112.a 1.a $4$ $86.054$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(230\) \(-4566\) \(67228\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(57+\beta _{1})q^{3}+(-1138-7\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
112.12.a.l 112.a 1.a $5$ $86.054$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(230\) \(8992\) \(-84035\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(46+\beta _{1})q^{3}+(1798+\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(112)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)