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Properties

Label 7.12.a
Level $7$
Weight $12$
Character orbit 7.a
Rep. character $\chi_{7}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $2$
Sturm bound $8$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(7))\).

Total New Old
Modular forms 9 5 4
Cusp forms 7 5 2
Eisenstein series 2 0 2

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

\(7\)Dim
\(+\)\(3\)
\(-\)\(2\)

Trace form

\( 5 q + 23 q^{2} - 20 q^{3} + 9593 q^{4} - 8474 q^{5} + 20606 q^{6} - 16807 q^{7} - 238857 q^{8} + 553993 q^{9} + O(q^{10}) \) \( 5 q + 23 q^{2} - 20 q^{3} + 9593 q^{4} - 8474 q^{5} + 20606 q^{6} - 16807 q^{7} - 238857 q^{8} + 553993 q^{9} + 1042644 q^{10} - 1789868 q^{11} + 1819958 q^{12} - 1890770 q^{13} - 2201717 q^{14} - 9435032 q^{15} + 2149889 q^{16} + 19957386 q^{17} - 1990249 q^{18} - 5658356 q^{19} + 6668648 q^{20} + 4369820 q^{21} + 80192256 q^{22} - 58884264 q^{23} - 198158526 q^{24} + 63067331 q^{25} + 67980080 q^{26} + 207253576 q^{27} - 23412151 q^{28} - 65337794 q^{29} - 265674448 q^{30} + 146902960 q^{31} - 473095673 q^{32} + 573117184 q^{33} - 212873502 q^{34} - 311366482 q^{35} + 375771185 q^{36} + 39902998 q^{37} + 468983810 q^{38} - 1868351576 q^{39} + 4382370240 q^{40} + 715406594 q^{41} - 1814181194 q^{42} + 2286943060 q^{43} - 4718958808 q^{44} - 4391788658 q^{45} - 4651377792 q^{46} + 3292066368 q^{47} + 4287400094 q^{48} + 1412376245 q^{49} + 10622634109 q^{50} - 6461937192 q^{51} - 15294285692 q^{52} - 4092389850 q^{53} + 32250303908 q^{54} + 1274569032 q^{55} - 8219681841 q^{56} + 5946205456 q^{57} - 17444657898 q^{58} + 9010519428 q^{59} - 34060585216 q^{60} + 17149402798 q^{61} + 13269410460 q^{62} - 12824497315 q^{63} + 13615008977 q^{64} - 2914537052 q^{65} + 891867824 q^{66} - 17170761380 q^{67} + 36598047990 q^{68} - 11808442704 q^{69} - 3006503388 q^{70} - 7747961400 q^{71} - 7237262385 q^{72} - 24853197998 q^{73} + 4955041758 q^{74} + 11893755908 q^{75} + 11061587194 q^{76} + 4844382452 q^{77} + 19699573880 q^{78} - 52902212480 q^{79} - 25423990864 q^{80} + 82069315789 q^{81} - 92408290926 q^{82} + 128838218460 q^{83} + 51064136662 q^{84} + 17293659084 q^{85} + 71981969896 q^{86} - 36276200504 q^{87} + 79695820872 q^{88} - 193189723934 q^{89} - 273768252412 q^{90} + 31457224918 q^{91} - 269180571312 q^{92} + 274848055968 q^{93} + 374073101844 q^{94} + 96812954440 q^{95} - 460410920270 q^{96} - 263465735750 q^{97} + 6496930727 q^{98} - 158745210908 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(7))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
7.12.a.a 7.a 1.a $2$ $5.378$ \(\Q(\sqrt{3369}) \) None \(-54\) \(120\) \(-13500\) \(33614\) $-$ $\mathrm{SU}(2)$ \(q+(-3^{3}-\beta )q^{2}+(60+6\beta )q^{3}+(2050+\cdots)q^{4}+\cdots\)
7.12.a.b 7.a 1.a $3$ $5.378$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(77\) \(-140\) \(5026\) \(-50421\) $+$ $\mathrm{SU}(2)$ \(q+(26+\beta _{2})q^{2}+(-47-11\beta _{1}+10\beta _{2})q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(7))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(7)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)