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Properties

Label 129.8.a
Level $129$
Weight $8$
Character orbit 129.a
Rep. character $\chi_{129}(1,\cdot)$
Character field $\Q$
Dimension $50$
Newform subspaces $4$
Sturm bound $117$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 129.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(117\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(129))\).

Total New Old
Modular forms 104 50 54
Cusp forms 100 50 50
Eisenstein series 4 0 4

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

\(3\)\(43\)FrickeDim
\(+\)\(+\)$+$\(13\)
\(+\)\(-\)$-$\(12\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(15\)
Plus space\(+\)\(28\)
Minus space\(-\)\(22\)

Trace form

\( 50 q + 16 q^{2} + 3276 q^{4} - 776 q^{5} + 324 q^{6} - 1348 q^{7} + 5076 q^{8} + 36450 q^{9} + O(q^{10}) \) \( 50 q + 16 q^{2} + 3276 q^{4} - 776 q^{5} + 324 q^{6} - 1348 q^{7} + 5076 q^{8} + 36450 q^{9} - 8272 q^{10} - 4738 q^{11} + 20698 q^{13} - 7588 q^{14} + 34776 q^{15} + 235772 q^{16} - 80430 q^{17} + 11664 q^{18} - 40484 q^{19} + 154784 q^{20} + 37044 q^{21} + 22816 q^{22} - 13518 q^{23} - 71280 q^{24} + 737242 q^{25} - 6744 q^{26} - 125036 q^{28} - 118204 q^{29} - 320112 q^{30} + 734798 q^{31} + 486244 q^{32} - 385236 q^{33} - 1493480 q^{34} + 74884 q^{35} + 2388204 q^{36} - 112784 q^{37} - 885660 q^{38} - 800456 q^{40} + 1625026 q^{41} + 432108 q^{42} + 318028 q^{43} - 2293924 q^{44} - 565704 q^{45} + 2145020 q^{46} - 1259192 q^{47} + 622944 q^{48} + 7210710 q^{49} + 3824848 q^{50} + 1593648 q^{51} + 3507824 q^{52} - 2889274 q^{53} + 236196 q^{54} - 1264500 q^{55} + 1950892 q^{56} + 3412260 q^{57} + 1955544 q^{58} - 1871796 q^{59} + 3508596 q^{60} - 3141696 q^{61} + 11981128 q^{62} - 982692 q^{63} + 19645236 q^{64} + 6684604 q^{65} - 1133784 q^{66} + 10214522 q^{67} + 8652896 q^{68} - 6421788 q^{69} + 1347128 q^{70} + 236548 q^{71} + 3700404 q^{72} - 5498896 q^{73} - 1538208 q^{74} - 8724024 q^{75} - 14063824 q^{76} - 5203484 q^{77} - 1648620 q^{78} + 14018840 q^{79} - 20047352 q^{80} + 26572050 q^{81} - 8929536 q^{82} - 24175410 q^{83} + 7440120 q^{84} - 4749684 q^{85} - 4763880 q^{87} + 576856 q^{88} + 21142504 q^{89} - 6030288 q^{90} - 5330756 q^{91} + 30444520 q^{92} - 12285216 q^{93} - 2466508 q^{94} - 6641532 q^{95} + 2850984 q^{96} + 4173746 q^{97} + 27677144 q^{98} - 3454002 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(129))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 43
129.8.a.a 129.a 1.a $10$ $40.298$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-1\) \(270\) \(-122\) \(-2052\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3^{3}q^{3}+(37+\beta _{1}+\beta _{2})q^{4}+\cdots\)
129.8.a.b 129.a 1.a $12$ $40.298$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-7\) \(-324\) \(-766\) \(-1366\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3^{3}q^{3}+(58+\beta _{2}+\cdots)q^{4}+\cdots\)
129.8.a.c 129.a 1.a $13$ $40.298$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(9\) \(-351\) \(-266\) \(6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3^{3}q^{3}+(73-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
129.8.a.d 129.a 1.a $15$ $40.298$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(15\) \(405\) \(378\) \(2064\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3^{3}q^{3}+(84-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(129))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(129)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 2}\)