Properties

Label 3.12.a
Level $3$
Weight $12$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $4$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(3))\).

Total New Old
Modular forms 5 1 4
Cusp forms 3 1 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.

\(3\)Dim.
\(+\)\(1\)

Trace form

\( q + 78 q^{2} - 243 q^{3} + 4036 q^{4} - 5370 q^{5} - 18954 q^{6} - 27760 q^{7} + 155064 q^{8} + 59049 q^{9} + O(q^{10}) \) \( q + 78 q^{2} - 243 q^{3} + 4036 q^{4} - 5370 q^{5} - 18954 q^{6} - 27760 q^{7} + 155064 q^{8} + 59049 q^{9} - 418860 q^{10} + 637836 q^{11} - 980748 q^{12} + 766214 q^{13} - 2165280 q^{14} + 1304910 q^{15} + 3829264 q^{16} + 3084354 q^{17} + 4605822 q^{18} - 19511404 q^{19} - 21673320 q^{20} + 6745680 q^{21} + 49751208 q^{22} + 15312360 q^{23} - 37680552 q^{24} - 19991225 q^{25} + 59764692 q^{26} - 14348907 q^{27} - 112039360 q^{28} + 10751262 q^{29} + 101782980 q^{30} - 50937400 q^{31} - 18888480 q^{32} - 154994148 q^{33} + 240579612 q^{34} + 149071200 q^{35} + 238321764 q^{36} + 664740830 q^{37} - 1521889512 q^{38} - 186190002 q^{39} - 832693680 q^{40} + 898833450 q^{41} + 526163040 q^{42} - 957947188 q^{43} + 2574306096 q^{44} - 317093130 q^{45} + 1194364080 q^{46} - 1555741344 q^{47} - 930511152 q^{48} - 1206709143 q^{49} - 1559315550 q^{50} - 749498022 q^{51} + 3092439704 q^{52} + 3792417030 q^{53} - 1119214746 q^{54} - 3425179320 q^{55} - 4304576640 q^{56} + 4741271172 q^{57} + 838598436 q^{58} + 555306924 q^{59} + 5266616760 q^{60} + 4950420998 q^{61} - 3973117200 q^{62} - 1639200240 q^{63} - 9315634112 q^{64} - 4114569180 q^{65} - 12089543544 q^{66} + 5292399284 q^{67} + 12448452744 q^{68} - 3720903480 q^{69} + 11627553600 q^{70} - 14831086248 q^{71} + 9156374136 q^{72} + 13971005210 q^{73} + 51849784740 q^{74} + 4857867675 q^{75} - 78748026544 q^{76} - 17706327360 q^{77} - 14522820156 q^{78} + 3720542360 q^{79} - 20563147680 q^{80} + 3486784401 q^{81} + 70109009100 q^{82} + 8768454036 q^{83} + 27225564480 q^{84} - 16562980980 q^{85} - 74719880664 q^{86} - 2612556666 q^{87} + 98905401504 q^{88} - 25472769174 q^{89} - 24733264140 q^{90} - 21270100640 q^{91} + 61800684960 q^{92} + 12377788200 q^{93} - 121347824832 q^{94} + 104776239480 q^{95} + 4589900640 q^{96} - 39092494846 q^{97} - 94123313154 q^{98} + 37663577964 q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(3))\) 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
3.12.a.a 3.a 1.a $1$ $2.305$ \(\Q\) None \(78\) \(-243\) \(-5370\) \(-27760\) $+$ $\mathrm{SU}(2)$ \(q+78q^{2}-3^{5}q^{3}+4036q^{4}-5370q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)