Properties

Label 3.16.a
Level $3$
Weight $16$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $5$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(5\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(3))\).

Total New Old
Modular forms 6 2 4
Cusp forms 4 2 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.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\( 2 q - 306 q^{2} - 5596 q^{4} + 59220 q^{5} + 354294 q^{6} - 3522344 q^{7} + 6867864 q^{8} + 9565938 q^{9} + O(q^{10}) \) \( 2 q - 306 q^{2} - 5596 q^{4} + 59220 q^{5} + 354294 q^{6} - 3522344 q^{7} + 6867864 q^{8} + 9565938 q^{9} - 49738860 q^{10} + 71200368 q^{11} - 108413964 q^{12} + 104047996 q^{13} + 476090496 q^{14} - 1098311400 q^{15} - 719764720 q^{16} + 3752120340 q^{17} - 1463588514 q^{18} - 7168853264 q^{19} + 12281831640 q^{20} - 1696359672 q^{21} - 10639456920 q^{22} - 7183099728 q^{23} + 3986516088 q^{24} + 66820767950 q^{25} - 69703062156 q^{26} + 29080928128 q^{28} - 164366907948 q^{29} + 178532289540 q^{30} + 74360242552 q^{31} + 39122031456 q^{32} + 6863383368 q^{33} - 474144293412 q^{34} + 90470615760 q^{35} - 26765494524 q^{36} + 414541609276 q^{37} + 921006940248 q^{38} - 1452160406736 q^{39} - 254353653360 q^{40} + 1229001516612 q^{41} - 364429642752 q^{42} - 703402249136 q^{43} - 277003641168 q^{44} + 283247424180 q^{45} - 1310959874448 q^{46} + 3596817155472 q^{47} + 4159193314896 q^{48} - 2990848277550 q^{49} - 12632540500350 q^{50} + 2698113202416 q^{51} + 16166691650200 q^{52} - 9402324365628 q^{53} + 1694577218886 q^{54} + 1320224806080 q^{55} - 12802431962880 q^{56} - 4747345446888 q^{57} + 19677397654116 q^{58} + 30487277052864 q^{59} - 137062176840 q^{60} - 51165426961940 q^{61} - 26502594515568 q^{62} - 16847262159336 q^{63} - 15529712956864 q^{64} + 169810389342360 q^{65} + 11562833934792 q^{66} - 68739036837440 q^{67} - 41077049005368 q^{68} - 65069301586464 q^{69} + 55938961532160 q^{70} - 74160484607376 q^{71} + 32848780608216 q^{72} + 99993522076852 q^{73} + 2589181575876 q^{74} - 65042001108000 q^{75} + 73861699830736 q^{76} - 126613201161888 q^{77} + 240612332578020 q^{78} + 104253009464920 q^{79} - 498849243588000 q^{80} + 45753584909922 q^{81} - 198509068516788 q^{82} + 435753307340208 q^{83} + 195682052168064 q^{84} - 198683084417400 q^{85} + 5368846365288 q^{86} - 147709960072056 q^{87} + 247357473555744 q^{88} - 142972544516364 q^{89} - 237899425475340 q^{90} + 74270028249872 q^{91} + 757550397685536 q^{92} - 408387889938024 q^{93} - 505319834071296 q^{94} + 332795843199360 q^{95} - 894490897204512 q^{96} + 163536037017700 q^{97} + 678902094335934 q^{98} + 340549152932592 q^{99} + O(q^{100}) \)

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
3.16.a.a $1$ $4.281$ \(\Q\) None \(-234\) \(-2187\) \(280710\) \(-1373344\) $+$ \(q-234q^{2}-3^{7}q^{3}+21988q^{4}+280710q^{5}+\cdots\)
3.16.a.b $1$ $4.281$ \(\Q\) None \(-72\) \(2187\) \(-221490\) \(-2149000\) $-$ \(q-72q^{2}+3^{7}q^{3}-27584q^{4}-221490q^{5}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)