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Properties

Label 3.20.a
Level $3$
Weight $20$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $6$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(3))\).

Total New Old
Modular forms 7 3 4
Cusp forms 5 3 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
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3 q - 402 q^{2} - 19683 q^{3} + 1467012 q^{4} + 9532410 q^{5} - 35547498 q^{6} - 81698520 q^{7} + 820556664 q^{8} + 1162261467 q^{9} + O(q^{10}) \) \( 3 q - 402 q^{2} - 19683 q^{3} + 1467012 q^{4} + 9532410 q^{5} - 35547498 q^{6} - 81698520 q^{7} + 820556664 q^{8} + 1162261467 q^{9} + 4780280340 q^{10} - 9396929220 q^{11} - 1534407948 q^{12} - 88472801406 q^{13} + 155230785312 q^{14} - 49204941210 q^{15} - 37037839344 q^{16} - 449701045866 q^{17} - 155743036578 q^{18} + 917529390276 q^{19} + 9364339167000 q^{20} - 6091546960584 q^{21} - 16615928152152 q^{22} + 7268812701720 q^{23} - 23549658112872 q^{24} + 491067463125 q^{25} + 96507401354196 q^{26} - 7625597484987 q^{27} - 162528726930240 q^{28} + 293165772642642 q^{29} - 246907577173500 q^{30} - 1103509347456 q^{31} + 59063350272480 q^{32} + 76826947856292 q^{33} + 68377350274524 q^{34} - 763971710819280 q^{35} + 568350506408868 q^{36} + 214541768601690 q^{37} - 1772527797374952 q^{38} - 6457777952130 q^{39} + 3082572607358160 q^{40} + 36117739384494 q^{41} + 5444971751141280 q^{42} + 3956619191676252 q^{43} - 16630934170959312 q^{44} + 3693050943548490 q^{45} - 11390312842270416 q^{46} - 4071719279051664 q^{47} - 5437331931370032 q^{48} + 16981598133894843 q^{49} + 49343632177008450 q^{50} - 22089522102612246 q^{51} - 3662935255751016 q^{52} + 9478548542794410 q^{53} - 13771829057886522 q^{54} - 101689996818710520 q^{55} + 83281437047844480 q^{56} - 5643282616526628 q^{57} + 61636794482767716 q^{58} + 37546964215826604 q^{59} - 88180690814996040 q^{60} - 65839606734261198 q^{61} + 28961995987437360 q^{62} - 31651680568976280 q^{63} - 253827091361167296 q^{64} - 26788465728644100 q^{65} + 446429936037796488 q^{66} + 767364718585772724 q^{67} - 1060003104102450936 q^{68} + 49843425642458472 q^{69} + 802706080735753920 q^{70} - 28358482901250744 q^{71} + 317900464019088696 q^{72} - 787389978479424690 q^{73} - 1150804422931511388 q^{74} - 273785229241636725 q^{75} - 696607169913035184 q^{76} + 1265570460418319520 q^{77} + 30091011686536644 q^{78} - 676018458956583120 q^{79} + 950313005686646880 q^{80} + 450283905890997363 q^{81} + 3675019929537285900 q^{82} + 3077702007691905156 q^{83} - 2148548003869326912 q^{84} - 5568478266829874220 q^{85} + 1640978749090426920 q^{86} - 5290568598145265586 q^{87} - 4713152322325163616 q^{88} + 7296838195168516446 q^{89} + 1851978546879886260 q^{90} + 2149235336559905904 q^{91} - 463273104918015840 q^{92} - 1659745510007603040 q^{93} + 3793350947439764160 q^{94} - 3884054656531697160 q^{95} + 9524122810436974176 q^{96} + 976515183972078054 q^{97} - 44582425294272288354 q^{98} - 3640562913510788580 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\) 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
3.20.a.a 3.a 1.a $1$ $6.865$ \(\Q\) None \(-1104\) \(19683\) \(3516270\) \(-195590584\) $-$ $\mathrm{SU}(2)$ \(q-1104q^{2}+3^{9}q^{3}+694528q^{4}+\cdots\)
3.20.a.b 3.a 1.a $2$ $6.865$ \(\Q(\sqrt{87481}) \) None \(702\) \(-39366\) \(6016140\) \(113892064\) $+$ $\mathrm{SU}(2)$ \(q+(351-\beta )q^{2}-3^{9}q^{3}+(386242-702\beta )q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)