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Properties

Label 3.30.a
Level $3$
Weight $30$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $2$
Sturm bound $10$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{30}(\Gamma_0(3))\).

Total New Old
Modular forms 11 5 6
Cusp forms 9 5 4
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
\(+\)\(2\)
\(-\)\(3\)

Trace form

\( 5 q - 33666 q^{2} + 4782969 q^{3} + 2317310516 q^{4} + 35680523406 q^{5} + 269788149414 q^{6} + 3367889703880 q^{7} + 7778720393448 q^{8} + 114383962274805 q^{9} + O(q^{10}) \) \( 5 q - 33666 q^{2} + 4782969 q^{3} + 2317310516 q^{4} + 35680523406 q^{5} + 269788149414 q^{6} + 3367889703880 q^{7} + 7778720393448 q^{8} + 114383962274805 q^{9} + 171907890247764 q^{10} + 1366144892304492 q^{11} + 9052308753172596 q^{12} - 12361315731947402 q^{13} - 226466432160780432 q^{14} + 168864135184609254 q^{15} + 1383871481050014224 q^{16} + 768762053759550042 q^{17} - 770170094788717026 q^{18} - 4254151285566720908 q^{19} + 21441848131981844664 q^{20} - 9183599600261630904 q^{21} - 4977228299716913976 q^{22} + 1139319855427683480 q^{23} - 31175841519160870104 q^{24} + 694299657773167892171 q^{25} - 1317631965027289742460 q^{26} + 109418989131512359209 q^{27} - 1111552332222024139040 q^{28} - 573727225266369721626 q^{29} - 4024708806292033917564 q^{30} + 874575436715357515744 q^{31} + 21632826338162898164640 q^{32} - 4786770519802149245316 q^{33} + 62778493618984917998172 q^{34} - 101496238049117183741520 q^{35} + 53012631728230581669876 q^{36} - 147536219660404225059890 q^{37} + 11173433641677487992696 q^{38} + 147700288454792470991694 q^{39} + 190619318209073462111088 q^{40} + 258354660989087410026354 q^{41} - 69036826488487354584720 q^{42} - 1139190453328713361541876 q^{43} - 2700808870924343843995920 q^{44} + 816255928643440161317166 q^{45} + 4307698246997613179416464 q^{46} - 2046083847613299533692752 q^{47} + 7731512664581516318537616 q^{48} + 7716540176957078860188093 q^{49} - 27779742358441462465090206 q^{50} + 6579623188098677211684930 q^{51} + 26920485456487672207829272 q^{52} - 12682341544407995141709570 q^{53} + 6171887500952086133542854 q^{54} + 9614686856738973448034376 q^{55} - 151546885676638941943728960 q^{56} + 54533630073316307689431396 q^{57} + 62309777277870418889785572 q^{58} - 141949687119373427106547812 q^{59} + 322471929035301163167005496 q^{60} + 229031641409934422545419238 q^{61} - 511290685975746871273745280 q^{62} + 77046513766862820526948680 q^{63} + 74885603748557304530177600 q^{64} - 192344861799374044671037212 q^{65} + 80607976862733238471344360 q^{66} + 134076035712199162011310372 q^{67} + 74014674281940772548762408 q^{68} + 16126077307043010049778808 q^{69} + 440687991555250165196710560 q^{70} + 103908489604981772087058120 q^{71} + 177952172006082467739495528 q^{72} - 667898688881482633186946990 q^{73} - 1114072846287045694094746092 q^{74} - 2296949022983924978296193001 q^{75} - 2630801882986850322878187632 q^{76} + 3232213895350288890376395360 q^{77} - 8025127102096756434155722572 q^{78} + 2176133411918622187383634960 q^{79} + 29515119880053891203760596064 q^{80} + 2616738165136802686067557605 q^{81} - 22434006880815823798209744660 q^{82} + 35005942823392994902796883252 q^{83} - 28025681310219792325631719968 q^{84} - 29038992201508758500243727108 q^{85} - 4181889596778973632312603768 q^{86} + 12078163448987889439785924702 q^{87} - 8315606904626810628324302112 q^{88} - 5975598877697204259248128158 q^{89} + 3932701126568311148100957204 q^{90} + 60722805575186877699993847664 q^{91} - 25992440511160073345040324000 q^{92} - 45419872256566033528410261120 q^{93} + 34317595302438574344228368544 q^{94} - 33252863635760230502810983176 q^{95} + 77106242152821220141026100896 q^{96} - 118517998798721882645128790678 q^{97} + 17295844568260655533195668942 q^{98} + 31253013164654910497407984812 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(3))\) 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
3.30.a.a 3.a 1.a $2$ $15.983$ \(\Q(\sqrt{77089}) \) None \(-45036\) \(-9565938\) \(187613820\) \(26\!\cdots\!48\) $+$ $\mathrm{SU}(2)$ \(q+(-22518-\beta )q^{2}-3^{14}q^{3}+(106174408+\cdots)q^{4}+\cdots\)
3.30.a.b 3.a 1.a $3$ $15.983$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(11370\) \(14348907\) \(35492909586\) \(723913582632\) $-$ $\mathrm{SU}(2)$ \(q+(3790-\beta _{1})q^{2}+3^{14}q^{3}+(701653900+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{30}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{30}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)