Properties

Label 2.30.a
Level 2
Weight 30
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 2
Sturm bound 7
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 30 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(7\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{30}(\Gamma_0(2))\).

Total New Old
Modular forms 8 2 6
Cusp forms 6 2 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.

\(2\)Dim.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\(2q \) \(\mathstrut +\mathstrut 1990440q^{3} \) \(\mathstrut +\mathstrut 536870912q^{4} \) \(\mathstrut +\mathstrut 12717698220q^{5} \) \(\mathstrut +\mathstrut 124117843968q^{6} \) \(\mathstrut +\mathstrut 2336537360080q^{7} \) \(\mathstrut -\mathstrut 106585334980614q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 1990440q^{3} \) \(\mathstrut +\mathstrut 536870912q^{4} \) \(\mathstrut +\mathstrut 12717698220q^{5} \) \(\mathstrut +\mathstrut 124117843968q^{6} \) \(\mathstrut +\mathstrut 2336537360080q^{7} \) \(\mathstrut -\mathstrut 106585334980614q^{9} \) \(\mathstrut -\mathstrut 9601265172480q^{10} \) \(\mathstrut +\mathstrut 1570988620558584q^{11} \) \(\mathstrut +\mathstrut 534304669040640q^{12} \) \(\mathstrut -\mathstrut 11092528397458820q^{13} \) \(\mathstrut -\mathstrut 8658937335250944q^{14} \) \(\mathstrut +\mathstrut 10437215130445680q^{15} \) \(\mathstrut +\mathstrut 144115188075855872q^{16} \) \(\mathstrut +\mathstrut 87866546144448420q^{17} \) \(\mathstrut +\mathstrut 247049121347665920q^{18} \) \(\mathstrut -\mathstrut 4477051823515048760q^{19} \) \(\mathstrut +\mathstrut 3413881120956088320q^{20} \) \(\mathstrut +\mathstrut 323530636305489984q^{21} \) \(\mathstrut +\mathstrut 51200554907593605120q^{22} \) \(\mathstrut -\mathstrut 95907354703812009360q^{23} \) \(\mathstrut +\mathstrut 33317630043286929408q^{24} \) \(\mathstrut -\mathstrut 291487399212667482850q^{25} \) \(\mathstrut +\mathstrut 706133570096965091328q^{26} \) \(\mathstrut -\mathstrut 185565836660000868720q^{27} \) \(\mathstrut +\mathstrut 627209471714110996480q^{28} \) \(\mathstrut -\mathstrut 2555257587077173639140q^{29} \) \(\mathstrut +\mathstrut 779691270526080122880q^{30} \) \(\mathstrut +\mathstrut 169166581849494746944q^{31} \) \(\mathstrut +\mathstrut 13400407582651610552160q^{33} \) \(\mathstrut -\mathstrut 18998464053346465480704q^{34} \) \(\mathstrut +\mathstrut 15012542783195751598560q^{35} \) \(\mathstrut -\mathstrut 28611282998433870249984q^{36} \) \(\mathstrut +\mathstrut 18088105620268138813420q^{37} \) \(\mathstrut -\mathstrut 74118708338195567738880q^{38} \) \(\mathstrut +\mathstrut 152209748625073150210992q^{39} \) \(\mathstrut -\mathstrut 2577319994751587450880q^{40} \) \(\mathstrut +\mathstrut 26593814015979466568724q^{41} \) \(\mathstrut +\mathstrut 136385442127117591511040q^{42} \) \(\mathstrut -\mathstrut 865229292465291154625480q^{43} \) \(\mathstrut +\mathstrut 421709046730454470754304q^{44} \) \(\mathstrut -\mathstrut 682178227204430521550340q^{45} \) \(\mathstrut +\mathstrut 1769728347835643449049088q^{46} \) \(\mathstrut +\mathstrut 998249982188035700256480q^{47} \) \(\mathstrut +\mathstrut 143426317476853280931840q^{48} \) \(\mathstrut -\mathstrut 3570452184086812928260686q^{49} \) \(\mathstrut -\mathstrut 122105992993796888985600q^{50} \) \(\mathstrut -\mathstrut 4304760864279158031767856q^{51} \) \(\mathstrut -\mathstrut 2977627918564807587921920q^{52} \) \(\mathstrut +\mathstrut 26074981958386252391645580q^{53} \) \(\mathstrut -\mathstrut 14886957230892313498091520q^{54} \) \(\mathstrut +\mathstrut 9074021667830402435695440q^{55} \) \(\mathstrut -\mathstrut 2324365792063512027070464q^{56} \) \(\mathstrut -\mathstrut 21590970399735234643559520q^{57} \) \(\mathstrut -\mathstrut 35513498882505347495362560q^{58} \) \(\mathstrut +\mathstrut 49041446699913876612175320q^{59} \) \(\mathstrut +\mathstrut 2801718602911285594030080q^{60} \) \(\mathstrut -\mathstrut 27848119893603351461348516q^{61} \) \(\mathstrut +\mathstrut 147333983768781360086384640q^{62} \) \(\mathstrut -\mathstrut 128504847187810963788735600q^{63} \) \(\mathstrut +\mathstrut 38685626227668133590597632q^{64} \) \(\mathstrut -\mathstrut 83164030562052343270687320q^{65} \) \(\mathstrut +\mathstrut 148449676496132250645037056q^{66} \) \(\mathstrut -\mathstrut 367875641163928062259190360q^{67} \) \(\mathstrut +\mathstrut 23586496381430053491179520q^{68} \) \(\mathstrut +\mathstrut 313690154866925031921776832q^{69} \) \(\mathstrut -\mathstrut 66277733357573469414359040q^{70} \) \(\mathstrut +\mathstrut 270407262189016411897648464q^{71} \) \(\mathstrut +\mathstrut 66316743543360035770859520q^{72} \) \(\mathstrut +\mathstrut 1107225994900076724981834580q^{73} \) \(\mathstrut -\mathstrut 1947095847566123849087975424q^{74} \) \(\mathstrut -\mathstrut 318323468699684350554335400q^{75} \) \(\mathstrut -\mathstrut 1201799447780893636752834560q^{76} \) \(\mathstrut +\mathstrut 1009547237855384554720201920q^{77} \) \(\mathstrut +\mathstrut 14367897208700143958753280q^{78} \) \(\mathstrut +\mathstrut 2042244860690974310098032160q^{79} \) \(\mathstrut +\mathstrut 916406735433638724155473920q^{80} \) \(\mathstrut +\mathstrut 3688634504504320503801826482q^{81} \) \(\mathstrut +\mathstrut 4715128091077280318367989760q^{82} \) \(\mathstrut -\mathstrut 19254028694951877925883196600q^{83} \) \(\mathstrut +\mathstrut 86847093886634359158472704q^{84} \) \(\mathstrut +\mathstrut 898493890263434618998446360q^{85} \) \(\mathstrut +\mathstrut 7292895364293136107970756608q^{86} \) \(\mathstrut -\mathstrut 10753320478710149773229042640q^{87} \) \(\mathstrut +\mathstrut 13744044304072927253071134720q^{88} \) \(\mathstrut +\mathstrut 24020566427012643326985883380q^{89} \) \(\mathstrut +\mathstrut 2082625117731129163052482560q^{90} \) \(\mathstrut -\mathstrut 24347948301454207535800248736q^{91} \) \(\mathstrut -\mathstrut 25744934493671521670827868160q^{92} \) \(\mathstrut +\mathstrut 34230133338133394847024656640q^{93} \) \(\mathstrut -\mathstrut 45362773861560443655614889984q^{94} \) \(\mathstrut -\mathstrut 27143376553139339894582408400q^{95} \) \(\mathstrut +\mathstrut 8943633213509026634476290048q^{96} \) \(\mathstrut +\mathstrut 130930662883674339980685698500q^{97} \) \(\mathstrut -\mathstrut 20231930582405390618087915520q^{98} \) \(\mathstrut -\mathstrut 60161478645516500791725626088q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.30.a.a \(1\) \(10.656\) \(\Q\) None \(-16384\) \(-2792556\) \(6651856470\) \(14\!\cdots\!48\) \(+\) \(q-2^{14}q^{2}-2792556q^{3}+2^{28}q^{4}+\cdots\)
2.30.a.b \(1\) \(10.656\) \(\Q\) None \(16384\) \(4782996\) \(6065841750\) \(904018883432\) \(-\) \(q+2^{14}q^{2}+4782996q^{3}+2^{28}q^{4}+\cdots\)

Decomposition of \(S_{30}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

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