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Properties

Label 2.30.a
Level $2$
Weight $30$
Character orbit 2.a
Rep. character $\chi_{2}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

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

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
2.30.a.a 2.a 1.a $1$ $10.656$ \(\Q\) None \(-16384\) \(-2792556\) \(6651856470\) \(14\!\cdots\!48\) $+$ $\mathrm{SU}(2)$ \(q-2^{14}q^{2}-2792556q^{3}+2^{28}q^{4}+\cdots\)
2.30.a.b 2.a 1.a $1$ $10.656$ \(\Q\) None \(16384\) \(4782996\) \(6065841750\) \(904018883432\) $-$ $\mathrm{SU}(2)$ \(q+2^{14}q^{2}+4782996q^{3}+2^{28}q^{4}+\cdots\)

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

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