Properties

Label 75.22.a
Level $75$
Weight $22$
Character orbit 75.a
Rep. character $\chi_{75}(1,\cdot)$
Character field $\Q$
Dimension $66$
Newform subspaces $14$
Sturm bound $220$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(220\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(75))\).

Total New Old
Modular forms 216 66 150
Cusp forms 204 66 138
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(18\)
\(-\)\(+\)\(-\)\(17\)
\(-\)\(-\)\(+\)\(16\)
Plus space\(+\)\(31\)
Minus space\(-\)\(35\)

Trace form

\( 66 q + 1598 q^{2} + 68549728 q^{4} - 105225318 q^{6} + 559598932 q^{7} + 6582404412 q^{8} + 230127770466 q^{9} + O(q^{10}) \) \( 66 q + 1598 q^{2} + 68549728 q^{4} - 105225318 q^{6} + 559598932 q^{7} + 6582404412 q^{8} + 230127770466 q^{9} - 140550939936 q^{11} + 212790629772 q^{12} + 476755719848 q^{13} + 5951926495932 q^{14} + 74654476857140 q^{16} + 2895711111680 q^{17} + 5571881472798 q^{18} + 20854110608092 q^{19} - 53270783960796 q^{21} - 305003876572940 q^{22} + 332917865015616 q^{23} - 146507777465724 q^{24} - 576566152075128 q^{26} + 5134394163382324 q^{28} - 4640335547701836 q^{29} + 1051949236026996 q^{31} - 11777670119768948 q^{32} - 5075328150422316 q^{33} + 5320941144924676 q^{34} + 239018122283192928 q^{36} + 43291387140114432 q^{37} + 1762108332088064 q^{38} - 41884285798323612 q^{39} - 8589059150896524 q^{41} - 276597377309387484 q^{42} + 337242287299800512 q^{43} - 518361176774084976 q^{44} - 520177620234876064 q^{46} + 41641218071661944 q^{47} + 757307119626960768 q^{48} + 6307843855416529270 q^{49} - 456369712891637472 q^{51} + 3678055801729764120 q^{52} - 4772255569432856824 q^{53} - 366897997392664518 q^{54} + 8305011149862147480 q^{56} - 1235155990591987416 q^{57} - 13003977897019306552 q^{58} - 9360986021400428352 q^{59} + 16978528007485279200 q^{61} + 23551546864273794264 q^{62} + 1951200826913859732 q^{63} + 79397858551386240332 q^{64} + 5266431451152350856 q^{66} + 48974675829821969080 q^{67} + 70650292973404157056 q^{68} - 31248085931695120968 q^{69} + 19836002127692491872 q^{71} + 22951425024835177212 q^{72} - 62504040788772941884 q^{73} - 320787328791694921068 q^{74} + 356693975412702702128 q^{76} + 562671241634060126784 q^{77} - 243727291343685212820 q^{78} - 454274786635617948080 q^{79} + 802405920297757300866 q^{81} + 861244093915636154684 q^{82} - 71078281851982135896 q^{83} - 304707361620478730472 q^{84} - 901966438206279205596 q^{86} + 674271097049404924308 q^{87} - 678426143664569738508 q^{88} - 2210353435062926891148 q^{89} - 1359502168448881585004 q^{91} + 4401063710813395972032 q^{92} + 103186520604806078808 q^{93} - 2381241412886494240592 q^{94} - 3754707195708424128996 q^{96} + 1109641272603148372380 q^{97} - 4335613966611832465538 q^{98} - 490070824914732738336 q^{99} + O(q^{100}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\) 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 5
75.22.a.a 75.a 1.a $1$ $209.608$ \(\Q\) None \(-1728\) \(59049\) \(0\) \(-538429808\) $-$ $+$ $\mathrm{SU}(2)$ \(q-12^{3}q^{2}+3^{10}q^{3}+888832q^{4}+\cdots\)
75.22.a.b 75.a 1.a $1$ $209.608$ \(\Q\) None \(-544\) \(-59049\) \(0\) \(-1277698380\) $+$ $+$ $\mathrm{SU}(2)$ \(q-544q^{2}-3^{10}q^{3}-1801216q^{4}+\cdots\)
75.22.a.c 75.a 1.a $1$ $209.608$ \(\Q\) None \(2844\) \(59049\) \(0\) \(-363303920\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2844q^{2}+3^{10}q^{3}+5991184q^{4}+\cdots\)
75.22.a.d 75.a 1.a $2$ $209.608$ \(\Q(\sqrt{649}) \) None \(-666\) \(-118098\) \(0\) \(-679896112\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-333-\beta )q^{2}-3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)
75.22.a.e 75.a 1.a $3$ $209.608$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-803\) \(-177147\) \(0\) \(1577598316\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-268+\beta _{1})q^{2}-3^{10}q^{3}+(2238318+\cdots)q^{4}+\cdots\)
75.22.a.f 75.a 1.a $3$ $209.608$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-702\) \(177147\) \(0\) \(2072418204\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-234-\beta _{1})q^{2}+3^{10}q^{3}+(1748076+\cdots)q^{4}+\cdots\)
75.22.a.g 75.a 1.a $3$ $209.608$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(2300\) \(-177147\) \(0\) \(-465666872\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(767+\beta _{1})q^{2}-3^{10}q^{3}+(421908+\cdots)q^{4}+\cdots\)
75.22.a.h 75.a 1.a $4$ $209.608$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(897\) \(236196\) \(0\) \(234577504\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(224+\beta _{1})q^{2}+3^{10}q^{3}+(-290854+\cdots)q^{4}+\cdots\)
75.22.a.i 75.a 1.a $6$ $209.608$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-92\) \(354294\) \(0\) \(-1327143454\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-15+\beta _{1})q^{2}+3^{10}q^{3}+(731822+\cdots)q^{4}+\cdots\)
75.22.a.j 75.a 1.a $6$ $209.608$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(92\) \(-354294\) \(0\) \(1327143454\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(15-\beta _{1})q^{2}-3^{10}q^{3}+(731822+\cdots)q^{4}+\cdots\)
75.22.a.k 75.a 1.a $8$ $209.608$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-666\) \(472392\) \(0\) \(-134034472\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-83-\beta _{1})q^{2}+3^{10}q^{3}+(1335201+\cdots)q^{4}+\cdots\)
75.22.a.l 75.a 1.a $8$ $209.608$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(666\) \(-472392\) \(0\) \(134034472\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(83+\beta _{1})q^{2}-3^{10}q^{3}+(1335201+\cdots)q^{4}+\cdots\)
75.22.a.m 75.a 1.a $10$ $209.608$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-645\) \(590490\) \(0\) \(-115357290\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-65+\beta _{1})q^{2}+3^{10}q^{3}+(1004337+\cdots)q^{4}+\cdots\)
75.22.a.n 75.a 1.a $10$ $209.608$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(645\) \(-590490\) \(0\) \(115357290\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(65-\beta _{1})q^{2}-3^{10}q^{3}+(1004337+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)