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Properties

Label 9.42.a
Level $9$
Weight $42$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $4$
Sturm bound $42$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(42\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_0(9))\).

Total New Old
Modular forms 43 17 26
Cusp forms 39 16 23
Eisenstein series 4 1 3

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
\(+\)\(6\)
\(-\)\(10\)

Trace form

\( 16 q + 703890 q^{2} + 15669452695636 q^{4} + 55114840313244 q^{5} - 404054215348451680 q^{7} - 4991827485626157960 q^{8} + O(q^{10}) \) \( 16 q + 703890 q^{2} + 15669452695636 q^{4} + 55114840313244 q^{5} - 404054215348451680 q^{7} - 4991827485626157960 q^{8} + 163211148629855731764 q^{10} + 176866704254081061504 q^{11} + 47122715641609379037800 q^{13} - 150347775316232906164944 q^{14} + 17040789779862682441409296 q^{16} + 13639238321984595808703100 q^{17} + 27731901886320922799259920 q^{19} - 82261029296156301454942584 q^{20} - 10286723561410505351864927160 q^{22} + 17248014880461358609368832560 q^{23} + 70451493273491440010163432376 q^{25} + 385551140291104523114310323580 q^{26} - 1536342726991536275887437754720 q^{28} + 1990390773840471115126212345948 q^{29} - 4978434427451554480647808651360 q^{31} + 4390782332660488525704670522080 q^{32} + 83088712462324021908552282282876 q^{34} - 89392686757381230360691382565840 q^{35} - 94121197338713928855336954919840 q^{37} - 699218370363215149125655755857400 q^{38} + 480351570085866995657976777895728 q^{40} + 1203379011994541296559785644821868 q^{41} + 2414090595432333982084551013763600 q^{43} + 13109945793401943098733180812346000 q^{44} - 40952416151605814594708057617349808 q^{46} + 7404252560718551567781557700809520 q^{47} + 116568960041912975161137562126333872 q^{49} - 98211789355697913435910402326454674 q^{50} + 161832173737680449043996601825861400 q^{52} - 210295495941724842395249014126502580 q^{53} + 253473529215267766531370996120631456 q^{55} + 135052615227713575064286735193786560 q^{56} - 3678641858273074608960433967196080700 q^{58} + 6199462728286021551760791869317918896 q^{59} + 13363296122188112068993700160883016576 q^{61} - 2369166319411175196732684379802133600 q^{62} - 19785134037439578080420637295814280128 q^{64} - 16239651318539517257247514680383617368 q^{65} + 15947975743355959050643981871675963600 q^{67} - 39655489300602076413495881940044419560 q^{68} + 3054777004348470256651106036347703520 q^{70} + 75618162139217821540101999150115310640 q^{71} - 133192862524186177739598364128123544600 q^{73} + 541137435440381527270129755772648964076 q^{74} - 455806005912645718520260647164615901424 q^{76} + 550649904189508906687498367531631569760 q^{77} - 1517697590252497803818848473905074481344 q^{79} + 167942846407394817290048053952123745696 q^{80} + 4064618793793323357647032021341338678220 q^{82} + 281207724683527620172914917481349048800 q^{83} - 880312264287177941558134619347690771608 q^{85} + 14795770612975446756150336372891174818424 q^{86} - 22930655145693860342418012855050883403680 q^{88} + 42643574700882764763262008799736757875484 q^{89} - 30426545232122682910907363592425358533408 q^{91} + 122556958895333291985705257860502073987360 q^{92} - 50784565610823579943120524695615306473248 q^{94} + 109880274566018640147286276996994551814496 q^{95} + 9777285769647952377382022571637741980680 q^{97} + 375063069815000313958824243931478834787330 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.42.a.a 9.a 1.a $3$ $95.825$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(289380\) \(0\) \(-38\!\cdots\!26\) \(-44\!\cdots\!68\) $-$ $\mathrm{SU}(2)$ \(q+(96460+\beta _{1})q^{2}+(-751422059600+\cdots)q^{4}+\cdots\)
9.42.a.b 9.a 1.a $3$ $95.825$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(344688\) \(0\) \(21\!\cdots\!50\) \(57\!\cdots\!92\) $-$ $\mathrm{SU}(2)$ \(q+(114896-\beta _{1})q^{2}+(2090568301312+\cdots)q^{4}+\cdots\)
9.42.a.c 9.a 1.a $4$ $95.825$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(69822\) \(0\) \(-11\!\cdots\!80\) \(15\!\cdots\!36\) $-$ $\mathrm{SU}(2)$ \(q+(17455+\beta _{1})q^{2}+(1338237117038+\cdots)q^{4}+\cdots\)
9.42.a.d 9.a 1.a $6$ $95.825$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-56\!\cdots\!40\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1049844396928+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{42}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{42}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{42}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)