LMFDB - Embedded newform 1.66.a.a.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1,66,Mod(1,1)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1.1"); S:= CuspForms(chi, 66); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 66, names="a")

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 66 $
Character orbit:	$[\chi]$	$=$	1.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7572356472$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} + \cdots - 10\!\cdots\!36 $ x^5 - x^4 - 64723040936454512x^3 - 84407031146177217128088x^2 + 970644106288906308951592802733936*x - 10693396258963037598644570162849209944336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.84762e8$ of defining polynomial
Character	$\chi$	$=$	1.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.07662e9 q^{2} +2.82746e15 q^{3} +2.83384e19 q^{4} -4.30990e22 q^{5} +2.28363e25 q^{6} -4.31488e27 q^{7} -6.90965e28 q^{8} -2.30651e30 q^{9} -3.48095e32 q^{10} -3.09454e33 q^{11} +8.01256e34 q^{12} -1.30257e36 q^{13} -3.48497e37 q^{14} -1.21861e38 q^{15} -1.60357e39 q^{16} +1.01238e40 q^{17} -1.86288e40 q^{18} +6.77371e41 q^{19} -1.22136e42 q^{20} -1.22002e43 q^{21} -2.49934e43 q^{22} +2.44678e44 q^{23} -1.95368e44 q^{24} -8.52980e44 q^{25} -1.05204e46 q^{26} -3.56474e46 q^{27} -1.22277e47 q^{28} +6.24422e46 q^{29} -9.84224e47 q^{30} -4.23499e48 q^{31} -1.04022e49 q^{32} -8.74970e48 q^{33} +8.17660e49 q^{34} +1.85967e50 q^{35} -6.53627e49 q^{36} -6.58889e50 q^{37} +5.47087e51 q^{38} -3.68296e51 q^{39} +2.97799e51 q^{40} -3.22294e51 q^{41} -9.85361e52 q^{42} -7.33033e52 q^{43} -8.76942e52 q^{44} +9.94083e52 q^{45} +1.97617e54 q^{46} +4.96764e53 q^{47} -4.53403e54 q^{48} +1.00799e55 q^{49} -6.88920e54 q^{50} +2.86246e55 q^{51} -3.69126e55 q^{52} -1.14911e56 q^{53} -2.87911e56 q^{54} +1.33372e56 q^{55} +2.98143e56 q^{56} +1.91524e57 q^{57} +5.04323e56 q^{58} -7.39266e56 q^{59} -3.45334e57 q^{60} +1.27879e58 q^{61} -3.42044e58 q^{62} +9.95231e57 q^{63} -2.48535e58 q^{64} +5.61394e58 q^{65} -7.06680e58 q^{66} -1.92730e59 q^{67} +2.86892e59 q^{68} +6.91818e59 q^{69} +1.50199e60 q^{70} -1.96706e60 q^{71} +1.59372e59 q^{72} -4.77567e60 q^{73} -5.32160e60 q^{74} -2.41177e60 q^{75} +1.91956e61 q^{76} +1.33526e61 q^{77} -2.97459e61 q^{78} +1.98178e61 q^{79} +6.91122e61 q^{80} -7.70322e61 q^{81} -2.60304e61 q^{82} +3.39503e62 q^{83} -3.45732e62 q^{84} -4.36325e62 q^{85} -5.92043e62 q^{86} +1.76553e62 q^{87} +2.13822e62 q^{88} +8.38014e62 q^{89} +8.02883e62 q^{90} +5.62042e63 q^{91} +6.93378e63 q^{92} -1.19743e64 q^{93} +4.01217e63 q^{94} -2.91940e64 q^{95} -2.94118e64 q^{96} +1.21487e64 q^{97} +8.14112e64 q^{98} +7.13759e63 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3959709648 q^{2} - 22\!\cdots\!04 q^{3} + 11\!\cdots\!60 q^{4} + 26\!\cdots\!50 q^{5} - 33\!\cdots\!40 q^{6} - 69\!\cdots\!08 q^{7} - 44\!\cdots\!20 q^{8} + 31\!\cdots\!65 q^{9} - 46\!\cdots\!00 q^{10}+ \cdots - 36\!\cdots\!20 q^{99}+O(q^{100}) $ 5 * q - 3959709648 * q^2 - 2231307434935404 * q^3 + 116912191997379733760 * q^4 + 26879673327548389029150 * q^5 - 33295729280564090947602240 * q^6 - 6923148241844308586296558808 * q^7 - 446887903591926627901545861120 * q^8 + 31912530537586467884680398342465 * q^9 - 46444835254758466872049375039200 * q^10 - 5436990447821613455266185060489540 * q^11 + 195800126486592722298045722544448512 * q^12 - 2890859300726790832888435764458467514 * q^13 + 41180574237900302034722115163160949120 * q^14 + 103408365234403422747890968078751873400 * q^15 - 1473868696955444538230489908858040811520 * q^16 + 13864680686430637673540501618888491948122 * q^17 - 52766923382337010938622933627608214498704 * q^18 - 532140170263349448171574488831326440156700 * q^19 + 3499008192062174379356405538388976603788800 * q^20 - 11546436905526557006526119132251889169971040 * q^21 + 22561892587607883403404755239099221639703104 * q^22 - 94948785027142700354971234970110107282841224 * q^23 - 2324009070474316353878252433966072715423334400 * q^24 - 89368364701600219577770390611457383673793125 * q^25 - 30521938596122123780659173092206061048452902240 * q^26 - 129706546221561841046313530503436938966984152120 * q^27 - 518698434811712968791028896222612623314185598976 * q^28 - 1144597713377055966543673581436050616842314015850 * q^29 - 8898165260141424207772350064014611180432376163200 * q^30 - 4708831653623167525869999868687533742455793996640 * q^31 - 23840662820937316611832895513659338085021120462848 * q^32 - 47088400041590886495026610585051639514556983630608 * q^33 + 65410975808473083189009824374957335825741658669920 * q^34 + 431016017471985629901399384019473069677000966146800 * q^35 + 1775126313486315490006860454334179833046020355380480 * q^36 + 2530086119390649596783488597048185172700211081307582 * q^37 + 8239199056511383470655384896825357549181759120989120 * q^38 + 4961423111454378211563271585245526388719534092320280 * q^39 - 40685898216651980081843388349290441658499772853248000 * q^40 - 32007461843289961215814846894363640801437553437785390 * q^41 - 195314045299892072857192155130437414288210017195751936 * q^42 - 402059303617352898401840600044858362031212461462830244 * q^43 - 612244818796766798372650999094717302459543241837614080 * q^44 + 1043713954525992259035161696571453857355706435519522950 * q^45 + 2514856669257653110995391415101692168449088734110815360 * q^46 + 1572646433047878010336915637334684433480129688652944112 * q^47 + 24675155031348794802454414815955354749687650397068722176 * q^48 + 16734108893933327602070337185312116893972489911225126685 * q^49 - 11063577880472461941859651947553302608752466885786430000 * q^50 - 95495399267422751891095337933943462124412320320925036440 * q^51 - 183526322178405888877826050411801068824199241263844049408 * q^52 - 95621521245111298200083441902514245466096845017659053554 * q^53 - 557318597848485774743582734276566483558675400940047747200 * q^54 - 383625033771696018967747534079220095868070927508130274200 * q^55 + 1279479797562181227760971870185499697207329786220906905600 * q^56 + 3318990221727917572672253796992679057503071099524130319760 * q^57 + 8032881606980455898794762567234097793264864123190024459680 * q^58 - 2559150002019539763980782190342674875710406916414141257300 * q^59 + 15929220555559091493107654286005818307326801014009906124800 * q^60 + 3946890902136488204413925902971677716173875604816310291510 * q^61 - 92928211608041968363364267171290588248132222513261182532096 * q^62 - 68802271924921645132530256442236118351318471682753406875384 * q^63 - 69150375570305937045593462943529104399703176808647979171840 * q^64 - 203428189337486180767895923031910965130055608932610312683100 * q^65 + 326824005648317890913619314213537102979279278284539154417920 * q^66 + 479747394632655989552085275852886843492252096722130444354772 * q^67 + 420712177041165958943716953506888102330291794585554871017984 * q^68 + 2085757498496947569861287804370568246775481786853310358067680 * q^69 + 2223344131713304350953089067854023542847735414462666460473600 * q^70 - 3493409458155820303614582574535715900391996656268214500281240 * q^71 - 6018736038355998608239490168867843174641985013841657017077760 * q^72 + 733760875924726020894973106598755498748960690384116338055026 * q^73 - 15514223525339201048939769580727994056556223964900985146362080 * q^74 - 14732126168569891490987332632351786074968338319638722054452500 * q^75 - 36133342560695069070126026304846041053110806476015652787276800 * q^76 + 25871798846989842274158113588872654076768674584213145553762784 * q^77 + 152853676099484832021645784710928259514016680275865958517430912 * q^78 + 59699536569481886874118825694486111435856996884515844586586000 * q^79 + 219177295509817100053456351931646916206927220422985017075302400 * q^80 - 65583962416587044196237908629906173582326728520698989687871395 * q^81 - 142296434381269688499177453649327032329978008843889237410699296 * q^82 + 209047534837132969695050943022419892956201321886275074830990116 * q^83 - 1354709036605046757778457263098823484206949849923331537372897280 * q^84 - 1670550938011517784167531235760024383135110452768829071189193700 * q^85 - 1010926193522040357819139524349238009908059943237053024421679040 * q^86 - 13125541831432259160108322296931402049512991915132402846472360 * q^87 + 3795986156893622389528985077855302896604937707291737426546933760 * q^88 + 1493974679319648583095502764787645056501586744645243776851520450 * q^89 + 11168691266815947461428091063256972957381716390895407392159178400 * q^90 + 2617522707396973663766320817347076504527878438540996722544653360 * q^91 + 5247747536363486457856850601737389388282095910031013129592297472 * q^92 + 18739223210125935355706186663397055877109210120967221009013809792 * q^93 - 12861091983377738108305257044091193029406256915207083515703514880 * q^94 - 82117654407511305658368168133560153272734117033702557537140601000 * q^95 - 56452805466036872845515504865736329476063459354106526340942397440 * q^96 - 31085940707578182510537462951801111142341367310530261899829858838 * q^97 - 46798198452208171360777266613142082443922483732249545388761625936 * q^98 - 36017030678524367090996061767134771004855794650938085175529135220 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.07662e9	1.32970	0.664852	−	0.746975i	$-0.268495\pi$
$2$	8.07662e9	1.32970	0.664852	+	0.746975i	$0.268495\pi$
$3$	2.82746e15	0.880960	0.440480	−	0.897763i	$-0.354808\pi$
$3$	2.82746e15	0.880960	0.440480	+	0.897763i	$0.354808\pi$
$4$	2.83384e19	0.768113
$5$	−4.30990e22	−0.827832	−0.413916	−	0.910315i	$-0.635839\pi$
$5$	−4.30990e22	−0.827832	−0.413916	+	0.910315i	$0.635839\pi$
$6$	2.28363e25	1.17142
$7$	−4.31488e27	−1.47667	−0.738333	−	0.674436i	$-0.764387\pi$
$7$	−4.31488e27	−1.47667	−0.738333	+	0.674436i	$0.764387\pi$
$8$	−6.90965e28	−0.308341
$9$	−2.30651e30	−0.223910
$10$	−3.48095e32	−1.10077
$11$	−3.09454e33	−0.441910	−0.220955	−	0.975284i	$-0.570917\pi$
$11$	−3.09454e33	−0.441910	−0.220955	+	0.975284i	$0.570917\pi$
$12$	8.01256e34	0.676676
$13$	−1.30257e36	−0.815908	−0.407954	−	0.913002i	$-0.633758\pi$
$13$	−1.30257e36	−0.815908	−0.407954	+	0.913002i	$0.633758\pi$
$14$	−3.48497e37	−1.96353
$15$	−1.21861e38	−0.729287
$16$	−1.60357e39	−1.17812
$17$	1.01238e40	1.03694	0.518469	−	0.855096i	$-0.326502\pi$
$17$	1.01238e40	1.03694	0.518469	+	0.855096i	$0.326502\pi$
$18$	−1.86288e40	−0.297734
$19$	6.77371e41	1.86782	0.933909	−	0.357512i	$-0.116375\pi$
$19$	6.77371e41	1.86782	0.933909	+	0.357512i	$0.116375\pi$
$20$	−1.22136e42	−0.635869
$21$	−1.22002e43	−1.30088
$22$	−2.49934e43	−0.587610
$23$	2.44678e44	1.35656	0.678282	−	0.734801i	$-0.262725\pi$
$23$	2.44678e44	1.35656	0.678282	+	0.734801i	$0.262725\pi$
$24$	−1.95368e44	−0.271636
$25$	−8.52980e44	−0.314694
$26$	−1.05204e46	−1.08492
$27$	−3.56474e46	−1.07822
$28$	−1.22277e47	−1.13425
$29$	6.24422e46	0.185158	0.0925792	−	0.995705i	$-0.470489\pi$
$29$	6.24422e46	0.185158	0.0925792	+	0.995705i	$0.470489\pi$
$30$	−9.84224e47	−0.969735
$31$	−4.23499e48	−1.43746	−0.718732	−	0.695287i	$-0.755277\pi$
$31$	−4.23499e48	−1.43746	−0.718732	+	0.695287i	$0.755277\pi$
$32$	−1.04022e49	−1.25820
$33$	−8.74970e48	−0.389305
$34$	8.17660e49	1.37882
$35$	1.85967e50	1.22243
$36$	−6.53627e49	−0.171988
$37$	−6.58889e50	−0.711633	−0.355817	−	0.934556i	$-0.615797\pi$
$37$	−6.58889e50	−0.711633	−0.355817	+	0.934556i	$0.615797\pi$
$38$	5.47087e51	2.48364
$39$	−3.68296e51	−0.718782
$40$	2.97799e51	0.255255
$41$	−3.22294e51	−0.123816	−0.0619079	−	0.998082i	$-0.519719\pi$
$41$	−3.22294e51	−0.123816	−0.0619079	+	0.998082i	$0.519719\pi$
$42$	−9.85361e52	−1.72979
$43$	−7.33033e52	−0.598961	−0.299481	−	0.954102i	$-0.596813\pi$
$43$	−7.33033e52	−0.598961	−0.299481	+	0.954102i	$0.596813\pi$
$44$	−8.76942e52	−0.339437
$45$	9.94083e52	0.185360
$46$	1.97617e54	1.80383
$47$	4.96764e53	0.225408	0.112704	−	0.993629i	$-0.464049\pi$
$47$	4.96764e53	0.225408	0.112704	+	0.993629i	$0.464049\pi$
$48$	−4.53403e54	−1.03787
$49$	1.00799e55	1.18054
$50$	−6.88920e54	−0.418450
$51$	2.86246e55	0.913501
$52$	−3.69126e55	−0.626710
$53$	−1.14911e56	−1.05050	−0.525249	−	0.850948i	$-0.676028\pi$
$53$	−1.14911e56	−1.05050	−0.525249	+	0.850948i	$0.676028\pi$
$54$	−2.87911e56	−1.43371
$55$	1.33372e56	0.365827
$56$	2.98143e56	0.455317
$57$	1.91524e57	1.64547
$58$	5.04323e56	0.246206
$59$	−7.39266e56	−0.207067	−0.103533	−	0.994626i	$-0.533015\pi$
$59$	−7.39266e56	−0.207067	−0.103533	+	0.994626i	$0.533015\pi$
$60$	−3.45334e57	−0.560174
$61$	1.27879e58	1.21222	0.606109	−	0.795381i	$-0.292729\pi$
$61$	1.27879e58	1.21222	0.606109	+	0.795381i	$0.292729\pi$
$62$	−3.42044e58	−1.91140
$63$	9.95231e57	0.330640
$64$	−2.48535e58	−0.494923
$65$	5.61394e58	0.675435
$66$	−7.06680e58	−0.517660
$67$	−1.92730e59	−0.866000	−0.433000	−	0.901394i	$-0.642545\pi$
$67$	−1.92730e59	−0.866000	−0.433000	+	0.901394i	$0.642545\pi$
$68$	2.86892e59	0.796486
$69$	6.91818e59	1.19508
$70$	1.50199e60	1.62547
$71$	−1.96706e60	−1.34252	−0.671259	−	0.741223i	$-0.734246\pi$
$71$	−1.96706e60	−1.34252	−0.671259	+	0.741223i	$0.734246\pi$
$72$	1.59372e59	0.0690407
$73$	−4.77567e60	−1.32142	−0.660711	−	0.750640i	$-0.729745\pi$
$73$	−4.77567e60	−1.32142	−0.660711	+	0.750640i	$0.729745\pi$
$74$	−5.32160e60	−0.946262
$75$	−2.41177e60	−0.277233
$76$	1.91956e61	1.43469
$77$	1.33526e61	0.652554
$78$	−2.97459e61	−0.955768
$79$	1.98178e61	0.420896	0.210448	−	0.977605i	$-0.432508\pi$
$79$	1.98178e61	0.420896	0.210448	+	0.977605i	$0.432508\pi$
$80$	6.91122e61	0.975282
$81$	−7.70322e61	−0.725954
$82$	−2.60304e61	−0.164638
$83$	3.39503e62	1.44812	0.724062	−	0.689735i	$-0.242273\pi$
$83$	3.39503e62	1.44812	0.724062	+	0.689735i	$0.242273\pi$
$84$	−3.45732e62	−0.999225
$85$	−4.36325e62	−0.858411
$86$	−5.92043e62	−0.796441
$87$	1.76553e62	0.163117
$88$	2.13822e62	0.136259
$89$	8.38014e62	0.369893	0.184947	−	0.982749i	$-0.440789\pi$
$89$	8.38014e62	0.369893	0.184947	+	0.982749i	$0.440789\pi$
$90$	8.02883e62	0.246474
$91$	5.62042e63	1.20482
$92$	6.93378e63	1.04199
$93$	−1.19743e64	−1.26635
$94$	4.01217e63	0.299726
$95$	−2.91940e64	−1.54624
$96$	−2.94118e64	−1.10843
$97$	1.21487e64	0.326925	0.163462	−	0.986550i	$-0.447734\pi$
$97$	1.21487e64	0.326925	0.163462	+	0.986550i	$0.447734\pi$
$98$	8.14112e64	1.56977
$99$	7.13759e63	0.0989481

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1.66.a.a.1.5	✓	5
3.2	odd	2		9.66.a.b.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.66.a.a.1.5	✓	5	1.1	even	1	trivial
9.66.a.b.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 66 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\(q+8.07662e9 q^{2} +2.82746e15 q^{3} +2.83384e19 q^{4} -4.30990e22 q^{5} +2.28363e25 q^{6} -4.31488e27 q^{7} -6.90965e28 q^{8} -2.30651e30 q^{9} -3.48095e32 q^{10} -3.09454e33 q^{11} +8.01256e34 q^{12} -1.30257e36 q^{13} -3.48497e37 q^{14} -1.21861e38 q^{15} -1.60357e39 q^{16} +1.01238e40 q^{17} -1.86288e40 q^{18} +6.77371e41 q^{19} -1.22136e42 q^{20} -1.22002e43 q^{21} -2.49934e43 q^{22} +2.44678e44 q^{23} -1.95368e44 q^{24} -8.52980e44 q^{25} -1.05204e46 q^{26} -3.56474e46 q^{27} -1.22277e47 q^{28} +6.24422e46 q^{29} -9.84224e47 q^{30} -4.23499e48 q^{31} -1.04022e49 q^{32} -8.74970e48 q^{33} +8.17660e49 q^{34} +1.85967e50 q^{35} -6.53627e49 q^{36} -6.58889e50 q^{37} +5.47087e51 q^{38} -3.68296e51 q^{39} +2.97799e51 q^{40} -3.22294e51 q^{41} -9.85361e52 q^{42} -7.33033e52 q^{43} -8.76942e52 q^{44} +9.94083e52 q^{45} +1.97617e54 q^{46} +4.96764e53 q^{47} -4.53403e54 q^{48} +1.00799e55 q^{49} -6.88920e54 q^{50} +2.86246e55 q^{51} -3.69126e55 q^{52} -1.14911e56 q^{53} -2.87911e56 q^{54} +1.33372e56 q^{55} +2.98143e56 q^{56} +1.91524e57 q^{57} +5.04323e56 q^{58} -7.39266e56 q^{59} -3.45334e57 q^{60} +1.27879e58 q^{61} -3.42044e58 q^{62} +9.95231e57 q^{63} -2.48535e58 q^{64} +5.61394e58 q^{65} -7.06680e58 q^{66} -1.92730e59 q^{67} +2.86892e59 q^{68} +6.91818e59 q^{69} +1.50199e60 q^{70} -1.96706e60 q^{71} +1.59372e59 q^{72} -4.77567e60 q^{73} -5.32160e60 q^{74} -2.41177e60 q^{75} +1.91956e61 q^{76} +1.33526e61 q^{77} -2.97459e61 q^{78} +1.98178e61 q^{79} +6.91122e61 q^{80} -7.70322e61 q^{81} -2.60304e61 q^{82} +3.39503e62 q^{83} -3.45732e62 q^{84} -4.36325e62 q^{85} -5.92043e62 q^{86} +1.76553e62 q^{87} +2.13822e62 q^{88} +8.38014e62 q^{89} +8.02883e62 q^{90} +5.62042e63 q^{91} +6.93378e63 q^{92} -1.19743e64 q^{93} +4.01217e63 q^{94} -2.91940e64 q^{95} -2.94118e64 q^{96} +1.21487e64 q^{97} +8.14112e64 q^{98} +7.13759e63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3959709648 q^{2} - 22\!\cdots\!04 q^{3} + 11\!\cdots\!60 q^{4} + 26\!\cdots\!50 q^{5} - 33\!\cdots\!40 q^{6} - 69\!\cdots\!08 q^{7} - 44\!\cdots\!20 q^{8} + 31\!\cdots\!65 q^{9} - 46\!\cdots\!00 q^{10}+ \cdots - 36\!\cdots\!20 q^{99}+O(q^{100}) \) 5 * q - 3959709648 * q^2 - 2231307434935404 * q^3 + 116912191997379733760 * q^4 + 26879673327548389029150 * q^5 - 33295729280564090947602240 * q^6 - 6923148241844308586296558808 * q^7 - 446887903591926627901545861120 * q^8 + 31912530537586467884680398342465 * q^9 - 46444835254758466872049375039200 * q^10 - 5436990447821613455266185060489540 * q^11 + 195800126486592722298045722544448512 * q^12 - 2890859300726790832888435764458467514 * q^13 + 41180574237900302034722115163160949120 * q^14 + 103408365234403422747890968078751873400 * q^15 - 1473868696955444538230489908858040811520 * q^16 + 13864680686430637673540501618888491948122 * q^17 - 52766923382337010938622933627608214498704 * q^18 - 532140170263349448171574488831326440156700 * q^19 + 3499008192062174379356405538388976603788800 * q^20 - 11546436905526557006526119132251889169971040 * q^21 + 22561892587607883403404755239099221639703104 * q^22 - 94948785027142700354971234970110107282841224 * q^23 - 2324009070474316353878252433966072715423334400 * q^24 - 89368364701600219577770390611457383673793125 * q^25 - 30521938596122123780659173092206061048452902240 * q^26 - 129706546221561841046313530503436938966984152120 * q^27 - 518698434811712968791028896222612623314185598976 * q^28 - 1144597713377055966543673581436050616842314015850 * q^29 - 8898165260141424207772350064014611180432376163200 * q^30 - 4708831653623167525869999868687533742455793996640 * q^31 - 23840662820937316611832895513659338085021120462848 * q^32 - 47088400041590886495026610585051639514556983630608 * q^33 + 65410975808473083189009824374957335825741658669920 * q^34 + 431016017471985629901399384019473069677000966146800 * q^35 + 1775126313486315490006860454334179833046020355380480 * q^36 + 2530086119390649596783488597048185172700211081307582 * q^37 + 8239199056511383470655384896825357549181759120989120 * q^38 + 4961423111454378211563271585245526388719534092320280 * q^39 - 40685898216651980081843388349290441658499772853248000 * q^40 - 32007461843289961215814846894363640801437553437785390 * q^41 - 195314045299892072857192155130437414288210017195751936 * q^42 - 402059303617352898401840600044858362031212461462830244 * q^43 - 612244818796766798372650999094717302459543241837614080 * q^44 + 1043713954525992259035161696571453857355706435519522950 * q^45 + 2514856669257653110995391415101692168449088734110815360 * q^46 + 1572646433047878010336915637334684433480129688652944112 * q^47 + 24675155031348794802454414815955354749687650397068722176 * q^48 + 16734108893933327602070337185312116893972489911225126685 * q^49 - 11063577880472461941859651947553302608752466885786430000 * q^50 - 95495399267422751891095337933943462124412320320925036440 * q^51 - 183526322178405888877826050411801068824199241263844049408 * q^52 - 95621521245111298200083441902514245466096845017659053554 * q^53 - 557318597848485774743582734276566483558675400940047747200 * q^54 - 383625033771696018967747534079220095868070927508130274200 * q^55 + 1279479797562181227760971870185499697207329786220906905600 * q^56 + 3318990221727917572672253796992679057503071099524130319760 * q^57 + 8032881606980455898794762567234097793264864123190024459680 * q^58 - 2559150002019539763980782190342674875710406916414141257300 * q^59 + 15929220555559091493107654286005818307326801014009906124800 * q^60 + 3946890902136488204413925902971677716173875604816310291510 * q^61 - 92928211608041968363364267171290588248132222513261182532096 * q^62 - 68802271924921645132530256442236118351318471682753406875384 * q^63 - 69150375570305937045593462943529104399703176808647979171840 * q^64 - 203428189337486180767895923031910965130055608932610312683100 * q^65 + 326824005648317890913619314213537102979279278284539154417920 * q^66 + 479747394632655989552085275852886843492252096722130444354772 * q^67 + 420712177041165958943716953506888102330291794585554871017984 * q^68 + 2085757498496947569861287804370568246775481786853310358067680 * q^69 + 2223344131713304350953089067854023542847735414462666460473600 * q^70 - 3493409458155820303614582574535715900391996656268214500281240 * q^71 - 6018736038355998608239490168867843174641985013841657017077760 * q^72 + 733760875924726020894973106598755498748960690384116338055026 * q^73 - 15514223525339201048939769580727994056556223964900985146362080 * q^74 - 14732126168569891490987332632351786074968338319638722054452500 * q^75 - 36133342560695069070126026304846041053110806476015652787276800 * q^76 + 25871798846989842274158113588872654076768674584213145553762784 * q^77 + 152853676099484832021645784710928259514016680275865958517430912 * q^78 + 59699536569481886874118825694486111435856996884515844586586000 * q^79 + 219177295509817100053456351931646916206927220422985017075302400 * q^80 - 65583962416587044196237908629906173582326728520698989687871395 * q^81 - 142296434381269688499177453649327032329978008843889237410699296 * q^82 + 209047534837132969695050943022419892956201321886275074830990116 * q^83 - 1354709036605046757778457263098823484206949849923331537372897280 * q^84 - 1670550938011517784167531235760024383135110452768829071189193700 * q^85 - 1010926193522040357819139524349238009908059943237053024421679040 * q^86 - 13125541831432259160108322296931402049512991915132402846472360 * q^87 + 3795986156893622389528985077855302896604937707291737426546933760 * q^88 + 1493974679319648583095502764787645056501586744645243776851520450 * q^89 + 11168691266815947461428091063256972957381716390895407392159178400 * q^90 + 2617522707396973663766320817347076504527878438540996722544653360 * q^91 + 5247747536363486457856850601737389388282095910031013129592297472 * q^92 + 18739223210125935355706186663397055877109210120967221009013809792 * q^93 - 12861091983377738108305257044091193029406256915207083515703514880 * q^94 - 82117654407511305658368168133560153272734117033702557537140601000 * q^95 - 56452805466036872845515504865736329476063459354106526340942397440 * q^96 - 31085940707578182510537462951801111142341367310530261899829858838 * q^97 - 46798198452208171360777266613142082443922483732249545388761625936 * q^98 - 36017030678524367090996061767134771004855794650938085175529135220 * q^99

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