LMFDB - Newform orbit 2.48.a.b

Newspace parameters

Level:	$ N $	$=$	$ 2 $
Weight:	$ k $	$=$	$ 48 $
Character orbit:	$[\chi]$	$=$	2.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$27.9815325310$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 8640\sqrt{23589383914321}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q -8388608 q^{2} + ( 61144922412 - 5 \beta ) q^{3} + 70368744177664 q^{4} + ( 9089248919597070 + 460908 \beta ) q^{5} + ( -512920785304682496 + 41943040 \beta ) q^{6} + ( 80509272728013413816 - 675210690 \beta ) q^{7} -590295810358705651712 q^{8} + ( 21173339014074419649957 - 611449224120 \beta ) q^{9} +O(q^{10})$	$ q -8388608 q^{2} +(61144922412 - 5 \beta) q^{3} +70368744177664 q^{4} +(9089248919597070 + 460908 \beta) q^{5} +(-512920785304682496 + 41943040 \beta) q^{6} +(80509272728013413816 - 675210690 \beta) q^{7} -$$59\!\cdots\!12$$ q^{8} +($$21\!\cdots\!57$$ - 611449224120 \beta) q^{9} +(-$$76\!\cdots\!60$$ - 3866376536064 \beta) q^{10} +($$57\!\cdots\!92$$ - 79914943998855 \beta) q^{11} +($$43\!\cdots\!68$$ - 351843720888320 \beta) q^{12} +(-$$45\!\cdots\!18$$ + 4479527100978540 \beta) q^{13} +(-$$67\!\cdots\!28$$ + 5664077795819520 \beta) q^{14} +(-$$35\!\cdots\!60$$ - 17264060698915254 \beta) q^{15} +$$49\!\cdots\!96$$ q^{16} +(-$$51\!\cdots\!94$$ - 2192118261102219960 \beta) q^{17} +(-$$17\!\cdots\!56$$ + 5129207853046824960 \beta) q^{18} +($$51\!\cdots\!60$$ + 15893727992186364855 \beta) q^{19} +($$63\!\cdots\!80$$ + 32433517141438758912 \beta) q^{20} +($$10\!\cdots\!92$$ - $$44\!\cdots\!60$$ \beta) q^{21} +(-$$48\!\cdots\!36$$ + $$67\!\cdots\!40$$ \beta) q^{22} +($$43\!\cdots\!12$$ - $$68\!\cdots\!30$$ \beta) q^{23} +(-$$36\!\cdots\!44$$ + $$29\!\cdots\!60$$ \beta) q^{24} +(-$$25\!\cdots\!25$$ + $$83\!\cdots\!20$$ \beta) q^{25} +($$38\!\cdots\!44$$ - $$37\!\cdots\!20$$ \beta) q^{26} +($$50\!\cdots\!40$$ - $$10\!\cdots\!90$$ \beta) q^{27} +($$56\!\cdots\!24$$ - $$47\!\cdots\!60$$ \beta) q^{28} +($$22\!\cdots\!30$$ + $$51\!\cdots\!60$$ \beta) q^{29} +($$29\!\cdots\!80$$ + $$14\!\cdots\!32$$ \beta) q^{30} +($$10\!\cdots\!52$$ - $$16\!\cdots\!60$$ \beta) q^{31} -$$41\!\cdots\!68$$ q^{32} +($$73\!\cdots\!04$$ - $$77\!\cdots\!20$$ \beta) q^{33} +($$43\!\cdots\!52$$ + $$18\!\cdots\!80$$ \beta) q^{34} +($$18\!\cdots\!20$$ + $$30\!\cdots\!28$$ \beta) q^{35} +($$14\!\cdots\!48$$ - $$43\!\cdots\!80$$ \beta) q^{36} +($$77\!\cdots\!26$$ - $$74\!\cdots\!60$$ \beta) q^{37} +(-$$42\!\cdots\!80$$ - $$13\!\cdots\!40$$ \beta) q^{38} +(-$$42\!\cdots\!16$$ + $$50\!\cdots\!70$$ \beta) q^{39} +(-$$53\!\cdots\!40$$ - $$27\!\cdots\!96$$ \beta) q^{40} +(-$$91\!\cdots\!58$$ - $$29\!\cdots\!40$$ \beta) q^{41} +(-$$91\!\cdots\!36$$ + $$37\!\cdots\!80$$ \beta) q^{42} +($$12\!\cdots\!52$$ - $$58\!\cdots\!15$$ \beta) q^{43} +($$40\!\cdots\!88$$ - $$56\!\cdots\!20$$ \beta) q^{44} +(-$$30\!\cdots\!10$$ + $$42\!\cdots\!56$$ \beta) q^{45} +(-$$36\!\cdots\!96$$ + $$57\!\cdots\!40$$ \beta) q^{46} +($$14\!\cdots\!56$$ + $$49\!\cdots\!60$$ \beta) q^{47} +($$30\!\cdots\!52$$ - $$24\!\cdots\!80$$ \beta) q^{48} +($$20\!\cdots\!13$$ - $$10\!\cdots\!80$$ \beta) q^{49} +($$21\!\cdots\!00$$ - $$70\!\cdots\!60$$ \beta) q^{50} +($$16\!\cdots\!72$$ + $$12\!\cdots\!50$$ \beta) q^{51} +(-$$32\!\cdots\!52$$ + $$31\!\cdots\!60$$ \beta) q^{52} +($$18\!\cdots\!82$$ - $$12\!\cdots\!60$$ \beta) q^{53} +(-$$42\!\cdots\!20$$ + $$86\!\cdots\!20$$ \beta) q^{54} +(-$$59\!\cdots\!60$$ - $$46\!\cdots\!14$$ \beta) q^{55} +(-$$47\!\cdots\!92$$ + $$39\!\cdots\!80$$ \beta) q^{56} +(-$$10\!\cdots\!80$$ - $$15\!\cdots\!40$$ \beta) q^{57} +(-$$18\!\cdots\!40$$ - $$42\!\cdots\!80$$ \beta) q^{58} +($$10\!\cdots\!60$$ + $$11\!\cdots\!45$$ \beta) q^{59} +(-$$24\!\cdots\!40$$ - $$12\!\cdots\!56$$ \beta) q^{60} +($$11\!\cdots\!02$$ + $$22\!\cdots\!80$$ \beta) q^{61} +(-$$87\!\cdots\!16$$ + $$13\!\cdots\!80$$ \beta) q^{62} +($$24\!\cdots\!12$$ - $$63\!\cdots\!50$$ \beta) q^{63} +$$34\!\cdots\!44$$ q^{64} +($$32\!\cdots\!40$$ + $$19\!\cdots\!56$$ \beta) q^{65} +(-$$61\!\cdots\!32$$ + $$65\!\cdots\!60$$ \beta) q^{66} +($$48\!\cdots\!96$$ + $$15\!\cdots\!55$$ \beta) q^{67} +(-$$36\!\cdots\!16$$ - $$15\!\cdots\!40$$ \beta) q^{68} +($$63\!\cdots\!44$$ - $$64\!\cdots\!20$$ \beta) q^{69} +(-$$15\!\cdots\!60$$ - $$25\!\cdots\!24$$ \beta) q^{70} +(-$$17\!\cdots\!28$$ - $$16\!\cdots\!70$$ \beta) q^{71} +(-$$12\!\cdots\!84$$ + $$36\!\cdots\!40$$ \beta) q^{72} +(-$$22\!\cdots\!18$$ - $$39\!\cdots\!60$$ \beta) q^{73} +(-$$64\!\cdots\!08$$ + $$62\!\cdots\!80$$ \beta) q^{74} +(-$$89\!\cdots\!00$$ + $$17\!\cdots\!65$$ \beta) q^{75} +($$35\!\cdots\!40$$ + $$11\!\cdots\!20$$ \beta) q^{76} +($$14\!\cdots\!72$$ - $$68\!\cdots\!60$$ \beta) q^{77} +($$35\!\cdots\!28$$ - $$42\!\cdots\!60$$ \beta) q^{78} +($$22\!\cdots\!00$$ + $$97\!\cdots\!80$$ \beta) q^{79} +($$45\!\cdots\!20$$ + $$22\!\cdots\!68$$ \beta) q^{80} +(-$$16\!\cdots\!79$$ - $$96\!\cdots\!40$$ \beta) q^{81} +($$77\!\cdots\!64$$ + $$24\!\cdots\!20$$ \beta) q^{82} +(-$$67\!\cdots\!08$$ + $$40\!\cdots\!75$$ \beta) q^{83} +($$76\!\cdots\!88$$ - $$31\!\cdots\!40$$ \beta) q^{84} +(-$$22\!\cdots\!80$$ - $$43\!\cdots\!52$$ \beta) q^{85} +(-$$10\!\cdots\!16$$ + $$48\!\cdots\!20$$ \beta) q^{86} +(-$$31\!\cdots\!40$$ - $$80\!\cdots\!30$$ \beta) q^{87} +(-$$33\!\cdots\!04$$ + $$47\!\cdots\!60$$ \beta) q^{88} +($$22\!\cdots\!10$$ + $$28\!\cdots\!00$$ \beta) q^{89} +($$25\!\cdots\!80$$ - $$35\!\cdots\!48$$ \beta) q^{90} +(-$$89\!\cdots\!88$$ + $$39\!\cdots\!60$$ \beta) q^{91} +($$30\!\cdots\!68$$ - $$48\!\cdots\!20$$ \beta) q^{92} +($$20\!\cdots\!24$$ - $$61\!\cdots\!80$$ \beta) q^{93} +(-$$12\!\cdots\!48$$ - $$41\!\cdots\!80$$ \beta) q^{94} +($$17\!\cdots\!00$$ + $$37\!\cdots\!30$$ \beta) q^{95} +(-$$25\!\cdots\!16$$ + $$20\!\cdots\!40$$ \beta) q^{96} +($$26\!\cdots\!46$$ + $$78\!\cdots\!00$$ \beta) q^{97} +(-$$17\!\cdots\!04$$ + $$91\!\cdots\!40$$ \beta) q^{98} +($$98\!\cdots\!44$$ - $$20\!\cdots\!75$$ \beta) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + 161018545456026827632q^{7} - 1180591620717411303424q^{8} + 42346678028148839299914q^{9} + O(q^{10}) $	$ 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + $$16\!\cdots\!32$$q^{7} - $$11\!\cdots\!24$$q^{8} + $$42\!\cdots\!14$$q^{9} - $$15\!\cdots\!20$$q^{10} + $$11\!\cdots\!84$$q^{11} + $$86\!\cdots\!36$$q^{12} - $$91\!\cdots\!36$$q^{13} - $$13\!\cdots\!56$$q^{14} - $$70\!\cdots\!20$$q^{15} + $$99\!\cdots\!92$$q^{16} - $$10\!\cdots\!88$$q^{17} - $$35\!\cdots\!12$$q^{18} + $$10\!\cdots\!20$$q^{19} + $$12\!\cdots\!60$$q^{20} + $$21\!\cdots\!84$$q^{21} - $$96\!\cdots\!72$$q^{22} + $$87\!\cdots\!24$$q^{23} - $$72\!\cdots\!88$$q^{24} - $$50\!\cdots\!50$$q^{25} + $$76\!\cdots\!88$$q^{26} + $$10\!\cdots\!80$$q^{27} + $$11\!\cdots\!48$$q^{28} + $$44\!\cdots\!60$$q^{29} + $$58\!\cdots\!60$$q^{30} + $$20\!\cdots\!04$$q^{31} - $$83\!\cdots\!36$$q^{32} + $$14\!\cdots\!08$$q^{33} + $$86\!\cdots\!04$$q^{34} + $$36\!\cdots\!40$$q^{35} + $$29\!\cdots\!96$$q^{36} + $$15\!\cdots\!52$$q^{37} - $$85\!\cdots\!60$$q^{38} - $$84\!\cdots\!32$$q^{39} - $$10\!\cdots\!80$$q^{40} - $$18\!\cdots\!16$$q^{41} - $$18\!\cdots\!72$$q^{42} + $$25\!\cdots\!04$$q^{43} + $$80\!\cdots\!76$$q^{44} - $$60\!\cdots\!20$$q^{45} - $$73\!\cdots\!92$$q^{46} + $$28\!\cdots\!12$$q^{47} + $$60\!\cdots\!04$$q^{48} + $$40\!\cdots\!26$$q^{49} + $$42\!\cdots\!00$$q^{50} + $$32\!\cdots\!44$$q^{51} - $$64\!\cdots\!04$$q^{52} + $$36\!\cdots\!64$$q^{53} - $$84\!\cdots\!40$$q^{54} - $$11\!\cdots\!20$$q^{55} - $$95\!\cdots\!84$$q^{56} - $$21\!\cdots\!60$$q^{57} - $$37\!\cdots\!80$$q^{58} + $$21\!\cdots\!20$$q^{59} - $$49\!\cdots\!80$$q^{60} + $$23\!\cdots\!04$$q^{61} - $$17\!\cdots\!32$$q^{62} + $$48\!\cdots\!24$$q^{63} + $$69\!\cdots\!88$$q^{64} + $$64\!\cdots\!80$$q^{65} - $$12\!\cdots\!64$$q^{66} + $$96\!\cdots\!92$$q^{67} - $$72\!\cdots\!32$$q^{68} + $$12\!\cdots\!88$$q^{69} - $$30\!\cdots\!20$$q^{70} - $$34\!\cdots\!56$$q^{71} - $$24\!\cdots\!68$$q^{72} - $$44\!\cdots\!36$$q^{73} - $$12\!\cdots\!16$$q^{74} - $$17\!\cdots\!00$$q^{75} + $$71\!\cdots\!80$$q^{76} + $$28\!\cdots\!44$$q^{77} + $$70\!\cdots\!56$$q^{78} + $$45\!\cdots\!00$$q^{79} + $$90\!\cdots\!40$$q^{80} - $$32\!\cdots\!58$$q^{81} + $$15\!\cdots\!28$$q^{82} - $$13\!\cdots\!16$$q^{83} + $$15\!\cdots\!76$$q^{84} - $$44\!\cdots\!60$$q^{85} - $$21\!\cdots\!32$$q^{86} - $$62\!\cdots\!80$$q^{87} - $$67\!\cdots\!08$$q^{88} + $$45\!\cdots\!20$$q^{89} + $$50\!\cdots\!60$$q^{90} - $$17\!\cdots\!76$$q^{91} + $$61\!\cdots\!36$$q^{92} + $$40\!\cdots\!48$$q^{93} - $$24\!\cdots\!96$$q^{94} + $$35\!\cdots\!00$$q^{95} - $$50\!\cdots\!32$$q^{96} + $$53\!\cdots\!92$$q^{97} - $$34\!\cdots\!08$$q^{98} + $$19\!\cdots\!88$$q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

2.42845e6
−2.42844e6

−8.38861e6

−1.48673e11

7.03687e13

2.84306e16

1.24716e18

5.21750e19

−5.90296e20

−4.48523e21

−2.38493e23

1.2

−8.38861e6

2.70963e11

7.03687e13

−1.02521e16

−2.27300e18

1.08843e20

−5.90296e20

4.68319e22

8.60007e22

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2.48.a.b	✓	2
4.b	odd	2	1		16.48.a.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.48.a.b	✓	2	1.a	even	1	1	trivial
16.48.a.b		2	4.b	odd	2	1

Atkin-Lehner signs

$ p $	Sign
$2$	$1$

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} - 122289844824 T_{3} - $$40\!\cdots\!56$ acting on $S_{48}^{\mathrm{new}}(\Gamma_0(2))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ ( 1 + 8388608 T )^{2} $
$3$	$ 1 - 122289844824 T + $$12\!\cdots\!18$$ T^{2} - $$32\!\cdots\!88$$ T^{3} + $$70\!\cdots\!69$$ T^{4} $
$5$	$ 1 - 18178497839194140 T + $$11\!\cdots\!50$$ T^{2} - $$12\!\cdots\!00$$ T^{3} + $$50\!\cdots\!25$$ T^{4} $
$7$	$ 1 - $$16\!\cdots\!32$$ T + $$16\!\cdots\!42$$ T^{2} - $$84\!\cdots\!76$$ T^{3} + $$27\!\cdots\!49$$ T^{4} $
$11$	$ 1 - $$11\!\cdots\!84$$ T + $$67\!\cdots\!06$$ T^{2} - $$10\!\cdots\!64$$ T^{3} + $$77\!\cdots\!41$$ T^{4} $
$13$	$ 1 + $$91\!\cdots\!36$$ T + $$12\!\cdots\!58$$ T^{2} + $$20\!\cdots\!12$$ T^{3} + $$51\!\cdots\!89$$ T^{4} $
$17$	$ 1 + $$10\!\cdots\!88$$ T + $$77\!\cdots\!82$$ T^{2} + $$69\!\cdots\!24$$ T^{3} + $$45\!\cdots\!29$$ T^{4} $
$19$	$ 1 - $$10\!\cdots\!20$$ T + $$23\!\cdots\!78$$ T^{2} - $$12\!\cdots\!80$$ T^{3} + $$15\!\cdots\!21$$ T^{4} $
$23$	$ 1 - $$87\!\cdots\!24$$ T + $$19\!\cdots\!38$$ T^{2} - $$87\!\cdots\!28$$ T^{3} + $$10\!\cdots\!09$$ T^{4} $
$29$	$ 1 - $$44\!\cdots\!60$$ T + $$11\!\cdots\!18$$ T^{2} - $$24\!\cdots\!40$$ T^{3} + $$29\!\cdots\!81$$ T^{4} $
$31$	$ 1 - $$20\!\cdots\!04$$ T + $$31\!\cdots\!26$$ T^{2} - $$25\!\cdots\!44$$ T^{3} + $$15\!\cdots\!21$$ T^{4} $
$37$	$ 1 - $$15\!\cdots\!52$$ T + $$15\!\cdots\!42$$ T^{2} - $$78\!\cdots\!16$$ T^{3} + $$25\!\cdots\!89$$ T^{4} $
$41$	$ 1 + $$18\!\cdots\!16$$ T + $$20\!\cdots\!26$$ T^{2} + $$11\!\cdots\!96$$ T^{3} + $$39\!\cdots\!61$$ T^{4} $
$43$	$ 1 - $$25\!\cdots\!04$$ T + $$74\!\cdots\!18$$ T^{2} - $$14\!\cdots\!28$$ T^{3} + $$35\!\cdots\!49$$ T^{4} $
$47$	$ 1 - $$28\!\cdots\!12$$ T + $$55\!\cdots\!62$$ T^{2} - $$11\!\cdots\!56$$ T^{3} + $$15\!\cdots\!69$$ T^{4} $
$53$	$ 1 - $$36\!\cdots\!64$$ T + $$24\!\cdots\!98$$ T^{2} - $$39\!\cdots\!68$$ T^{3} + $$12\!\cdots\!69$$ T^{4} $
$59$	$ 1 - $$21\!\cdots\!20$$ T + $$11\!\cdots\!38$$ T^{2} - $$36\!\cdots\!80$$ T^{3} + $$28\!\cdots\!61$$ T^{4} $
$61$	$ 1 - $$23\!\cdots\!04$$ T + $$29\!\cdots\!46$$ T^{2} - $$19\!\cdots\!84$$ T^{3} + $$66\!\cdots\!41$$ T^{4} $
$67$	$ 1 - $$96\!\cdots\!92$$ T + $$11\!\cdots\!62$$ T^{2} - $$64\!\cdots\!16$$ T^{3} + $$44\!\cdots\!29$$ T^{4} $
$71$	$ 1 + $$34\!\cdots\!56$$ T + $$23\!\cdots\!66$$ T^{2} + $$35\!\cdots\!96$$ T^{3} + $$10\!\cdots\!81$$ T^{4} $
$73$	$ 1 + $$44\!\cdots\!36$$ T + $$77\!\cdots\!18$$ T^{2} + $$16\!\cdots\!92$$ T^{3} + $$14\!\cdots\!09$$ T^{4} $
$79$	$ 1 - $$45\!\cdots\!00$$ T + $$19\!\cdots\!18$$ T^{2} - $$70\!\cdots\!00$$ T^{3} + $$23\!\cdots\!81$$ T^{4} $
$83$	$ 1 + $$13\!\cdots\!16$$ T + $$73\!\cdots\!18$$ T^{2} + $$21\!\cdots\!32$$ T^{3} + $$24\!\cdots\!29$$ T^{4} $
$89$	$ 1 - $$45\!\cdots\!20$$ T + $$87\!\cdots\!58$$ T^{2} - $$19\!\cdots\!80$$ T^{3} + $$17\!\cdots\!41$$ T^{4} $
$97$	$ 1 - $$53\!\cdots\!92$$ T + $$44\!\cdots\!42$$ T^{2} - $$12\!\cdots\!96$$ T^{3} + $$57\!\cdots\!69$$ T^{4} $
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 48 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)

\(f(q)\)	\(=\)	\( q -8388608 q^{2} + ( 61144922412 - 5 \beta ) q^{3} + 70368744177664 q^{4} + ( 9089248919597070 + 460908 \beta ) q^{5} + ( -512920785304682496 + 41943040 \beta ) q^{6} + ( 80509272728013413816 - 675210690 \beta ) q^{7} -590295810358705651712 q^{8} + ( 21173339014074419649957 - 611449224120 \beta ) q^{9} +O(q^{10})\)	\( q -8388608 q^{2} +(61144922412 - 5 \beta) q^{3} +70368744177664 q^{4} +(9089248919597070 + 460908 \beta) q^{5} +(-512920785304682496 + 41943040 \beta) q^{6} +(80509272728013413816 - 675210690 \beta) q^{7} -\)\(59\!\cdots\!12\)\( q^{8} +(\)\(21\!\cdots\!57\)\( - 611449224120 \beta) q^{9} +(-\)\(76\!\cdots\!60\)\( - 3866376536064 \beta) q^{10} +(\)\(57\!\cdots\!92\)\( - 79914943998855 \beta) q^{11} +(\)\(43\!\cdots\!68\)\( - 351843720888320 \beta) q^{12} +(-\)\(45\!\cdots\!18\)\( + 4479527100978540 \beta) q^{13} +(-\)\(67\!\cdots\!28\)\( + 5664077795819520 \beta) q^{14} +(-\)\(35\!\cdots\!60\)\( - 17264060698915254 \beta) q^{15} +\)\(49\!\cdots\!96\)\( q^{16} +(-\)\(51\!\cdots\!94\)\( - 2192118261102219960 \beta) q^{17} +(-\)\(17\!\cdots\!56\)\( + 5129207853046824960 \beta) q^{18} +(\)\(51\!\cdots\!60\)\( + 15893727992186364855 \beta) q^{19} +(\)\(63\!\cdots\!80\)\( + 32433517141438758912 \beta) q^{20} +(\)\(10\!\cdots\!92\)\( - \)\(44\!\cdots\!60\)\( \beta) q^{21} +(-\)\(48\!\cdots\!36\)\( + \)\(67\!\cdots\!40\)\( \beta) q^{22} +(\)\(43\!\cdots\!12\)\( - \)\(68\!\cdots\!30\)\( \beta) q^{23} +(-\)\(36\!\cdots\!44\)\( + \)\(29\!\cdots\!60\)\( \beta) q^{24} +(-\)\(25\!\cdots\!25\)\( + \)\(83\!\cdots\!20\)\( \beta) q^{25} +(\)\(38\!\cdots\!44\)\( - \)\(37\!\cdots\!20\)\( \beta) q^{26} +(\)\(50\!\cdots\!40\)\( - \)\(10\!\cdots\!90\)\( \beta) q^{27} +(\)\(56\!\cdots\!24\)\( - \)\(47\!\cdots\!60\)\( \beta) q^{28} +(\)\(22\!\cdots\!30\)\( + \)\(51\!\cdots\!60\)\( \beta) q^{29} +(\)\(29\!\cdots\!80\)\( + \)\(14\!\cdots\!32\)\( \beta) q^{30} +(\)\(10\!\cdots\!52\)\( - \)\(16\!\cdots\!60\)\( \beta) q^{31} -\)\(41\!\cdots\!68\)\( q^{32} +(\)\(73\!\cdots\!04\)\( - \)\(77\!\cdots\!20\)\( \beta) q^{33} +(\)\(43\!\cdots\!52\)\( + \)\(18\!\cdots\!80\)\( \beta) q^{34} +(\)\(18\!\cdots\!20\)\( + \)\(30\!\cdots\!28\)\( \beta) q^{35} +(\)\(14\!\cdots\!48\)\( - \)\(43\!\cdots\!80\)\( \beta) q^{36} +(\)\(77\!\cdots\!26\)\( - \)\(74\!\cdots\!60\)\( \beta) q^{37} +(-\)\(42\!\cdots\!80\)\( - \)\(13\!\cdots\!40\)\( \beta) q^{38} +(-\)\(42\!\cdots\!16\)\( + \)\(50\!\cdots\!70\)\( \beta) q^{39} +(-\)\(53\!\cdots\!40\)\( - \)\(27\!\cdots\!96\)\( \beta) q^{40} +(-\)\(91\!\cdots\!58\)\( - \)\(29\!\cdots\!40\)\( \beta) q^{41} +(-\)\(91\!\cdots\!36\)\( + \)\(37\!\cdots\!80\)\( \beta) q^{42} +(\)\(12\!\cdots\!52\)\( - \)\(58\!\cdots\!15\)\( \beta) q^{43} +(\)\(40\!\cdots\!88\)\( - \)\(56\!\cdots\!20\)\( \beta) q^{44} +(-\)\(30\!\cdots\!10\)\( + \)\(42\!\cdots\!56\)\( \beta) q^{45} +(-\)\(36\!\cdots\!96\)\( + \)\(57\!\cdots\!40\)\( \beta) q^{46} +(\)\(14\!\cdots\!56\)\( + \)\(49\!\cdots\!60\)\( \beta) q^{47} +(\)\(30\!\cdots\!52\)\( - \)\(24\!\cdots\!80\)\( \beta) q^{48} +(\)\(20\!\cdots\!13\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{49} +(\)\(21\!\cdots\!00\)\( - \)\(70\!\cdots\!60\)\( \beta) q^{50} +(\)\(16\!\cdots\!72\)\( + \)\(12\!\cdots\!50\)\( \beta) q^{51} +(-\)\(32\!\cdots\!52\)\( + \)\(31\!\cdots\!60\)\( \beta) q^{52} +(\)\(18\!\cdots\!82\)\( - \)\(12\!\cdots\!60\)\( \beta) q^{53} +(-\)\(42\!\cdots\!20\)\( + \)\(86\!\cdots\!20\)\( \beta) q^{54} +(-\)\(59\!\cdots\!60\)\( - \)\(46\!\cdots\!14\)\( \beta) q^{55} +(-\)\(47\!\cdots\!92\)\( + \)\(39\!\cdots\!80\)\( \beta) q^{56} +(-\)\(10\!\cdots\!80\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{57} +(-\)\(18\!\cdots\!40\)\( - \)\(42\!\cdots\!80\)\( \beta) q^{58} +(\)\(10\!\cdots\!60\)\( + \)\(11\!\cdots\!45\)\( \beta) q^{59} +(-\)\(24\!\cdots\!40\)\( - \)\(12\!\cdots\!56\)\( \beta) q^{60} +(\)\(11\!\cdots\!02\)\( + \)\(22\!\cdots\!80\)\( \beta) q^{61} +(-\)\(87\!\cdots\!16\)\( + \)\(13\!\cdots\!80\)\( \beta) q^{62} +(\)\(24\!\cdots\!12\)\( - \)\(63\!\cdots\!50\)\( \beta) q^{63} +\)\(34\!\cdots\!44\)\( q^{64} +(\)\(32\!\cdots\!40\)\( + \)\(19\!\cdots\!56\)\( \beta) q^{65} +(-\)\(61\!\cdots\!32\)\( + \)\(65\!\cdots\!60\)\( \beta) q^{66} +(\)\(48\!\cdots\!96\)\( + \)\(15\!\cdots\!55\)\( \beta) q^{67} +(-\)\(36\!\cdots\!16\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{68} +(\)\(63\!\cdots\!44\)\( - \)\(64\!\cdots\!20\)\( \beta) q^{69} +(-\)\(15\!\cdots\!60\)\( - \)\(25\!\cdots\!24\)\( \beta) q^{70} +(-\)\(17\!\cdots\!28\)\( - \)\(16\!\cdots\!70\)\( \beta) q^{71} +(-\)\(12\!\cdots\!84\)\( + \)\(36\!\cdots\!40\)\( \beta) q^{72} +(-\)\(22\!\cdots\!18\)\( - \)\(39\!\cdots\!60\)\( \beta) q^{73} +(-\)\(64\!\cdots\!08\)\( + \)\(62\!\cdots\!80\)\( \beta) q^{74} +(-\)\(89\!\cdots\!00\)\( + \)\(17\!\cdots\!65\)\( \beta) q^{75} +(\)\(35\!\cdots\!40\)\( + \)\(11\!\cdots\!20\)\( \beta) q^{76} +(\)\(14\!\cdots\!72\)\( - \)\(68\!\cdots\!60\)\( \beta) q^{77} +(\)\(35\!\cdots\!28\)\( - \)\(42\!\cdots\!60\)\( \beta) q^{78} +(\)\(22\!\cdots\!00\)\( + \)\(97\!\cdots\!80\)\( \beta) q^{79} +(\)\(45\!\cdots\!20\)\( + \)\(22\!\cdots\!68\)\( \beta) q^{80} +(-\)\(16\!\cdots\!79\)\( - \)\(96\!\cdots\!40\)\( \beta) q^{81} +(\)\(77\!\cdots\!64\)\( + \)\(24\!\cdots\!20\)\( \beta) q^{82} +(-\)\(67\!\cdots\!08\)\( + \)\(40\!\cdots\!75\)\( \beta) q^{83} +(\)\(76\!\cdots\!88\)\( - \)\(31\!\cdots\!40\)\( \beta) q^{84} +(-\)\(22\!\cdots\!80\)\( - \)\(43\!\cdots\!52\)\( \beta) q^{85} +(-\)\(10\!\cdots\!16\)\( + \)\(48\!\cdots\!20\)\( \beta) q^{86} +(-\)\(31\!\cdots\!40\)\( - \)\(80\!\cdots\!30\)\( \beta) q^{87} +(-\)\(33\!\cdots\!04\)\( + \)\(47\!\cdots\!60\)\( \beta) q^{88} +(\)\(22\!\cdots\!10\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{89} +(\)\(25\!\cdots\!80\)\( - \)\(35\!\cdots\!48\)\( \beta) q^{90} +(-\)\(89\!\cdots\!88\)\( + \)\(39\!\cdots\!60\)\( \beta) q^{91} +(\)\(30\!\cdots\!68\)\( - \)\(48\!\cdots\!20\)\( \beta) q^{92} +(\)\(20\!\cdots\!24\)\( - \)\(61\!\cdots\!80\)\( \beta) q^{93} +(-\)\(12\!\cdots\!48\)\( - \)\(41\!\cdots\!80\)\( \beta) q^{94} +(\)\(17\!\cdots\!00\)\( + \)\(37\!\cdots\!30\)\( \beta) q^{95} +(-\)\(25\!\cdots\!16\)\( + \)\(20\!\cdots\!40\)\( \beta) q^{96} +(\)\(26\!\cdots\!46\)\( + \)\(78\!\cdots\!00\)\( \beta) q^{97} +(-\)\(17\!\cdots\!04\)\( + \)\(91\!\cdots\!40\)\( \beta) q^{98} +(\)\(98\!\cdots\!44\)\( - \)\(20\!\cdots\!75\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + 161018545456026827632q^{7} - 1180591620717411303424q^{8} + 42346678028148839299914q^{9} + O(q^{10}) \)	\( 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + \)\(16\!\cdots\!32\)\(q^{7} - \)\(11\!\cdots\!24\)\(q^{8} + \)\(42\!\cdots\!14\)\(q^{9} - \)\(15\!\cdots\!20\)\(q^{10} + \)\(11\!\cdots\!84\)\(q^{11} + \)\(86\!\cdots\!36\)\(q^{12} - \)\(91\!\cdots\!36\)\(q^{13} - \)\(13\!\cdots\!56\)\(q^{14} - \)\(70\!\cdots\!20\)\(q^{15} + \)\(99\!\cdots\!92\)\(q^{16} - \)\(10\!\cdots\!88\)\(q^{17} - \)\(35\!\cdots\!12\)\(q^{18} + \)\(10\!\cdots\!20\)\(q^{19} + \)\(12\!\cdots\!60\)\(q^{20} + \)\(21\!\cdots\!84\)\(q^{21} - \)\(96\!\cdots\!72\)\(q^{22} + \)\(87\!\cdots\!24\)\(q^{23} - \)\(72\!\cdots\!88\)\(q^{24} - \)\(50\!\cdots\!50\)\(q^{25} + \)\(76\!\cdots\!88\)\(q^{26} + \)\(10\!\cdots\!80\)\(q^{27} + \)\(11\!\cdots\!48\)\(q^{28} + \)\(44\!\cdots\!60\)\(q^{29} + \)\(58\!\cdots\!60\)\(q^{30} + \)\(20\!\cdots\!04\)\(q^{31} - \)\(83\!\cdots\!36\)\(q^{32} + \)\(14\!\cdots\!08\)\(q^{33} + \)\(86\!\cdots\!04\)\(q^{34} + \)\(36\!\cdots\!40\)\(q^{35} + \)\(29\!\cdots\!96\)\(q^{36} + \)\(15\!\cdots\!52\)\(q^{37} - \)\(85\!\cdots\!60\)\(q^{38} - \)\(84\!\cdots\!32\)\(q^{39} - \)\(10\!\cdots\!80\)\(q^{40} - \)\(18\!\cdots\!16\)\(q^{41} - \)\(18\!\cdots\!72\)\(q^{42} + \)\(25\!\cdots\!04\)\(q^{43} + \)\(80\!\cdots\!76\)\(q^{44} - \)\(60\!\cdots\!20\)\(q^{45} - \)\(73\!\cdots\!92\)\(q^{46} + \)\(28\!\cdots\!12\)\(q^{47} + \)\(60\!\cdots\!04\)\(q^{48} + \)\(40\!\cdots\!26\)\(q^{49} + \)\(42\!\cdots\!00\)\(q^{50} + \)\(32\!\cdots\!44\)\(q^{51} - \)\(64\!\cdots\!04\)\(q^{52} + \)\(36\!\cdots\!64\)\(q^{53} - \)\(84\!\cdots\!40\)\(q^{54} - \)\(11\!\cdots\!20\)\(q^{55} - \)\(95\!\cdots\!84\)\(q^{56} - \)\(21\!\cdots\!60\)\(q^{57} - \)\(37\!\cdots\!80\)\(q^{58} + \)\(21\!\cdots\!20\)\(q^{59} - \)\(49\!\cdots\!80\)\(q^{60} + \)\(23\!\cdots\!04\)\(q^{61} - \)\(17\!\cdots\!32\)\(q^{62} + \)\(48\!\cdots\!24\)\(q^{63} + \)\(69\!\cdots\!88\)\(q^{64} + \)\(64\!\cdots\!80\)\(q^{65} - \)\(12\!\cdots\!64\)\(q^{66} + \)\(96\!\cdots\!92\)\(q^{67} - \)\(72\!\cdots\!32\)\(q^{68} + \)\(12\!\cdots\!88\)\(q^{69} - \)\(30\!\cdots\!20\)\(q^{70} - \)\(34\!\cdots\!56\)\(q^{71} - \)\(24\!\cdots\!68\)\(q^{72} - \)\(44\!\cdots\!36\)\(q^{73} - \)\(12\!\cdots\!16\)\(q^{74} - \)\(17\!\cdots\!00\)\(q^{75} + \)\(71\!\cdots\!80\)\(q^{76} + \)\(28\!\cdots\!44\)\(q^{77} + \)\(70\!\cdots\!56\)\(q^{78} + \)\(45\!\cdots\!00\)\(q^{79} + \)\(90\!\cdots\!40\)\(q^{80} - \)\(32\!\cdots\!58\)\(q^{81} + \)\(15\!\cdots\!28\)\(q^{82} - \)\(13\!\cdots\!16\)\(q^{83} + \)\(15\!\cdots\!76\)\(q^{84} - \)\(44\!\cdots\!60\)\(q^{85} - \)\(21\!\cdots\!32\)\(q^{86} - \)\(62\!\cdots\!80\)\(q^{87} - \)\(67\!\cdots\!08\)\(q^{88} + \)\(45\!\cdots\!20\)\(q^{89} + \)\(50\!\cdots\!60\)\(q^{90} - \)\(17\!\cdots\!76\)\(q^{91} + \)\(61\!\cdots\!36\)\(q^{92} + \)\(40\!\cdots\!48\)\(q^{93} - \)\(24\!\cdots\!96\)\(q^{94} + \)\(35\!\cdots\!00\)\(q^{95} - \)\(50\!\cdots\!32\)\(q^{96} + \)\(53\!\cdots\!92\)\(q^{97} - \)\(34\!\cdots\!08\)\(q^{98} + \)\(19\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

\(n\):		e.g. 2-40 or 990-1000
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