LMFDB - Newform orbit 4000.1.cq.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,1,Mod(33,4000)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(100))

chi = DirichletCharacter(H, H._module([0, 0, 83]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4000.33");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	4000.cq (of order $100$, degree $40$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.99626005053$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{100})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{30} + x^{20} - x^{10} + 1 $ x^40 - x^30 + x^20 - x^10 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{100}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{100} - \cdots)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{100}^{13} q^{5} - \zeta_{100}^{49} q^{9} +O(q^{10}) $ q + z^13 * q^5 - z^49 * q^9	$ q + \zeta_{100}^{13} q^{5} - \zeta_{100}^{49} q^{9} + ( - \zeta_{100}^{21} + \zeta_{100}^{2}) q^{13} + ( - \zeta_{100}^{44} + \zeta_{100}^{17}) q^{17} + \zeta_{100}^{26} q^{25} + ( - \zeta_{100}^{28} - \zeta_{100}^{6}) q^{29} + (\zeta_{100}^{35} - \zeta_{100}^{18}) q^{37} + (\zeta_{100}^{47} - \zeta_{100}^{11}) q^{41} + \zeta_{100}^{12} q^{45} + \zeta_{100}^{45} q^{49} + ( - \zeta_{100}^{14} + \zeta_{100}^{5}) q^{53} + ( - \zeta_{100}^{33} + \zeta_{100}^{9}) q^{61} + ( - \zeta_{100}^{34} + \zeta_{100}^{15}) q^{65} + (\zeta_{100}^{29} + \zeta_{100}^{8}) q^{73} - \zeta_{100}^{48} q^{81} + (\zeta_{100}^{30} + \zeta_{100}^{7}) q^{85} + ( - \zeta_{100}^{40} - \zeta_{100}^{22}) q^{89} + (\zeta_{100}^{36} + \zeta_{100}^{23}) q^{97} +O(q^{100}) $ q + z^13 * q^5 - z^49 * q^9 + (-z^21 + z^2) * q^13 + (-z^44 + z^17) * q^17 + z^26 * q^25 + (-z^28 - z^6) * q^29 + (z^35 - z^18) * q^37 + (z^47 - z^11) * q^41 + z^12 * q^45 + z^45 * q^49 + (-z^14 + z^5) * q^53 + (-z^33 + z^9) * q^61 + (-z^34 + z^15) * q^65 + (z^29 + z^8) * q^73 - z^48 * q^81 + (z^30 + z^7) * q^85 + (-z^40 - z^22) * q^89 + (z^36 + z^23) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 10 q^{85} + 10 q^{89}+O(q^{100}) $ 40 * q + 10 * q^85 + 10 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$\zeta_{100}^{7}$	$1$	$1$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

33.1

−0.368125	−	0.929776i
0.844328	−	0.535827i
−0.770513	−	0.637424i
0.998027	−	0.0627905i
0.125333	−	0.992115i
0.125333	+	0.992115i
−0.125333	−	0.992115i
−0.125333	+	0.992115i
−0.998027	+	0.0627905i
0.770513	+	0.637424i
−0.844328	+	0.535827i
0.368125	+	0.929776i
−0.904827	−	0.425779i
−0.904827	+	0.425779i
−0.982287	−	0.187381i
−0.982287	+	0.187381i
−0.248690	+	0.968583i
−0.368125	+	0.929776i
0.248690	+	0.968583i
−0.770513	+	0.637424i

0.982287

−

0.187381i

−0.368125

0.929776i

97.1

0.481754

−

0.876307i

0.844328

0.535827i

353.1

0.904827

−

0.425779i

−0.770513

0.637424i

417.1

0.684547

−

0.728969i

0.998027

0.0627905i

513.1

0.998027

0.0627905i

0.125333

0.992115i

577.1

0.998027

−

0.0627905i

0.125333

−

0.992115i

673.1

−0.998027

0.0627905i

−0.125333

0.992115i

737.1

−0.998027

−

0.0627905i

−0.125333

−

0.992115i

833.1

−0.684547

0.728969i

−0.998027

−

0.0627905i

897.1

−0.904827

0.425779i

0.770513

−

0.637424i

1153.1

−0.481754

0.876307i

−0.844328

−

0.535827i

1217.1

−0.982287

0.187381i

0.368125

−

0.929776i

1313.1

−0.844328

0.535827i

−0.904827

0.425779i

1377.1

−0.844328

−

0.535827i

−0.904827

−

0.425779i

1473.1

0.770513

−

0.637424i

−0.982287

0.187381i

1537.1

0.770513

0.637424i

−0.982287

−

0.187381i

1633.1

0.125333

−

0.992115i

−0.248690

−

0.968583i

1697.1

0.982287

0.187381i

−0.368125

−

0.929776i

1953.1

−0.125333

−

0.992115i

0.248690

−

0.968583i

2017.1

0.904827

0.425779i

−0.770513

−

0.637424i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
125.i	odd	100	1	inner
500.r	even	100	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.1.cq.a	✓	40
4.b	odd	2	1	CM	4000.1.cq.a	✓	40
125.i	odd	100	1	inner	4000.1.cq.a	✓	40
500.r	even	100	1	inner	4000.1.cq.a	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.1.cq.a	✓	40	1.a	even	1	1	trivial
4000.1.cq.a	✓	40	4.b	odd	2	1	CM
4000.1.cq.a	✓	40	125.i	odd	100	1	inner
4000.1.cq.a	✓	40	500.r	even	100	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(4000, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{40} $ T^40
$3$	$ T^{40} $ T^40
$5$	$ T^{40} - T^{30} + \cdots + 1 $ T^40 - T^30 + T^20 - T^10 + 1
$7$	$ T^{40} $ T^40
$11$	$ T^{40} $ T^40
$13$	$ T^{40} - 2 T^{35} + \cdots + 1 $ T^40 - 2T^35 + 100T^34 - 50T^33 - 35T^32 - 10T^31 + 2T^30 - 150T^29 + 3450T^28 - 3610T^27 - 85T^26 + 2750T^25 + 2275T^24 - 4450T^23 + 47845T^22 - 76830T^21 + 57871T^20 + 23100T^19 - 153300T^18 + 96580T^17 + 172880T^16 - 459740T^15 + 713900T^14 - 944950T^13 + 1132140T^12 - 860560T^11 + 637263T^10 - 472300T^9 + 319400T^8 - 166590T^7 + 36835T^6 + 5004T^5 - 4675T^4 + 650T^3 + 295T^2 + 20*T + 1
$17$	$ T^{40} - 2 T^{35} + \cdots + 1 $ T^40 - 2T^35 - 25T^34 + 200T^33 - 35T^32 - 10T^31 + 2T^30 + 100T^29 + 325T^28 - 3610T^27 + 12315T^26 - 4750T^25 - 100T^24 - 2950T^23 - 5805T^22 + 45820T^21 - 130254T^20 + 248850T^19 - 149675T^18 + 14980T^17 + 227355T^16 - 480990T^15 + 716775T^14 - 948950T^13 + 1132465T^12 - 844310T^11 + 449138T^10 - 301800T^9 + 234775T^8 - 160340T^7 + 92560T^6 - 31246T^5 + 10325T^4 - 1100T^3 - 30T^2 + 20*T + 1
$19$	$ T^{40} $ T^40
$23$	$ T^{40} $ T^40
$29$	$ (T^{20} - 5 T^{17} + \cdots + 5)^{2} $ (T^20 - 5T^17 - 5T^16 + 25T^14 + 50T^13 + 25T^12 - 125T^11 - 375T^10 - 375T^9 + 500T^8 + 2525T^7 + 4400T^6 + 4380T^5 + 2750T^4 + 1125T^3 + 300T^2 + 50T + 5)^2
$31$	$ T^{40} $ T^40
$37$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 + 2T^35 - 35T^34 - 40T^33 + 70T^32 + 320T^31 - 118T^30 + 460T^29 + 330T^28 - 4120T^27 + 435T^26 + 11010T^25 - 10455T^24 - 16250T^23 + 30835T^22 + 2500T^21 + 53641T^20 - 4680T^19 - 32015T^18 - 99190T^17 - 177505T^16 - 36990T^15 + 316140T^14 + 415150T^13 + 104845T^12 + 126150T^11 + 5248T^10 + 7990T^9 + 28540T^8 - 36530T^7 + 42705T^6 - 34254T^5 + 15970T^4 - 3460T^3 + 410T^2 - 20*T + 1
$41$	$ T^{40} - 10 T^{36} + \cdots + 25 $ T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 7550T^26 + 20125T^24 - 28525T^22 + 23135T^20 - 32625T^18 - 21300T^16 + 461625T^14 + 622500T^12 + 331900T^10 + 100000T^8 + 9625T^6 + 5000T^4 - 500T^2 + 25
$43$	$ T^{40} $ T^40
$47$	$ T^{40} $ T^40
$53$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 + 2T^35 - 35T^34 - 40T^33 + 70T^32 + 320T^31 - 118T^30 + 460T^29 + 330T^28 - 4120T^27 + 435T^26 + 11010T^25 - 10455T^24 - 16250T^23 + 30835T^22 + 2500T^21 + 53641T^20 - 4680T^19 - 32015T^18 - 99190T^17 - 177505T^16 - 36990T^15 + 316140T^14 + 415150T^13 + 104845T^12 + 126150T^11 + 5248T^10 + 7990T^9 + 28540T^8 - 36530T^7 + 42705T^6 - 34254T^5 + 15970T^4 - 3460T^3 + 410T^2 - 20*T + 1
$59$	$ T^{40} $ T^40
$61$	$ T^{40} - 10 T^{36} + \cdots + 25 $ T^40 - 10T^36 + 100T^34 + 75T^32 - 750T^30 + 2750T^28 + 3075T^26 - 10625T^24 + 38525T^22 - 6865T^20 - 196500T^18 + 470950T^16 - 277750T^14 + 375250T^12 - 233725T^10 + 1250T^8 + 8875T^6 + 6875T^4 + 375T^2 + 25
$67$	$ T^{40} $ T^40
$71$	$ T^{40} $ T^40
$73$	$ T^{40} + 2 T^{35} + \cdots + 1 $ T^40 + 2T^35 - 25T^34 + 50T^33 - 35T^32 + 10T^31 + 2T^30 - 100T^29 + 825T^28 - 2640T^27 + 4290T^26 - 4000T^25 + 2025T^24 + 3450T^23 - 30805T^22 + 119330T^21 - 277129T^20 + 419400T^19 - 421925T^18 + 141570T^17 + 1061830T^16 - 5170260T^15 + 14842775T^14 - 29709300T^13 + 43428440T^12 - 47601040T^11 + 39782888T^10 - 25593200T^9 + 12712650T^8 - 4861110T^7 + 1418285T^6 - 311254T^5 + 50450T^4 - 5900T^3 + 470T^2 - 20*T + 1
$79$	$ T^{40} $ T^40
$83$	$ T^{40} $ T^40
$89$	$ (T^{20} - 5 T^{19} + \cdots + 5)^{2} $ (T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5)^2
$97$	$ T^{40} + 2 T^{35} + \cdots + 1 $ T^40 + 2T^35 - 25T^34 - 200T^33 - 35T^32 + 10T^31 + 2T^30 - 100T^29 + 325T^28 + 3610T^27 + 12315T^26 + 4750T^25 - 100T^24 + 2950T^23 - 5805T^22 - 45820T^21 - 130254T^20 - 248850T^19 - 149675T^18 - 14980T^17 + 227355T^16 + 480990T^15 + 716775T^14 + 948950T^13 + 1132465T^12 + 844310T^11 + 449138T^10 + 301800T^9 + 234775T^8 + 160340T^7 + 92560T^6 + 31246T^5 + 10325T^4 + 1100T^3 - 30T^2 - 20*T + 1
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.cq (of order \(100\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{100}^{13} q^{5} - \zeta_{100}^{49} q^{9} +O(q^{10}) \) q + z^13 * q^5 - z^49 * q^9	\( q + \zeta_{100}^{13} q^{5} - \zeta_{100}^{49} q^{9} + ( - \zeta_{100}^{21} + \zeta_{100}^{2}) q^{13} + ( - \zeta_{100}^{44} + \zeta_{100}^{17}) q^{17} + \zeta_{100}^{26} q^{25} + ( - \zeta_{100}^{28} - \zeta_{100}^{6}) q^{29} + (\zeta_{100}^{35} - \zeta_{100}^{18}) q^{37} + (\zeta_{100}^{47} - \zeta_{100}^{11}) q^{41} + \zeta_{100}^{12} q^{45} + \zeta_{100}^{45} q^{49} + ( - \zeta_{100}^{14} + \zeta_{100}^{5}) q^{53} + ( - \zeta_{100}^{33} + \zeta_{100}^{9}) q^{61} + ( - \zeta_{100}^{34} + \zeta_{100}^{15}) q^{65} + (\zeta_{100}^{29} + \zeta_{100}^{8}) q^{73} - \zeta_{100}^{48} q^{81} + (\zeta_{100}^{30} + \zeta_{100}^{7}) q^{85} + ( - \zeta_{100}^{40} - \zeta_{100}^{22}) q^{89} + (\zeta_{100}^{36} + \zeta_{100}^{23}) q^{97} +O(q^{100}) \) q + z^13 * q^5 - z^49 * q^9 + (-z^21 + z^2) * q^13 + (-z^44 + z^17) * q^17 + z^26 * q^25 + (-z^28 - z^6) * q^29 + (z^35 - z^18) * q^37 + (z^47 - z^11) * q^41 + z^12 * q^45 + z^45 * q^49 + (-z^14 + z^5) * q^53 + (-z^33 + z^9) * q^61 + (-z^34 + z^15) * q^65 + (z^29 + z^8) * q^73 - z^48 * q^81 + (z^30 + z^7) * q^85 + (-z^40 - z^22) * q^89 + (z^36 + z^23) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 10 q^{85} + 10 q^{89}+O(q^{100}) \) 40 * q + 10 * q^85 + 10 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.1.cq.a

Level	$4000$
Weight	$1$
Character orbit	4000.cq
Analytic conductor	$1.996$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{100}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.99626005053\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Coefficient field:	\(\Q(\zeta_{100})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{30} + x^{20} - x^{10} + 1 \) x^40 - x^30 + x^20 - x^10 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{100}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{100} - \cdots)\)

\(n\)	\(1377\)	\(2501\)	\(2751\)
\(\chi(n)\)	\(\zeta_{100}^{7}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 33.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
125.i	odd	100	1	inner
500.r	even	100	1	inner

$p$	$F_p(T)$
$2$	\( T^{40} \) T^40
$3$	\( T^{40} \) T^40
$5$	\( T^{40} - T^{30} + \cdots + 1 \) T^40 - T^30 + T^20 - T^10 + 1
$7$	\( T^{40} \) T^40
$11$	\( T^{40} \) T^40
$13$	\( T^{40} - 2 T^{35} + \cdots + 1 \) T^40 - 2T^35 + 100T^34 - 50T^33 - 35T^32 - 10T^31 + 2T^30 - 150T^29 + 3450T^28 - 3610T^27 - 85T^26 + 2750T^25 + 2275T^24 - 4450T^23 + 47845T^22 - 76830T^21 + 57871T^20 + 23100T^19 - 153300T^18 + 96580T^17 + 172880T^16 - 459740T^15 + 713900T^14 - 944950T^13 + 1132140T^12 - 860560T^11 + 637263T^10 - 472300T^9 + 319400T^8 - 166590T^7 + 36835T^6 + 5004T^5 - 4675T^4 + 650T^3 + 295T^2 + 20*T + 1
$17$	\( T^{40} - 2 T^{35} + \cdots + 1 \) T^40 - 2T^35 - 25T^34 + 200T^33 - 35T^32 - 10T^31 + 2T^30 + 100T^29 + 325T^28 - 3610T^27 + 12315T^26 - 4750T^25 - 100T^24 - 2950T^23 - 5805T^22 + 45820T^21 - 130254T^20 + 248850T^19 - 149675T^18 + 14980T^17 + 227355T^16 - 480990T^15 + 716775T^14 - 948950T^13 + 1132465T^12 - 844310T^11 + 449138T^10 - 301800T^9 + 234775T^8 - 160340T^7 + 92560T^6 - 31246T^5 + 10325T^4 - 1100T^3 - 30T^2 + 20*T + 1
$19$	\( T^{40} \) T^40
$23$	\( T^{40} \) T^40
$29$	\( (T^{20} - 5 T^{17} + \cdots + 5)^{2} \) (T^20 - 5T^17 - 5T^16 + 25T^14 + 50T^13 + 25T^12 - 125T^11 - 375T^10 - 375T^9 + 500T^8 + 2525T^7 + 4400T^6 + 4380T^5 + 2750T^4 + 1125T^3 + 300T^2 + 50T + 5)^2
$31$	\( T^{40} \) T^40
$37$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 + 2T^35 - 35T^34 - 40T^33 + 70T^32 + 320T^31 - 118T^30 + 460T^29 + 330T^28 - 4120T^27 + 435T^26 + 11010T^25 - 10455T^24 - 16250T^23 + 30835T^22 + 2500T^21 + 53641T^20 - 4680T^19 - 32015T^18 - 99190T^17 - 177505T^16 - 36990T^15 + 316140T^14 + 415150T^13 + 104845T^12 + 126150T^11 + 5248T^10 + 7990T^9 + 28540T^8 - 36530T^7 + 42705T^6 - 34254T^5 + 15970T^4 - 3460T^3 + 410T^2 - 20*T + 1
$41$	\( T^{40} - 10 T^{36} + \cdots + 25 \) T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 7550T^26 + 20125T^24 - 28525T^22 + 23135T^20 - 32625T^18 - 21300T^16 + 461625T^14 + 622500T^12 + 331900T^10 + 100000T^8 + 9625T^6 + 5000T^4 - 500T^2 + 25
$43$	\( T^{40} \) T^40
$47$	\( T^{40} \) T^40
$53$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 + 2T^35 - 35T^34 - 40T^33 + 70T^32 + 320T^31 - 118T^30 + 460T^29 + 330T^28 - 4120T^27 + 435T^26 + 11010T^25 - 10455T^24 - 16250T^23 + 30835T^22 + 2500T^21 + 53641T^20 - 4680T^19 - 32015T^18 - 99190T^17 - 177505T^16 - 36990T^15 + 316140T^14 + 415150T^13 + 104845T^12 + 126150T^11 + 5248T^10 + 7990T^9 + 28540T^8 - 36530T^7 + 42705T^6 - 34254T^5 + 15970T^4 - 3460T^3 + 410T^2 - 20*T + 1
$59$	\( T^{40} \) T^40
$61$	\( T^{40} - 10 T^{36} + \cdots + 25 \) T^40 - 10T^36 + 100T^34 + 75T^32 - 750T^30 + 2750T^28 + 3075T^26 - 10625T^24 + 38525T^22 - 6865T^20 - 196500T^18 + 470950T^16 - 277750T^14 + 375250T^12 - 233725T^10 + 1250T^8 + 8875T^6 + 6875T^4 + 375T^2 + 25
$67$	\( T^{40} \) T^40
$71$	\( T^{40} \) T^40
$73$	\( T^{40} + 2 T^{35} + \cdots + 1 \) T^40 + 2T^35 - 25T^34 + 50T^33 - 35T^32 + 10T^31 + 2T^30 - 100T^29 + 825T^28 - 2640T^27 + 4290T^26 - 4000T^25 + 2025T^24 + 3450T^23 - 30805T^22 + 119330T^21 - 277129T^20 + 419400T^19 - 421925T^18 + 141570T^17 + 1061830T^16 - 5170260T^15 + 14842775T^14 - 29709300T^13 + 43428440T^12 - 47601040T^11 + 39782888T^10 - 25593200T^9 + 12712650T^8 - 4861110T^7 + 1418285T^6 - 311254T^5 + 50450T^4 - 5900T^3 + 470T^2 - 20*T + 1
$79$	\( T^{40} \) T^40
$83$	\( T^{40} \) T^40
$89$	\( (T^{20} - 5 T^{19} + \cdots + 5)^{2} \) (T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5)^2
$97$	\( T^{40} + 2 T^{35} + \cdots + 1 \) T^40 + 2T^35 - 25T^34 - 200T^33 - 35T^32 + 10T^31 + 2T^30 - 100T^29 + 325T^28 + 3610T^27 + 12315T^26 + 4750T^25 - 100T^24 + 2950T^23 - 5805T^22 - 45820T^21 - 130254T^20 - 248850T^19 - 149675T^18 - 14980T^17 + 227355T^16 + 480990T^15 + 716775T^14 + 948950T^13 + 1132465T^12 + 844310T^11 + 449138T^10 + 301800T^9 + 234775T^8 + 160340T^7 + 92560T^6 + 31246T^5 + 10325T^4 + 1100T^3 - 30T^2 - 20*T + 1
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