LMFDB - Newform orbit 3700.1.dx.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,1,Mod(71,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(90))

chi = DirichletCharacter(H, H._module([45, 54, 20]))

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

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

chi := DirichletCharacter("3700.71");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3700.dx (of order $90$, degree $24$, 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:	$1.84654054674$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{45})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{45}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{45} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$\zeta_{90}^{20}$	$\zeta_{90}^{18}$	$-1$

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} $

71.1

−0.374607	+	0.927184i
0.990268	+	0.139173i
0.848048	+	0.529919i
0.0348995	−	0.999391i
−0.241922	−	0.970296i
−0.882948	+	0.469472i
−0.997564	+	0.0697565i
0.559193	−	0.829038i
0.438371	+	0.898794i
0.559193	+	0.829038i
0.438371	−	0.898794i
0.0348995	+	0.999391i
−0.882948	−	0.469472i
−0.997564	−	0.0697565i
−0.241922	+	0.970296i
−0.615661	−	0.788011i
0.990268	−	0.139173i
0.961262	−	0.275637i
0.848048	−	0.529919i
−0.719340	−	0.694658i

0.848048

−

0.529919i

0.438371

−

0.898794i

0.913545

0.406737i

−0.104528

−

0.994522i

0.990268

−

0.139173i

0.990268

−

0.139173i

231.1

−0.882948

0.469472i

0.559193

−

0.829038i

0.913545

−

0.406737i

−0.104528

0.994522i

−0.615661

0.788011i

−0.615661

0.788011i

271.1

−0.374607

−

0.927184i

−0.719340

0.694658i

−0.104528

−

0.994522i

0.913545

0.406737i

−0.882948

0.469472i

−0.882948

0.469472i

571.1

−0.615661

0.788011i

−0.241922

−

0.970296i

−0.104528

−

0.994522i

0.913545

0.406737i

0.848048

0.529919i

0.848048

0.529919i

811.1

−0.997564

−

0.0697565i

0.990268

0.139173i

0.669131

−

0.743145i

−0.978148

−

0.207912i

−0.719340

0.694658i

−0.719340

0.694658i

971.1

0.990268

0.139173i

0.961262

0.275637i

−0.104528

−

0.994522i

0.913545

0.406737i

0.0348995

−

0.999391i

0.0348995

−

0.999391i

1011.1

−0.241922

0.970296i

−0.882948

−

0.469472i

−0.978148

−

0.207912i

0.669131

−

0.743145i

0.438371

−

0.898794i

0.438371

−

0.898794i

1191.1

0.961262

0.275637i

0.848048

0.529919i

−0.978148

0.207912i

0.669131

0.743145i

−0.997564

−

0.0697565i

−0.997564

−

0.0697565i

1291.1

−0.719340

0.694658i

0.0348995

−

0.999391i

−0.978148

0.207912i

0.669131

0.743145i

0.559193

−

0.829038i

0.559193

−

0.829038i

1311.1

0.961262

−

0.275637i

0.848048

−

0.529919i

−0.978148

−

0.207912i

0.669131

−

0.743145i

−0.997564

0.0697565i

−0.997564

0.0697565i

1711.1

−0.719340

−

0.694658i

0.0348995

0.999391i

−0.978148

−

0.207912i

0.669131

−

0.743145i

0.559193

0.829038i

0.559193

0.829038i

1931.1

−0.615661

−

0.788011i

−0.241922

0.970296i

−0.104528

0.994522i

0.913545

−

0.406737i

0.848048

−

0.529919i

0.848048

−

0.529919i

2031.1

0.990268

−

0.139173i

0.961262

−

0.275637i

−0.104528

0.994522i

0.913545

−

0.406737i

0.0348995

0.999391i

0.0348995

0.999391i

2291.1

−0.241922

−

0.970296i

−0.882948

0.469472i

−0.978148

0.207912i

0.669131

0.743145i

0.438371

0.898794i

0.438371

0.898794i

2491.1

−0.997564

0.0697565i

0.990268

−

0.139173i

0.669131

0.743145i

−0.978148

0.207912i

−0.719340

−

0.694658i

−0.719340

−

0.694658i

2671.1

0.0348995

0.999391i

−0.997564

0.0697565i

0.913545

0.406737i

−0.104528

−

0.994522i

−0.374607

0.927184i

−0.374607

0.927184i

2771.1

−0.882948

−

0.469472i

0.559193

0.829038i

0.913545

0.406737i

−0.104528

−

0.994522i

−0.615661

−

0.788011i

−0.615661

−

0.788011i

2791.1

0.559193

0.829038i

−0.374607

0.927184i

0.669131

0.743145i

−0.978148

0.207912i

−0.241922

0.970296i

−0.241922

0.970296i

3031.1

−0.374607

0.927184i

−0.719340

−

0.694658i

−0.104528

0.994522i

0.913545

−

0.406737i

−0.882948

−

0.469472i

−0.882948

−

0.469472i

3191.1

0.438371

−

0.898794i

−0.615661

−

0.788011i

0.669131

0.743145i

−0.978148

0.207912i

0.961262

−

0.275637i

0.961262

−

0.275637i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
925.bs	even	45	1	inner
3700.dx	odd	90	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3700.1.dx.b	yes	24
4.b	odd	2	1	CM	3700.1.dx.b	yes	24
25.d	even	5	1		3700.1.dx.a	✓	24
37.f	even	9	1		3700.1.dx.a	✓	24
100.j	odd	10	1		3700.1.dx.a	✓	24
148.p	odd	18	1		3700.1.dx.a	✓	24
925.bs	even	45	1	inner	3700.1.dx.b	yes	24
3700.dx	odd	90	1	inner	3700.1.dx.b	yes	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3700.1.dx.a	✓	24	25.d	even	5	1
3700.1.dx.a	✓	24	37.f	even	9	1
3700.1.dx.a	✓	24	100.j	odd	10	1
3700.1.dx.a	✓	24	148.p	odd	18	1
3700.1.dx.b	yes	24	1.a	even	1	1	trivial
3700.1.dx.b	yes	24	4.b	odd	2	1	CM
3700.1.dx.b	yes	24	925.bs	even	45	1	inner
3700.1.dx.b	yes	24	3700.dx	odd	90	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{13}^{24} - 29 T_{13}^{21} + 345 T_{13}^{18} - 2141 T_{13}^{15} + 7314 T_{13}^{12} - 13421 T_{13}^{9} + \cdots + 1 $ T13^24 - 29*T13^21 + 345*T13^18 - 2141*T13^15 + 7314*T13^12 - 13421*T13^9 + 12005*T13^6 - 24*T13^3 + 1 acting on $S_{1}^{\mathrm{new}}(3700, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} - T^{21} + \cdots + 1 $ T^24 - T^21 + T^15 - T^12 + T^9 - T^3 + 1
$3$	$ T^{24} $ T^24
$5$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$7$	$ T^{24} $ T^24
$11$	$ T^{24} $ T^24
$13$	$ T^{24} - 29 T^{21} + \cdots + 1 $ T^24 - 29T^21 + 345T^18 - 2141T^15 + 7314T^12 - 13421T^9 + 12005T^6 - 24*T^3 + 1
$17$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 18T^18 - 105T^17 + 327T^16 - 311T^15 - 282T^14 + 555T^13 + 1026T^12 - 2163T^11 + 714T^10 + 436T^9 + 531T^8 - 384T^7 - 1249T^6 + 255T^5 + 555T^4 + 81T^3 + 201T^2 - 12*T + 1
$19$	$ T^{24} $ T^24
$23$	$ T^{24} $ T^24
$29$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 - 2T^21 + 9T^19 + 30T^18 - 129T^16 + T^15 + 468T^14 - 744T^13 + 1764T^11 - 2628T^10 + T^9 + 6345T^8 - 11979T^7 + 13253T^6 - 10062T^5 + 5478T^4 - 1995T^3 + 405T^2 - 33*T + 1
$31$	$ T^{24} $ T^24
$37$	$ T^{24} - T^{21} + \cdots + 1 $ T^24 - T^21 + T^15 - T^12 + T^9 - T^3 + 1
$41$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 18T^18 - 105T^17 + 327T^16 - 311T^15 - 282T^14 + 555T^13 + 1026T^12 - 2163T^11 + 714T^10 + 436T^9 + 531T^8 - 384T^7 - 1249T^6 + 255T^5 + 555T^4 + 81T^3 + 201T^2 - 12*T + 1
$43$	$ T^{24} $ T^24
$47$	$ T^{24} $ T^24
$53$	$ T^{24} + 6 T^{21} + \cdots + 1 $ T^24 + 6T^21 - 10T^18 + 74T^15 + 2004T^12 + 1304T^9 + 11955T^6 - 214*T^3 + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} + 6 T^{23} + \cdots + 1 $ T^24 + 6T^23 + 21T^22 + 55T^21 + 132T^20 + 291T^19 + 576T^18 + 996T^17 + 1590T^16 + 2443T^15 + 3576T^14 + 4506T^13 + 3945T^12 + 1572T^11 - 1527T^10 - 3593T^9 - 3333T^8 - 1551T^7 + 632T^6 + 903T^5 + 225T^4 + 69T^3 + 192T^2 - 27*T + 1
$67$	$ T^{24} $ T^24
$71$	$ T^{24} $ T^24
$73$	$ (T^{8} - 2 T^{7} + 3 T^{6} + \cdots + 1)^{3} $ (T^8 - 2T^7 + 3T^6 + T^5 + T^3 + 23T^2 + 3T + 1)^3
$79$	$ T^{24} $ T^24
$83$	$ T^{24} $ T^24
$89$	$ T^{24} - 24 T^{23} + \cdots + 1 $ T^24 - 24T^23 + 276T^22 - 2025T^21 + 10647T^20 - 42714T^19 + 135926T^18 - 352089T^17 + 755820T^16 - 1361767T^15 + 2077521T^14 - 2699529T^13 + 2997630T^12 - 2847483T^11 + 2310903T^10 - 1596208T^9 + 932022T^8 - 455121T^7 + 182847T^6 - 58932T^5 + 14625T^4 - 2596T^3 + 282T^2 - 12*T + 1
$97$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 - 2T^21 + 9T^19 + 30T^18 - 84T^16 + T^15 + 333T^14 + 516T^13 - 1881T^11 - 3933T^10 + T^9 + 13905T^8 + 28476T^7 + 30263T^6 + 19773T^5 + 8178T^4 + 2055T^3 + 270T^2 + 12*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:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3700.dx (of order \(90\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{90}^{19} q^{2} + \zeta_{90}^{38} q^{4} + \zeta_{90}^{42} q^{5} + \zeta_{90}^{12} q^{8} + \zeta_{90}^{16} q^{9} +O(q^{10}) \) q - z^19 * q^2 + z^38 * q^4 + z^42 * q^5 + z^12 * q^8 + z^16 * q^9	\( q - \zeta_{90}^{19} q^{2} + \zeta_{90}^{38} q^{4} + \zeta_{90}^{42} q^{5} + \zeta_{90}^{12} q^{8} + \zeta_{90}^{16} q^{9} + \zeta_{90}^{16} q^{10} + (\zeta_{90}^{44} - \zeta_{90}^{23}) q^{13} - \zeta_{90}^{31} q^{16} + ( - \zeta_{90}^{11} - \zeta_{90}^{3}) q^{17} - \zeta_{90}^{35} q^{18} - \zeta_{90}^{35} q^{20} - \zeta_{90}^{39} q^{25} + (\zeta_{90}^{42} + \zeta_{90}^{18}) q^{26} + ( - \zeta_{90}^{37} + \zeta_{90}^{14}) q^{29} - \zeta_{90}^{5} q^{32} + (\zeta_{90}^{30} + \zeta_{90}^{22}) q^{34} - \zeta_{90}^{9} q^{36} + \zeta_{90}^{2} q^{37} - \zeta_{90}^{9} q^{40} + ( - \zeta_{90}^{43} + \zeta_{90}^{24}) q^{41} - \zeta_{90}^{13} q^{45} + \zeta_{90}^{10} q^{49} - \zeta_{90}^{13} q^{50} + ( - \zeta_{90}^{37} + \zeta_{90}^{16}) q^{52} + (\zeta_{90}^{16} + \zeta_{90}^{10}) q^{53} + ( - \zeta_{90}^{33} - \zeta_{90}^{11}) q^{58} + ( - \zeta_{90}^{31} - \zeta_{90}^{27}) q^{61} + \zeta_{90}^{24} q^{64} + ( - \zeta_{90}^{41} + \zeta_{90}^{20}) q^{65} + ( - \zeta_{90}^{41} + \zeta_{90}^{4}) q^{68} + \zeta_{90}^{28} q^{72} + (\zeta_{90}^{12} + \zeta_{90}^{6}) q^{73} - \zeta_{90}^{21} q^{74} + \zeta_{90}^{28} q^{80} + \zeta_{90}^{32} q^{81} + ( - \zeta_{90}^{43} - \zeta_{90}^{17}) q^{82} + (\zeta_{90}^{8} + 1) q^{85} + (\zeta_{90}^{8} + 1) q^{89} + \zeta_{90}^{32} q^{90} + (\zeta_{90}^{44} + \zeta_{90}^{22}) q^{97} - \zeta_{90}^{29} q^{98} +O(q^{100}) \) q - z^19 * q^2 + z^38 * q^4 + z^42 * q^5 + z^12 * q^8 + z^16 * q^9 + z^16 * q^10 + (z^44 - z^23) * q^13 - z^31 * q^16 + (-z^11 - z^3) * q^17 - z^35 * q^18 - z^35 * q^20 - z^39 * q^25 + (z^42 + z^18) * q^26 + (-z^37 + z^14) * q^29 - z^5 * q^32 + (z^30 + z^22) * q^34 - z^9 * q^36 + z^2 * q^37 - z^9 * q^40 + (-z^43 + z^24) * q^41 - z^13 * q^45 + z^10 * q^49 - z^13 * q^50 + (-z^37 + z^16) * q^52 + (z^16 + z^10) * q^53 + (-z^33 - z^11) * q^58 + (-z^31 - z^27) * q^61 + z^24 * q^64 + (-z^41 + z^20) * q^65 + (-z^41 + z^4) * q^68 + z^28 * q^72 + (z^12 + z^6) * q^73 - z^21 * q^74 + z^28 * q^80 + z^32 * q^81 + (-z^43 - z^17) * q^82 + (z^8 + 1) * q^85 + (z^8 + 1) * q^89 + z^32 * q^90 + (z^44 + z^22) * q^97 - z^29 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{5} + 3 q^{8}+O(q^{10}) \) 24 * q + 3 * q^5 + 3 * q^8	\( 24 q + 3 q^{5} + 3 q^{8} + 3 q^{17} + 3 q^{25} - 3 q^{26} - 12 q^{34} - 6 q^{36} - 6 q^{40} + 3 q^{41} + 3 q^{58} - 6 q^{61} + 3 q^{64} + 6 q^{73} + 3 q^{74} + 24 q^{85} + 24 q^{89}+O(q^{100}) \) 24 * q + 3 * q^5 + 3 * q^8 + 3 * q^17 + 3 * q^25 - 3 * q^26 - 12 * q^34 - 6 * q^36 - 6 * q^40 + 3 * q^41 + 3 * q^58 - 6 * q^61 + 3 * q^64 + 6 * q^73 + 3 * q^74 + 24 * q^85 + 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3700.1.dx.b

Level	$3700$
Weight	$1$
Character orbit	3700.dx
Analytic conductor	$1.847$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{45}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.84654054674\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{45})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{45}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

\(n\)	\(1001\)	\(1777\)	\(1851\)
\(\chi(n)\)	\(\zeta_{90}^{20}\)	\(\zeta_{90}^{18}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 71.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
925.bs	even	45	1	inner
3700.dx	odd	90	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} - T^{21} + \cdots + 1 \) T^24 - T^21 + T^15 - T^12 + T^9 - T^3 + 1
$3$	\( T^{24} \) T^24
$5$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$7$	\( T^{24} \) T^24
$11$	\( T^{24} \) T^24
$13$	\( T^{24} - 29 T^{21} + \cdots + 1 \) T^24 - 29T^21 + 345T^18 - 2141T^15 + 7314T^12 - 13421T^9 + 12005T^6 - 24*T^3 + 1
$17$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 18T^18 - 105T^17 + 327T^16 - 311T^15 - 282T^14 + 555T^13 + 1026T^12 - 2163T^11 + 714T^10 + 436T^9 + 531T^8 - 384T^7 - 1249T^6 + 255T^5 + 555T^4 + 81T^3 + 201T^2 - 12*T + 1
$19$	\( T^{24} \) T^24
$23$	\( T^{24} \) T^24
$29$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 - 2T^21 + 9T^19 + 30T^18 - 129T^16 + T^15 + 468T^14 - 744T^13 + 1764T^11 - 2628T^10 + T^9 + 6345T^8 - 11979T^7 + 13253T^6 - 10062T^5 + 5478T^4 - 1995T^3 + 405T^2 - 33*T + 1
$31$	\( T^{24} \) T^24
$37$	\( T^{24} - T^{21} + \cdots + 1 \) T^24 - T^21 + T^15 - T^12 + T^9 - T^3 + 1
$41$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 18T^18 - 105T^17 + 327T^16 - 311T^15 - 282T^14 + 555T^13 + 1026T^12 - 2163T^11 + 714T^10 + 436T^9 + 531T^8 - 384T^7 - 1249T^6 + 255T^5 + 555T^4 + 81T^3 + 201T^2 - 12*T + 1
$43$	\( T^{24} \) T^24
$47$	\( T^{24} \) T^24
$53$	\( T^{24} + 6 T^{21} + \cdots + 1 \) T^24 + 6T^21 - 10T^18 + 74T^15 + 2004T^12 + 1304T^9 + 11955T^6 - 214*T^3 + 1
$59$	\( T^{24} \) T^24
$61$	\( T^{24} + 6 T^{23} + \cdots + 1 \) T^24 + 6T^23 + 21T^22 + 55T^21 + 132T^20 + 291T^19 + 576T^18 + 996T^17 + 1590T^16 + 2443T^15 + 3576T^14 + 4506T^13 + 3945T^12 + 1572T^11 - 1527T^10 - 3593T^9 - 3333T^8 - 1551T^7 + 632T^6 + 903T^5 + 225T^4 + 69T^3 + 192T^2 - 27*T + 1
$67$	\( T^{24} \) T^24
$71$	\( T^{24} \) T^24
$73$	\( (T^{8} - 2 T^{7} + 3 T^{6} + \cdots + 1)^{3} \) (T^8 - 2T^7 + 3T^6 + T^5 + T^3 + 23T^2 + 3T + 1)^3
$79$	\( T^{24} \) T^24
$83$	\( T^{24} \) T^24
$89$	\( T^{24} - 24 T^{23} + \cdots + 1 \) T^24 - 24T^23 + 276T^22 - 2025T^21 + 10647T^20 - 42714T^19 + 135926T^18 - 352089T^17 + 755820T^16 - 1361767T^15 + 2077521T^14 - 2699529T^13 + 2997630T^12 - 2847483T^11 + 2310903T^10 - 1596208T^9 + 932022T^8 - 455121T^7 + 182847T^6 - 58932T^5 + 14625T^4 - 2596T^3 + 282T^2 - 12*T + 1
$97$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 - 2T^21 + 9T^19 + 30T^18 - 84T^16 + T^15 + 333T^14 + 516T^13 - 1881T^11 - 3933T^10 + T^9 + 13905T^8 + 28476T^7 + 30263T^6 + 19773T^5 + 8178T^4 + 2055T^3 + 270T^2 + 12*T + 1
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