LMFDB - Newform orbit 2028.1.bm.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,1,Mod(17,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(78))

chi = DirichletCharacter(H, H._module([0, 39, 73]))

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

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

chi := DirichletCharacter("2028.17");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2028.bm (of order $78$, 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:	$1.01210384562$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{39})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 $ x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{78}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{78} - \cdots)$

$q$-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_{78}^{29} q^{3} + (\zeta_{78}^{15} + \zeta_{78}^{8}) q^{7} - \zeta_{78}^{19} q^{9} +O(q^{10}) $ q + z^29 * q^3 + (z^15 + z^8) * q^7 - z^19 * q^9	$ q + \zeta_{78}^{29} q^{3} + (\zeta_{78}^{15} + \zeta_{78}^{8}) q^{7} - \zeta_{78}^{19} q^{9} + \zeta_{78}^{13} q^{13} + (\zeta_{78}^{19} - \zeta_{78}^{7}) q^{19} + (\zeta_{78}^{37} - \zeta_{78}^{5}) q^{21} - \zeta_{78}^{30} q^{25} + \zeta_{78}^{9} q^{27} + (\zeta_{78}^{10} - \zeta_{78}^{8}) q^{31} + ( - \zeta_{78}^{17} - \zeta_{78}^{14}) q^{37} - \zeta_{78}^{3} q^{39} + ( - \zeta_{78}^{35} + \zeta_{78}^{12}) q^{43} + (\zeta_{78}^{30} + \cdots + \zeta_{78}^{16}) q^{49} + \cdots + ( - \zeta_{78}^{24} - \zeta_{78}^{11}) q^{97} +O(q^{100}) $ q + z^29 * q^3 + (z^15 + z^8) * q^7 - z^19 * q^9 + z^13 * q^13 + (z^19 - z^7) * q^19 + (z^37 - z^5) * q^21 - z^30 * q^25 + z^9 * q^27 + (z^10 - z^8) * q^31 + (-z^17 - z^14) * q^37 - z^3 * q^39 + (-z^35 + z^12) * q^43 + (z^30 + z^23 + z^16) * q^49 + (-z^36 - z^9) * q^57 + (z^35 + z^33) * q^61 + (-z^34 - z^27) * q^63 + (z^36 + z^31) * q^67 + (z^22 + z^5) * q^73 + z^20 * q^75 + (z^11 - z^4) * q^79 + z^38 * q^81 + (z^28 + z^21) * q^91 + (-z^37 - 1) * q^93 + (-z^24 - z^11) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{3} + 3 q^{7} + q^{9}+O(q^{10}) $ 24 * q - q^3 + 3 * q^7 + q^9	$ 24 q - q^{3} + 3 q^{7} + q^{9} + 12 q^{13} + 2 q^{25} + 2 q^{27} - 2 q^{39} - q^{43} - 2 q^{49} + q^{61} - 3 q^{63} - 3 q^{67} + q^{75} - 2 q^{79} + q^{81} + 3 q^{91} - 23 q^{93} + 3 q^{97}+O(q^{100}) $ 24 * q - q^3 + 3 * q^7 + q^9 + 12 * q^13 + 2 * q^25 + 2 * q^27 - 2 * q^39 - q^43 - 2 * q^49 + q^61 - 3 * q^63 - 3 * q^67 + q^75 - 2 * q^79 + q^81 + 3 * q^91 - 23 * q^93 + 3 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$-1$	$1$	$\zeta_{78}^{25}$

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} $

17.1

0.799443	−	0.600742i
−0.200026	−	0.979791i
0.428693	−	0.903450i
−0.996757	+	0.0804666i
−0.0402659	−	0.999189i
−0.0402659	+	0.999189i
0.987050	+	0.160411i
−0.845190	−	0.534466i
0.278217	−	0.960518i
−0.996757	−	0.0804666i
−0.919979	−	0.391967i
−0.919979	+	0.391967i
−0.632445	+	0.774605i
0.799443	+	0.600742i
−0.200026	+	0.979791i
0.692724	−	0.721202i
0.278217	+	0.960518i
−0.632445	−	0.774605i
0.692724	+	0.721202i
−0.845190	+	0.534466i

−0.987050

−

0.160411i

1.39963

0.664135i

0.948536

0.316668i

101.1

−0.428693

0.903450i

−0.160803

−

0.00648012i

−0.632445

−

0.774605i

173.1

−0.278217

0.960518i

−0.565375

−

1.32698i

−0.845190

−

0.534466i

257.1

−0.692724

−

0.721202i

1.15405

−

1.53576i

−0.0402659

0.999189i

329.1

0.919979

0.391967i

0.380472

−

1.13965i

0.692724

0.721202i

413.1

0.919979

−

0.391967i

0.380472

1.13965i

0.692724

−

0.721202i

569.1

0.0402659

0.999189i

1.02673

0.297395i

−0.996757

0.0804666i

641.1

−0.799443

−

0.600742i

−0.768090

−

0.156807i

0.278217

0.960518i

725.1

−0.948536

−

0.316668i

−1.51790

1.23933i

0.799443

0.600742i

797.1

−0.692724

0.721202i

1.15405

1.53576i

−0.0402659

−

0.999189i

881.1

0.632445

−

0.774605i

−0.0258155

−

0.319782i

−0.200026

−

0.979791i

953.1

0.632445

0.774605i

−0.0258155

0.319782i

−0.200026

0.979791i

1109.1

0.845190

−

0.534466i

1.44124

−

1.38433i

0.428693

−

0.903450i

1193.1

−0.987050

0.160411i

1.39963

−

0.664135i

0.948536

−

0.316668i

1265.1

−0.428693

−

0.903450i

−0.160803

0.00648012i

−0.632445

0.774605i

1349.1

0.200026

−

0.979791i

0.101594

−

0.625134i

−0.919979

−

0.391967i

1421.1

−0.948536

0.316668i

−1.51790

−

1.23933i

0.799443

−

0.600742i

1505.1

0.845190

0.534466i

1.44124

1.38433i

0.428693

0.903450i

1577.1

0.200026

0.979791i

0.101594

0.625134i

−0.919979

0.391967i

1661.1

−0.799443

0.600742i

−0.768090

0.156807i

0.278217

−

0.960518i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
169.k	even	78	1	inner
507.v	odd	78	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2028.1.bm.a	✓	24
3.b	odd	2	1	CM	2028.1.bm.a	✓	24
169.k	even	78	1	inner	2028.1.bm.a	✓	24
507.v	odd	78	1	inner	2028.1.bm.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2028.1.bm.a	✓	24	1.a	even	1	1	trivial
2028.1.bm.a	✓	24	3.b	odd	2	1	CM
2028.1.bm.a	✓	24	169.k	even	78	1	inner
2028.1.bm.a	✓	24	507.v	odd	78	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2028, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} $ T^24
$3$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 - T^21 - T^20 + T^18 + T^17 - T^15 - T^14 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$5$	$ T^{24} $ T^24
$7$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 6T^22 - 9T^21 + 35T^20 - 78T^19 + 129T^18 - 192T^17 + 436T^16 - 732T^15 + 875T^14 - 897T^13 + 1106T^12 - 624T^11 - 94T^10 + 1140T^9 - 1279T^8 - 51T^7 + 441T^6 - 156T^5 + 328T^4 + 108T^3 + 44T^2 + 12*T + 1
$11$	$ T^{24} $ T^24
$13$	$ (T^{2} - T + 1)^{12} $ (T^2 - T + 1)^12
$17$	$ T^{24} $ T^24
$19$	$ T^{24} - 13 T^{22} + \cdots + 169 $ T^24 - 13T^22 + 104T^20 - 533T^18 + 2015T^16 - 5499T^14 + 11349T^12 - 16731T^10 + 18083T^8 - 12506T^6 + 5915T^4 - 1183*T^2 + 169
$23$	$ T^{24} $ T^24
$29$	$ T^{24} $ T^24
$31$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 + 13T^21 + 9T^20 - 39T^19 + 90T^18 + 117T^17 - 244T^16 + 195T^15 + 732T^14 - 442T^13 - 350T^12 + 1326T^11 - 302T^10 - 936T^9 + 1036T^8 - 234T^7 + 441T^6 + 208T^5 + 198T^4 + 221T^3 + 82T^2 + 13T + 1
$37$	$ T^{24} - 13 T^{20} + \cdots + 169 $ T^24 - 13T^20 - 39T^17 + 130T^16 - 13T^14 + 1053T^13 - 481T^12 + 1859T^10 - 1014T^9 + 1352T^8 + 1014T^7 + 1014T^6 - 3549T^5 - 338T^4 + 507T^3 + 1352T^2 + 507T + 169
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 - T^21 - 27T^20 - 26T^19 + T^18 + 66T^17 + 312T^16 + 246T^15 - 53T^14 - 767T^13 - 1754T^12 - 988T^11 + 2118T^10 + 4120T^9 + 3861T^8 - 753T^7 - 1065T^6 + 702T^5 + 844T^4 - 404T^3 + 104T^2 + 14*T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + T^21 - T^20 - 26T^19 + 40T^18 - 14T^17 - 26T^16 + 27T^15 + 610T^14 - 1365T^13 + 911T^12 + 455T^11 - 1704T^10 - 272T^9 + 1469T^8 - 859T^7 - 324T^6 + 2899T^5 + 3002T^4 + 1197T^3 + 208T^2 + 12*T + 1
$67$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 6T^22 + 9T^21 + 9T^20 - 27T^18 - 94T^17 - 19T^16 + 43T^15 + 30T^14 + 91T^13 + 170T^12 + 234T^11 + 361T^10 - 802T^9 + 2426T^8 - 2822T^7 + 2339T^6 - 1248T^5 + 601T^4 - 134T^3 - 8*T^2 + T + 1
$71$	$ T^{24} $ T^24
$73$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 + 9T^20 - 27T^18 + 13T^17 - 10T^16 + 143T^15 - 126T^14 - 364T^13 + 365T^12 + 1092T^11 - 926T^10 - 5642T^9 + 15791T^8 - 20254T^7 + 15833T^6 - 8190T^5 + 2902T^4 - 715T^3 + 121T^2 - 13T + 1
$79$	$ T^{24} + 2 T^{23} + \cdots + 1 $ T^24 + 2T^23 + 3T^22 - 9T^21 - 21T^20 - 33T^19 + 72T^18 + 177T^17 + 256T^16 - 211T^15 - 678T^14 - 1002T^13 + 520T^12 + 2040T^11 + 4912T^10 + 4742T^9 + 4754T^8 + 1724T^7 + 2243T^6 + 3152T^5 + 2540T^4 + 1109T^3 + 224T^2 + 15*T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} - 3 T^{23} + \cdots + 531441 $ T^24 - 3T^23 + 6T^22 - 9T^21 + 9T^20 - 27T^18 + 81T^17 - 162T^16 + 243T^15 - 243T^14 + 729T^12 - 2187T^10 + 6561T^9 - 13122T^8 + 19683T^7 - 19683T^6 + 59049T^4 - 177147T^3 + 354294T^2 - 531441*T + 531441
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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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2028.bm (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{78}^{29} q^{3} + (\zeta_{78}^{15} + \zeta_{78}^{8}) q^{7} - \zeta_{78}^{19} q^{9} +O(q^{10}) \) q + z^29 * q^3 + (z^15 + z^8) * q^7 - z^19 * q^9	\( q + \zeta_{78}^{29} q^{3} + (\zeta_{78}^{15} + \zeta_{78}^{8}) q^{7} - \zeta_{78}^{19} q^{9} + \zeta_{78}^{13} q^{13} + (\zeta_{78}^{19} - \zeta_{78}^{7}) q^{19} + (\zeta_{78}^{37} - \zeta_{78}^{5}) q^{21} - \zeta_{78}^{30} q^{25} + \zeta_{78}^{9} q^{27} + (\zeta_{78}^{10} - \zeta_{78}^{8}) q^{31} + ( - \zeta_{78}^{17} - \zeta_{78}^{14}) q^{37} - \zeta_{78}^{3} q^{39} + ( - \zeta_{78}^{35} + \zeta_{78}^{12}) q^{43} + (\zeta_{78}^{30} + \cdots + \zeta_{78}^{16}) q^{49} + \cdots + ( - \zeta_{78}^{24} - \zeta_{78}^{11}) q^{97} +O(q^{100}) \) q + z^29 * q^3 + (z^15 + z^8) * q^7 - z^19 * q^9 + z^13 * q^13 + (z^19 - z^7) * q^19 + (z^37 - z^5) * q^21 - z^30 * q^25 + z^9 * q^27 + (z^10 - z^8) * q^31 + (-z^17 - z^14) * q^37 - z^3 * q^39 + (-z^35 + z^12) * q^43 + (z^30 + z^23 + z^16) * q^49 + (-z^36 - z^9) * q^57 + (z^35 + z^33) * q^61 + (-z^34 - z^27) * q^63 + (z^36 + z^31) * q^67 + (z^22 + z^5) * q^73 + z^20 * q^75 + (z^11 - z^4) * q^79 + z^38 * q^81 + (z^28 + z^21) * q^91 + (-z^37 - 1) * q^93 + (-z^24 - z^11) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{3} + 3 q^{7} + q^{9}+O(q^{10}) \) 24 * q - q^3 + 3 * q^7 + q^9	\( 24 q - q^{3} + 3 q^{7} + q^{9} + 12 q^{13} + 2 q^{25} + 2 q^{27} - 2 q^{39} - q^{43} - 2 q^{49} + q^{61} - 3 q^{63} - 3 q^{67} + q^{75} - 2 q^{79} + q^{81} + 3 q^{91} - 23 q^{93} + 3 q^{97}+O(q^{100}) \) 24 * q - q^3 + 3 * q^7 + q^9 + 12 * q^13 + 2 * q^25 + 2 * q^27 - 2 * q^39 - q^43 - 2 * q^49 + q^61 - 3 * q^63 - 3 * q^67 + q^75 - 2 * q^79 + q^81 + 3 * q^91 - 23 * q^93 + 3 * q^97

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	2028.1.bm.a

Level	$2028$
Weight	$1$
Character orbit	2028.bm
Analytic conductor	$1.012$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{78}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.01210384562\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{39})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 \) x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{78}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{78} - \cdots)\)

\(n\)	\(677\)	\(1015\)	\(1861\)
\(\chi(n)\)	\(-1\)	\(1\)	\(\zeta_{78}^{25}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 17.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
169.k	even	78	1	inner
507.v	odd	78	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} \) T^24
$3$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 - T^21 - T^20 + T^18 + T^17 - T^15 - T^14 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$5$	\( T^{24} \) T^24
$7$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 6T^22 - 9T^21 + 35T^20 - 78T^19 + 129T^18 - 192T^17 + 436T^16 - 732T^15 + 875T^14 - 897T^13 + 1106T^12 - 624T^11 - 94T^10 + 1140T^9 - 1279T^8 - 51T^7 + 441T^6 - 156T^5 + 328T^4 + 108T^3 + 44T^2 + 12*T + 1
$11$	\( T^{24} \) T^24
$13$	\( (T^{2} - T + 1)^{12} \) (T^2 - T + 1)^12
$17$	\( T^{24} \) T^24
$19$	\( T^{24} - 13 T^{22} + \cdots + 169 \) T^24 - 13T^22 + 104T^20 - 533T^18 + 2015T^16 - 5499T^14 + 11349T^12 - 16731T^10 + 18083T^8 - 12506T^6 + 5915T^4 - 1183*T^2 + 169
$23$	\( T^{24} \) T^24
$29$	\( T^{24} \) T^24
$31$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 + 13T^21 + 9T^20 - 39T^19 + 90T^18 + 117T^17 - 244T^16 + 195T^15 + 732T^14 - 442T^13 - 350T^12 + 1326T^11 - 302T^10 - 936T^9 + 1036T^8 - 234T^7 + 441T^6 + 208T^5 + 198T^4 + 221T^3 + 82T^2 + 13T + 1
$37$	\( T^{24} - 13 T^{20} + \cdots + 169 \) T^24 - 13T^20 - 39T^17 + 130T^16 - 13T^14 + 1053T^13 - 481T^12 + 1859T^10 - 1014T^9 + 1352T^8 + 1014T^7 + 1014T^6 - 3549T^5 - 338T^4 + 507T^3 + 1352T^2 + 507T + 169
$41$	\( T^{24} \) T^24
$43$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 - T^21 - 27T^20 - 26T^19 + T^18 + 66T^17 + 312T^16 + 246T^15 - 53T^14 - 767T^13 - 1754T^12 - 988T^11 + 2118T^10 + 4120T^9 + 3861T^8 - 753T^7 - 1065T^6 + 702T^5 + 844T^4 - 404T^3 + 104T^2 + 14*T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} - T^{23} + \cdots + 1 \) T^24 - T^23 + T^21 - T^20 - 26T^19 + 40T^18 - 14T^17 - 26T^16 + 27T^15 + 610T^14 - 1365T^13 + 911T^12 + 455T^11 - 1704T^10 - 272T^9 + 1469T^8 - 859T^7 - 324T^6 + 2899T^5 + 3002T^4 + 1197T^3 + 208T^2 + 12*T + 1
$67$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 6T^22 + 9T^21 + 9T^20 - 27T^18 - 94T^17 - 19T^16 + 43T^15 + 30T^14 + 91T^13 + 170T^12 + 234T^11 + 361T^10 - 802T^9 + 2426T^8 - 2822T^7 + 2339T^6 - 1248T^5 + 601T^4 - 134T^3 - 8*T^2 + T + 1
$71$	\( T^{24} \) T^24
$73$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 + 9T^20 - 27T^18 + 13T^17 - 10T^16 + 143T^15 - 126T^14 - 364T^13 + 365T^12 + 1092T^11 - 926T^10 - 5642T^9 + 15791T^8 - 20254T^7 + 15833T^6 - 8190T^5 + 2902T^4 - 715T^3 + 121T^2 - 13T + 1
$79$	\( T^{24} + 2 T^{23} + \cdots + 1 \) T^24 + 2T^23 + 3T^22 - 9T^21 - 21T^20 - 33T^19 + 72T^18 + 177T^17 + 256T^16 - 211T^15 - 678T^14 - 1002T^13 + 520T^12 + 2040T^11 + 4912T^10 + 4742T^9 + 4754T^8 + 1724T^7 + 2243T^6 + 3152T^5 + 2540T^4 + 1109T^3 + 224T^2 + 15*T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} - 3 T^{23} + \cdots + 531441 \) T^24 - 3T^23 + 6T^22 - 9T^21 + 9T^20 - 27T^18 + 81T^17 - 162T^16 + 243T^15 - 243T^14 + 729T^12 - 2187T^10 + 6561T^9 - 13122T^8 + 19683T^7 - 19683T^6 + 59049T^4 - 177147T^3 + 354294T^2 - 531441*T + 531441
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