LMFDB - Newform orbit 2852.1.bn.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2852,1,Mod(551,2852)] 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("2852.551"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2852, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 15, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2852 = 2^{2} \cdot 23 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2852.bn (of order $30$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$1.42333341603$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$3$ over $\Q(\zeta_{30})$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{90}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{90} - \cdots)$

$q$-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_{90}^{10} q^{2} + (\zeta_{90}^{26} + \zeta_{90}^{22}) q^{3} + \zeta_{90}^{20} q^{4} + ( - \zeta_{90}^{36} - \zeta_{90}^{32}) q^{6} - \zeta_{90}^{30} q^{8} + (\zeta_{90}^{44} + \cdots - \zeta_{90}^{3}) q^{9} + \cdots + \zeta_{90}^{4} q^{98} +O(q^{100}) $ q - z^10 * q^2 + (z^26 + z^22) * q^3 + z^20 * q^4 + (-z^36 - z^32) * q^6 - z^30 * q^8 + (z^44 - z^7 - z^3) * q^9 + (z^42 - z) * q^12 + (-z^31 - z^2) * q^13 + z^40 * q^16 + (z^17 + z^13 + z^9) * q^18 - z^18 * q^23 + (z^11 + z^7) * q^24 - z^15 * q^25 + (z^41 + z^12) * q^26 + (-z^33 - z^29 - z^25 - z^21) * q^27 + (-z^14 - z^13) * q^29 + z * q^31 + z^5 * q^32 + (-z^27 - z^23 - z^19) * q^36 + (-z^28 - z^24 + z^12 + z^8) * q^39 + (-z^38 - z^4) * q^41 + z^28 * q^46 + (z^10 - z^8) * q^47 + (-z^21 - z^17) * q^48 + z^39 * q^49 + z^25 * q^50 + (-z^22 + z^6) * q^52 + (z^43 + z^39 + z^35 + z^31) * q^54 + (z^24 + z^23) * q^58 + (z^42 - z^6) * q^59 - z^11 * q^62 - z^15 * q^64 + (-z^44 - z^40) * q^69 + (z^35 + z^16) * q^71 + (z^37 + z^33 + z^29) * q^72 + (z^19 - z^5) * q^73 + (-z^41 - z^37) * q^75 + (z^38 + z^34 - z^22 - z^18) * q^78 + (-z^43 + z^14 + z^10 + z^6 + z^2) * q^81 + (z^14 - z^3) * q^82 + (-z^40 - z^39 - z^36 - z^35) * q^87 - z^38 * q^92 + (z^27 + z^23) * q^93 + (-z^20 + z^18) * q^94 + (z^31 + z^27) * q^96 + z^4 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 6 q^{6} + 12 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{18} + 6 q^{23} - 12 q^{25} + 3 q^{26} + 6 q^{27} - 6 q^{36} + 3 q^{48} - 3 q^{49} + 3 q^{52} - 3 q^{54} + 3 q^{58} - 12 q^{64} - 3 q^{72} + 6 q^{78}+ \cdots + 6 q^{96}+O(q^{100}) $ 24 * q + 6 * q^6 + 12 * q^8 + 3 * q^9 + 3 * q^12 + 6 * q^18 + 6 * q^23 - 12 * q^25 + 3 * q^26 + 6 * q^27 - 6 * q^36 + 3 * q^48 - 3 * q^49 + 3 * q^52 - 3 * q^54 + 3 * q^58 - 12 * q^64 - 3 * q^72 + 6 * q^78 + 3 * q^81 + 3 * q^82 + 9 * q^87 + 6 * q^93 - 6 * q^94 + 6 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2852\mathbb{Z}\right)^\times$.

$n$	$373$	$1427$	$2669$
$\chi(n)$	$-1$	$-1$	$\zeta_{90}^{3}$

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} $

551.1

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

−0.766044

−

0.642788i

1.40724

−

0.299118i

0.173648

0.984808i

−1.27028

−

0.675419i

0.500000

−

0.866025i

0.977310

−

0.435126i

551.2

−0.173648

0.984808i

−1.88051

0.399715i

−0.939693

−

0.342020i

−0.0670951

−

1.92135i

0.500000

−

0.866025i

2.46301

−

1.09660i

551.3

0.939693

−

0.342020i

0.473271

−

0.100597i

0.766044

−

0.642788i

0.410323

−

0.256398i

0.500000

−

0.866025i

−0.699680

0.311518i

735.1

−0.766044

0.642788i

1.40724

0.299118i

0.173648

−

0.984808i

−1.27028

0.675419i

0.500000

0.866025i

0.977310

0.435126i

735.2

−0.173648

−

0.984808i

−1.88051

−

0.399715i

−0.939693

0.342020i

−0.0670951

1.92135i

0.500000

0.866025i

2.46301

1.09660i

735.3

0.939693

0.342020i

0.473271

0.100597i

0.766044

0.642788i

0.410323

0.256398i

0.500000

0.866025i

−0.699680

−

0.311518i

827.1

−0.766044

0.642788i

−0.207022

−

1.96969i

0.173648

−

0.984808i

1.42468

1.37580i

0.500000

0.866025i

−2.85866

0.607627i

827.2

−0.173648

−

0.984808i

0.128708

1.22458i

−0.939693

0.342020i

1.18362

−

0.339399i

0.500000

0.866025i

−0.504877

0.107315i

827.3

0.939693

0.342020i

0.0783141

0.745109i

0.766044

0.642788i

−0.181251

0.726958i

0.500000

0.866025i

0.429093

−

0.0912066i

1195.1

−0.766044

−

0.642788i

0.586655

−

0.651546i

0.173648

0.984808i

−0.868210

0.122019i

0.500000

−

0.866025i

0.0241798

0.230056i

1195.2

−0.173648

0.984808i

0.748346

−

0.831123i

−0.939693

−

0.342020i

0.688547

0.881300i

0.500000

−

0.866025i

−0.0262144

−

0.249413i

1195.3

0.939693

−

0.342020i

−1.33500

1.48267i

0.766044

−

0.642788i

−0.747388

1.84985i

0.500000

−

0.866025i

−0.311551

−

2.96421i

1563.1

−0.766044

0.642788i

−1.61323

0.718254i

0.173648

−

0.984808i

0.774117

−

1.58718i

0.500000

0.866025i

1.41748

−

1.57427i

1563.2

−0.173648

−

0.984808i

0.0637646

−

0.0283898i

−0.939693

0.342020i

−0.0390311

−

0.0578660i

0.500000

0.866025i

−0.665871

0.739524i

1563.3

0.939693

0.342020i

1.54946

−

0.689864i

0.766044

0.642788i

1.69196

−

0.118314i

0.500000

0.866025i

1.25579

−

1.39469i

1655.1

−0.766044

−

0.642788i

−1.61323

−

0.718254i

0.173648

0.984808i

0.774117

1.58718i

0.500000

−

0.866025i

1.41748

1.57427i

1655.2

−0.173648

0.984808i

0.0637646

0.0283898i

−0.939693

−

0.342020i

−0.0390311

0.0578660i

0.500000

−

0.866025i

−0.665871

−

0.739524i

1655.3

0.939693

−

0.342020i

1.54946

0.689864i

0.766044

−

0.642788i

1.69196

0.118314i

0.500000

−

0.866025i

1.25579

1.39469i

1747.1

−0.766044

0.642788i

0.586655

0.651546i

0.173648

−

0.984808i

−0.868210

−

0.122019i

0.500000

0.866025i

0.0241798

−

0.230056i

1747.2

−0.173648

−

0.984808i

0.748346

0.831123i

−0.939693

0.342020i

0.688547

−

0.881300i

0.500000

0.866025i

−0.0262144

0.249413i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
124.p	even	30	1	inner
2852.bn	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2852.1.bn.b	yes	24
4.b	odd	2	1		2852.1.bn.a	✓	24
23.b	odd	2	1	CM	2852.1.bn.b	yes	24
31.h	odd	30	1		2852.1.bn.a	✓	24
92.b	even	2	1		2852.1.bn.a	✓	24
124.p	even	30	1	inner	2852.1.bn.b	yes	24
713.s	even	30	1		2852.1.bn.a	✓	24
2852.bn	odd	30	1	inner	2852.1.bn.b	yes	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2852.1.bn.a	✓	24	4.b	odd	2	1
2852.1.bn.a	✓	24	31.h	odd	30	1
2852.1.bn.a	✓	24	92.b	even	2	1
2852.1.bn.a	✓	24	713.s	even	30	1
2852.1.bn.b	yes	24	1.a	even	1	1	trivial
2852.1.bn.b	yes	24	23.b	odd	2	1	CM
2852.1.bn.b	yes	24	124.p	even	30	1	inner
2852.1.bn.b	yes	24	2852.bn	odd	30	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{24} - 3 T_{3}^{22} - 2 T_{3}^{21} + 9 T_{3}^{19} + 30 T_{3}^{18} - 129 T_{3}^{16} + T_{3}^{15} + \cdots + 1 $ T3^24 - 3*T3^22 - 2*T3^21 + 9*T3^19 + 30*T3^18 - 129*T3^16 + T3^15 + 468*T3^14 - 744*T3^13 + 1764*T3^11 - 2628*T3^10 + T3^9 + 6345*T3^8 - 11979*T3^7 + 13253*T3^6 - 10062*T3^5 + 5478*T3^4 - 1995*T3^3 + 405*T3^2 - 33*T3 + 1 acting on $S_{1}^{\mathrm{new}}(2852, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{3} + 1)^{4} $ (T^6 - T^3 + 1)^4
$3$	$ 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
$5$	$ T^{24} $ T^24
$7$	$ T^{24} $ T^24
$11$	$ T^{24} $ T^24
$13$	$ T^{24} + 3 T^{22} + \cdots + 1 $ T^24 + 3T^22 - 9T^19 - 30T^18 - 54T^17 - 69T^16 + 59T^15 - 234T^14 + 60T^13 + 220T^12 + 522T^11 + 2472T^10 + 2955T^9 + 5823T^8 + 4107T^7 + 3793T^6 + 1332T^5 + 720T^4 + 211T^3 - 9T^2 - 3T + 1
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{6} $ (T^4 - T^3 + T^2 - T + 1)^6
$29$	$ T^{24} - 6 T^{22} + \cdots + 1 $ T^24 - 6T^22 + 27T^20 - 111T^18 + 426T^16 - 140T^15 - 882T^14 + 390T^13 + 1516T^12 - 900T^11 - 2451T^10 + 2310T^9 + 3933T^8 - 9030T^7 + 7843T^6 - 3330T^5 + 657T^4 - 130T^3 + 72T^2 - 15*T + 1
$31$	$ T^{24} + T^{21} + \cdots + 1 $ T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$37$	$ T^{24} $ T^24
$41$	$ 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
$43$	$ T^{24} $ T^24
$47$	$ T^{24} - 6 T^{22} + \cdots + 1 $ T^24 - 6T^22 + 5T^21 + 27T^20 - 30T^19 - 96T^18 + 90T^17 + 351T^16 - 255T^15 - 792T^14 + 780T^13 + 1926T^12 - 1035T^11 - 3546T^10 + 5T^9 + 4788T^8 + 60T^7 - 1882T^6 - 1035T^5 + 372T^4 + 410T^3 + 117T^2 + 15T + 1
$53$	$ T^{24} $ T^24
$59$	$ (T^{8} + 10 T^{5} + \cdots + 25)^{3} $ (T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25)^3
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ T^{24} + 3 T^{22} + \cdots + 1 $ T^24 + 3T^22 - 5T^21 - 24T^19 - 15T^18 - 9T^17 - 9T^16 + 174T^15 - 279T^14 + 795T^13 - 585T^12 + 522T^11 + 1377T^10 - 3650T^9 + 6813T^8 - 6108T^7 + 3788T^6 - 963T^5 + 300T^4 + 76T^3 + 36T^2 + 12*T + 1
$73$	$ T^{24} + 3 T^{22} + \cdots + 1 $ T^24 + 3T^22 + 5T^21 + 24T^19 - 15T^18 + 9T^17 - 9T^16 - 174T^15 - 279T^14 - 795T^13 - 585T^12 - 522T^11 + 1377T^10 + 3650T^9 + 6813T^8 + 6108T^7 + 3788T^6 + 963T^5 + 300T^4 - 76T^3 + 36T^2 - 12*T + 1
$79$	$ T^{24} $ T^24
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Overview	Random
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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:	\( N \)	\(=\)	\( 2852 = 2^{2} \cdot 23 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2852.bn (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{90}^{10} q^{2} + (\zeta_{90}^{26} + \zeta_{90}^{22}) q^{3} + \zeta_{90}^{20} q^{4} + ( - \zeta_{90}^{36} - \zeta_{90}^{32}) q^{6} - \zeta_{90}^{30} q^{8} + (\zeta_{90}^{44} + \cdots - \zeta_{90}^{3}) q^{9} + \cdots + \zeta_{90}^{4} q^{98} +O(q^{100}) \) q - z^10 * q^2 + (z^26 + z^22) * q^3 + z^20 * q^4 + (-z^36 - z^32) * q^6 - z^30 * q^8 + (z^44 - z^7 - z^3) * q^9 + (z^42 - z) * q^12 + (-z^31 - z^2) * q^13 + z^40 * q^16 + (z^17 + z^13 + z^9) * q^18 - z^18 * q^23 + (z^11 + z^7) * q^24 - z^15 * q^25 + (z^41 + z^12) * q^26 + (-z^33 - z^29 - z^25 - z^21) * q^27 + (-z^14 - z^13) * q^29 + z * q^31 + z^5 * q^32 + (-z^27 - z^23 - z^19) * q^36 + (-z^28 - z^24 + z^12 + z^8) * q^39 + (-z^38 - z^4) * q^41 + z^28 * q^46 + (z^10 - z^8) * q^47 + (-z^21 - z^17) * q^48 + z^39 * q^49 + z^25 * q^50 + (-z^22 + z^6) * q^52 + (z^43 + z^39 + z^35 + z^31) * q^54 + (z^24 + z^23) * q^58 + (z^42 - z^6) * q^59 - z^11 * q^62 - z^15 * q^64 + (-z^44 - z^40) * q^69 + (z^35 + z^16) * q^71 + (z^37 + z^33 + z^29) * q^72 + (z^19 - z^5) * q^73 + (-z^41 - z^37) * q^75 + (z^38 + z^34 - z^22 - z^18) * q^78 + (-z^43 + z^14 + z^10 + z^6 + z^2) * q^81 + (z^14 - z^3) * q^82 + (-z^40 - z^39 - z^36 - z^35) * q^87 - z^38 * q^92 + (z^27 + z^23) * q^93 + (-z^20 + z^18) * q^94 + (z^31 + z^27) * q^96 + z^4 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{6} + 12 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{18} + 6 q^{23} - 12 q^{25} + 3 q^{26} + 6 q^{27} - 6 q^{36} + 3 q^{48} - 3 q^{49} + 3 q^{52} - 3 q^{54} + 3 q^{58} - 12 q^{64} - 3 q^{72} + 6 q^{78}+ \cdots + 6 q^{96}+O(q^{100}) \) 24 * q + 6 * q^6 + 12 * q^8 + 3 * q^9 + 3 * q^12 + 6 * q^18 + 6 * q^23 - 12 * q^25 + 3 * q^26 + 6 * q^27 - 6 * q^36 + 3 * q^48 - 3 * q^49 + 3 * q^52 - 3 * q^54 + 3 * q^58 - 12 * q^64 - 3 * q^72 + 6 * q^78 + 3 * q^81 + 3 * q^82 + 9 * q^87 + 6 * q^93 - 6 * q^94 + 6 * q^96

Introduction

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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	2852.1.bn.b

Level	$2852$
Weight	$1$
Character orbit	2852.bn
Analytic conductor	$1.423$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{90}$
CM discriminant	-23
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.42333341603\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(3\) over \(\Q(\zeta_{30})\)
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:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{90}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{90} - \cdots)\)

\(n\)	\(373\)	\(1427\)	\(2669\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{90}^{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 551.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{3} + 1)^{4} \) (T^6 - T^3 + 1)^4
$3$	\( T^{24} - 3 T^{22} + \cdots + 1 \) 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
$5$	\( T^{24} \) T^24
$7$	\( T^{24} \) T^24
$11$	\( T^{24} \) T^24
$13$	\( T^{24} + 3 T^{22} + \cdots + 1 \) T^24 + 3T^22 - 9T^19 - 30T^18 - 54T^17 - 69T^16 + 59T^15 - 234T^14 + 60T^13 + 220T^12 + 522T^11 + 2472T^10 + 2955T^9 + 5823T^8 + 4107T^7 + 3793T^6 + 1332T^5 + 720T^4 + 211T^3 - 9T^2 - 3T + 1
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{6} \) (T^4 - T^3 + T^2 - T + 1)^6
$29$	\( T^{24} - 6 T^{22} + \cdots + 1 \) T^24 - 6T^22 + 27T^20 - 111T^18 + 426T^16 - 140T^15 - 882T^14 + 390T^13 + 1516T^12 - 900T^11 - 2451T^10 + 2310T^9 + 3933T^8 - 9030T^7 + 7843T^6 - 3330T^5 + 657T^4 - 130T^3 + 72T^2 - 15*T + 1
$31$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$37$	\( T^{24} \) T^24
$41$	\( T^{24} - 3 T^{22} + \cdots + 1 \) 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
$43$	\( T^{24} \) T^24
$47$	\( T^{24} - 6 T^{22} + \cdots + 1 \) T^24 - 6T^22 + 5T^21 + 27T^20 - 30T^19 - 96T^18 + 90T^17 + 351T^16 - 255T^15 - 792T^14 + 780T^13 + 1926T^12 - 1035T^11 - 3546T^10 + 5T^9 + 4788T^8 + 60T^7 - 1882T^6 - 1035T^5 + 372T^4 + 410T^3 + 117T^2 + 15T + 1
$53$	\( T^{24} \) T^24
$59$	\( (T^{8} + 10 T^{5} + \cdots + 25)^{3} \) (T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25)^3
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( T^{24} + 3 T^{22} + \cdots + 1 \) T^24 + 3T^22 - 5T^21 - 24T^19 - 15T^18 - 9T^17 - 9T^16 + 174T^15 - 279T^14 + 795T^13 - 585T^12 + 522T^11 + 1377T^10 - 3650T^9 + 6813T^8 - 6108T^7 + 3788T^6 - 963T^5 + 300T^4 + 76T^3 + 36T^2 + 12*T + 1
$73$	\( T^{24} + 3 T^{22} + \cdots + 1 \) T^24 + 3T^22 + 5T^21 + 24T^19 - 15T^18 + 9T^17 - 9T^16 - 174T^15 - 279T^14 - 795T^13 - 585T^12 - 522T^11 + 1377T^10 + 3650T^9 + 6813T^8 + 6108T^7 + 3788T^6 + 963T^5 + 300T^4 - 76T^3 + 36T^2 - 12*T + 1
$79$	\( T^{24} \) T^24
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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