LMFDB - Newform orbit 3620.1.dy.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3620,1,Mod(219,3620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3620, base_ring=CyclotomicField(90))

chi = DirichletCharacter(H, H._module([45, 45, 68]))

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

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

chi := DirichletCharacter("3620.219");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3620 = 2^{2} \cdot 5 \cdot 181 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3620.dy (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:	$1.80661534573$
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 $\left(\mathbb{Z}/3620\mathbb{Z}\right)^\times$.

$n$	$1811$	$2897$	$3441$
$\chi(n)$	$-1$	$-1$	$-\zeta_{90}^{41}$

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} $

219.1

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

0.719340

−

0.694658i

0.116903

0.173316i

0.0348995

−

0.999391i

0.913545

−

0.406737i

0.204489

0.0434654i

0.766044

1.32683i

−0.669131

−

0.743145i

0.358234

−

0.886661i

0.374607

−

0.927184i

539.1

−0.0348995

0.999391i

−0.732841

1.81385i

−0.997564

−

0.0697565i

0.669131

−

0.743145i

−1.78716

−

0.795697i

0.173648

−

0.300767i

0.104528

−

0.994522i

−2.03364

−

1.96386i

0.719340

0.694658i

559.1

0.997564

0.0697565i

1.31430

1.26920i

0.990268

0.139173i

−0.104528

−

0.994522i

1.22256

1.35779i

−0.939693

−

1.62760i

0.978148

0.207912i

0.0816041

2.33684i

−0.0348995

−

0.999391i

739.1

−0.848048

−

0.529919i

1.93726

−

0.272264i

0.438371

0.898794i

0.669131

−

0.743145i

−1.78716

−

0.795697i

−0.939693

1.62760i

0.104528

−

0.994522i

2.71757

−

0.779252i

−0.961262

0.275637i

799.1

−0.990268

−

0.139173i

−0.0467046

−

1.33745i

0.961262

0.275637i

−0.978148

0.207912i

−0.139886

1.33093i

0.766044

−

1.32683i

−0.913545

−

0.406737i

−0.789016

0.0551734i

0.997564

−

0.0697565i

919.1

−0.961262

−

0.275637i

−0.208548

0.0145831i

0.848048

0.529919i

0.913545

−

0.406737i

0.204489

0.0434654i

0.173648

0.300767i

−0.669131

−

0.743145i

−0.946989

0.133091i

−0.990268

0.139173i

939.1

0.997564

−

0.0697565i

1.31430

−

1.26920i

0.990268

−

0.139173i

−0.104528

0.994522i

1.22256

−

1.35779i

−0.939693

1.62760i

0.978148

−

0.207912i

0.0816041

−

2.33684i

−0.0348995

0.999391i

1099.1

−0.961262

0.275637i

−0.208548

−

0.0145831i

0.848048

−

0.529919i

0.913545

0.406737i

0.204489

−

0.0434654i

0.173648

−

0.300767i

−0.669131

0.743145i

−0.946989

−

0.133091i

−0.990268

−

0.139173i

1699.1

−0.990268

0.139173i

−0.0467046

1.33745i

0.961262

−

0.275637i

−0.978148

−

0.207912i

−0.139886

−

1.33093i

0.766044

1.32683i

−0.913545

0.406737i

−0.789016

−

0.0551734i

0.997564

0.0697565i

1799.1

0.882948

−

0.469472i

−1.20442

−

1.54158i

0.559193

−

0.829038i

0.669131

−

0.743145i

−1.78716

−

0.795697i

0.766044

−

1.32683i

0.104528

−

0.994522i

−0.683935

2.74311i

0.241922

−

0.970296i

1819.1

0.374607

−

0.927184i

1.18161

−

0.628276i

−0.719340

−

0.694658i

−0.978148

−

0.207912i

−0.139886

−

1.33093i

−0.939693

−

1.62760i

−0.913545

0.406737i

0.442290

−

0.655722i

−0.559193

0.829038i

1939.1

0.241922

0.970296i

0.0916445

−

0.187899i

−0.882948

0.469472i

0.913545

−

0.406737i

0.204489

0.0434654i

−0.939693

−

1.62760i

−0.669131

−

0.743145i

0.588754

0.753571i

0.615661

0.788011i

1979.1

−0.848048

0.529919i

1.93726

0.272264i

0.438371

−

0.898794i

0.669131

0.743145i

−1.78716

0.795697i

−0.939693

−

1.62760i

0.104528

0.994522i

2.71757

0.779252i

−0.961262

−

0.275637i

2139.1

0.882948

0.469472i

−1.20442

1.54158i

0.559193

0.829038i

0.669131

0.743145i

−1.78716

0.795697i

0.766044

1.32683i

0.104528

0.994522i

−0.683935

−

2.74311i

0.241922

0.970296i

2259.1

0.241922

−

0.970296i

0.0916445

0.187899i

−0.882948

−

0.469472i

0.913545

0.406737i

0.204489

−

0.0434654i

−0.939693

1.62760i

−0.669131

0.743145i

0.588754

−

0.753571i

0.615661

−

0.788011i

2479.1

0.615661

−

0.788011i

−1.13491

0.709170i

−0.241922

−

0.970296i

−0.978148

0.207912i

−0.139886

1.33093i

0.173648

−

0.300767i

−0.913545

−

0.406737i

0.346727

−

0.710895i

−0.438371

0.898794i

2579.1

−0.0348995

−

0.999391i

−0.732841

−

1.81385i

−0.997564

0.0697565i

0.669131

0.743145i

−1.78716

0.795697i

0.173648

0.300767i

0.104528

0.994522i

−2.03364

1.96386i

0.719340

−

0.694658i

2759.1

−0.438371

−

0.898794i

−1.75631

0.503615i

−0.615661

0.788011i

−0.104528

−

0.994522i

1.22256

1.35779i

0.766044

1.32683i

0.978148

0.207912i

1.98296

−

1.23909i

−0.848048

0.529919i

2859.1

−0.438371

0.898794i

−1.75631

−

0.503615i

−0.615661

−

0.788011i

−0.104528

0.994522i

1.22256

−

1.35779i

0.766044

−

1.32683i

0.978148

−

0.207912i

1.98296

1.23909i

−0.848048

−

0.529919i

2899.1

−0.559193

−

0.829038i

0.442013

1.77282i

−0.374607

0.927184i

−0.104528

0.994522i

1.22256

−

1.35779i

0.173648

−

0.300767i

0.978148

−

0.207912i

−2.06456

1.09775i

0.882948

−

0.469472i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
181.o	even	45	1	inner
3620.dy	odd	90	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3620.1.dy.a	✓	24
4.b	odd	2	1		3620.1.dy.b	yes	24
5.b	even	2	1		3620.1.dy.b	yes	24
20.d	odd	2	1	CM	3620.1.dy.a	✓	24
181.o	even	45	1	inner	3620.1.dy.a	✓	24
724.bg	odd	90	1		3620.1.dy.b	yes	24
905.cb	even	90	1		3620.1.dy.b	yes	24
3620.dy	odd	90	1	inner	3620.1.dy.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3620.1.dy.a	✓	24	1.a	even	1	1	trivial
3620.1.dy.a	✓	24	20.d	odd	2	1	CM
3620.1.dy.a	✓	24	181.o	even	45	1	inner
3620.1.dy.a	✓	24	3620.dy	odd	90	1	inner
3620.1.dy.b	yes	24	4.b	odd	2	1
3620.1.dy.b	yes	24	5.b	even	2	1
3620.1.dy.b	yes	24	724.bg	odd	90	1
3620.1.dy.b	yes	24	905.cb	even	90	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{24} - 6T_{3}^{21} - 10T_{3}^{18} - 74T_{3}^{15} + 2004T_{3}^{12} - 1304T_{3}^{9} + 11955T_{3}^{6} + 214T_{3}^{3} + 1 $ T3^24 - 6*T3^21 - 10*T3^18 - 74*T3^15 + 2004*T3^12 - 1304*T3^9 + 11955*T3^6 + 214*T3^3 + 1 acting on $S_{1}^{\mathrm{new}}(3620, [\chi])$.

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} - 6 T^{21} + \cdots + 1 $ T^24 - 6T^21 - 10T^18 - 74T^15 + 2004T^12 - 1304T^9 + 11955T^6 + 214*T^3 + 1
$5$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$7$	$ (T^{6} + 3 T^{4} - 2 T^{3} + \cdots + 1)^{4} $ (T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1)^4
$11$	$ T^{24} $ T^24
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ 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
$29$	$ 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^24
$41$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 - 30T^19 + 117T^18 - 126T^17 + 15T^16 + 188T^15 + 330T^14 - 1398T^13 + 1169T^12 + 21T^11 - 996T^10 - 2134T^9 + 6240T^8 - 3816T^7 + 1737T^6 - 672T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$43$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$47$	$ 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
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 68T^18 - 90T^17 - 18T^16 + 94T^15 + 303T^14 - 465T^13 + 991T^12 - 1278T^11 + 1209T^10 - 1669T^9 + 3231T^8 - 5364T^7 + 7256T^6 - 7125T^5 + 4470T^4 - 1579T^3 + 261T^2 - 12*T + 1
$67$	$ (T^{8} - T^{7} - 4 T^{5} + \cdots + 1)^{3} $ (T^8 - T^7 - 4T^5 - 6T^4 + 11T^3 + 20T^2 - 6*T + 1)^3
$71$	$ T^{24} $ T^24
$73$	$ T^{24} $ T^24
$79$	$ T^{24} $ T^24
$83$	$ T^{24} - 7 T^{21} + \cdots + 1 $ T^24 - 7T^21 + 30T^18 - 127T^15 + 339T^12 - 43T^9 - 10T^6 - 3*T^3 + 1
$89$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 68T^18 - 90T^17 - 18T^16 + 94T^15 + 303T^14 - 465T^13 + 991T^12 - 1278T^11 + 1209T^10 - 1669T^9 + 3231T^8 - 5364T^7 + 7256T^6 - 7125T^5 + 4470T^4 - 1579T^3 + 261T^2 - 12*T + 1
$97$	$ T^{24} $ T^24
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3620 = 2^{2} \cdot 5 \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3620.dy (of order \(90\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{90}^{43} q^{2} + (\zeta_{90}^{35} + \zeta_{90}^{11}) q^{3} - \zeta_{90}^{41} q^{4} - \zeta_{90}^{3} q^{5} + ( - \zeta_{90}^{33} - \zeta_{90}^{9}) q^{6} + (\zeta_{90}^{25} + \zeta_{90}^{5}) q^{7} + \zeta_{90}^{39} q^{8} + ( - \zeta_{90}^{25} + \cdots - \zeta_{90}) q^{9} +O(q^{10}) \) q + z^43 * q^2 + (z^35 + z^11) * q^3 - z^41 * q^4 - z^3 * q^5 + (-z^33 - z^9) * q^6 + (z^25 + z^5) * q^7 + z^39 * q^8 + (-z^25 + z^22 - z) * q^9	\( q + \zeta_{90}^{43} q^{2} + (\zeta_{90}^{35} + \zeta_{90}^{11}) q^{3} - \zeta_{90}^{41} q^{4} - \zeta_{90}^{3} q^{5} + ( - \zeta_{90}^{33} - \zeta_{90}^{9}) q^{6} + (\zeta_{90}^{25} + \zeta_{90}^{5}) q^{7} + \zeta_{90}^{39} q^{8} + ( - \zeta_{90}^{25} + \cdots - \zeta_{90}) q^{9} + \cdots + ( - \zeta_{90}^{28} + \cdots + \zeta_{90}^{3}) q^{98} +O(q^{100}) \) q + z^43 * q^2 + (z^35 + z^11) * q^3 - z^41 * q^4 - z^3 * q^5 + (-z^33 - z^9) * q^6 + (z^25 + z^5) * q^7 + z^39 * q^8 + (-z^25 + z^22 - z) * q^9 + z * q^10 + (z^31 + z^7) * q^12 + (-z^23 - z^3) * q^14 + (-z^38 - z^14) * q^15 - z^37 * q^16 + (-z^44 + z^23 - z^20) * q^18 + z^44 * q^20 + (z^40 + z^36 + z^16 - z^15) * q^21 + (z^7 - z^6) * q^23 + (-z^29 - z^5) * q^24 + z^6 * q^25 + (-z^36 + z^33 + z^15 - z^12) * q^27 + (z^21 + z) * q^28 + (z^32 - z) * q^29 + (z^36 + z^12) * q^30 + z^35 * q^32 + (-z^28 - z^8) * q^35 + (z^42 - z^21 + z^18) * q^36 - z^42 * q^40 + (-z^39 + z^34) * q^41 + (-z^38 - z^34 - z^14 + z^13) * q^42 + (z^33 - z^22) * q^43 + (z^28 - z^25 + z^4) * q^45 + (-z^5 + z^4) * q^46 + (z^25 + z^13) * q^47 + (z^27 + z^3) * q^48 + (z^30 + z^10 - z^5) * q^49 - z^4 * q^50 + (z^34 - z^31 - z^13 + z^10) * q^54 + (z^44 - z^19) * q^56 + (-z^44 - z^30) * q^58 + (-z^34 - z^10) * q^60 + (z^34 + z^6) * q^61 + (-z^30 + z^27 - z^26 - z^6 + z^5 - z^2) * q^63 - z^33 * q^64 + (-z^24 - z^18) * q^67 + (z^42 - z^41 + z^18 - z^17) * q^69 + (z^26 + z^6) * q^70 + (-z^40 + z^19 - z^16) * q^72 + (z^41 + z^17) * q^75 + z^40 * q^80 + (z^44 + z^26 - z^23 - z^5 + z^2) * q^81 + (z^37 - z^32) * q^82 + (z^37 + z^19) * q^83 + (z^36 + z^32 + z^12 - z^11) * q^84 + (-z^31 + z^20) * q^86 + (z^43 - z^36 - z^22 - z^12) * q^87 + (-z^29 - z^21) * q^89 + (-z^26 + z^23 - z^2) * q^90 + (z^3 - z^2) * q^92 + (-z^23 - z^11) * q^94 + (-z^25 - z) * q^96 + (-z^28 - z^8 + z^3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{5} - 3 q^{6} - 3 q^{8}+O(q^{10}) \) 24 * q + 3 * q^5 - 3 * q^6 - 3 * q^8	\( 24 q + 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{14} - 18 q^{21} - 3 q^{23} + 3 q^{25} + 12 q^{27} - 3 q^{28} - 3 q^{30} - 3 q^{40} + 3 q^{41} - 3 q^{43} + 3 q^{48} - 12 q^{49} + 12 q^{58} + 3 q^{61} + 15 q^{63} + 3 q^{64} + 3 q^{67} - 3 q^{69} + 3 q^{70} - 3 q^{84} + 3 q^{87} + 3 q^{89} - 3 q^{92} - 3 q^{98}+O(q^{100}) \) 24 * q + 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^14 - 18 * q^21 - 3 * q^23 + 3 * q^25 + 12 * q^27 - 3 * q^28 - 3 * q^30 - 3 * q^40 + 3 * q^41 - 3 * q^43 + 3 * q^48 - 12 * q^49 + 12 * q^58 + 3 * q^61 + 15 * q^63 + 3 * q^64 + 3 * q^67 - 3 * q^69 + 3 * q^70 - 3 * q^84 + 3 * q^87 + 3 * q^89 - 3 * q^92 - 3 * q^98

Introduction

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Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Inner twists

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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	3620.1.dy.a

Level	$3620$
Weight	$1$
Character orbit	3620.dy
Analytic conductor	$1.807$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{45}$
CM discriminant	-20
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.80661534573\)
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 \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

\(n\)	\(1811\)	\(2897\)	\(3441\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{90}^{41}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 219.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$3$	\( T^{24} - 6 T^{21} + \cdots + 1 \) T^24 - 6T^21 - 10T^18 - 74T^15 + 2004T^12 - 1304T^9 + 11955T^6 + 214*T^3 + 1
$5$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$7$	\( (T^{6} + 3 T^{4} - 2 T^{3} + \cdots + 1)^{4} \) (T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1)^4
$11$	\( T^{24} \) T^24
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( T^{24} + 3 T^{23} + \cdots + 1 \) 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
$29$	\( T^{24} - 3 T^{22} + \cdots + 1 \) 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^24
$41$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 - 30T^19 + 117T^18 - 126T^17 + 15T^16 + 188T^15 + 330T^14 - 1398T^13 + 1169T^12 + 21T^11 - 996T^10 - 2134T^9 + 6240T^8 - 3816T^7 + 1737T^6 - 672T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$43$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$47$	\( T^{24} - 6 T^{21} + \cdots + 1 \) T^24 - 6T^21 - 10T^18 - 74T^15 + 2004T^12 - 1304T^9 + 11955T^6 + 214*T^3 + 1
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 68T^18 - 90T^17 - 18T^16 + 94T^15 + 303T^14 - 465T^13 + 991T^12 - 1278T^11 + 1209T^10 - 1669T^9 + 3231T^8 - 5364T^7 + 7256T^6 - 7125T^5 + 4470T^4 - 1579T^3 + 261T^2 - 12*T + 1
$67$	\( (T^{8} - T^{7} - 4 T^{5} + \cdots + 1)^{3} \) (T^8 - T^7 - 4T^5 - 6T^4 + 11T^3 + 20T^2 - 6*T + 1)^3
$71$	\( T^{24} \) T^24
$73$	\( T^{24} \) T^24
$79$	\( T^{24} \) T^24
$83$	\( T^{24} - 7 T^{21} + \cdots + 1 \) T^24 - 7T^21 + 30T^18 - 127T^15 + 339T^12 - 43T^9 - 10T^6 - 3*T^3 + 1
$89$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 15T^19 + 68T^18 - 90T^17 - 18T^16 + 94T^15 + 303T^14 - 465T^13 + 991T^12 - 1278T^11 + 1209T^10 - 1669T^9 + 3231T^8 - 5364T^7 + 7256T^6 - 7125T^5 + 4470T^4 - 1579T^3 + 261T^2 - 12*T + 1
$97$	\( T^{24} \) T^24
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