LMFDB - Newform orbit 1917.1.s.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1917,1,Mod(70,1917)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1917, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([4, 9]))

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

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

chi := DirichletCharacter("1917.70");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1917 = 3^{3} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1917.s (of order $18$, degree $6$, 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:	$0.956707629217$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$6$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1917\mathbb{Z}\right)^\times$.

$n$	$433$	$569$
$\chi(n)$	$-1$	$\zeta_{126}^{14}$

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} $

70.1

−0.124344	−	0.992239i
−0.318487	+	0.947927i
0.542546	+	0.840026i
0.698237	−	0.715867i
−0.853291	−	0.521435i
0.995031	+	0.0995678i
−0.0249307	−	0.999689i
−0.411287	+	0.911506i
0.456211	+	0.889872i
−0.797133	−	0.603804i
−0.969077	+	0.246757i
0.980172	+	0.198146i
−0.998757	+	0.0498459i
0.921476	+	0.388435i
0.878222	−	0.478254i
−0.661686	−	0.749781i
−0.583744	+	0.811938i
0.270840	+	0.962624i
−0.998757	−	0.0498459i
0.921476	−	0.388435i

−1.48471

−

1.24582i

0.542546

−

0.840026i

0.478651

2.71457i

−0.686617

−

0.249908i

−1.85205

0.571281i

1.70213

−

2.94817i

−0.411287

−

0.911506i

0.708087

1.22644i

70.2

−1.22128

−

1.02477i

−0.853291

0.521435i

0.267710

1.51826i

−1.55282

−

0.565181i

1.57646

0.237613i

0.431791

−

0.747885i

0.456211

−

0.889872i

1.31724

2.28153i

70.3

−0.630128

−

0.528741i

−0.124344

0.992239i

−0.0561529

−

0.318459i

1.85839

0.676400i

0.602990

−

0.559493i

−0.544286

0.942730i

−0.969077

−

0.246757i

−0.813387

−

1.40883i

70.4

−0.0381960

−

0.0320503i

0.995031

−

0.0995678i

−0.173216

−

0.982359i

1.37769

0.501437i

−0.0411974

−

0.0280879i

−0.0497994

0.0862551i

0.980172

−

0.198146i

−0.0365510

−

0.0633082i

70.5

0.698955

0.586493i

−0.318487

−

0.947927i

−0.0290839

−

0.164943i

−0.140447

−

0.0511184i

0.333345

−

0.849349i

0.532620

−

0.922525i

−0.797133

0.603804i

−0.0681853

−

0.118100i

70.6

1.50171

1.26009i

0.698237

0.715867i

0.493674

2.79976i

−1.79589

−

0.653650i

0.146497

1.95486i

−1.80641

3.12880i

−0.0249307

0.999689i

−1.87325

−

3.24457i

283.1

−0.346865

−

1.96717i

0.980172

−

0.198146i

−2.80974

1.02266i

0.114493

0.0960712i

−0.729774

−

1.85943i

1.98759

3.44260i

0.921476

−

0.388435i

0.149274

−

0.258551i

283.2

−0.229801

−

1.30327i

−0.969077

−

0.246757i

−0.706003

0.256964i

1.46402

1.22846i

−0.0988957

1.31967i

−0.164553

−

0.285014i

0.878222

0.478254i

1.26458

−

2.19031i

283.3

−0.202732

−

1.14975i

−0.797133

0.603804i

−0.341134

0.124162i

−1.51498

−

1.27122i

0.855829

0.794093i

−0.371829

−

0.644027i

0.270840

−

0.962624i

−1.15445

1.99956i

283.4

0.0940619

0.533452i

0.456211

−

0.889872i

0.663970

−

0.241665i

0.559735

0.469673i

0.517616

0.159663i

0.462211

0.800574i

−0.583744

−

0.811938i

−0.197898

0.342770i

283.5

0.305003

1.72976i

−0.411287

−

0.911506i

−1.95935

0.713144i

−1.12310

−

0.942393i

1.45124

−

0.989440i

−0.952952

−

1.65056i

−0.661686

0.749781i

1.28756

−

2.23013i

283.6

0.320025

1.81495i

−0.0249307

0.999689i

−2.25195

0.819642i

1.26587

1.06219i

−1.82237

0.274678i

−1.28682

−

2.22883i

−0.998757

−

0.0498459i

−1.52272

2.63743i

709.1

−1.87005

0.680641i

0.921476

−

0.388435i

2.26776

−

1.90287i

−0.343417

−

1.94762i

−1.45882

1.35359i

−1.95060

3.37854i

0.698237

−

0.715867i

1.96783

3.40839i

709.2

−1.31226

0.477622i

−0.998757

−

0.0498459i

0.727848

−

0.610737i

0.126882

0.719581i

1.33443

−

0.411618i

0.0348151

−

0.0603014i

0.995031

0.0995678i

−0.510189

−

0.883673i

709.3

−1.01965

0.371124i

−0.661686

0.749781i

0.135916

−

0.114047i

0.0259535

0.147190i

0.396429

−

1.01008i

0.446285

−

0.772988i

−0.124344

−

0.992239i

−0.0810891

−

0.140450i

709.4

0.233690

−

0.0850561i

0.878222

0.478254i

−0.718668

0.603034i

0.286950

1.62737i

0.245910

0.0370649i

−0.240997

0.417420i

0.542546

0.840026i

0.205475

0.355893i

709.5

0.598559

−

0.217858i

0.270840

−

0.962624i

−0.455233

0.381986i

0.331867

1.88211i

−0.0476011

−

0.635192i

−0.507752

0.879452i

−0.853291

−

0.521435i

0.608674

1.05425i

709.6

1.60366

−

0.583685i

−0.583744

−

0.811938i

1.46500

−

1.22928i

−0.254586

−

1.44383i

−1.41004

−

0.961352i

0.778561

−

1.34851i

−0.318487

0.947927i

−1.25101

−

2.16682i

922.1

−1.87005

−

0.680641i

0.921476

0.388435i

2.26776

1.90287i

−0.343417

1.94762i

−1.45882

−

1.35359i

−1.95060

−

3.37854i

0.698237

0.715867i

1.96783

−

3.40839i

922.2

−1.31226

−

0.477622i

−0.998757

0.0498459i

0.727848

0.610737i

0.126882

−

0.719581i

1.33443

0.411618i

0.0348151

0.0603014i

0.995031

−

0.0995678i

−0.510189

0.883673i

See all 36 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.b	odd	2	1	CM by $\Q(\sqrt{-71}) $
27.e	even	9	1	inner
1917.s	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1917.1.s.b	✓	36
27.e	even	9	1	inner	1917.1.s.b	✓	36
71.b	odd	2	1	CM	1917.1.s.b	✓	36
1917.s	odd	18	1	inner	1917.1.s.b	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1917.1.s.b	✓	36	1.a	even	1	1	trivial
1917.1.s.b	✓	36	27.e	even	9	1	inner
1917.1.s.b	✓	36	71.b	odd	2	1	CM
1917.1.s.b	✓	36	1917.s	odd	18	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{36} + 6 T_{2}^{35} + 21 T_{2}^{34} + 55 T_{2}^{33} + 117 T_{2}^{32} + 207 T_{2}^{31} + 367 T_{2}^{30} + \cdots + 1 $ T2^36 + 6*T2^35 + 21*T2^34 + 55*T2^33 + 117*T2^32 + 207*T2^31 + 367*T2^30 + 711*T2^29 + 1434*T2^28 + 2724*T2^27 + 4443*T2^26 + 5196*T2^25 + 4912*T2^24 + 6894*T2^23 + 19845*T2^22 + 57241*T2^21 + 128979*T2^20 + 212070*T2^19 + 261114*T2^18 + 239910*T2^17 + 155685*T2^16 + 81463*T2^15 + 95211*T2^14 + 128250*T2^13 + 119125*T2^12 + 50904*T2^11 - 35025*T2^10 - 46607*T2^9 + 7104*T2^8 + 14991*T2^7 + 5249*T2^6 + 1665*T2^5 + 4725*T2^4 - 2350*T2^3 + 207*T2^2 + 24*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1917, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 367T^30 + 711T^29 + 1434T^28 + 2724T^27 + 4443T^26 + 5196T^25 + 4912T^24 + 6894T^23 + 19845T^22 + 57241T^21 + 128979T^20 + 212070T^19 + 261114T^18 + 239910T^17 + 155685T^16 + 81463T^15 + 95211T^14 + 128250T^13 + 119125T^12 + 50904T^11 - 35025T^10 - 46607T^9 + 7104T^8 + 14991T^7 + 5249T^6 + 1665T^5 + 4725T^4 - 2350T^3 + 207T^2 + 24*T + 1
$3$	$ T^{36} - T^{33} + \cdots + 1 $ T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$5$	$ T^{36} + T^{33} + \cdots + 1 $ T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$7$	$ T^{36} $ T^36
$11$	$ T^{36} $ T^36
$13$	$ T^{36} $ T^36
$17$	$ T^{36} $ T^36
$19$	$ T^{36} + 18 T^{34} + \cdots + 1 $ T^36 + 18T^34 - 2T^33 + 189T^32 - 33T^31 + 1339T^30 - 315T^29 + 7125T^28 - 2001T^27 + 29295T^26 - 9396T^25 + 95863T^24 - 33453T^23 + 250527T^22 - 92687T^21 + 527049T^20 - 200109T^19 + 883533T^18 - 337716T^17 + 1177281T^16 - 441395T^15 + 1217052T^14 - 441402T^13 + 967096T^12 - 331659T^11 + 561066T^10 - 178262T^9 + 234036T^8 - 65241T^7 + 60398T^6 - 13797T^5 + 9798T^4 - 2068T^3 + 504T^2 - 24T + 1
$23$	$ T^{36} $ T^36
$29$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 18T^31 + 97T^30 - 171T^29 + 75T^28 + 78T^27 + 339T^26 - 1158T^25 + 4048T^24 - 8460T^23 + 9504T^22 - 5723T^21 + 11889T^20 - 23685T^19 + 44790T^18 - 55722T^17 + 24411T^16 + 15232T^15 + 50265T^14 - 45432T^13 + 69904T^12 - 27270T^11 - 44574T^10 + 15331T^9 + 29964T^8 - 23853T^7 + 28496T^6 - 19359T^5 + 8271T^4 - 1189T^3 + 468T^2 - 39*T + 1
$31$	$ T^{36} $ T^36
$37$	$ (T^{12} + T^{11} + 7 T^{10} + \cdots + 1)^{3} $ (T^12 + T^11 + 7T^10 + 6T^9 + 34T^8 + 28T^7 + 78T^6 + 49T^5 + 118T^4 + 76T^3 + 56T^2 + 8T + 1)^3
$41$	$ T^{36} $ T^36
$43$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 + 16T^30 - 162T^29 + 273T^28 - 246T^27 + 51T^26 + 1065T^25 + 214T^24 - 5292T^23 + 6669T^22 - 2807T^21 + 1692T^20 + 11631T^19 - 13854T^18 - 58233T^17 + 154056T^16 - 148847T^15 + 92394T^14 - 41832T^13 - 4535T^12 + 13599T^11 + 17958T^10 - 6431T^9 + 2406T^8 - 7257T^7 - 2527T^6 + 11808T^5 + 11385T^4 + 3455T^3 + 486T^2 + 15T + 1
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} $ T^36
$61$	$ T^{36} $ T^36
$67$	$ T^{36} $ T^36
$71$	$ (T^{2} + T + 1)^{18} $ (T^2 + T + 1)^18
$73$	$ (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{6} $ (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^6
$79$	$ T^{36} + T^{33} + \cdots + 1 $ T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$83$	$ T^{36} + T^{33} + \cdots + 1 $ T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$89$	$ T^{36} + 18 T^{34} + \cdots + 1 $ T^36 + 18T^34 - 2T^33 + 189T^32 - 33T^31 + 1339T^30 - 315T^29 + 7125T^28 - 2001T^27 + 29295T^26 - 9396T^25 + 95863T^24 - 33453T^23 + 250527T^22 - 92687T^21 + 527049T^20 - 200109T^19 + 883533T^18 - 337716T^17 + 1177281T^16 - 441395T^15 + 1217052T^14 - 441402T^13 + 967096T^12 - 331659T^11 + 561066T^10 - 178262T^9 + 234036T^8 - 65241T^7 + 60398T^6 - 13797T^5 + 9798T^4 - 2068T^3 + 504T^2 - 24T + 1
$97$	$ T^{36} $ T^36
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1917 = 3^{3} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1917.s (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{126}^{9} - \zeta_{126}^{5}) q^{2} + \zeta_{126}^{8} q^{3} + (\zeta_{126}^{18} + \cdots + \zeta_{126}^{10}) q^{4}+ \cdots + \zeta_{126}^{16} q^{9}+O(q^{10}) \) q + (-z^9 - z^5) * q^2 + z^8 * q^3 + (z^18 + z^14 + z^10) * q^4 + (z^38 + z^32) * q^5 + (-z^17 - z^13) * q^6 + (-z^27 - z^23 - z^19 - z^15) * q^8 + z^16 * q^9	\( q + ( - \zeta_{126}^{9} - \zeta_{126}^{5}) q^{2} + \zeta_{126}^{8} q^{3} + (\zeta_{126}^{18} + \cdots + \zeta_{126}^{10}) q^{4}+ \cdots + (\zeta_{126}^{44} + \zeta_{126}^{40}) q^{98}+O(q^{100}) \) q + (-z^9 - z^5) * q^2 + z^8 * q^3 + (z^18 + z^14 + z^10) * q^4 + (z^38 + z^32) * q^5 + (-z^17 - z^13) * q^6 + (-z^27 - z^23 - z^19 - z^15) * q^8 + z^16 * q^9 + (-z^47 - z^43 - z^41 - z^37) * q^10 + (z^26 + z^22 + z^18) * q^12 + (z^46 + z^40) * q^15 + (z^36 + z^32 + z^28 + z^24 + z^20) * q^16 + (-z^25 - z^21) * q^18 + (-z^61 + z^44) * q^19 + (z^56 + z^52 + z^50 + z^48 + z^46 + z^42) * q^20 + (-z^35 - z^31 - z^27 - z^23) * q^24 + (-z^13 - z^7 - z) * q^25 + z^24 * q^27 + (-z^57 + z^20) * q^29 + (-z^55 - z^51 - z^49 - z^45) * q^30 + (-z^45 - z^41 - z^37 - z^33 - z^29 - z^25) * q^32 + (z^34 + z^30 + z^26) * q^36 + (z^12 - z^9) * q^37 + (-z^53 - z^49 - z^7 - z^3) * q^38 + (-z^61 - z^59 - z^57 - z^55 - z^53 - z^51 - z^47 + z^2) * q^40 + (z^60 - z^59) * q^43 + (z^54 + z^48) * q^45 + (z^44 + z^40 + z^36 + z^32 + z^28) * q^48 - z^35 * q^49 + (z^22 + z^18 + z^16 + z^12 + z^10 + z^6) * q^50 + (-z^33 - z^29) * q^54 + (z^52 + z^6) * q^57 + (z^62 - z^29 - z^25 - z^3) * q^58 + (z^60 + z^58 + z^56 + z^54 + z^50 - z) * q^60 + (z^54 + z^50 + z^46 + z^42 + z^38 + z^34 + z^30) * q^64 + z^42 * q^71 + (-z^43 - z^39 - z^35 - z^31) * q^72 + (-z^39 - z^3) * q^73 + (-z^21 + z^18 - z^17 + z^14) * q^74 + (-z^21 - z^15 - z^9) * q^75 + (z^62 + z^58 + z^54 + z^16 + z^12 + z^8) * q^76 + (z^46 - z^31) * q^79 + (z^62 + z^60 + z^58 + z^56 + z^52 - z^11 - z^7 - z^5 - z^3 - z) * q^80 + z^32 * q^81 + (z^58 - z^19) * q^83 + (z^6 - z^5 + z^2 - z) * q^86 + (z^28 + z^2) * q^87 + (z^62 - z^43) * q^89 + (-z^59 - z^57 - z^53 + 1) * q^90 + (z^36 + z^30 - z^19 - z^13) * q^95 + (-z^53 - z^49 - z^45 - z^41 - z^37 - z^33) * q^96 + (z^44 + z^40) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 6 q^{2} - 6 q^{4} - 3 q^{8}+O(q^{10}) \) 36 * q - 6 * q^2 - 6 * q^4 - 3 * q^8	\( 36 q - 6 q^{2} - 6 q^{4} - 3 q^{8} - 6 q^{12} - 3 q^{16} - 18 q^{18} - 15 q^{20} - 6 q^{24} + 3 q^{27} + 3 q^{29} - 3 q^{30} - 3 q^{32} + 3 q^{36} - 3 q^{37} + 3 q^{38} + 6 q^{40} + 3 q^{43} - 3 q^{45} - 6 q^{48} + 3 q^{54} + 3 q^{57} + 3 q^{58} - 3 q^{60} - 21 q^{64} - 18 q^{71} + 3 q^{72} + 6 q^{73} - 24 q^{74} - 21 q^{75} - 3 q^{76} + 6 q^{80} + 3 q^{86} + 39 q^{90} - 3 q^{95} - 3 q^{96}+O(q^{100}) \) 36 * q - 6 * q^2 - 6 * q^4 - 3 * q^8 - 6 * q^12 - 3 * q^16 - 18 * q^18 - 15 * q^20 - 6 * q^24 + 3 * q^27 + 3 * q^29 - 3 * q^30 - 3 * q^32 + 3 * q^36 - 3 * q^37 + 3 * q^38 + 6 * q^40 + 3 * q^43 - 3 * q^45 - 6 * q^48 + 3 * q^54 + 3 * q^57 + 3 * q^58 - 3 * q^60 - 21 * q^64 - 18 * q^71 + 3 * q^72 + 6 * q^73 - 24 * q^74 - 21 * q^75 - 3 * q^76 + 6 * q^80 + 3 * q^86 + 39 * q^90 - 3 * q^95 - 3 * q^96

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	1917.1.s.b

Level	$1917$
Weight	$1$
Character orbit	1917.s
Analytic conductor	$0.957$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{63}$
CM discriminant	-71
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.956707629217\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{63})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 \) x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{63}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{63} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 70.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 367T^30 + 711T^29 + 1434T^28 + 2724T^27 + 4443T^26 + 5196T^25 + 4912T^24 + 6894T^23 + 19845T^22 + 57241T^21 + 128979T^20 + 212070T^19 + 261114T^18 + 239910T^17 + 155685T^16 + 81463T^15 + 95211T^14 + 128250T^13 + 119125T^12 + 50904T^11 - 35025T^10 - 46607T^9 + 7104T^8 + 14991T^7 + 5249T^6 + 1665T^5 + 4725T^4 - 2350T^3 + 207T^2 + 24*T + 1
$3$	\( T^{36} - T^{33} + \cdots + 1 \) T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$5$	\( T^{36} + T^{33} + \cdots + 1 \) T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$7$	\( T^{36} \) T^36
$11$	\( T^{36} \) T^36
$13$	\( T^{36} \) T^36
$17$	\( T^{36} \) T^36
$19$	\( T^{36} + 18 T^{34} + \cdots + 1 \) T^36 + 18T^34 - 2T^33 + 189T^32 - 33T^31 + 1339T^30 - 315T^29 + 7125T^28 - 2001T^27 + 29295T^26 - 9396T^25 + 95863T^24 - 33453T^23 + 250527T^22 - 92687T^21 + 527049T^20 - 200109T^19 + 883533T^18 - 337716T^17 + 1177281T^16 - 441395T^15 + 1217052T^14 - 441402T^13 + 967096T^12 - 331659T^11 + 561066T^10 - 178262T^9 + 234036T^8 - 65241T^7 + 60398T^6 - 13797T^5 + 9798T^4 - 2068T^3 + 504T^2 - 24T + 1
$23$	\( T^{36} \) T^36
$29$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 18T^31 + 97T^30 - 171T^29 + 75T^28 + 78T^27 + 339T^26 - 1158T^25 + 4048T^24 - 8460T^23 + 9504T^22 - 5723T^21 + 11889T^20 - 23685T^19 + 44790T^18 - 55722T^17 + 24411T^16 + 15232T^15 + 50265T^14 - 45432T^13 + 69904T^12 - 27270T^11 - 44574T^10 + 15331T^9 + 29964T^8 - 23853T^7 + 28496T^6 - 19359T^5 + 8271T^4 - 1189T^3 + 468T^2 - 39*T + 1
$31$	\( T^{36} \) T^36
$37$	\( (T^{12} + T^{11} + 7 T^{10} + \cdots + 1)^{3} \) (T^12 + T^11 + 7T^10 + 6T^9 + 34T^8 + 28T^7 + 78T^6 + 49T^5 + 118T^4 + 76T^3 + 56T^2 + 8T + 1)^3
$41$	\( T^{36} \) T^36
$43$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 + 16T^30 - 162T^29 + 273T^28 - 246T^27 + 51T^26 + 1065T^25 + 214T^24 - 5292T^23 + 6669T^22 - 2807T^21 + 1692T^20 + 11631T^19 - 13854T^18 - 58233T^17 + 154056T^16 - 148847T^15 + 92394T^14 - 41832T^13 - 4535T^12 + 13599T^11 + 17958T^10 - 6431T^9 + 2406T^8 - 7257T^7 - 2527T^6 + 11808T^5 + 11385T^4 + 3455T^3 + 486T^2 + 15T + 1
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} \) T^36
$61$	\( T^{36} \) T^36
$67$	\( T^{36} \) T^36
$71$	\( (T^{2} + T + 1)^{18} \) (T^2 + T + 1)^18
$73$	\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{6} \) (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^6
$79$	\( T^{36} + T^{33} + \cdots + 1 \) T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$83$	\( T^{36} + T^{33} + \cdots + 1 \) T^36 + T^33 + 70T^30 + 111T^27 + 4108T^24 + 5026T^21 + 59067T^18 - 25193T^15 + 579298T^12 + 224566T^9 + 90461T^6 + 302T^3 + 1
$89$	\( T^{36} + 18 T^{34} + \cdots + 1 \) T^36 + 18T^34 - 2T^33 + 189T^32 - 33T^31 + 1339T^30 - 315T^29 + 7125T^28 - 2001T^27 + 29295T^26 - 9396T^25 + 95863T^24 - 33453T^23 + 250527T^22 - 92687T^21 + 527049T^20 - 200109T^19 + 883533T^18 - 337716T^17 + 1177281T^16 - 441395T^15 + 1217052T^14 - 441402T^13 + 967096T^12 - 331659T^11 + 561066T^10 - 178262T^9 + 234036T^8 - 65241T^7 + 60398T^6 - 13797T^5 + 9798T^4 - 2068T^3 + 504T^2 - 24T + 1
$97$	\( T^{36} \) T^36
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\(n\)	\(433\)	\(569\)
\(\chi(n)\)	\(-1\)	\(\zeta_{126}^{14}\)