LMFDB - Newform orbit 2271.1.bm.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2271,1,Mod(176,2271)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2271, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 16]))

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

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

chi := DirichletCharacter("2271.176");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2271 = 3 \cdot 757 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2271.bm (of order $126$, degree $36$, 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:	$1.13337664369$
Analytic rank:	$0$
Dimension:	$36$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

$q$-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_{126}^{21} q^{3} - \zeta_{126}^{53} q^{4} + ( - \zeta_{126}^{55} + \zeta_{126}^{28}) q^{7} + \zeta_{126}^{42} q^{9} +O(q^{10}) $ q - z^21 * q^3 - z^53 * q^4 + (-z^55 + z^28) * q^7 + z^42 * q^9	$ q - \zeta_{126}^{21} q^{3} - \zeta_{126}^{53} q^{4} + ( - \zeta_{126}^{55} + \zeta_{126}^{28}) q^{7} + \zeta_{126}^{42} q^{9} - \zeta_{126}^{11} q^{12} + ( - \zeta_{126}^{19} + \zeta_{126}^{12}) q^{13} - \zeta_{126}^{43} q^{16} + (\zeta_{126}^{58} + \zeta_{126}^{18}) q^{19} + ( - \zeta_{126}^{49} - \zeta_{126}^{13}) q^{21} - \zeta_{126}^{59} q^{25} + q^{27} + ( - \zeta_{126}^{45} + \zeta_{126}^{18}) q^{28} + ( - \zeta_{126}^{15} + \zeta_{126}^{4}) q^{31} + \zeta_{126}^{32} q^{36} + (\zeta_{126}^{58} - \zeta_{126}^{45}) q^{37} + (\zeta_{126}^{40} - \zeta_{126}^{33}) q^{39} + (\zeta_{126}^{62} + \zeta_{126}^{6}) q^{43} - \zeta_{126} q^{48} + (\zeta_{126}^{56} + \cdots + \zeta_{126}^{20}) q^{49} + \cdots + ( - \zeta_{126}^{29} + \zeta_{126}^{14}) q^{97} +O(q^{100}) $ q - z^21 * q^3 - z^53 * q^4 + (-z^55 + z^28) * q^7 + z^42 * q^9 - z^11 * q^12 + (-z^19 + z^12) * q^13 - z^43 * q^16 + (z^58 + z^18) * q^19 + (-z^49 - z^13) * q^21 - z^59 * q^25 + q^27 + (-z^45 + z^18) * q^28 + (-z^15 + z^4) * q^31 + z^32 * q^36 + (z^58 - z^45) * q^37 + (z^40 - z^33) * q^39 + (z^62 + z^6) * q^43 - z * q^48 + (z^56 - z^47 + z^20) * q^49 + (-z^9 + z^2) * q^52 + (-z^39 + z^16) * q^57 + (z^52 + z^44) * q^61 + (z^34 - z^7) * q^63 - z^33 * q^64 + (-z^41 + z^36) * q^67 + (-z^13 - z) * q^73 - z^17 * q^75 + (z^48 + z^8) * q^76 + z^9 * q^79 - z^21 * q^81 + (-z^39 - z^3) * q^84 + (-z^47 + z^40 - z^11 + z^4) * q^91 + (z^36 - z^25) * q^93 + (-z^29 + z^14) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 18 q^{3} - 18 q^{9}+O(q^{10}) $ 36 * q - 18 * q^3 - 18 * q^9	$ 36 q - 18 q^{3} - 18 q^{9} + 3 q^{13} - 6 q^{19} + 36 q^{27} - 12 q^{28} + 3 q^{31} - 6 q^{37} + 3 q^{39} + 3 q^{43} - 6 q^{52} + 3 q^{57} + 3 q^{64} - 6 q^{67} + 3 q^{76} + 6 q^{79} - 18 q^{81} + 6 q^{84} - 6 q^{93}+O(q^{100}) $ 36 * q - 18 * q^3 - 18 * q^9 + 3 * q^13 - 6 * q^19 + 36 * q^27 - 12 * q^28 + 3 * q^31 - 6 * q^37 + 3 * q^39 + 3 * q^43 - 6 * q^52 + 3 * q^57 + 3 * q^64 - 6 * q^67 + 3 * q^76 + 6 * q^79 - 18 * q^81 + 6 * q^84 - 6 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2271\mathbb{Z}\right)^\times$.

$n$	$758$	$1516$
$\chi(n)$	$-1$	$-\zeta_{126}^{53}$

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} $

176.1

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

−0.500000

0.866025i

0.698237

0.715867i

−1.25818

1.28995i

−0.500000

−

0.866025i

194.1

−0.500000

−

0.866025i

0.878222

−

0.478254i

1.09512

0.596373i

−0.500000

0.866025i

272.1

−0.500000

0.866025i

−0.797133

−

0.603804i

0.354757

−

0.268718i

−0.500000

−

0.866025i

278.1

−0.500000

0.866025i

0.456211

0.889872i

−0.203033

0.396030i

−0.500000

−

0.866025i

311.1

−0.500000

−

0.866025i

0.995031

0.0995678i

−1.79298

0.179415i

−0.500000

0.866025i

314.1

−0.500000

0.866025i

0.995031

−

0.0995678i

−1.79298

−

0.179415i

−0.500000

−

0.866025i

371.1

−0.500000

0.866025i

0.270840

−

0.962624i

−0.488038

−

1.73459i

−0.500000

−

0.866025i

602.1

−0.500000

−

0.866025i

−0.969077

−

0.246757i

1.74622

−

0.444641i

−0.500000

0.866025i

683.1

−0.500000

0.866025i

−0.853291

0.521435i

−1.06404

−

0.650219i

−0.500000

−

0.866025i

827.1

−0.500000

−

0.866025i

−0.583744

0.811938i

1.05187

1.46306i

−0.500000

0.866025i

848.1

−0.500000

−

0.866025i

−0.661686

−

0.749781i

−0.825109

0.934962i

−0.500000

0.866025i

908.1

−0.500000

0.866025i

−0.411287

0.911506i

0.741114

1.64248i

−0.500000

−

0.866025i

938.1

−0.500000

−

0.866025i

−0.124344

−

0.992239i

0.0553382

−

0.441588i

−0.500000

0.866025i

1049.1

−0.500000

0.866025i

−0.583744

−

0.811938i

1.05187

−

1.46306i

−0.500000

−

0.866025i

1100.1

−0.500000

−

0.866025i

−0.998757

0.0498459i

0.444489

0.0221835i

−0.500000

0.866025i

1115.1

−0.500000

0.866025i

−0.318487

−

0.947927i

−0.397146

1.18205i

−0.500000

−

0.866025i

1133.1

−0.500000

−

0.866025i

−0.411287

−

0.911506i

0.741114

−

1.64248i

−0.500000

0.866025i

1331.1

−0.500000

0.866025i

−0.661686

0.749781i

−0.825109

−

0.934962i

−0.500000

−

0.866025i

1691.1

−0.500000

−

0.866025i

0.456211

−

0.889872i

−0.203033

−

0.396030i

−0.500000

0.866025i

1700.1

−0.500000

−

0.866025i

0.980172

−

0.198146i

1.22226

0.247084i

−0.500000

0.866025i

See all 36 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
757.q	even	63	1	inner
2271.bm	odd	126	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2271.1.bm.a	✓	36
3.b	odd	2	1	CM	2271.1.bm.a	✓	36
757.q	even	63	1	inner	2271.1.bm.a	✓	36
2271.bm	odd	126	1	inner	2271.1.bm.a	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2271.1.bm.a	✓	36	1.a	even	1	1	trivial
2271.1.bm.a	✓	36	3.b	odd	2	1	CM
2271.1.bm.a	✓	36	757.q	even	63	1	inner
2271.1.bm.a	✓	36	2271.bm	odd	126	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2271, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{36} $ T^36
$3$	$ (T^{2} + T + 1)^{18} $ (T^2 + T + 1)^18
$5$	$ T^{36} $ T^36
$7$	$ T^{36} - T^{33} + \cdots + 1 $ T^36 - T^33 - 28T^30 + 197T^27 + 587T^24 - 3010T^21 + 8667T^18 - 15064T^15 + 18157T^12 - 1889T^9 + 84T^6 - 15T^3 + 1
$11$	$ T^{36} $ T^36
$13$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$17$	$ T^{36} $ T^36
$19$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 249T^31 + 549T^30 + 1173T^29 + 2274T^28 + 3788T^27 + 6270T^26 + 11328T^25 + 21705T^24 + 39654T^23 + 61446T^22 + 82203T^21 + 92208T^20 + 89136T^19 + 75292T^18 + 24303T^17 - 51900T^16 - 120627T^15 - 111114T^14 + 8046T^13 + 111453T^12 + 80640T^11 - 10560T^10 - 69203T^9 - 6063T^8 + 58965T^7 + 44302T^6 + 6747T^5 - 4263T^4 - 418T^3 + 417T^2 - 18*T + 1
$23$	$ T^{36} $ T^36
$29$	$ T^{36} $ T^36
$31$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 + 24T^31 - 99T^30 + 165T^29 - 93T^28 + 8T^27 + 906T^26 - 2838T^25 + 4398T^24 - 3924T^23 + 5493T^22 - 39T^21 - 24063T^20 + 45846T^19 - 26309T^18 + 16140T^17 - 31344T^16 - 2016T^15 + 56061T^14 - 63387T^13 + 105324T^12 - 127692T^11 + 114018T^10 - 79463T^9 + 79482T^8 - 4575T^7 + 27103T^6 + 6954T^5 + 4386T^4 + 2444T^3 + 615T^2 + 45*T + 1
$37$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 297T^30 + 291T^29 + 111T^28 - 48T^27 + 474T^26 + 2865T^25 + 8412T^24 + 17121T^23 + 25053T^22 + 20176T^21 - 2124T^20 - 53454T^19 - 128513T^18 - 197646T^17 - 212361T^16 - 136342T^15 - 4140T^14 + 85956T^13 + 169287T^12 + 326655T^11 + 511164T^10 + 624602T^9 + 599493T^8 + 443412T^7 + 246721T^6 + 97089T^5 + 24906T^4 + 3243T^3 + 39T^2 - 18*T + 1
$41$	$ T^{36} $ T^36
$43$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} $ T^36
$61$	$ T^{36} - 3 T^{34} + \cdots + 1 $ T^36 - 3T^34 - 2T^33 - 33T^31 + 30T^30 + 126T^29 - 15T^28 + 57T^27 + 315T^26 - 1332T^25 - 1332T^24 + 2898T^23 + 585T^22 - 728T^21 + 8118T^20 - 19782T^19 + 49T^18 + 48159T^17 - 7896T^16 + 9839T^15 - 45657T^14 - 2985T^13 + 132801T^12 - 63027T^11 + 8577T^10 - 19432T^9 + 24750T^8 + 30687T^7 - 15041T^6 - 189T^5 + 5346T^4 + 774T^3 + 441T^2 + 39*T + 1
$67$	$ 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
$71$	$ T^{36} $ T^36
$73$	$ 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
$79$	$ (T^{6} - T^{5} + T^{4} + \cdots + 1)^{6} $ (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^6
$83$	$ T^{36} $ T^36
$89$	$ T^{36} $ T^36
$97$	$ T^{36} + 8 T^{33} + \cdots + 1 $ T^36 + 8T^33 + 35T^30 + 188T^27 + 2288T^24 - 3577T^21 - 41985T^18 - 111958T^15 + 590386T^12 - 55511T^9 + 90048T^6 - 573*T^3 + 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:	\( N \)	\(=\)	\( 2271 = 3 \cdot 757 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2271.bm (of order \(126\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{126}^{21} q^{3} - \zeta_{126}^{53} q^{4} + ( - \zeta_{126}^{55} + \zeta_{126}^{28}) q^{7} + \zeta_{126}^{42} q^{9} +O(q^{10}) \) q - z^21 * q^3 - z^53 * q^4 + (-z^55 + z^28) * q^7 + z^42 * q^9	\( q - \zeta_{126}^{21} q^{3} - \zeta_{126}^{53} q^{4} + ( - \zeta_{126}^{55} + \zeta_{126}^{28}) q^{7} + \zeta_{126}^{42} q^{9} - \zeta_{126}^{11} q^{12} + ( - \zeta_{126}^{19} + \zeta_{126}^{12}) q^{13} - \zeta_{126}^{43} q^{16} + (\zeta_{126}^{58} + \zeta_{126}^{18}) q^{19} + ( - \zeta_{126}^{49} - \zeta_{126}^{13}) q^{21} - \zeta_{126}^{59} q^{25} + q^{27} + ( - \zeta_{126}^{45} + \zeta_{126}^{18}) q^{28} + ( - \zeta_{126}^{15} + \zeta_{126}^{4}) q^{31} + \zeta_{126}^{32} q^{36} + (\zeta_{126}^{58} - \zeta_{126}^{45}) q^{37} + (\zeta_{126}^{40} - \zeta_{126}^{33}) q^{39} + (\zeta_{126}^{62} + \zeta_{126}^{6}) q^{43} - \zeta_{126} q^{48} + (\zeta_{126}^{56} + \cdots + \zeta_{126}^{20}) q^{49} + \cdots + ( - \zeta_{126}^{29} + \zeta_{126}^{14}) q^{97} +O(q^{100}) \) q - z^21 * q^3 - z^53 * q^4 + (-z^55 + z^28) * q^7 + z^42 * q^9 - z^11 * q^12 + (-z^19 + z^12) * q^13 - z^43 * q^16 + (z^58 + z^18) * q^19 + (-z^49 - z^13) * q^21 - z^59 * q^25 + q^27 + (-z^45 + z^18) * q^28 + (-z^15 + z^4) * q^31 + z^32 * q^36 + (z^58 - z^45) * q^37 + (z^40 - z^33) * q^39 + (z^62 + z^6) * q^43 - z * q^48 + (z^56 - z^47 + z^20) * q^49 + (-z^9 + z^2) * q^52 + (-z^39 + z^16) * q^57 + (z^52 + z^44) * q^61 + (z^34 - z^7) * q^63 - z^33 * q^64 + (-z^41 + z^36) * q^67 + (-z^13 - z) * q^73 - z^17 * q^75 + (z^48 + z^8) * q^76 + z^9 * q^79 - z^21 * q^81 + (-z^39 - z^3) * q^84 + (-z^47 + z^40 - z^11 + z^4) * q^91 + (z^36 - z^25) * q^93 + (-z^29 + z^14) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 18 q^{3} - 18 q^{9}+O(q^{10}) \) 36 * q - 18 * q^3 - 18 * q^9	\( 36 q - 18 q^{3} - 18 q^{9} + 3 q^{13} - 6 q^{19} + 36 q^{27} - 12 q^{28} + 3 q^{31} - 6 q^{37} + 3 q^{39} + 3 q^{43} - 6 q^{52} + 3 q^{57} + 3 q^{64} - 6 q^{67} + 3 q^{76} + 6 q^{79} - 18 q^{81} + 6 q^{84} - 6 q^{93}+O(q^{100}) \) 36 * q - 18 * q^3 - 18 * q^9 + 3 * q^13 - 6 * q^19 + 36 * q^27 - 12 * q^28 + 3 * q^31 - 6 * q^37 + 3 * q^39 + 3 * q^43 - 6 * q^52 + 3 * q^57 + 3 * q^64 - 6 * q^67 + 3 * q^76 + 6 * q^79 - 18 * q^81 + 6 * q^84 - 6 * q^93

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	2271.1.bm.a

Level	$2271$
Weight	$1$
Character orbit	2271.bm
Analytic conductor	$1.133$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{63}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.13337664369\)
Analytic rank:	\(0\)
Dimension:	\(36\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
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 176.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
757.q	even	63	1	inner
2271.bm	odd	126	1	inner

$p$	$F_p(T)$
$2$	\( T^{36} \) T^36
$3$	\( (T^{2} + T + 1)^{18} \) (T^2 + T + 1)^18
$5$	\( T^{36} \) T^36
$7$	\( T^{36} - T^{33} + \cdots + 1 \) T^36 - T^33 - 28T^30 + 197T^27 + 587T^24 - 3010T^21 + 8667T^18 - 15064T^15 + 18157T^12 - 1889T^9 + 84T^6 - 15T^3 + 1
$11$	\( T^{36} \) T^36
$13$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$17$	\( T^{36} \) T^36
$19$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 249T^31 + 549T^30 + 1173T^29 + 2274T^28 + 3788T^27 + 6270T^26 + 11328T^25 + 21705T^24 + 39654T^23 + 61446T^22 + 82203T^21 + 92208T^20 + 89136T^19 + 75292T^18 + 24303T^17 - 51900T^16 - 120627T^15 - 111114T^14 + 8046T^13 + 111453T^12 + 80640T^11 - 10560T^10 - 69203T^9 - 6063T^8 + 58965T^7 + 44302T^6 + 6747T^5 - 4263T^4 - 418T^3 + 417T^2 - 18*T + 1
$23$	\( T^{36} \) T^36
$29$	\( T^{36} \) T^36
$31$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 + 24T^31 - 99T^30 + 165T^29 - 93T^28 + 8T^27 + 906T^26 - 2838T^25 + 4398T^24 - 3924T^23 + 5493T^22 - 39T^21 - 24063T^20 + 45846T^19 - 26309T^18 + 16140T^17 - 31344T^16 - 2016T^15 + 56061T^14 - 63387T^13 + 105324T^12 - 127692T^11 + 114018T^10 - 79463T^9 + 79482T^8 - 4575T^7 + 27103T^6 + 6954T^5 + 4386T^4 + 2444T^3 + 615T^2 + 45*T + 1
$37$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 297T^30 + 291T^29 + 111T^28 - 48T^27 + 474T^26 + 2865T^25 + 8412T^24 + 17121T^23 + 25053T^22 + 20176T^21 - 2124T^20 - 53454T^19 - 128513T^18 - 197646T^17 - 212361T^16 - 136342T^15 - 4140T^14 + 85956T^13 + 169287T^12 + 326655T^11 + 511164T^10 + 624602T^9 + 599493T^8 + 443412T^7 + 246721T^6 + 97089T^5 + 24906T^4 + 3243T^3 + 39T^2 - 18*T + 1
$41$	\( T^{36} \) T^36
$43$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} \) T^36
$61$	\( T^{36} - 3 T^{34} + \cdots + 1 \) T^36 - 3T^34 - 2T^33 - 33T^31 + 30T^30 + 126T^29 - 15T^28 + 57T^27 + 315T^26 - 1332T^25 - 1332T^24 + 2898T^23 + 585T^22 - 728T^21 + 8118T^20 - 19782T^19 + 49T^18 + 48159T^17 - 7896T^16 + 9839T^15 - 45657T^14 - 2985T^13 + 132801T^12 - 63027T^11 + 8577T^10 - 19432T^9 + 24750T^8 + 30687T^7 - 15041T^6 - 189T^5 + 5346T^4 + 774T^3 + 441T^2 + 39*T + 1
$67$	\( T^{36} + 6 T^{35} + \cdots + 1 \) 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
$71$	\( T^{36} \) T^36
$73$	\( T^{36} + T^{33} + \cdots + 1 \) 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
$79$	\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{6} \) (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^6
$83$	\( T^{36} \) T^36
$89$	\( T^{36} \) T^36
$97$	\( T^{36} + 8 T^{33} + \cdots + 1 \) T^36 + 8T^33 + 35T^30 + 188T^27 + 2288T^24 - 3577T^21 - 41985T^18 - 111958T^15 + 590386T^12 - 55511T^9 + 90048T^6 - 573*T^3 + 1
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\(n\)	\(758\)	\(1516\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{126}^{53}\)