LMFDB - Newform orbit 2061.1.bn.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2061,1,Mod(109,2061)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2061, base_ring=CyclotomicField(76))

chi = DirichletCharacter(H, H._module([0, 9]))

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

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

chi := DirichletCharacter("2061.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2061 = 3^{2} \cdot 229 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2061.bn (of order $76$, 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:	$1.02857299104$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{76})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} + \cdots + 1 $ x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{76}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{76} - \cdots)$

$q$-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_{76}^{23} q^{4} + (\zeta_{76}^{26} - \zeta_{76}^{17}) q^{7} +O(q^{10}) $ q - z^23 * q^4 + (z^26 - z^17) * q^7	$ q - \zeta_{76}^{23} q^{4} + (\zeta_{76}^{26} - \zeta_{76}^{17}) q^{7} + ( - \zeta_{76}^{33} - \zeta_{76}^{30}) q^{13} - \zeta_{76}^{8} q^{16} + (\zeta_{76}^{28} + \zeta_{76}^{4}) q^{19} - \zeta_{76}^{6} q^{25} + (\zeta_{76}^{11} - \zeta_{76}^{2}) q^{28} + ( - \zeta_{76}^{20} - \zeta_{76}^{19}) q^{31} + (\zeta_{76}^{12} - \zeta_{76}^{10}) q^{37} + ( - \zeta_{76}^{13} + \zeta_{76}^{9}) q^{43} + (\zeta_{76}^{34} + \cdots + \zeta_{76}^{5}) q^{49} + \cdots + (\zeta_{76}^{24} + \zeta_{76}^{2}) q^{97} +O(q^{100}) $ q - z^23 * q^4 + (z^26 - z^17) * q^7 + (-z^33 - z^30) * q^13 - z^8 * q^16 + (z^28 + z^4) * q^19 - z^6 * q^25 + (z^11 - z^2) * q^28 + (-z^20 - z^19) * q^31 + (z^12 - z^10) * q^37 + (-z^13 + z^9) * q^43 + (z^34 - z^14 + z^5) * q^49 + (-z^18 - z^15) * q^52 + (-z^31 + z^3) * q^61 + z^31 * q^64 + (z^14 - z) * q^67 + (-z^24 - z^7) * q^73 + (-z^27 + z^13) * q^76 + (z^32 + z) * q^79 + (z^21 + z^18 - z^12 - z^9) * q^91 + (z^24 + z^2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2 q^{7}+O(q^{10}) $ 36 * q + 2 * q^7	$ 36 q + 2 q^{7} - 2 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{25} - 2 q^{28} + 2 q^{31} - 4 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} - 2 q^{79} + 4 q^{91}+O(q^{100}) $ 36 * q + 2 * q^7 - 2 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^25 - 2 * q^28 + 2 * q^31 - 4 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 - 2 * q^79 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2061\mathbb{Z}\right)^\times$.

$n$	$235$	$1604$
$\chi(n)$	$-\zeta_{76}^{21}$	$1$

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} $

109.1

0.915773	−	0.401695i
0.614213	+	0.789141i
−0.837166	+	0.546948i
−0.837166	−	0.546948i
0.915773	+	0.401695i
−0.475947	+	0.879474i
−0.969400	+	0.245485i
−0.735724	−	0.677282i
0.164595	+	0.986361i
−0.324699	+	0.945817i
0.475947	+	0.879474i
−0.324699	−	0.945817i
0.324699	+	0.945817i
−0.475947	−	0.879474i
0.324699	−	0.945817i
−0.164595	−	0.986361i
0.735724	+	0.677282i
0.969400	−	0.245485i
0.475947	−	0.879474i
−0.915773	−	0.401695i

0.996584

−

0.0825793i

−0.981209

1.64668i

136.1

0.475947

−

0.879474i

1.05198

−

1.24207i

145.1

0.735724

−

0.677282i

−1.70491

−

0.212517i

199.1

0.735724

0.677282i

−1.70491

0.212517i

208.1

0.996584

0.0825793i

−0.981209

−

1.64668i

343.1

0.915773

0.401695i

−0.108651

0.222249i

352.1

0.837166

0.546948i

0.510414

0.714879i

370.1

−0.164595

−

0.986361i

1.87606

0.558527i

406.1

−0.614213

−

0.789141i

0.0769960

0.0300439i

424.1

−0.969400

0.245485i

0.0630689

−

1.52486i

460.1

−0.915773

0.401695i

−1.78298

0.871648i

559.1

−0.969400

−

0.245485i

0.0630689

1.52486i

586.1

0.969400

0.245485i

1.29149

−

0.0534166i

685.1

0.915773

−

0.401695i

−0.108651

−

0.222249i

721.1

0.969400

−

0.245485i

1.29149

0.0534166i

739.1

0.614213

0.789141i

0.726395

−

1.86159i

775.1

0.164595

0.986361i

−0.117111

0.393368i

793.1

−0.837166

−

0.546948i

1.46231

−

1.04407i

802.1

−0.915773

−

0.401695i

−1.78298

−

0.871648i

937.1

−0.996584

−

0.0825793i

0.490238

−

0.292119i

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}) $
229.j	odd	76	1	inner
687.r	even	76	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2061.1.bn.a	✓	36
3.b	odd	2	1	CM	2061.1.bn.a	✓	36
229.j	odd	76	1	inner	2061.1.bn.a	✓	36
687.r	even	76	1	inner	2061.1.bn.a	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2061.1.bn.a	✓	36	1.a	even	1	1	trivial
2061.1.bn.a	✓	36	3.b	odd	2	1	CM
2061.1.bn.a	✓	36	229.j	odd	76	1	inner
2061.1.bn.a	✓	36	687.r	even	76	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2061, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{36} $ T^36
$3$	$ T^{36} $ T^36
$5$	$ T^{36} $ T^36
$7$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 2T^34 - 4T^32 + 8T^31 - 8T^30 + 149T^28 - 298T^27 + 298T^26 - 596T^24 + 4954T^23 - 8716T^22 + 7524T^21 + 6906T^20 - 28860T^19 + 50368T^18 - 43014T^17 - 14708T^16 - 22458T^15 + 74332T^14 - 100822T^13 + 83304T^12 + 35036T^11 - 64483T^10 + 58666T^9 + 12147T^8 - 15276T^7 + 6258T^6 - 2636T^5 + 2655T^4 + 105T^2 - 20*T + 1
$11$	$ T^{36} $ T^36
$13$	$ T^{36} + 2 T^{35} + \cdots + 1 $ T^36 + 2T^35 + 2T^34 - 4T^32 - 8T^31 - 8T^30 + 38T^29 + 92T^28 + 108T^27 + 32T^26 - 1292T^25 - 2648T^24 - 2712T^23 + 347T^22 + 6384T^21 + 12074T^20 + 11380T^19 + 47670T^18 + 72578T^17 + 49816T^16 - 43358T^15 - 78770T^14 - 70824T^13 + 15892T^12 + 36974T^11 + 75338T^10 + 76728T^9 + 5497T^8 + 5548T^7 + 3047T^6 - 5002T^5 + 261T^4 + 304T^3 + 181T^2 + 20T + 1
$17$	$ T^{36} $ T^36
$19$	$ (T^{18} + 2 T^{17} + \cdots + 1)^{2} $ (T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1)^2
$23$	$ T^{36} $ T^36
$29$	$ T^{36} $ T^36
$31$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 21T^34 - 38T^33 + 205T^32 - 334T^31 + 1227T^30 - 1786T^29 + 4994T^28 - 6416T^27 + 14472T^26 - 16112T^25 + 30412T^24 - 28600T^23 + 46764T^22 - 36328T^21 + 54710T^20 - 36764T^19 + 56486T^18 - 39442T^17 + 58290T^16 - 38038T^15 + 35078T^14 + 13672T^13 - 47112T^12 + 12616T^11 + 96124T^10 - 66316T^9 - 47988T^8 + 43852T^7 + 12148T^6 - 11224T^5 - 879T^4 + 950T^3 + 29T^2 - 20*T + 1
$37$	$ (T^{18} + 2 T^{17} + \cdots + 1)^{2} $ (T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 33T^11 + 66T^10 + 132T^9 + 36T^8 - 23T^7 - 46T^6 - 206T^5 + 25T^4 + 31T^3 + 43T^2 + 10*T + 1)^2
$41$	$ T^{36} $ T^36
$43$	$ T^{36} + 38 T^{28} + \cdots + 361 $ T^36 + 38T^28 + 798T^26 + 38T^24 + 437T^20 - 36860T^18 + 80503T^16 - 15162T^14 + 1805T^12 + 79059T^10 + 222376T^8 + 128155T^6 + 25992T^4 + 361
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} $ T^36
$61$	$ T^{36} - 456 T^{26} + \cdots + 361 $ T^36 - 456T^26 + 8151T^22 + 38T^20 - 18088T^18 + 30324T^16 + 12635T^14 + 235372T^12 + 722T^10 + 180861T^8 - 33573T^6 + 19855T^4 + 5054T^2 + 361
$67$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 2T^34 - 4T^32 + 8T^31 - 8T^30 + 16T^28 - 32T^27 + 203T^26 - 1938T^25 + 3679T^24 - 3482T^23 - 394T^22 + 7752T^21 - 14716T^20 + 13928T^19 + 1576T^18 - 31006T^17 + 64275T^16 + 28766T^15 + 6331T^14 + 1284T^13 - 10214T^12 + 17898T^11 - 15368T^10 - 5060T^9 + 40856T^8 - 71782T^7 + 70003T^6 - 39914T^5 + 13466T^4 - 2698T^3 + 314T^2 - 20*T + 1
$71$	$ T^{36} $ T^36
$73$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 2T^34 - 38T^33 + 72T^32 - 68T^31 + 619T^30 - 1102T^29 + 966T^28 - 5618T^27 + 9304T^26 - 7372T^25 + 30849T^24 - 46954T^23 + 32210T^22 - 103892T^21 + 143364T^20 - 78944T^19 + 207650T^18 - 257410T^17 + 99520T^16 - 232218T^15 + 265719T^14 - 67002T^13 + 154896T^12 - 170468T^11 + 31144T^10 - 40476T^9 + 37683T^8 + 5586T^7 + 5840T^6 - 6550T^5 + 1420T^4 - 114T^3 + 105T^2 + 18*T + 1
$79$	$ T^{36} + 2 T^{35} + \cdots + 1 $ T^36 + 2T^35 + 2T^34 - 4T^32 - 8T^31 - 8T^30 + 38T^29 + 92T^28 + 108T^27 + 32T^26 - 1292T^25 - 2648T^24 - 2712T^23 + 347T^22 + 6384T^21 + 12074T^20 + 11380T^19 + 47670T^18 + 72578T^17 + 49816T^16 - 43358T^15 - 78770T^14 - 70824T^13 + 15892T^12 + 36974T^11 + 75338T^10 + 76728T^9 + 5497T^8 + 5548T^7 + 3047T^6 - 5002T^5 + 261T^4 + 304T^3 + 181T^2 + 20T + 1
$83$	$ T^{36} $ T^36
$89$	$ T^{36} $ T^36
$97$	$ (T^{18} - 38 T^{13} + \cdots + 19)^{2} $ (T^18 - 38T^13 + 19T^10 + 323T^8 + 19T^6 + 323T^5 - 323T^3 + 38T^2 + 38T + 19)^2
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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:	\( N \)	\(=\)	\( 2061 = 3^{2} \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2061.bn (of order \(76\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{76}^{23} q^{4} + (\zeta_{76}^{26} - \zeta_{76}^{17}) q^{7} +O(q^{10}) \) q - z^23 * q^4 + (z^26 - z^17) * q^7	\( q - \zeta_{76}^{23} q^{4} + (\zeta_{76}^{26} - \zeta_{76}^{17}) q^{7} + ( - \zeta_{76}^{33} - \zeta_{76}^{30}) q^{13} - \zeta_{76}^{8} q^{16} + (\zeta_{76}^{28} + \zeta_{76}^{4}) q^{19} - \zeta_{76}^{6} q^{25} + (\zeta_{76}^{11} - \zeta_{76}^{2}) q^{28} + ( - \zeta_{76}^{20} - \zeta_{76}^{19}) q^{31} + (\zeta_{76}^{12} - \zeta_{76}^{10}) q^{37} + ( - \zeta_{76}^{13} + \zeta_{76}^{9}) q^{43} + (\zeta_{76}^{34} + \cdots + \zeta_{76}^{5}) q^{49} + \cdots + (\zeta_{76}^{24} + \zeta_{76}^{2}) q^{97} +O(q^{100}) \) q - z^23 * q^4 + (z^26 - z^17) * q^7 + (-z^33 - z^30) * q^13 - z^8 * q^16 + (z^28 + z^4) * q^19 - z^6 * q^25 + (z^11 - z^2) * q^28 + (-z^20 - z^19) * q^31 + (z^12 - z^10) * q^37 + (-z^13 + z^9) * q^43 + (z^34 - z^14 + z^5) * q^49 + (-z^18 - z^15) * q^52 + (-z^31 + z^3) * q^61 + z^31 * q^64 + (z^14 - z) * q^67 + (-z^24 - z^7) * q^73 + (-z^27 + z^13) * q^76 + (z^32 + z) * q^79 + (z^21 + z^18 - z^12 - z^9) * q^91 + (z^24 + z^2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2 q^{7}+O(q^{10}) \) 36 * q + 2 * q^7	\( 36 q + 2 q^{7} - 2 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{25} - 2 q^{28} + 2 q^{31} - 4 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} - 2 q^{79} + 4 q^{91}+O(q^{100}) \) 36 * q + 2 * q^7 - 2 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^25 - 2 * q^28 + 2 * q^31 - 4 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 - 2 * q^79 + 4 * q^91

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	2061.1.bn.a

Level	$2061$
Weight	$1$
Character orbit	2061.bn
Analytic conductor	$1.029$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{76}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.02857299104\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Coefficient field:	\(\Q(\zeta_{76})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} + \cdots + 1 \) x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{76}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{76} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
229.j	odd	76	1	inner
687.r	even	76	1	inner

$p$	$F_p(T)$
$2$	\( T^{36} \) T^36
$3$	\( T^{36} \) T^36
$5$	\( T^{36} \) T^36
$7$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 2T^34 - 4T^32 + 8T^31 - 8T^30 + 149T^28 - 298T^27 + 298T^26 - 596T^24 + 4954T^23 - 8716T^22 + 7524T^21 + 6906T^20 - 28860T^19 + 50368T^18 - 43014T^17 - 14708T^16 - 22458T^15 + 74332T^14 - 100822T^13 + 83304T^12 + 35036T^11 - 64483T^10 + 58666T^9 + 12147T^8 - 15276T^7 + 6258T^6 - 2636T^5 + 2655T^4 + 105T^2 - 20*T + 1
$11$	\( T^{36} \) T^36
$13$	\( T^{36} + 2 T^{35} + \cdots + 1 \) T^36 + 2T^35 + 2T^34 - 4T^32 - 8T^31 - 8T^30 + 38T^29 + 92T^28 + 108T^27 + 32T^26 - 1292T^25 - 2648T^24 - 2712T^23 + 347T^22 + 6384T^21 + 12074T^20 + 11380T^19 + 47670T^18 + 72578T^17 + 49816T^16 - 43358T^15 - 78770T^14 - 70824T^13 + 15892T^12 + 36974T^11 + 75338T^10 + 76728T^9 + 5497T^8 + 5548T^7 + 3047T^6 - 5002T^5 + 261T^4 + 304T^3 + 181T^2 + 20T + 1
$17$	\( T^{36} \) T^36
$19$	\( (T^{18} + 2 T^{17} + \cdots + 1)^{2} \) (T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1)^2
$23$	\( T^{36} \) T^36
$29$	\( T^{36} \) T^36
$31$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 21T^34 - 38T^33 + 205T^32 - 334T^31 + 1227T^30 - 1786T^29 + 4994T^28 - 6416T^27 + 14472T^26 - 16112T^25 + 30412T^24 - 28600T^23 + 46764T^22 - 36328T^21 + 54710T^20 - 36764T^19 + 56486T^18 - 39442T^17 + 58290T^16 - 38038T^15 + 35078T^14 + 13672T^13 - 47112T^12 + 12616T^11 + 96124T^10 - 66316T^9 - 47988T^8 + 43852T^7 + 12148T^6 - 11224T^5 - 879T^4 + 950T^3 + 29T^2 - 20*T + 1
$37$	\( (T^{18} + 2 T^{17} + \cdots + 1)^{2} \) (T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 33T^11 + 66T^10 + 132T^9 + 36T^8 - 23T^7 - 46T^6 - 206T^5 + 25T^4 + 31T^3 + 43T^2 + 10*T + 1)^2
$41$	\( T^{36} \) T^36
$43$	\( T^{36} + 38 T^{28} + \cdots + 361 \) T^36 + 38T^28 + 798T^26 + 38T^24 + 437T^20 - 36860T^18 + 80503T^16 - 15162T^14 + 1805T^12 + 79059T^10 + 222376T^8 + 128155T^6 + 25992T^4 + 361
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} \) T^36
$61$	\( T^{36} - 456 T^{26} + \cdots + 361 \) T^36 - 456T^26 + 8151T^22 + 38T^20 - 18088T^18 + 30324T^16 + 12635T^14 + 235372T^12 + 722T^10 + 180861T^8 - 33573T^6 + 19855T^4 + 5054T^2 + 361
$67$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 2T^34 - 4T^32 + 8T^31 - 8T^30 + 16T^28 - 32T^27 + 203T^26 - 1938T^25 + 3679T^24 - 3482T^23 - 394T^22 + 7752T^21 - 14716T^20 + 13928T^19 + 1576T^18 - 31006T^17 + 64275T^16 + 28766T^15 + 6331T^14 + 1284T^13 - 10214T^12 + 17898T^11 - 15368T^10 - 5060T^9 + 40856T^8 - 71782T^7 + 70003T^6 - 39914T^5 + 13466T^4 - 2698T^3 + 314T^2 - 20*T + 1
$71$	\( T^{36} \) T^36
$73$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 2T^34 - 38T^33 + 72T^32 - 68T^31 + 619T^30 - 1102T^29 + 966T^28 - 5618T^27 + 9304T^26 - 7372T^25 + 30849T^24 - 46954T^23 + 32210T^22 - 103892T^21 + 143364T^20 - 78944T^19 + 207650T^18 - 257410T^17 + 99520T^16 - 232218T^15 + 265719T^14 - 67002T^13 + 154896T^12 - 170468T^11 + 31144T^10 - 40476T^9 + 37683T^8 + 5586T^7 + 5840T^6 - 6550T^5 + 1420T^4 - 114T^3 + 105T^2 + 18*T + 1
$79$	\( T^{36} + 2 T^{35} + \cdots + 1 \) T^36 + 2T^35 + 2T^34 - 4T^32 - 8T^31 - 8T^30 + 38T^29 + 92T^28 + 108T^27 + 32T^26 - 1292T^25 - 2648T^24 - 2712T^23 + 347T^22 + 6384T^21 + 12074T^20 + 11380T^19 + 47670T^18 + 72578T^17 + 49816T^16 - 43358T^15 - 78770T^14 - 70824T^13 + 15892T^12 + 36974T^11 + 75338T^10 + 76728T^9 + 5497T^8 + 5548T^7 + 3047T^6 - 5002T^5 + 261T^4 + 304T^3 + 181T^2 + 20T + 1
$83$	\( T^{36} \) T^36
$89$	\( T^{36} \) T^36
$97$	\( (T^{18} - 38 T^{13} + \cdots + 19)^{2} \) (T^18 - 38T^13 + 19T^10 + 323T^8 + 19T^6 + 323T^5 - 323T^3 + 38T^2 + 38T + 19)^2
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\(n\)	\(235\)	\(1604\)
\(\chi(n)\)	\(-\zeta_{76}^{21}\)	\(1\)