LMFDB - Newform orbit 3789.1.du.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3789,1,Mod(10,3789)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3789, base_ring=CyclotomicField(140))

chi = DirichletCharacter(H, H._module([0, 93]))

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

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

chi := DirichletCharacter("3789.10");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3789 = 3^{2} \cdot 421 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3789.du (of order $140$, degree $48$, 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:	$1.89095733287$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{140})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 $ x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{140}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{140} - \cdots)$

$q$-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_{140}^{33} q^{4} + ( - \zeta_{140}^{27} - \zeta_{140}^{11}) q^{7}+O(q^{10}) $ q + z^33 * q^4 + (-z^27 - z^11) * q^7	$ q + \zeta_{140}^{33} q^{4} + ( - \zeta_{140}^{27} - \zeta_{140}^{11}) q^{7} + ( - \zeta_{140}^{61} + \zeta_{140}^{58}) q^{13} + \zeta_{140}^{66} q^{16} + ( - \zeta_{140}^{68} + \zeta_{140}^{53}) q^{19} + \zeta_{140}^{4} q^{25} + ( - \zeta_{140}^{60} - \zeta_{140}^{44}) q^{28} + (\zeta_{140}^{36} + \zeta_{140}^{12}) q^{31} + ( - \zeta_{140}^{67} - \zeta_{140}^{56}) q^{37} + (\zeta_{140}^{26} + \zeta_{140}) q^{43} + (\zeta_{140}^{54} + \cdots + \zeta_{140}^{22}) q^{49}+ \cdots + ( - \zeta_{140}^{45} + \zeta_{140}^{9}) q^{97}+O(q^{100}) $ q + z^33 * q^4 + (-z^27 - z^11) * q^7 + (-z^61 + z^58) * q^13 + z^66 * q^16 + (-z^68 + z^53) * q^19 + z^4 * q^25 + (-z^60 - z^44) * q^28 + (z^36 + z^12) * q^31 + (-z^67 - z^56) * q^37 + (z^26 + z) * q^43 + (z^54 + z^38 + z^22) * q^49 + (z^24 - z^21) * q^52 + (-z^57 - z^40) * q^61 - z^29 * q^64 + (-z^50 + z^6) * q^67 + (z^35 + z^28) * q^73 + (z^31 - z^16) * q^76 + (-z^5 - z^3) * q^79 + (-z^69 - z^18 + z^15 - z^2) * q^91 + (-z^45 + z^9) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) $ 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3789\mathbb{Z}\right)^\times$.

$n$	$1685$	$2107$
$\chi(n)$	$1$	$-\zeta_{140}^{33}$

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} $

10.1

0.266037	−	0.963963i
0.834573	+	0.550897i
−0.657939	−	0.753071i
0.657939	−	0.753071i
0.990950	+	0.134233i
0.998993	−	0.0448648i
0.178557	−	0.983930i
0.880596	−	0.473869i
0.266037	+	0.963963i
−0.0896393	+	0.995974i
0.351375	+	0.936235i
−0.351375	+	0.936235i
−0.722795	+	0.691063i
−0.919528	−	0.393025i
−0.998993	−	0.0448648i
0.919528	−	0.393025i
−0.512899	+	0.858449i
0.880596	+	0.473869i
0.834573	−	0.550897i
−0.834573	+	0.550897i

0.512899

0.858449i

1.01313

0.433033i

19.1

0.919528

0.393025i

0.00804330

−

0.0893684i

37.1

0.990950

−

0.134233i

−1.51189

−

0.813584i

91.1

−0.990950

−

0.134233i

1.51189

−

0.813584i

127.1

−0.266037

−

0.963963i

0.790956

−

0.522106i

172.1

0.0896393

−

0.995974i

−1.23197

1.41010i

298.1

−0.351375

−

0.936235i

−0.0714220

0.258792i

352.1

−0.834573

0.550897i

−1.38073

−

0.0620088i

379.1

0.512899

−

0.858449i

1.01313

−

0.433033i

550.1

−0.178557

−

0.983930i

−1.49251

−

0.202174i

685.1

−0.657939

0.753071i

−0.988832

−

1.65503i

901.1

0.657939

0.753071i

0.988832

−

1.65503i

1000.1

−0.998993

0.0448648i

−0.691456

−

1.84238i

1045.1

−0.722795

−

0.691063i

−0.355676

−

1.95994i

1099.1

−0.0896393

−

0.995974i

1.23197

1.41010i

1207.1

0.722795

−

0.691063i

0.355676

−

1.95994i

1216.1

0.880596

0.473869i

−0.568153

0.543210i

1324.1

−0.834573

−

0.550897i

−1.38073

0.0620088i

1396.1

0.919528

−

0.393025i

0.00804330

0.0893684i

1972.1

−0.919528

0.393025i

−0.00804330

−

0.0893684i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
421.v	odd	140	1	inner
1263.bq	even	140	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3789.1.du.a	✓	48
3.b	odd	2	1	CM	3789.1.du.a	✓	48
421.v	odd	140	1	inner	3789.1.du.a	✓	48
1263.bq	even	140	1	inner	3789.1.du.a	✓	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3789.1.du.a	✓	48	1.a	even	1	1	trivial
3789.1.du.a	✓	48	3.b	odd	2	1	CM
3789.1.du.a	✓	48	421.v	odd	140	1	inner
3789.1.du.a	✓	48	1263.bq	even	140	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3789, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{48} $ T^48
$3$	$ T^{48} $ T^48
$5$	$ T^{48} $ T^48
$7$	$ T^{48} + 4 T^{46} + \cdots + 1 $ T^48 + 4T^46 + 10T^44 - 20T^42 - 47T^40 - 159T^38 + 860T^36 - 2403T^34 + 428T^32 - 41068T^30 + 196946T^28 - 252778T^26 + 1798141T^24 - 10636817T^22 + 28435358T^20 - 45220969T^18 + 48114859T^16 - 35683179T^14 + 19312739T^12 - 6427120T^10 + 1805548T^8 + 347439T^6 + 20582T^4 + 265*T^2 + 1
$11$	$ T^{48} $ T^48
$13$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 + 10T^45 - T^44 - 24T^43 + 29T^42 - 66T^41 + 14T^40 + 690T^39 + 59T^38 + 632T^37 + 1448T^36 - 7020T^35 - 5265T^34 - 3782T^33 - 4178T^32 + 48440T^31 + 66876T^30 + 83454T^29 + 99675T^28 - 115728T^27 - 131961T^26 + 73970T^25 + 679530T^24 + 2955482T^23 + 5249415T^22 + 7018520T^21 + 8217249T^20 + 3755544T^19 - 1330562T^18 - 543230T^17 - 569250T^16 - 1078224T^15 - 119959T^14 - 58654T^13 + 1018447T^12 - 127984T^11 - 934913T^10 + 1207130T^9 - 363992T^8 - 682570T^7 + 768392T^6 - 334712T^5 + 73427T^4 - 9058T^3 + 737T^2 - 22T + 1
$17$	$ T^{48} $ T^48
$19$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 + 64T^43 + 93T^42 + 56T^41 - 74T^40 - 4T^39 + 1796T^38 + 1940T^37 + 1350T^36 - 1026T^35 + 2434T^34 + 29488T^33 + 26121T^32 + 25850T^31 - 33651T^30 - 10332T^29 + 221874T^28 + 256242T^27 + 246375T^26 - 414736T^25 + 1023513T^24 + 3522498T^23 + 4300273T^22 + 799428T^21 - 855879T^20 - 4564182T^19 - 9106981T^18 - 9535136T^17 - 3213502T^16 + 1045944T^15 + 3410389T^14 + 4657140T^13 + 4518079T^12 + 2859404T^11 + 1550082T^10 + 738608T^9 + 254981T^8 + 22386T^7 - 8202T^6 - 3484T^5 - 609T^4 - 28T^3 + 25T^2 + 8*T + 1
$23$	$ T^{48} $ T^48
$29$	$ T^{48} $ T^48
$31$	$ (T^{24} - 2 T^{23} + \cdots + 1)^{2} $ (T^24 - 2T^23 + 12T^21 - 19T^20 - T^19 + 84T^18 - 117T^17 + 18T^16 + 426T^15 - 580T^14 + 450T^13 + 1573T^12 - 2547T^11 + 5582T^10 - 3989T^9 + 5781T^8 - 3225T^7 + 4117T^6 - 1324T^5 + 546T^4 + 569T^3 + 12T^2 - 17*T + 1)^2
$37$	$ T^{48} - 12 T^{47} + \cdots + 1 $ T^48 - 12T^47 + 79T^46 - 368T^45 + 1350T^44 - 4126T^43 + 10907T^42 - 25644T^41 + 54908T^40 - 109140T^39 + 204060T^38 - 360610T^37 + 600970T^36 - 940680T^35 + 1386664T^34 - 1955872T^33 + 2705914T^32 - 3702818T^31 + 4832147T^30 - 5564006T^29 + 5165267T^28 - 3909294T^27 + 4473353T^26 - 11021310T^25 + 24017656T^24 - 34272240T^23 + 25699390T^22 + 9404610T^21 - 54247160T^20 + 73064084T^19 - 42681340T^18 - 19616830T^17 + 66408024T^16 - 64626576T^15 + 26802895T^14 + 7817060T^13 - 16520620T^12 + 8841242T^11 - 1949178T^10 + 138916T^9 + 203246T^8 - 410382T^7 + 277076T^6 - 45150T^5 - 9061T^4 + 3822T^3 + 702T^2 + 34*T + 1
$41$	$ T^{48} $ T^48
$43$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 + 14T^45 + 28T^44 + 36T^43 + 149T^42 + 280T^41 + 374T^40 + 1298T^39 + 2636T^38 + 3732T^37 + 10674T^36 + 21136T^35 + 30574T^34 + 68240T^33 + 125297T^32 + 189832T^31 + 383773T^30 + 602182T^29 + 875646T^28 + 1737288T^27 + 1847443T^26 + 2135406T^25 + 4950989T^24 + 6109586T^23 + 6269121T^22 + 6133036T^21 + 5850597T^20 + 5744410T^19 + 4381767T^18 + 3390070T^17 + 883654T^16 - 1385548T^15 + 1328281T^14 + 1366174T^13 + 72659T^12 - 192568T^11 + 1962T^10 + 255132T^9 + 59401T^8 - 36022T^7 + 10782T^6 - 2210T^5 + 35T^4 - 70T^3 + 25T^2 - 6*T + 1
$47$	$ T^{48} $ T^48
$53$	$ T^{48} $ T^48
$59$	$ T^{48} $ T^48
$61$	$ T^{48} - 8 T^{47} + \cdots + 1 $ T^48 - 8T^47 + 37T^46 - 116T^45 + 273T^44 - 494T^43 + 744T^42 - 1100T^41 + 2129T^40 - 4762T^39 + 9271T^38 - 12288T^37 + 7224T^36 + 8676T^35 - 13391T^34 - 49420T^33 + 242592T^32 - 522828T^31 + 616608T^30 - 75438T^29 - 1315764T^28 + 3067668T^27 - 4072622T^26 + 3612786T^25 - 1965236T^24 - 90744T^23 + 3475251T^22 - 9566804T^21 + 17419727T^20 - 19184480T^19 + 9770207T^18 + 7185780T^17 - 16700661T^16 + 10464262T^15 + 4620271T^14 - 10931266T^13 + 5723009T^12 + 2618142T^11 - 4003783T^10 + 379152T^9 + 1869041T^8 - 1424952T^7 + 414317T^6 - 17970T^5 - 17240T^4 + 2070T^3 + 565T^2 + 24*T + 1
$67$	$ (T^{24} + 5 T^{23} + \cdots + 1)^{2} $ (T^24 + 5T^23 + 8T^22 + 55T^19 + 81T^18 - 150T^17 - 373T^16 + 205T^15 + 1184T^14 + 715T^13 - 528T^12 + 1025T^11 + 3756T^10 + 1880T^9 - 1928T^8 - 1885T^7 + 33T^6 + 625T^5 + 333T^4 + 15T^3 - 12T^2 - 5*T + 1)^2
$71$	$ T^{48} $ T^48
$73$	$ (T^{8} + 2 T^{7} + 7 T^{6} + \cdots + 1)^{6} $ (T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1)^6
$79$	$ T^{48} - 5 T^{46} + \cdots + 1 $ T^48 - 5T^46 + T^44 + 59T^42 - 183T^40 - 40T^38 + 780T^36 + 4634T^34 - 4884T^32 + 6492T^30 - 14642T^28 - 516220T^26 + 1459000T^24 - 657275T^22 - 365962T^20 - 1516826T^18 + 775731T^16 + 2434561T^14 + 1349130T^12 + 334195T^10 + 52450T^8 + 1712T^6 + 813T^4 - 42T^2 + 1
$83$	$ T^{48} $ T^48
$89$	$ T^{48} $ T^48
$97$	$ T^{48} - 5 T^{46} + \cdots + 1 $ T^48 - 5T^46 + 15T^44 - 60T^42 + 27T^40 + 541T^38 - 2174T^36 + 13825T^34 - 35026T^32 - 39806T^30 + 255320T^28 - 350838T^26 + 615934T^24 - 1971413T^22 + 3339873T^20 - 2408017T^18 + 3775315T^16 - 860815T^14 + 459717T^12 + 86381T^10 + 37659T^8 - 8487T^6 + 344T^4 + 49*T^2 + 1
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3789 = 3^{2} \cdot 421 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3789.du (of order \(140\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{140}^{33} q^{4} + ( - \zeta_{140}^{27} - \zeta_{140}^{11}) q^{7}+O(q^{10}) \) q + z^33 * q^4 + (-z^27 - z^11) * q^7	\( q + \zeta_{140}^{33} q^{4} + ( - \zeta_{140}^{27} - \zeta_{140}^{11}) q^{7} + ( - \zeta_{140}^{61} + \zeta_{140}^{58}) q^{13} + \zeta_{140}^{66} q^{16} + ( - \zeta_{140}^{68} + \zeta_{140}^{53}) q^{19} + \zeta_{140}^{4} q^{25} + ( - \zeta_{140}^{60} - \zeta_{140}^{44}) q^{28} + (\zeta_{140}^{36} + \zeta_{140}^{12}) q^{31} + ( - \zeta_{140}^{67} - \zeta_{140}^{56}) q^{37} + (\zeta_{140}^{26} + \zeta_{140}) q^{43} + (\zeta_{140}^{54} + \cdots + \zeta_{140}^{22}) q^{49}+ \cdots + ( - \zeta_{140}^{45} + \zeta_{140}^{9}) q^{97}+O(q^{100}) \) q + z^33 * q^4 + (-z^27 - z^11) * q^7 + (-z^61 + z^58) * q^13 + z^66 * q^16 + (-z^68 + z^53) * q^19 + z^4 * q^25 + (-z^60 - z^44) * q^28 + (z^36 + z^12) * q^31 + (-z^67 - z^56) * q^37 + (z^26 + z) * q^43 + (z^54 + z^38 + z^22) * q^49 + (z^24 - z^21) * q^52 + (-z^57 - z^40) * q^61 - z^29 * q^64 + (-z^50 + z^6) * q^67 + (z^35 + z^28) * q^73 + (z^31 - z^16) * q^76 + (-z^5 - z^3) * q^79 + (-z^69 - z^18 + z^15 - z^2) * q^91 + (-z^45 + z^9) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) \) 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3789.1.du.a

Level	$3789$
Weight	$1$
Character orbit	3789.du
Analytic conductor	$1.891$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{140}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.89095733287\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Coefficient field:	\(\Q(\zeta_{140})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 \) x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{140}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{140} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 10.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
421.v	odd	140	1	inner
1263.bq	even	140	1	inner

$p$	$F_p(T)$
$2$	\( T^{48} \) T^48
$3$	\( T^{48} \) T^48
$5$	\( T^{48} \) T^48
$7$	\( T^{48} + 4 T^{46} + \cdots + 1 \) T^48 + 4T^46 + 10T^44 - 20T^42 - 47T^40 - 159T^38 + 860T^36 - 2403T^34 + 428T^32 - 41068T^30 + 196946T^28 - 252778T^26 + 1798141T^24 - 10636817T^22 + 28435358T^20 - 45220969T^18 + 48114859T^16 - 35683179T^14 + 19312739T^12 - 6427120T^10 + 1805548T^8 + 347439T^6 + 20582T^4 + 265*T^2 + 1
$11$	\( T^{48} \) T^48
$13$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 + 10T^45 - T^44 - 24T^43 + 29T^42 - 66T^41 + 14T^40 + 690T^39 + 59T^38 + 632T^37 + 1448T^36 - 7020T^35 - 5265T^34 - 3782T^33 - 4178T^32 + 48440T^31 + 66876T^30 + 83454T^29 + 99675T^28 - 115728T^27 - 131961T^26 + 73970T^25 + 679530T^24 + 2955482T^23 + 5249415T^22 + 7018520T^21 + 8217249T^20 + 3755544T^19 - 1330562T^18 - 543230T^17 - 569250T^16 - 1078224T^15 - 119959T^14 - 58654T^13 + 1018447T^12 - 127984T^11 - 934913T^10 + 1207130T^9 - 363992T^8 - 682570T^7 + 768392T^6 - 334712T^5 + 73427T^4 - 9058T^3 + 737T^2 - 22T + 1
$17$	\( T^{48} \) T^48
$19$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 + 64T^43 + 93T^42 + 56T^41 - 74T^40 - 4T^39 + 1796T^38 + 1940T^37 + 1350T^36 - 1026T^35 + 2434T^34 + 29488T^33 + 26121T^32 + 25850T^31 - 33651T^30 - 10332T^29 + 221874T^28 + 256242T^27 + 246375T^26 - 414736T^25 + 1023513T^24 + 3522498T^23 + 4300273T^22 + 799428T^21 - 855879T^20 - 4564182T^19 - 9106981T^18 - 9535136T^17 - 3213502T^16 + 1045944T^15 + 3410389T^14 + 4657140T^13 + 4518079T^12 + 2859404T^11 + 1550082T^10 + 738608T^9 + 254981T^8 + 22386T^7 - 8202T^6 - 3484T^5 - 609T^4 - 28T^3 + 25T^2 + 8*T + 1
$23$	\( T^{48} \) T^48
$29$	\( T^{48} \) T^48
$31$	\( (T^{24} - 2 T^{23} + \cdots + 1)^{2} \) (T^24 - 2T^23 + 12T^21 - 19T^20 - T^19 + 84T^18 - 117T^17 + 18T^16 + 426T^15 - 580T^14 + 450T^13 + 1573T^12 - 2547T^11 + 5582T^10 - 3989T^9 + 5781T^8 - 3225T^7 + 4117T^6 - 1324T^5 + 546T^4 + 569T^3 + 12T^2 - 17*T + 1)^2
$37$	\( T^{48} - 12 T^{47} + \cdots + 1 \) T^48 - 12T^47 + 79T^46 - 368T^45 + 1350T^44 - 4126T^43 + 10907T^42 - 25644T^41 + 54908T^40 - 109140T^39 + 204060T^38 - 360610T^37 + 600970T^36 - 940680T^35 + 1386664T^34 - 1955872T^33 + 2705914T^32 - 3702818T^31 + 4832147T^30 - 5564006T^29 + 5165267T^28 - 3909294T^27 + 4473353T^26 - 11021310T^25 + 24017656T^24 - 34272240T^23 + 25699390T^22 + 9404610T^21 - 54247160T^20 + 73064084T^19 - 42681340T^18 - 19616830T^17 + 66408024T^16 - 64626576T^15 + 26802895T^14 + 7817060T^13 - 16520620T^12 + 8841242T^11 - 1949178T^10 + 138916T^9 + 203246T^8 - 410382T^7 + 277076T^6 - 45150T^5 - 9061T^4 + 3822T^3 + 702T^2 + 34*T + 1
$41$	\( T^{48} \) T^48
$43$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 + 14T^45 + 28T^44 + 36T^43 + 149T^42 + 280T^41 + 374T^40 + 1298T^39 + 2636T^38 + 3732T^37 + 10674T^36 + 21136T^35 + 30574T^34 + 68240T^33 + 125297T^32 + 189832T^31 + 383773T^30 + 602182T^29 + 875646T^28 + 1737288T^27 + 1847443T^26 + 2135406T^25 + 4950989T^24 + 6109586T^23 + 6269121T^22 + 6133036T^21 + 5850597T^20 + 5744410T^19 + 4381767T^18 + 3390070T^17 + 883654T^16 - 1385548T^15 + 1328281T^14 + 1366174T^13 + 72659T^12 - 192568T^11 + 1962T^10 + 255132T^9 + 59401T^8 - 36022T^7 + 10782T^6 - 2210T^5 + 35T^4 - 70T^3 + 25T^2 - 6*T + 1
$47$	\( T^{48} \) T^48
$53$	\( T^{48} \) T^48
$59$	\( T^{48} \) T^48
$61$	\( T^{48} - 8 T^{47} + \cdots + 1 \) T^48 - 8T^47 + 37T^46 - 116T^45 + 273T^44 - 494T^43 + 744T^42 - 1100T^41 + 2129T^40 - 4762T^39 + 9271T^38 - 12288T^37 + 7224T^36 + 8676T^35 - 13391T^34 - 49420T^33 + 242592T^32 - 522828T^31 + 616608T^30 - 75438T^29 - 1315764T^28 + 3067668T^27 - 4072622T^26 + 3612786T^25 - 1965236T^24 - 90744T^23 + 3475251T^22 - 9566804T^21 + 17419727T^20 - 19184480T^19 + 9770207T^18 + 7185780T^17 - 16700661T^16 + 10464262T^15 + 4620271T^14 - 10931266T^13 + 5723009T^12 + 2618142T^11 - 4003783T^10 + 379152T^9 + 1869041T^8 - 1424952T^7 + 414317T^6 - 17970T^5 - 17240T^4 + 2070T^3 + 565T^2 + 24*T + 1
$67$	\( (T^{24} + 5 T^{23} + \cdots + 1)^{2} \) (T^24 + 5T^23 + 8T^22 + 55T^19 + 81T^18 - 150T^17 - 373T^16 + 205T^15 + 1184T^14 + 715T^13 - 528T^12 + 1025T^11 + 3756T^10 + 1880T^9 - 1928T^8 - 1885T^7 + 33T^6 + 625T^5 + 333T^4 + 15T^3 - 12T^2 - 5*T + 1)^2
$71$	\( T^{48} \) T^48
$73$	\( (T^{8} + 2 T^{7} + 7 T^{6} + \cdots + 1)^{6} \) (T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1)^6
$79$	\( T^{48} - 5 T^{46} + \cdots + 1 \) T^48 - 5T^46 + T^44 + 59T^42 - 183T^40 - 40T^38 + 780T^36 + 4634T^34 - 4884T^32 + 6492T^30 - 14642T^28 - 516220T^26 + 1459000T^24 - 657275T^22 - 365962T^20 - 1516826T^18 + 775731T^16 + 2434561T^14 + 1349130T^12 + 334195T^10 + 52450T^8 + 1712T^6 + 813T^4 - 42T^2 + 1
$83$	\( T^{48} \) T^48
$89$	\( T^{48} \) T^48
$97$	\( T^{48} - 5 T^{46} + \cdots + 1 \) T^48 - 5T^46 + 15T^44 - 60T^42 + 27T^40 + 541T^38 - 2174T^36 + 13825T^34 - 35026T^32 - 39806T^30 + 255320T^28 - 350838T^26 + 615934T^24 - 1971413T^22 + 3339873T^20 - 2408017T^18 + 3775315T^16 - 860815T^14 + 459717T^12 + 86381T^10 + 37659T^8 - 8487T^6 + 344T^4 + 49*T^2 + 1
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\(n\)	\(1685\)	\(2107\)
\(\chi(n)\)	\(1\)	\(-\zeta_{140}^{33}\)