LMFDB - Newform orbit 2299.1.bj.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2299,1,Mod(37,2299)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2299, base_ring=CyclotomicField(110))

chi = DirichletCharacter(H, H._module([42, 55]))

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

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

chi := DirichletCharacter("2299.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2299.bj (of order $110$, degree $40$, 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:	$1.14735046404$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{55})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 $ x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{55}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{55} - \cdots)$

$q$-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_{110}^{13} q^{4} + (\zeta_{110}^{54} + \zeta_{110}^{28}) q^{5} + (\zeta_{110}^{30} + \zeta_{110}^{6}) q^{7} + \zeta_{110}^{44} q^{9} + \zeta_{110}^{22} q^{11} + \zeta_{110}^{26} q^{16} + ( - \zeta_{110}^{25} - \zeta_{110}^{7}) q^{17} + \cdots - \zeta_{110}^{11} q^{99} +O(q^{100}) $ q - z^13 * q^4 + (z^54 + z^28) * q^5 + (z^30 + z^6) * q^7 + z^44 * q^9 + z^22 * q^11 + z^26 * q^16 + (-z^25 - z^7) * q^17 - z^17 * q^19 + (-z^41 + z^12) * q^20 + (-z^21 - z^9) * q^23 + (-z^53 - z^27 - z) * q^25 + (-z^43 - z^19) * q^28 + (z^34 - z^29 - z^5 - z^3) * q^35 + z^2 * q^36 + (-z^41 - z^9) * q^43 - z^35 * q^44 + (-z^43 - z^17) * q^45 + (z^46 + z^18) * q^47 + (z^36 + z^12 - z^5) * q^49 + (z^50 - z^21) * q^55 + (z^32 + z^10) * q^61 + (z^50 - z^19) * q^63 - z^39 * q^64 + (z^38 + z^20) * q^68 + (z^38 - z^3) * q^73 + z^30 * q^76 + (z^52 + z^28) * q^77 + (z^54 - z^25) * q^80 - z^33 * q^81 + (z^24 - z^23) * q^83 + (-z^53 - z^35 + z^24 + z^6) * q^85 + (z^34 + z^22) * q^92 + (-z^45 + z^16) * q^95 - z^11 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + q^{4} + 2 q^{5} - 3 q^{7} - 10 q^{9} - 10 q^{11} + q^{16} - 3 q^{17} + q^{19} + 2 q^{20} + 2 q^{23} + 3 q^{25} + 2 q^{28} - q^{35} + q^{36} + 2 q^{43} - 4 q^{44} + 2 q^{45} + 2 q^{47} - 2 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) $ 40 * q + q^4 + 2 * q^5 - 3 * q^7 - 10 * q^9 - 10 * q^11 + q^16 - 3 * q^17 + q^19 + 2 * q^20 + 2 * q^23 + 3 * q^25 + 2 * q^28 - q^35 + q^36 + 2 * q^43 - 4 * q^44 + 2 * q^45 + 2 * q^47 - 2 * q^49 - 3 * q^55 - 3 * q^61 - 3 * q^63 + q^64 - 3 * q^68 + 2 * q^73 - 4 * q^76 + 2 * q^77 - 3 * q^80 - 10 * q^81 + 2 * q^83 - q^85 - 9 * q^92 - 3 * q^95 - 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2299\mathbb{Z}\right)^\times$.

$n$	$970$	$1332$
$\chi(n)$	$-\zeta_{110}^{13}$	$-1$

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} $

37.1

−0.998369	+	0.0570888i
−0.985354	+	0.170522i
0.974012	−	0.226497i
0.516397	+	0.856349i
−0.362808	+	0.931864i
0.516397	−	0.856349i
0.941844	+	0.336049i
−0.736741	−	0.676175i
0.696938	+	0.717132i
0.610648	+	0.791902i
0.897398	+	0.441221i
0.941844	−	0.336049i
−0.921124	−	0.389270i
0.0855750	+	0.996332i
−0.985354	−	0.170522i
0.0855750	−	0.996332i
0.897398	−	0.441221i
−0.466667	+	0.884433i
0.993482	−	0.113991i
−0.870746	−	0.491733i

−0.736741

0.676175i

−1.02693

−

1.05668i

0.799530

−

1.32587i

−0.809017

−

0.587785i

75.1

0.610648

0.791902i

−0.899779

0.825810i

0.931812

0.0532830i

0.309017

−

0.951057i

113.1

−0.985354

−

0.170522i

1.96749

0.112505i

1.03984

−

1.52072i

−0.809017

0.587785i

170.1

0.696938

0.717132i

−0.354349

−

1.34808i

1.83474

−

0.654632i

0.309017

0.951057i

246.1

0.993482

0.113991i

−0.927251

−

1.75734i

0.468333

0.197919i

−0.809017

−

0.587785i

284.1

0.696938

−

0.717132i

−0.354349

1.34808i

1.83474

0.654632i

0.309017

−

0.951057i

322.1

−0.254218

−

0.967147i

−0.0435095

−

0.506572i

−1.12153

0.128683i

−0.809017

0.587785i

379.1

0.974012

0.226497i

−1.09955

1.60804i

−1.21371

−

1.24888i

0.309017

0.951057i

455.1

−0.564443

−

0.825472i

−0.224186

−

1.10640i

0.500990

−

1.90596i

−0.809017

−

0.587785i

531.1

0.774142

−

0.633012i

1.50805

−

0.350681i

0.0420768

0.0386178i

−0.809017

0.587785i

588.1

0.941844

−

0.336049i

1.87141

−

0.214724i

−0.505709

1.29890i

0.309017

0.951057i

664.1

−0.254218

0.967147i

−0.0435095

0.506572i

−1.12153

−

0.128683i

−0.809017

−

0.587785i

702.1

−0.466667

0.884433i

−0.722533

−

0.590812i

0.104512

0.135534i

0.309017

−

0.951057i

740.1

0.897398

0.441221i

−0.651166

−

1.67251i

−0.0294925

1.03237i

−0.809017

0.587785i

797.1

0.610648

−

0.791902i

−0.899779

−

0.825810i

0.931812

−

0.0532830i

0.309017

0.951057i

873.1

0.897398

−

0.441221i

−0.651166

1.67251i

−0.0294925

−

1.03237i

−0.809017

−

0.587785i

911.1

0.941844

0.336049i

1.87141

0.214724i

−0.505709

−

1.29890i

0.309017

−

0.951057i

949.1

−0.0285561

0.999592i

0.0497301

−

0.0280839i

1.38943

−

1.13613i

−0.809017

0.587785i

1006.1

0.0855750

−

0.996332i

−0.00488737

0.171080i

−0.185351

−

0.351280i

0.309017

0.951057i

1082.1

−0.921124

−

0.389270i

−1.12496

1.45888i

−1.95786

0.338821i

−0.809017

−

0.587785i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by $\Q(\sqrt{-19}) $
121.g	even	55	1	inner
2299.bj	odd	110	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2299.1.bj.a	✓	40
19.b	odd	2	1	CM	2299.1.bj.a	✓	40
121.g	even	55	1	inner	2299.1.bj.a	✓	40
2299.bj	odd	110	1	inner	2299.1.bj.a	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2299.1.bj.a	✓	40	1.a	even	1	1	trivial
2299.1.bj.a	✓	40	19.b	odd	2	1	CM
2299.1.bj.a	✓	40	121.g	even	55	1	inner
2299.1.bj.a	✓	40	2299.bj	odd	110	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2299, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{40} $ T^40
$3$	$ T^{40} $ T^40
$5$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 - 39T^37 + 83T^36 - 8T^35 + 659T^34 - 1485T^33 + 275T^32 - 6347T^31 + 15038T^30 - 4026T^29 + 38661T^28 - 95266T^27 + 32983T^26 - 156955T^25 + 394872T^24 - 167103T^23 + 438647T^22 - 1092190T^21 + 546503T^20 - 858286T^19 + 2021096T^18 - 1173549T^17 + 1170031T^16 - 2443549T^15 + 1636587T^14 - 1138468T^13 + 1877553T^12 - 1385065T^11 + 720765T^10 - 777755T^9 + 669581T^8 - 382954T^7 + 208163T^6 - 86942T^5 + 21945T^4 - 3360T^3 + 522T^2 - 36T + 1
$7$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 16T^37 + 33T^36 + 42T^35 + 109T^34 + 220T^33 + 275T^32 + 748T^31 + 1728T^30 + 2519T^29 + 6101T^28 + 14444T^27 + 22348T^26 + 35770T^25 + 79062T^24 + 128747T^23 + 178222T^22 + 239800T^21 + 272163T^20 + 274219T^19 + 153846T^18 - 217314T^17 - 431184T^16 - 635584T^15 - 856763T^14 - 184893T^13 + 321383T^12 + 346775T^11 + 577545T^10 + 143660T^9 - 242924T^8 - 55869T^7 + 35793T^6 + 11833T^5 + 5385T^4 - 2095T^3 + 462T^2 - 31*T + 1
$11$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{10} $ (T^4 + T^3 + T^2 + T + 1)^10
$13$	$ T^{40} $ T^40
$17$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$19$	$ T^{40} - T^{39} + \cdots + 1 $ T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$23$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 38T^35 + 82T^34 - 97T^33 + 257T^32 - 585T^31 + 1162T^30 - 2296T^29 + 4448T^28 - 10098T^27 + 23609T^26 - 39106T^25 + 76088T^24 - 142931T^23 + 270480T^22 - 507063T^21 + 883914T^20 - 1388644T^19 + 1937365T^18 - 2418694T^17 + 3007555T^16 - 3411704T^15 + 3522302T^14 - 3006355T^13 + 2478909T^12 - 1659419T^11 + 968095T^10 - 285734T^9 + 80898T^8 + 108073T^7 - 15901T^6 + 15196T^5 + 4860T^4 - 2872T^3 + 511T^2 - 31*T + 1
$29$	$ T^{40} $ T^40
$31$	$ T^{40} $ T^40
$37$	$ T^{40} $ T^40
$41$	$ T^{40} $ T^40
$43$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^38 - 10T^37 + 31T^36 - 49T^35 + 104T^34 - 185T^33 + 444T^32 - 541T^31 + 1162T^30 - 1790T^29 + 4063T^28 - 2816T^27 + 9430T^26 - 10627T^25 + 28172T^24 + 7824T^23 + 83238T^22 - 29707T^21 + 159740T^20 - 125184T^19 + 235511T^18 - 326153T^17 + 530586T^16 - 798500T^15 + 1047159T^14 - 1314654T^13 + 1644592T^12 - 2150008T^11 + 2725829T^10 - 2990183T^9 + 2621018T^8 - 1692693T^7 + 759841T^6 - 229136T^5 + 45538T^4 - 5941T^3 + 511T^2 - 20*T + 1
$47$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 - 39T^37 + 83T^36 - 8T^35 + 659T^34 - 1485T^33 + 275T^32 - 6347T^31 + 15038T^30 - 4026T^29 + 38661T^28 - 95266T^27 + 32983T^26 - 156955T^25 + 394872T^24 - 167103T^23 + 438647T^22 - 1092190T^21 + 546503T^20 - 858286T^19 + 2021096T^18 - 1173549T^17 + 1170031T^16 - 2443549T^15 + 1636587T^14 - 1138468T^13 + 1877553T^12 - 1385065T^11 + 720765T^10 - 777755T^9 + 669581T^8 - 382954T^7 + 208163T^6 - 86942T^5 + 21945T^4 - 3360T^3 + 522T^2 - 36T + 1
$53$	$ T^{40} $ T^40
$59$	$ T^{40} $ T^40
$61$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 34T^34 - 55T^33 - 55T^32 + 144T^30 + 55T^29 - 356T^28 - 1000T^27 - 1610T^26 - 1597T^25 + 5T^24 + 4183T^23 + 10945T^22 + 17710T^21 + 17711T^20 - 6765T^18 - 2583T^17 + 1600T^16 + 1602T^15 + 5T^14 - 610T^13 - 246T^12 + 110T^11 + 89T^10 - 55T^9 - 55T^8 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$67$	$ T^{40} $ T^40
$71$	$ T^{40} $ T^40
$73$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 11T^37 - 22T^36 + 32T^35 + 24T^34 - 165T^33 + 330T^32 - 77T^31 - 637T^30 + 1584T^29 - 1049T^28 - 1661T^27 + 5203T^26 - 3895T^25 - 2673T^24 + 3447T^23 + 10802T^22 - 330T^21 + 4093T^20 + 40744T^19 - 3179T^18 - 4034T^17 + 97856T^16 - 59929T^15 - 17578T^14 + 47102T^13 + 12328T^12 + 12815T^11 - 33175T^10 + 12925T^9 + 19591T^8 - 10549T^7 + 3383T^6 - 672T^5 + 55T^4 - 55T^3 + 22T^2 - 6T + 1
$79$	$ T^{40} $ T^40
$83$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^37 - 5T^36 - 8T^35 + 32T^34 - 242T^33 + 462T^32 - 1143T^30 + 1144T^29 + 1943T^28 - 6848T^27 + 19244T^26 - 29806T^25 + 148T^24 + 73049T^23 - 71885T^22 - 133903T^21 + 417539T^20 - 553509T^19 + 376310T^18 + 100581T^17 - 569410T^16 + 492131T^15 + 616777T^14 - 2361690T^13 + 4078224T^12 - 5203979T^11 + 5270750T^10 - 4133459T^9 + 2457763T^8 - 1124882T^7 + 395339T^6 - 103189T^5 + 19250T^4 - 1237T^3 - 94T^2 + 19*T + 1
$89$	$ T^{40} $ T^40
$97$	$ T^{40} $ T^40
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2299.bj (of order \(110\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{110}^{13} q^{4} + (\zeta_{110}^{54} + \zeta_{110}^{28}) q^{5} + (\zeta_{110}^{30} + \zeta_{110}^{6}) q^{7} + \zeta_{110}^{44} q^{9} + \zeta_{110}^{22} q^{11} + \zeta_{110}^{26} q^{16} + ( - \zeta_{110}^{25} - \zeta_{110}^{7}) q^{17} + \cdots - \zeta_{110}^{11} q^{99} +O(q^{100}) \) q - z^13 * q^4 + (z^54 + z^28) * q^5 + (z^30 + z^6) * q^7 + z^44 * q^9 + z^22 * q^11 + z^26 * q^16 + (-z^25 - z^7) * q^17 - z^17 * q^19 + (-z^41 + z^12) * q^20 + (-z^21 - z^9) * q^23 + (-z^53 - z^27 - z) * q^25 + (-z^43 - z^19) * q^28 + (z^34 - z^29 - z^5 - z^3) * q^35 + z^2 * q^36 + (-z^41 - z^9) * q^43 - z^35 * q^44 + (-z^43 - z^17) * q^45 + (z^46 + z^18) * q^47 + (z^36 + z^12 - z^5) * q^49 + (z^50 - z^21) * q^55 + (z^32 + z^10) * q^61 + (z^50 - z^19) * q^63 - z^39 * q^64 + (z^38 + z^20) * q^68 + (z^38 - z^3) * q^73 + z^30 * q^76 + (z^52 + z^28) * q^77 + (z^54 - z^25) * q^80 - z^33 * q^81 + (z^24 - z^23) * q^83 + (-z^53 - z^35 + z^24 + z^6) * q^85 + (z^34 + z^22) * q^92 + (-z^45 + z^16) * q^95 - z^11 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + q^{4} + 2 q^{5} - 3 q^{7} - 10 q^{9} - 10 q^{11} + q^{16} - 3 q^{17} + q^{19} + 2 q^{20} + 2 q^{23} + 3 q^{25} + 2 q^{28} - q^{35} + q^{36} + 2 q^{43} - 4 q^{44} + 2 q^{45} + 2 q^{47} - 2 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) \) 40 * q + q^4 + 2 * q^5 - 3 * q^7 - 10 * q^9 - 10 * q^11 + q^16 - 3 * q^17 + q^19 + 2 * q^20 + 2 * q^23 + 3 * q^25 + 2 * q^28 - q^35 + q^36 + 2 * q^43 - 4 * q^44 + 2 * q^45 + 2 * q^47 - 2 * q^49 - 3 * q^55 - 3 * q^61 - 3 * q^63 + q^64 - 3 * q^68 + 2 * q^73 - 4 * q^76 + 2 * q^77 - 3 * q^80 - 10 * q^81 + 2 * q^83 - q^85 - 9 * q^92 - 3 * q^95 - 10 * q^99

Introduction

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Modular forms

Varieties

Fields

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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	2299.1.bj.a

Level	$2299$
Weight	$1$
Character orbit	2299.bj
Analytic conductor	$1.147$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{55}$
CM discriminant	-19
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.14735046404\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Coefficient field:	\(\Q(\zeta_{55})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 \) x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{55}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{55} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 37.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{40} \) T^40
$3$	\( T^{40} \) T^40
$5$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 - 39T^37 + 83T^36 - 8T^35 + 659T^34 - 1485T^33 + 275T^32 - 6347T^31 + 15038T^30 - 4026T^29 + 38661T^28 - 95266T^27 + 32983T^26 - 156955T^25 + 394872T^24 - 167103T^23 + 438647T^22 - 1092190T^21 + 546503T^20 - 858286T^19 + 2021096T^18 - 1173549T^17 + 1170031T^16 - 2443549T^15 + 1636587T^14 - 1138468T^13 + 1877553T^12 - 1385065T^11 + 720765T^10 - 777755T^9 + 669581T^8 - 382954T^7 + 208163T^6 - 86942T^5 + 21945T^4 - 3360T^3 + 522T^2 - 36T + 1
$7$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 16T^37 + 33T^36 + 42T^35 + 109T^34 + 220T^33 + 275T^32 + 748T^31 + 1728T^30 + 2519T^29 + 6101T^28 + 14444T^27 + 22348T^26 + 35770T^25 + 79062T^24 + 128747T^23 + 178222T^22 + 239800T^21 + 272163T^20 + 274219T^19 + 153846T^18 - 217314T^17 - 431184T^16 - 635584T^15 - 856763T^14 - 184893T^13 + 321383T^12 + 346775T^11 + 577545T^10 + 143660T^9 - 242924T^8 - 55869T^7 + 35793T^6 + 11833T^5 + 5385T^4 - 2095T^3 + 462T^2 - 31*T + 1
$11$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{10} \) (T^4 + T^3 + T^2 + T + 1)^10
$13$	\( T^{40} \) T^40
$17$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$19$	\( T^{40} - T^{39} + \cdots + 1 \) T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$23$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 38T^35 + 82T^34 - 97T^33 + 257T^32 - 585T^31 + 1162T^30 - 2296T^29 + 4448T^28 - 10098T^27 + 23609T^26 - 39106T^25 + 76088T^24 - 142931T^23 + 270480T^22 - 507063T^21 + 883914T^20 - 1388644T^19 + 1937365T^18 - 2418694T^17 + 3007555T^16 - 3411704T^15 + 3522302T^14 - 3006355T^13 + 2478909T^12 - 1659419T^11 + 968095T^10 - 285734T^9 + 80898T^8 + 108073T^7 - 15901T^6 + 15196T^5 + 4860T^4 - 2872T^3 + 511T^2 - 31*T + 1
$29$	\( T^{40} \) T^40
$31$	\( T^{40} \) T^40
$37$	\( T^{40} \) T^40
$41$	\( T^{40} \) T^40
$43$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^38 - 10T^37 + 31T^36 - 49T^35 + 104T^34 - 185T^33 + 444T^32 - 541T^31 + 1162T^30 - 1790T^29 + 4063T^28 - 2816T^27 + 9430T^26 - 10627T^25 + 28172T^24 + 7824T^23 + 83238T^22 - 29707T^21 + 159740T^20 - 125184T^19 + 235511T^18 - 326153T^17 + 530586T^16 - 798500T^15 + 1047159T^14 - 1314654T^13 + 1644592T^12 - 2150008T^11 + 2725829T^10 - 2990183T^9 + 2621018T^8 - 1692693T^7 + 759841T^6 - 229136T^5 + 45538T^4 - 5941T^3 + 511T^2 - 20*T + 1
$47$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 - 39T^37 + 83T^36 - 8T^35 + 659T^34 - 1485T^33 + 275T^32 - 6347T^31 + 15038T^30 - 4026T^29 + 38661T^28 - 95266T^27 + 32983T^26 - 156955T^25 + 394872T^24 - 167103T^23 + 438647T^22 - 1092190T^21 + 546503T^20 - 858286T^19 + 2021096T^18 - 1173549T^17 + 1170031T^16 - 2443549T^15 + 1636587T^14 - 1138468T^13 + 1877553T^12 - 1385065T^11 + 720765T^10 - 777755T^9 + 669581T^8 - 382954T^7 + 208163T^6 - 86942T^5 + 21945T^4 - 3360T^3 + 522T^2 - 36T + 1
$53$	\( T^{40} \) T^40
$59$	\( T^{40} \) T^40
$61$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 34T^34 - 55T^33 - 55T^32 + 144T^30 + 55T^29 - 356T^28 - 1000T^27 - 1610T^26 - 1597T^25 + 5T^24 + 4183T^23 + 10945T^22 + 17710T^21 + 17711T^20 - 6765T^18 - 2583T^17 + 1600T^16 + 1602T^15 + 5T^14 - 610T^13 - 246T^12 + 110T^11 + 89T^10 - 55T^9 - 55T^8 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$67$	\( T^{40} \) T^40
$71$	\( T^{40} \) T^40
$73$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 11T^37 - 22T^36 + 32T^35 + 24T^34 - 165T^33 + 330T^32 - 77T^31 - 637T^30 + 1584T^29 - 1049T^28 - 1661T^27 + 5203T^26 - 3895T^25 - 2673T^24 + 3447T^23 + 10802T^22 - 330T^21 + 4093T^20 + 40744T^19 - 3179T^18 - 4034T^17 + 97856T^16 - 59929T^15 - 17578T^14 + 47102T^13 + 12328T^12 + 12815T^11 - 33175T^10 + 12925T^9 + 19591T^8 - 10549T^7 + 3383T^6 - 672T^5 + 55T^4 - 55T^3 + 22T^2 - 6T + 1
$79$	\( T^{40} \) T^40
$83$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^37 - 5T^36 - 8T^35 + 32T^34 - 242T^33 + 462T^32 - 1143T^30 + 1144T^29 + 1943T^28 - 6848T^27 + 19244T^26 - 29806T^25 + 148T^24 + 73049T^23 - 71885T^22 - 133903T^21 + 417539T^20 - 553509T^19 + 376310T^18 + 100581T^17 - 569410T^16 + 492131T^15 + 616777T^14 - 2361690T^13 + 4078224T^12 - 5203979T^11 + 5270750T^10 - 4133459T^9 + 2457763T^8 - 1124882T^7 + 395339T^6 - 103189T^5 + 19250T^4 - 1237T^3 - 94T^2 + 19*T + 1
$89$	\( T^{40} \) T^40
$97$	\( T^{40} \) T^40
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\(n\)	\(970\)	\(1332\)
\(\chi(n)\)	\(-\zeta_{110}^{13}\)	\(-1\)