LMFDB - Newform orbit 1340.1.bl.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1340,1,Mod(19,1340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1340, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 33, 10]))

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

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

chi := DirichletCharacter("1340.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1340 = 2^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1340.bl (of order $66$, degree $20$, 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:	$0.668747116928$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times$.

$n$	$537$	$671$	$1141$
$\chi(n)$	$-1$	$-1$	$-\zeta_{66}^{29}$

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} $

19.1

0.235759	+	0.971812i
0.981929	+	0.189251i
0.723734	+	0.690079i
0.0475819	+	0.998867i
0.0475819	−	0.998867i
0.928368	−	0.371662i
−0.786053	+	0.618159i
−0.995472	+	0.0950560i
−0.327068	−	0.945001i
0.580057	+	0.814576i
−0.888835	−	0.458227i
−0.786053	−	0.618159i
0.981929	−	0.189251i
−0.995472	−	0.0950560i
−0.327068	+	0.945001i
0.928368	+	0.371662i
0.235759	−	0.971812i
0.723734	−	0.690079i
0.580057	−	0.814576i
−0.888835	+	0.458227i

0.888835

0.458227i

1.11312

−

0.326842i

0.580057

0.814576i

−0.654861

−

0.755750i

1.13915

0.219553i

−0.0688733

−

1.44583i

0.142315

0.989821i

0.290959

−

0.186988i

−0.235759

−

0.971812i

39.1

−0.928368

0.371662i

0.947890

1.09392i

0.723734

−

0.690079i

0.841254

0.540641i

−1.28656

−

0.663268i

−0.514186

−

0.404360i

−0.415415

0.909632i

−0.155858

1.08402i

−0.981929

−

0.189251i

199.1

−0.0475819

0.998867i

−1.91030

−

0.560914i

−0.995472

−

0.0950560i

−0.654861

0.755750i

0.651174

−

1.88144i

0.419102

−

0.216062i

0.142315

−

0.989821i

2.49336

1.60238i

−0.723734

−

0.690079i

339.1

0.995472

0.0950560i

−1.65210

−

1.06174i

0.981929

0.189251i

−0.142315

−

0.989821i

−1.54370

−

1.21398i

1.03115

−

1.44805i

0.959493

0.281733i

1.18673

2.59858i

−0.0475819

−

0.998867i

419.1

0.995472

−

0.0950560i

−1.65210

1.06174i

0.981929

−

0.189251i

−0.142315

0.989821i

−1.54370

1.21398i

1.03115

1.44805i

0.959493

−

0.281733i

1.18673

−

2.59858i

−0.0475819

0.998867i

479.1

−0.723734

−

0.690079i

0.0135432

0.0941952i

0.0475819

0.998867i

0.415415

−

0.909632i

0.0552004

−

0.0775182i

0.370638

−

1.52779i

0.654861

−

0.755750i

0.950804

−

0.279181i

−0.928368

0.371662i

559.1

−0.235759

−

0.971812i

−0.252989

1.75958i

−0.888835

0.458227i

0.415415

0.909632i

1.76962

−

0.168978i

−1.34378

1.28129i

0.654861

0.755750i

−2.07261

−

0.608574i

0.786053

−

0.618159i

619.1

−0.981929

−

0.189251i

−0.771316

−

1.68895i

0.928368

0.371662i

−0.959493

0.281733i

0.437742

1.80440i

0.379436

1.09631i

−0.841254

−

0.540641i

−1.60275

1.84967i

0.995472

−

0.0950560i

639.1

0.786053

0.618159i

0.308779

0.356349i

0.235759

0.971812i

0.841254

0.540641i

0.0224357

0.470984i

−1.82318

0.729892i

−0.415415

0.909632i

0.110674

−

0.769755i

0.327068

0.945001i

659.1

0.327068

0.945001i

0.653077

1.43004i

−0.786053

0.618159i

−0.959493

0.281733i

−1.13779

1.08488i

1.95496

0.376789i

−0.841254

−

0.540641i

−0.963639

1.11210i

−0.580057

−

0.814576i

719.1

−0.580057

0.814576i

0.550294

0.353653i

−0.327068

−

0.945001i

−0.142315

−

0.989821i

−0.607279

0.243118i

0.0947329

0.00904590i

0.959493

0.281733i

−0.237662

−

0.520406i

0.888835

0.458227i

839.1

−0.235759

0.971812i

−0.252989

−

1.75958i

−0.888835

−

0.458227i

0.415415

−

0.909632i

1.76962

0.168978i

−1.34378

−

1.28129i

0.654861

−

0.755750i

−2.07261

0.608574i

0.786053

0.618159i

859.1

−0.928368

−

0.371662i

0.947890

−

1.09392i

0.723734

0.690079i

0.841254

−

0.540641i

−1.28656

0.663268i

−0.514186

0.404360i

−0.415415

−

0.909632i

−0.155858

−

1.08402i

−0.981929

0.189251i

959.1

−0.981929

0.189251i

−0.771316

1.68895i

0.928368

−

0.371662i

−0.959493

−

0.281733i

0.437742

−

1.80440i

0.379436

−

1.09631i

−0.841254

0.540641i

−1.60275

−

1.84967i

0.995472

0.0950560i

1059.1

0.786053

−

0.618159i

0.308779

−

0.356349i

0.235759

−

0.971812i

0.841254

−

0.540641i

0.0224357

−

0.470984i

−1.82318

−

0.729892i

−0.415415

−

0.909632i

0.110674

0.769755i

0.327068

−

0.945001i

1119.1

−0.723734

0.690079i

0.0135432

−

0.0941952i

0.0475819

−

0.998867i

0.415415

0.909632i

0.0552004

0.0775182i

0.370638

1.52779i

0.654861

0.755750i

0.950804

0.279181i

−0.928368

−

0.371662i

1199.1

0.888835

−

0.458227i

1.11312

0.326842i

0.580057

−

0.814576i

−0.654861

0.755750i

1.13915

−

0.219553i

−0.0688733

1.44583i

0.142315

−

0.989821i

0.290959

0.186988i

−0.235759

0.971812i

1239.1

−0.0475819

−

0.998867i

−1.91030

0.560914i

−0.995472

0.0950560i

−0.654861

−

0.755750i

0.651174

1.88144i

0.419102

0.216062i

0.142315

0.989821i

2.49336

−

1.60238i

−0.723734

0.690079i

1279.1

0.327068

−

0.945001i

0.653077

−

1.43004i

−0.786053

−

0.618159i

−0.959493

−

0.281733i

−1.13779

−

1.08488i

1.95496

−

0.376789i

−0.841254

0.540641i

−0.963639

−

1.11210i

−0.580057

0.814576i

1299.1

−0.580057

−

0.814576i

0.550294

−

0.353653i

−0.327068

0.945001i

−0.142315

0.989821i

−0.607279

−

0.243118i

0.0947329

−

0.00904590i

0.959493

−

0.281733i

−0.237662

0.520406i

0.888835

−

0.458227i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
67.g	even	33	1	inner
1340.bl	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1340.1.bl.a	✓	20
4.b	odd	2	1		1340.1.bl.b	yes	20
5.b	even	2	1		1340.1.bl.b	yes	20
20.d	odd	2	1	CM	1340.1.bl.a	✓	20
67.g	even	33	1	inner	1340.1.bl.a	✓	20
268.o	odd	66	1		1340.1.bl.b	yes	20
335.u	even	66	1		1340.1.bl.b	yes	20
1340.bl	odd	66	1	inner	1340.1.bl.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1340.1.bl.a	✓	20	1.a	even	1	1	trivial
1340.1.bl.a	✓	20	20.d	odd	2	1	CM
1340.1.bl.a	✓	20	67.g	even	33	1	inner
1340.1.bl.a	✓	20	1340.bl	odd	66	1	inner
1340.1.bl.b	yes	20	4.b	odd	2	1
1340.1.bl.b	yes	20	5.b	even	2	1
1340.1.bl.b	yes	20	268.o	odd	66	1
1340.1.bl.b	yes	20	335.u	even	66	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{20} + 2 T_{3}^{19} + 3 T_{3}^{18} + 4 T_{3}^{17} + 5 T_{3}^{16} + 6 T_{3}^{15} + 18 T_{3}^{14} + \cdots + 1 $ T3^20 + 2*T3^19 + 3*T3^18 + 4*T3^17 + 5*T3^16 + 6*T3^15 + 18*T3^14 - 25*T3^13 + 9*T3^12 - T3^11 - T3^9 + 119*T3^8 - 971*T3^7 + 2658*T3^6 - 3151*T3^5 + 2051*T3^4 - 744*T3^3 + 146*T3^2 - 9*T3 + 1 acting on $S_{1}^{\mathrm{new}}(1340, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$3$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 3T^18 + 4T^17 + 5T^16 + 6T^15 + 18T^14 - 25T^13 + 9T^12 - T^11 - T^9 + 119T^8 - 971T^7 + 2658T^6 - 3151T^5 + 2051T^4 - 744T^3 + 146T^2 - 9T + 1
$5$	$ (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$7$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + T^17 - T^16 - 22T^15 + 23T^14 - 34T^13 + 22T^12 + T^11 + 450T^10 - 450T^9 + 1452T^8 - 760T^7 + 881T^6 - 12T^4 - 230T^3 + 154T^2 - 23*T + 1
$11$	$ T^{20} $ T^20
$13$	$ T^{20} $ T^20
$17$	$ T^{20} $ T^20
$19$	$ T^{20} $ T^20
$23$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$29$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 12T^14 + 67T^13 - 99T^12 - 122T^11 - 12T^10 + 109T^9 + 242T^8 - 472T^7 + 1464T^6 - 1331T^5 + 593T^4 - 243T^3 + 143T^2 - 21T + 1
$43$	$ T^{20} - 9 T^{19} + \cdots + 1 $ T^20 - 9T^19 + 47T^18 - 172T^17 + 489T^16 - 1127T^15 + 2174T^14 - 3567T^13 + 5047T^12 - 6194T^11 + 6633T^10 - 6194T^9 + 5069T^8 - 3578T^7 + 1965T^6 - 874T^5 + 742T^4 - 381T^3 + 36T^2 + 13*T + 1
$47$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + T^17 - T^16 - 22T^15 + 23T^14 - 34T^13 + 22T^12 + T^11 + 450T^10 - 450T^9 + 1452T^8 - 760T^7 + 881T^6 - 12T^4 - 230T^3 + 154T^2 - 23*T + 1
$53$	$ T^{20} $ T^20
$59$	$ T^{20} $ T^20
$61$	$ T^{20} + 9 T^{19} + \cdots + 1 $ T^20 + 9T^19 + 44T^18 + 151T^17 + 402T^16 + 869T^15 + 1571T^14 + 2424T^13 + 3223T^12 + 3713T^11 + 3750T^10 + 3302T^9 + 2453T^8 + 1631T^7 + 1116T^6 + 616T^5 + 145T^4 - 15T^3 + 5T + 1
$67$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 - 10T^17 + 10T^16 + 78T^14 - 89T^13 + 11T^12 - 241T^11 + 87T^10 + 155T^9 + 605T^8 + 87T^7 - 560T^6 - 253T^5 + 934T^4 - 626T^3 + 154T^2 - 12T + 1
$89$	$ (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$97$	$ T^{20} $ T^20
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1340.bl (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{66}^{31} q^{2} + (\zeta_{66}^{25} + \zeta_{66}^{17}) q^{3} - \zeta_{66}^{29} q^{4} - \zeta_{66}^{3} q^{5} + ( - \zeta_{66}^{23} - \zeta_{66}^{15}) q^{6} + ( - \zeta_{66}^{30} - \zeta_{66}^{10}) q^{7} + \zeta_{66}^{27} q^{8} + ( - \zeta_{66}^{17} + \cdots - \zeta_{66}) q^{9} +O(q^{10}) \) q + z^31 * q^2 + (z^25 + z^17) * q^3 - z^29 * q^4 - z^3 * q^5 + (-z^23 - z^15) * q^6 + (-z^30 - z^10) * q^7 + z^27 * q^8 + (-z^17 - z^9 - z) * q^9	\( q + \zeta_{66}^{31} q^{2} + (\zeta_{66}^{25} + \zeta_{66}^{17}) q^{3} - \zeta_{66}^{29} q^{4} - \zeta_{66}^{3} q^{5} + ( - \zeta_{66}^{23} - \zeta_{66}^{15}) q^{6} + ( - \zeta_{66}^{30} - \zeta_{66}^{10}) q^{7} + \zeta_{66}^{27} q^{8} + ( - \zeta_{66}^{17} + \cdots - \zeta_{66}) q^{9} + \cdots + (\zeta_{66}^{25} + \cdots + \zeta_{66}^{5}) q^{98} +O(q^{100}) \) q + z^31 * q^2 + (z^25 + z^17) * q^3 - z^29 * q^4 - z^3 * q^5 + (-z^23 - z^15) * q^6 + (-z^30 - z^10) * q^7 + z^27 * q^8 + (-z^17 - z^9 - z) * q^9 + z * q^10 + (z^21 + z^13) * q^12 + (z^28 + z^8) * q^14 + (-z^28 - z^20) * q^15 - z^25 * q^16 + (-z^32 + z^15 + z^7) * q^18 + z^32 * q^20 + (-z^27 + z^22 + z^14 + z^2) * q^21 + z^10 * q^23 + (-z^19 - z^11) * q^24 + z^6 * q^25 + (-z^26 - z^18 + z^9 - z) * q^27 + (-z^26 - z^6) * q^28 + (z^12 + z^10) * q^29 + (z^26 + z^18) * q^30 + z^23 * q^32 + (z^13 - 1) * q^35 + (z^30 - z^13 - z^5) * q^36 - z^30 * q^40 + (-z^31 - z^21) * q^41 + (z^25 - z^20 - z^12 - 1) * q^42 + (-z^22 - z^8) * q^43 + (z^20 + z^12 + z^4) * q^45 - z^8 * q^46 + (-z^16 + z^15) * q^47 + (z^17 + z^9) * q^48 + (-z^27 + z^20 - z^7) * q^49 - z^4 * q^50 + (z^32 + z^24 + z^16 - z^7) * q^54 + (z^24 + z^4) * q^56 + (-z^10 - z^8) * q^58 + (-z^24 - z^16) * q^60 + (z^22 + z^16) * q^61 + (z^31 + z^27 + z^19 - z^14 + z^11 - z^6) * q^63 - z^21 * q^64 + z^7 * q^67 + (z^27 - z^2) * q^69 + (-z^31 - z^11) * q^70 + (-z^28 + z^11 + z^3) * q^72 + (z^31 + z^23) * q^75 + z^28 * q^80 + (z^26 + z^18 + z^10 + z^2 - z) * q^81 + (z^29 + z^19) * q^82 + (-z^12 + z^5) * q^83 + (-z^31 - z^23 + z^18 + z^10) * q^84 + (z^20 + z^6) * q^86 + (z^29 + z^27 - z^4 - z^2) * q^87 + (-z^21 - z^3) * q^89 + (-z^18 - z^10 - z^2) * q^90 + z^6 * q^92 + (z^14 - z^13) * q^94 + (-z^15 - z^7) * q^96 + (z^25 - z^18 + z^5) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8}+O(q^{10}) \) 20 * q - q^2 - 2 * q^3 + q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8	\( 20 q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} - q^{10} + q^{12} + 2 q^{14} - 2 q^{15} + q^{16} + q^{20} - 10 q^{21} + q^{23} - 9 q^{24} - 2 q^{25} + 2 q^{27} + q^{28} - q^{29} - q^{30} - q^{32} - 21 q^{35} + 2 q^{40} - q^{41} - 20 q^{42} + 9 q^{43} - q^{46} + q^{47} + q^{48} - q^{50} + q^{54} - q^{56} - 2 q^{58} + q^{60} - 9 q^{61} + 11 q^{63} - 2 q^{64} - q^{67} + q^{69} - 9 q^{70} + 11 q^{72} - 2 q^{75} + q^{80} + 2 q^{81} - 2 q^{82} + q^{83} + q^{84} - q^{86} - q^{87} - 4 q^{89} - 2 q^{92} + 2 q^{94} - q^{96}+O(q^{100}) \) 20 * q - q^2 - 2 * q^3 + q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8 - q^10 + q^12 + 2 * q^14 - 2 * q^15 + q^16 + q^20 - 10 * q^21 + q^23 - 9 * q^24 - 2 * q^25 + 2 * q^27 + q^28 - q^29 - q^30 - q^32 - 21 * q^35 + 2 * q^40 - q^41 - 20 * q^42 + 9 * q^43 - q^46 + q^47 + q^48 - q^50 + q^54 - q^56 - 2 * q^58 + q^60 - 9 * q^61 + 11 * q^63 - 2 * q^64 - q^67 + q^69 - 9 * q^70 + 11 * q^72 - 2 * q^75 + q^80 + 2 * q^81 - 2 * q^82 + q^83 + q^84 - q^86 - q^87 - 4 * q^89 - 2 * q^92 + 2 * q^94 - q^96

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1340.1.bl.a

Level	$1340$
Weight	$1$
Character orbit	1340.bl
Analytic conductor	$0.669$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-20
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.668747116928\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{33})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{33}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

\(n\)	\(537\)	\(671\)	\(1141\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{66}^{29}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$3$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 3T^18 + 4T^17 + 5T^16 + 6T^15 + 18T^14 - 25T^13 + 9T^12 - T^11 - T^9 + 119T^8 - 971T^7 + 2658T^6 - 3151T^5 + 2051T^4 - 744T^3 + 146T^2 - 9T + 1
$5$	\( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$7$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + T^17 - T^16 - 22T^15 + 23T^14 - 34T^13 + 22T^12 + T^11 + 450T^10 - 450T^9 + 1452T^8 - 760T^7 + 881T^6 - 12T^4 - 230T^3 + 154T^2 - 23*T + 1
$11$	\( T^{20} \) T^20
$13$	\( T^{20} \) T^20
$17$	\( T^{20} \) T^20
$19$	\( T^{20} \) T^20
$23$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$29$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 12T^14 + 67T^13 - 99T^12 - 122T^11 - 12T^10 + 109T^9 + 242T^8 - 472T^7 + 1464T^6 - 1331T^5 + 593T^4 - 243T^3 + 143T^2 - 21T + 1
$43$	\( T^{20} - 9 T^{19} + \cdots + 1 \) T^20 - 9T^19 + 47T^18 - 172T^17 + 489T^16 - 1127T^15 + 2174T^14 - 3567T^13 + 5047T^12 - 6194T^11 + 6633T^10 - 6194T^9 + 5069T^8 - 3578T^7 + 1965T^6 - 874T^5 + 742T^4 - 381T^3 + 36T^2 + 13*T + 1
$47$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + T^17 - T^16 - 22T^15 + 23T^14 - 34T^13 + 22T^12 + T^11 + 450T^10 - 450T^9 + 1452T^8 - 760T^7 + 881T^6 - 12T^4 - 230T^3 + 154T^2 - 23*T + 1
$53$	\( T^{20} \) T^20
$59$	\( T^{20} \) T^20
$61$	\( T^{20} + 9 T^{19} + \cdots + 1 \) T^20 + 9T^19 + 44T^18 + 151T^17 + 402T^16 + 869T^15 + 1571T^14 + 2424T^13 + 3223T^12 + 3713T^11 + 3750T^10 + 3302T^9 + 2453T^8 + 1631T^7 + 1116T^6 + 616T^5 + 145T^4 - 15T^3 + 5T + 1
$67$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 - 10T^17 + 10T^16 + 78T^14 - 89T^13 + 11T^12 - 241T^11 + 87T^10 + 155T^9 + 605T^8 + 87T^7 - 560T^6 - 253T^5 + 934T^4 - 626T^3 + 154T^2 - 12T + 1
$89$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$97$	\( T^{20} \) T^20
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