LMFDB - Newform orbit 1412.1.r.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1412,1,Mod(35,1412)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1412, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 7]))

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

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

chi := DirichletCharacter("1412.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1412 = 2^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1412.r (of order $44$, 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:	$0.704679797838$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} + \cdots)$

$q$-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_{44}^{14} q^{2} - \zeta_{44}^{6} q^{4} + (\zeta_{44}^{4} - \zeta_{44}) q^{5} - \zeta_{44}^{20} q^{8} - \zeta_{44}^{5} q^{9} +O(q^{10}) $ q + z^14 * q^2 - z^6 * q^4 + (z^4 - z) * q^5 - z^20 * q^8 - z^5 * q^9	$ q + \zeta_{44}^{14} q^{2} - \zeta_{44}^{6} q^{4} + (\zeta_{44}^{4} - \zeta_{44}) q^{5} - \zeta_{44}^{20} q^{8} - \zeta_{44}^{5} q^{9} + (\zeta_{44}^{18} - \zeta_{44}^{15}) q^{10} + (\zeta_{44}^{20} + \zeta_{44}^{15}) q^{13} + \zeta_{44}^{12} q^{16} + (\zeta_{44}^{14} - \zeta_{44}^{4}) q^{17} - \zeta_{44}^{19} q^{18} + ( - \zeta_{44}^{10} + \zeta_{44}^{7}) q^{20} + (\zeta_{44}^{8} + \cdots + \zeta_{44}^{2}) q^{25} + \cdots + \zeta_{44}^{3} q^{98} +O(q^{100}) $ q + z^14 * q^2 - z^6 * q^4 + (z^4 - z) * q^5 - z^20 * q^8 - z^5 * q^9 + (z^18 - z^15) * q^10 + (z^20 + z^15) * q^13 + z^12 * q^16 + (z^14 - z^4) * q^17 - z^19 * q^18 + (-z^10 + z^7) * q^20 + (z^8 - z^5 + z^2) * q^25 + (-z^12 - z^7) * q^26 + (z^17 + z^15) * q^29 - z^4 * q^32 + (-z^18 - z^6) * q^34 + z^11 * q^36 + (z^7 - 1) * q^37 + (z^21 + z^2) * q^40 + (z^9 + z^5) * q^41 + (-z^9 + z^6) * q^45 - z^11 * q^49 + (-z^19 + z^16 - 1) * q^50 + (-z^21 + z^4) * q^52 + (-z^3 - z^2) * q^53 + (-z^9 - z^7) * q^58 + (-z^13 - z^11) * q^61 - z^18 * q^64 + (-z^21 + z^19 - z^16 - z^2) * q^65 + (-z^20 + z^10) * q^68 - z^3 * q^72 + (z^21 - z^14) * q^74 + (z^16 - z^13) * q^80 + z^10 * q^81 + (z^19 - z) * q^82 + (z^18 - z^15 - z^8 + z^5) * q^85 + (z^20 - z^17) * q^89 + (z^20 + z) * q^90 + (z^16 - z^10) * q^97 + z^3 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) $ 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8	$ 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} + 2 q^{10} - 2 q^{13} - 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{26} + 2 q^{32} - 4 q^{34} - 20 q^{37} + 2 q^{40} + 2 q^{45} - 22 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{64} + 4 q^{68} - 2 q^{74} - 2 q^{80} + 2 q^{81} + 4 q^{85} - 2 q^{89} - 2 q^{90} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8 + 2 * q^10 - 2 * q^13 - 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^26 + 2 * q^32 - 4 * q^34 - 20 * q^37 + 2 * q^40 + 2 * q^45 - 22 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^64 + 4 * q^68 - 2 * q^74 - 2 * q^80 + 2 * q^81 + 4 * q^85 - 2 * q^89 - 2 * q^90 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times$.

$n$	$707$	$709$
$\chi(n)$	$-1$	$-\zeta_{44}^{5}$

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} $

35.1

0.909632	−	0.415415i
−0.989821	+	0.142315i
−0.540641	+	0.841254i
−0.281733	+	0.959493i
−0.281733	−	0.959493i
0.989821	+	0.142315i
−0.989821	−	0.142315i
0.281733	+	0.959493i
0.281733	−	0.959493i
0.540641	−	0.841254i
0.989821	−	0.142315i
−0.909632	+	0.415415i
−0.909632	−	0.415415i
0.755750	+	0.654861i
0.540641	+	0.841254i
−0.755750	+	0.654861i
0.755750	−	0.654861i
−0.540641	−	0.841254i
−0.755750	−	0.654861i
0.909632	+	0.415415i

0.959493

0.281733i

0.841254

0.540641i

−1.05195

−

0.574406i

0.654861

0.755750i

0.540641

0.841254i

−0.847507

−

0.847507i

135.1

−0.415415

−

0.909632i

−0.654861

0.755750i

1.83107

−

0.682956i

0.959493

0.281733i

0.755750

−

0.654861i

−1.38189

−

1.38189i

171.1

0.142315

−

0.989821i

−0.959493

−

0.281733i

−0.114220

−

0.0855040i

−0.415415

0.909632i

0.281733

0.959493i

−0.100889

0.100889i

191.1

0.654861

0.755750i

−0.142315

0.989821i

0.697148

−

0.0498610i

−0.841254

0.540641i

0.989821

−

0.142315i

0.494217

0.494217i

207.1

0.654861

−

0.755750i

−0.142315

−

0.989821i

0.697148

0.0498610i

−0.841254

−

0.540641i

0.989821

0.142315i

0.494217

−

0.494217i

319.1

−0.415415

0.909632i

−0.654861

−

0.755750i

−0.148568

0.398326i

0.959493

−

0.281733i

−0.755750

−

0.654861i

−0.300613

−

0.300613i

387.1

−0.415415

0.909632i

−0.654861

−

0.755750i

1.83107

0.682956i

0.959493

−

0.281733i

0.755750

0.654861i

−1.38189

1.38189i

499.1

0.654861

−

0.755750i

−0.142315

−

0.989821i

0.133682

−

1.86912i

−0.841254

−

0.540641i

−0.989821

−

0.142315i

−1.32505

−

1.32505i

515.1

0.654861

0.755750i

−0.142315

0.989821i

0.133682

1.86912i

−0.841254

0.540641i

−0.989821

0.142315i

−1.32505

1.32505i

535.1

0.142315

−

0.989821i

−0.959493

−

0.281733i

−1.19550

1.59700i

−0.415415

0.909632i

−0.281733

−

0.959493i

1.41061

1.41061i

571.1

−0.415415

−

0.909632i

−0.654861

0.755750i

−0.148568

−

0.398326i

0.959493

0.281733i

−0.755750

0.654861i

−0.300613

0.300613i

671.1

0.959493

0.281733i

0.841254

0.540641i

0.767317

−

1.40524i

0.654861

0.755750i

−0.540641

−

0.841254i

1.13214

−

1.13214i

827.1

0.959493

−

0.281733i

0.841254

−

0.540641i

0.767317

1.40524i

0.654861

−

0.755750i

−0.540641

0.841254i

1.13214

1.13214i

971.1

−0.841254

−

0.540641i

0.415415

0.909632i

−1.71524

−

0.373128i

0.142315

−

0.989821i

0.909632

0.415415i

1.24123

1.24123i

995.1

0.142315

0.989821i

−0.959493

0.281733i

−1.19550

−

1.59700i

−0.415415

−

0.909632i

−0.281733

0.959493i

1.41061

−

1.41061i

1055.1

−0.841254

0.540641i

0.415415

−

0.909632i

−0.203743

−

0.936593i

0.142315

0.989821i

−0.909632

0.415415i

0.677760

0.677760i

1063.1

−0.841254

0.540641i

0.415415

−

0.909632i

−1.71524

0.373128i

0.142315

0.989821i

0.909632

−

0.415415i

1.24123

−

1.24123i

1123.1

0.142315

0.989821i

−0.959493

0.281733i

−0.114220

0.0855040i

−0.415415

−

0.909632i

0.281733

−

0.959493i

−0.100889

−

0.100889i

1147.1

−0.841254

−

0.540641i

0.415415

0.909632i

−0.203743

0.936593i

0.142315

−

0.989821i

−0.909632

−

0.415415i

0.677760

−

0.677760i

1291.1

0.959493

−

0.281733i

0.841254

−

0.540641i

−1.05195

0.574406i

0.654861

−

0.755750i

0.540641

−

0.841254i

−0.847507

0.847507i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
353.i	even	44	1	inner
1412.r	odd	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1412.1.r.a	✓	20
4.b	odd	2	1	CM	1412.1.r.a	✓	20
353.i	even	44	1	inner	1412.1.r.a	✓	20
1412.r	odd	44	1	inner	1412.1.r.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1412.1.r.a	✓	20	1.a	even	1	1	trivial
1412.1.r.a	✓	20	4.b	odd	2	1	CM
1412.1.r.a	✓	20	353.i	even	44	1	inner
1412.1.r.a	✓	20	1412.r	odd	44	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1412, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (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
$3$	$ T^{20} $ T^20
$5$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 66T^13 + 148T^12 + 164T^11 + 494T^10 + 658T^9 + 328T^8 - 176T^7 - 403T^6 - 454T^5 + 63T^4 + 66T^3 + 61T^2 + 12T + 1
$7$	$ T^{20} $ T^20
$11$	$ T^{20} $ T^20
$13$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 + 47T^14 + 132T^13 + 170T^12 + 76T^11 - 188T^10 - 530T^9 - 79T^8 - 66T^7 + 125T^6 + 382T^5 + 514T^4 + 330T^3 + 94T^2 + 12T + 1
$17$	$ (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1)^2
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} + 22 T^{16} + \cdots + 121 $ T^20 + 22T^16 + 165T^12 + 99T^10 + 484T^8 - 968T^6 + 484T^4 + 605*T^2 + 121
$31$	$ T^{20} $ T^20
$37$	$ T^{20} + 20 T^{19} + \cdots + 1 $ T^20 + 20T^19 + 189T^18 + 1122T^17 + 4693T^16 + 14704T^15 + 35819T^14 + 69498T^13 + 109136T^12 + 140152T^11 + 148070T^10 + 128988T^9 + 92530T^8 + 54384T^7 + 25942T^6 + 9892T^5 + 2945T^4 + 660T^3 + 105T^2 + 10*T + 1
$41$	$ T^{20} - 4 T^{18} + \cdots + 1 $ T^20 - 4T^18 + 16T^16 - 64T^14 + 234T^12 - 474T^10 + 444T^8 - 203T^6 + 119T^4 + 19*T^2 + 1
$43$	$ T^{20} $ T^20
$47$	$ T^{20} $ T^20
$53$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$59$	$ T^{20} $ T^20
$61$	$ T^{20} + 11 T^{18} + \cdots + 121 $ T^20 + 11T^18 + 55T^16 + 165T^14 + 330T^12 + 462T^10 + 484T^8 + 1089T^4 - 605T^2 + 121
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 30T^15 - 52T^14 - 44T^13 + 93T^12 + 274T^11 + 505T^10 + 460T^9 - 90T^8 + 110T^7 + 400T^6 + 316T^5 + 338T^4 + 44T^3 + 17T^2 - 10T + 1
$97$	$ (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1)^2
show more
show less

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:	\( N \)	\(=\)	\( 1412 = 2^{2} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1412.r (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{44}^{14} q^{2} - \zeta_{44}^{6} q^{4} + (\zeta_{44}^{4} - \zeta_{44}) q^{5} - \zeta_{44}^{20} q^{8} - \zeta_{44}^{5} q^{9} +O(q^{10}) \) q + z^14 * q^2 - z^6 * q^4 + (z^4 - z) * q^5 - z^20 * q^8 - z^5 * q^9	\( q + \zeta_{44}^{14} q^{2} - \zeta_{44}^{6} q^{4} + (\zeta_{44}^{4} - \zeta_{44}) q^{5} - \zeta_{44}^{20} q^{8} - \zeta_{44}^{5} q^{9} + (\zeta_{44}^{18} - \zeta_{44}^{15}) q^{10} + (\zeta_{44}^{20} + \zeta_{44}^{15}) q^{13} + \zeta_{44}^{12} q^{16} + (\zeta_{44}^{14} - \zeta_{44}^{4}) q^{17} - \zeta_{44}^{19} q^{18} + ( - \zeta_{44}^{10} + \zeta_{44}^{7}) q^{20} + (\zeta_{44}^{8} + \cdots + \zeta_{44}^{2}) q^{25} + \cdots + \zeta_{44}^{3} q^{98} +O(q^{100}) \) q + z^14 * q^2 - z^6 * q^4 + (z^4 - z) * q^5 - z^20 * q^8 - z^5 * q^9 + (z^18 - z^15) * q^10 + (z^20 + z^15) * q^13 + z^12 * q^16 + (z^14 - z^4) * q^17 - z^19 * q^18 + (-z^10 + z^7) * q^20 + (z^8 - z^5 + z^2) * q^25 + (-z^12 - z^7) * q^26 + (z^17 + z^15) * q^29 - z^4 * q^32 + (-z^18 - z^6) * q^34 + z^11 * q^36 + (z^7 - 1) * q^37 + (z^21 + z^2) * q^40 + (z^9 + z^5) * q^41 + (-z^9 + z^6) * q^45 - z^11 * q^49 + (-z^19 + z^16 - 1) * q^50 + (-z^21 + z^4) * q^52 + (-z^3 - z^2) * q^53 + (-z^9 - z^7) * q^58 + (-z^13 - z^11) * q^61 - z^18 * q^64 + (-z^21 + z^19 - z^16 - z^2) * q^65 + (-z^20 + z^10) * q^68 - z^3 * q^72 + (z^21 - z^14) * q^74 + (z^16 - z^13) * q^80 + z^10 * q^81 + (z^19 - z) * q^82 + (z^18 - z^15 - z^8 + z^5) * q^85 + (z^20 - z^17) * q^89 + (z^20 + z) * q^90 + (z^16 - z^10) * q^97 + z^3 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} + 2 q^{10} - 2 q^{13} - 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{26} + 2 q^{32} - 4 q^{34} - 20 q^{37} + 2 q^{40} + 2 q^{45} - 22 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{64} + 4 q^{68} - 2 q^{74} - 2 q^{80} + 2 q^{81} + 4 q^{85} - 2 q^{89} - 2 q^{90} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8 + 2 * q^10 - 2 * q^13 - 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^26 + 2 * q^32 - 4 * q^34 - 20 * q^37 + 2 * q^40 + 2 * q^45 - 22 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^64 + 4 * q^68 - 2 * q^74 - 2 * q^80 + 2 * q^81 + 4 * q^85 - 2 * q^89 - 2 * q^90 - 4 * q^97

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	1412.1.r.a

Level	$1412$
Weight	$1$
Character orbit	1412.r
Analytic conductor	$0.705$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{44}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.704679797838\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{44})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{44}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{44} + \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 35.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
353.i	even	44	1	inner
1412.r	odd	44	1	inner

$p$	$F_p(T)$
$2$	\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{20} \) T^20
$5$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 66T^13 + 148T^12 + 164T^11 + 494T^10 + 658T^9 + 328T^8 - 176T^7 - 403T^6 - 454T^5 + 63T^4 + 66T^3 + 61T^2 + 12T + 1
$7$	\( T^{20} \) T^20
$11$	\( T^{20} \) T^20
$13$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 + 47T^14 + 132T^13 + 170T^12 + 76T^11 - 188T^10 - 530T^9 - 79T^8 - 66T^7 + 125T^6 + 382T^5 + 514T^4 + 330T^3 + 94T^2 + 12T + 1
$17$	\( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1)^2
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} + 22 T^{16} + \cdots + 121 \) T^20 + 22T^16 + 165T^12 + 99T^10 + 484T^8 - 968T^6 + 484T^4 + 605*T^2 + 121
$31$	\( T^{20} \) T^20
$37$	\( T^{20} + 20 T^{19} + \cdots + 1 \) T^20 + 20T^19 + 189T^18 + 1122T^17 + 4693T^16 + 14704T^15 + 35819T^14 + 69498T^13 + 109136T^12 + 140152T^11 + 148070T^10 + 128988T^9 + 92530T^8 + 54384T^7 + 25942T^6 + 9892T^5 + 2945T^4 + 660T^3 + 105T^2 + 10*T + 1
$41$	\( T^{20} - 4 T^{18} + \cdots + 1 \) T^20 - 4T^18 + 16T^16 - 64T^14 + 234T^12 - 474T^10 + 444T^8 - 203T^6 + 119T^4 + 19*T^2 + 1
$43$	\( T^{20} \) T^20
$47$	\( T^{20} \) T^20
$53$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$59$	\( T^{20} \) T^20
$61$	\( T^{20} + 11 T^{18} + \cdots + 121 \) T^20 + 11T^18 + 55T^16 + 165T^14 + 330T^12 + 462T^10 + 484T^8 + 1089T^4 - 605T^2 + 121
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 30T^15 - 52T^14 - 44T^13 + 93T^12 + 274T^11 + 505T^10 + 460T^9 - 90T^8 + 110T^7 + 400T^6 + 316T^5 + 338T^4 + 44T^3 + 17T^2 - 10T + 1
$97$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1)^2
show more
show less

\(n\)	\(707\)	\(709\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{44}^{5}\)