LMFDB - Newform orbit 500.1.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,1,Mod(11,500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 38]))

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

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

chi := DirichletCharacter("500.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 500 = 2^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	500.p (of order $50$, 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:	$0.249532506317$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

$q$-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_{50}^{13} q^{2} - \zeta_{50} q^{4} - \zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} - \zeta_{50}^{7} q^{9} +O(q^{10}) $ q - z^13 * q^2 - z * q^4 - z^9 * q^5 + z^14 * q^8 - z^7 * q^9	\( q - \zeta_{50}^{13} q^{2} - \zeta_{50} q^{4} - \zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} - \zeta_{50}^{7} q^{9} + \zeta_{50}^{22} q^{10} + ( - \zeta_{50}^{11} - \zeta_{50}^{3}) q^{13} + \zeta_{50}^{2} q^{16} + ( - \zeta_{50}^{17} + \zeta_{50}^{6}) q^{17} + \zeta_{50}^{20} q^{18} + \zeta_{50}^{10} q^{20} + \zeta_{50}^{18} q^{25} + (\zeta_{50}^{24} + \zeta_{50}^{16}) q^{26} + (\zeta_{50}^{8} + \zeta_{50}^{4}) q^{29} - \zeta_{50}^{15} q^{32} + ( - \zeta_{50}^{19} - \zeta_{50}^{5}) q^{34} + \zeta_{50}^{8} q^{36} + (\zeta_{50}^{24} - \zeta_{50}^{5}) q^{37} - \zeta_{50}^{23} q^{40} + ( - \zeta_{50}^{23} - \zeta_{50}^{21}) q^{41} + \zeta_{50}^{16} q^{45} + \zeta_{50}^{10} q^{49} + \zeta_{50}^{6} q^{50} + (\zeta_{50}^{12} + \zeta_{50}^{4}) q^{52} + ( - \zeta_{50}^{15} + \zeta_{50}^{2}) q^{53} + ( - \zeta_{50}^{21} - \zeta_{50}^{17}) q^{58} + ( - \zeta_{50}^{19} + \zeta_{50}^{12}) q^{61} - \zeta_{50}^{3} q^{64} + (\zeta_{50}^{20} + \zeta_{50}^{12}) q^{65} + (\zeta_{50}^{18} - \zeta_{50}^{7}) q^{68} - \zeta_{50}^{21} q^{72} + (\zeta_{50}^{22} - \zeta_{50}^{19}) q^{73} + (\zeta_{50}^{18} + \zeta_{50}^{12}) q^{74} - \zeta_{50}^{11} q^{80} + \zeta_{50}^{14} q^{81} + ( - \zeta_{50}^{11} - \zeta_{50}^{9}) q^{82} + ( - \zeta_{50}^{15} - \zeta_{50}) q^{85} + ( - \zeta_{50}^{21} + \zeta_{50}^{20}) q^{89} + \zeta_{50}^{4} q^{90} + ( - \zeta_{50}^{23} + \zeta_{50}^{14}) q^{97} - \zeta_{50}^{23} q^{98} +O(q^{100}) \) q - z^13 * q^2 - z * q^4 - z^9 * q^5 + z^14 * q^8 - z^7 * q^9 + z^22 * q^10 + (-z^11 - z^3) * q^13 + z^2 * q^16 + (-z^17 + z^6) * q^17 + z^20 * q^18 + z^10 * q^20 + z^18 * q^25 + (z^24 + z^16) * q^26 + (z^8 + z^4) * q^29 - z^15 * q^32 + (-z^19 - z^5) * q^34 + z^8 * q^36 + (z^24 - z^5) * q^37 - z^23 * q^40 + (-z^23 - z^21) * q^41 + z^16 * q^45 + z^10 * q^49 + z^6 * q^50 + (z^12 + z^4) * q^52 + (-z^15 + z^2) * q^53 + (-z^21 - z^17) * q^58 + (-z^19 + z^12) * q^61 - z^3 * q^64 + (z^20 + z^12) * q^65 + (z^18 - z^7) * q^68 - z^21 * q^72 + (z^22 - z^19) * q^73 + (z^18 + z^12) * q^74 - z^11 * q^80 + z^14 * q^81 + (-z^11 - z^9) * q^82 + (-z^15 - z) * q^85 + (-z^21 + z^20) * q^89 + z^4 * q^90 + (-z^23 + z^14) * q^97 - z^23 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 5 q^{18} - 5 q^{20} - 5 q^{32} - 5 q^{34} - 5 q^{37} - 5 q^{49} - 5 q^{53} - 5 q^{65} - 5 q^{85} - 5 q^{89}+O(q^{100}) $ 20 * q - 5 * q^18 - 5 * q^20 - 5 * q^32 - 5 * q^34 - 5 * q^37 - 5 * q^49 - 5 * q^53 - 5 * q^65 - 5 * q^85 - 5 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$.

$n$	$251$	$377$
$\chi(n)$	$-1$	$-\zeta_{50}$

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} $

11.1

−0.0627905	+	0.998027i
0.992115	−	0.125333i
0.187381	+	0.982287i
−0.0627905	−	0.998027i
0.637424	−	0.770513i
−0.876307	−	0.481754i
0.425779	−	0.904827i
−0.535827	+	0.844328i
−0.968583	+	0.248690i
0.425779	+	0.904827i
−0.876307	+	0.481754i
0.929776	−	0.368125i
0.929776	+	0.368125i
0.187381	−	0.982287i
0.992115	+	0.125333i
−0.968583	−	0.248690i
−0.535827	−	0.844328i
−0.728969	+	0.684547i
−0.728969	−	0.684547i
0.637424	+	0.770513i

0.728969

−

0.684547i

0.0627905

−

0.998027i

0.535827

−

0.844328i

−0.637424

−

0.770513i

−0.425779

0.904827i

−0.187381

−

0.982287i

31.1

0.0627905

0.998027i

−0.992115

0.125333i

−0.425779

0.904827i

−0.187381

−

0.982287i

−0.637424

0.770513i

−0.929776

−

0.368125i

71.1

−0.637424

0.770513i

−0.187381

−

0.982287i

−0.992115

0.125333i

0.876307

0.481754i

0.968583

0.248690i

0.535827

−

0.844328i

91.1

0.728969

0.684547i

0.0627905

0.998027i

0.535827

0.844328i

−0.637424

0.770513i

−0.425779

−

0.904827i

−0.187381

0.982287i

111.1

−0.425779

−

0.904827i

−0.637424

0.770513i

0.0627905

0.998027i

0.968583

0.248690i

−0.992115

−

0.125333i

0.876307

−

0.481754i

131.1

0.968583

0.248690i

0.876307

0.481754i

−0.187381

−

0.982287i

0.728969

0.684547i

−0.929776

−

0.368125i

0.0627905

−

0.998027i

171.1

0.535827

0.844328i

−0.425779

0.904827i

0.728969

−

0.684547i

−0.992115

0.125333i

0.0627905

0.998027i

0.968583

0.248690i

191.1

0.876307

−

0.481754i

0.535827

−

0.844328i

−0.929776

−

0.368125i

0.0627905

−

0.998027i

0.728969

−

0.684547i

−0.992115

0.125333i

211.1

−0.992115

0.125333i

0.968583

−

0.248690i

−0.637424

−

0.770513i

−0.929776

0.368125i

−0.187381

−

0.982287i

0.728969

0.684547i

231.1

0.535827

−

0.844328i

−0.425779

−

0.904827i

0.728969

0.684547i

−0.992115

−

0.125333i

0.0627905

−

0.998027i

0.968583

−

0.248690i

271.1

0.968583

−

0.248690i

0.876307

−

0.481754i

−0.187381

0.982287i

0.728969

−

0.684547i

−0.929776

0.368125i

0.0627905

0.998027i

291.1

−0.187381

−

0.982287i

−0.929776

0.368125i

0.968583

−

0.248690i

0.535827

0.844328i

0.876307

0.481754i

−0.425779

−

0.904827i

311.1

−0.187381

0.982287i

−0.929776

−

0.368125i

0.968583

0.248690i

0.535827

−

0.844328i

0.876307

−

0.481754i

−0.425779

0.904827i

331.1

−0.637424

−

0.770513i

−0.187381

0.982287i

−0.992115

−

0.125333i

0.876307

−

0.481754i

0.968583

−

0.248690i

0.535827

0.844328i

371.1

0.0627905

−

0.998027i

−0.992115

−

0.125333i

−0.425779

−

0.904827i

−0.187381

0.982287i

−0.637424

−

0.770513i

−0.929776

0.368125i

391.1

−0.992115

−

0.125333i

0.968583

0.248690i

−0.637424

0.770513i

−0.929776

−

0.368125i

−0.187381

0.982287i

0.728969

−

0.684547i

411.1

0.876307

0.481754i

0.535827

0.844328i

−0.929776

0.368125i

0.0627905

0.998027i

0.728969

0.684547i

−0.992115

−

0.125333i

431.1

−0.929776

0.368125i

0.728969

−

0.684547i

0.876307

−

0.481754i

−0.425779

0.904827i

0.535827

0.844328i

−0.637424

0.770513i

471.1

−0.929776

−

0.368125i

0.728969

0.684547i

0.876307

0.481754i

−0.425779

−

0.904827i

0.535827

−

0.844328i

−0.637424

−

0.770513i

491.1

−0.425779

0.904827i

−0.637424

−

0.770513i

0.0627905

−

0.998027i

0.968583

−

0.248690i

−0.992115

0.125333i

0.876307

0.481754i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
125.g	even	25	1	inner
500.p	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	500.1.p.a	✓	20
4.b	odd	2	1	CM	500.1.p.a	✓	20
5.b	even	2	1		2500.1.p.a		20
5.c	odd	4	2		2500.1.n.a		40
20.d	odd	2	1		2500.1.p.a		20
20.e	even	4	2		2500.1.n.a		40
125.g	even	25	1	inner	500.1.p.a	✓	20
125.h	even	50	1		2500.1.p.a		20
125.i	odd	100	2		2500.1.n.a		40
500.n	odd	50	1		2500.1.p.a		20
500.p	odd	50	1	inner	500.1.p.a	✓	20
500.r	even	100	2		2500.1.n.a		40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
500.1.p.a	✓	20	1.a	even	1	1	trivial
500.1.p.a	✓	20	4.b	odd	2	1	CM
500.1.p.a	✓	20	125.g	even	25	1	inner
500.1.p.a	✓	20	500.p	odd	50	1	inner
2500.1.n.a		40	5.c	odd	4	2
2500.1.n.a		40	20.e	even	4	2
2500.1.n.a		40	125.i	odd	100	2
2500.1.n.a		40	500.r	even	100	2
2500.1.p.a		20	5.b	even	2	1
2500.1.p.a		20	20.d	odd	2	1
2500.1.p.a		20	125.h	even	50	1
2500.1.p.a		20	500.n	odd	50	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(500, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} + T^{15} + \cdots + 1 $ T^20 + T^15 + T^10 + T^5 + 1
$3$	$ T^{20} $ T^20
$5$	$ T^{20} + T^{15} + \cdots + 1 $ T^20 + T^15 + T^10 + T^5 + 1
$7$	$ T^{20} $ T^20
$11$	$ T^{20} $ T^20
$13$	$ T^{20} - 20 T^{17} + \cdots + 1 $ T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$17$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 - 75T^13 + 20T^12 + 145T^11 - 246T^10 - 225T^9 + 675T^8 - 615T^7 + 385T^6 - 122T^5 - 25T^4 + 200T^3 + 85T^2 + 10*T + 1
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	$ T^{20} $ T^20
$37$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$41$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$43$	$ T^{20} $ T^20
$47$	$ T^{20} $ T^20
$53$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$59$	$ T^{20} $ T^20
$61$	$ T^{20} - 20 T^{17} + \cdots + 1 $ T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 - 75T^13 + 20T^12 + 145T^11 - 246T^10 - 225T^9 + 675T^8 - 615T^7 + 385T^6 - 122T^5 - 25T^4 + 200T^3 + 85T^2 + 10*T + 1
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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:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	500.p (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{50}^{13} q^{2} - \zeta_{50} q^{4} - \zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} - \zeta_{50}^{7} q^{9} +O(q^{10}) \) q - z^13 * q^2 - z * q^4 - z^9 * q^5 + z^14 * q^8 - z^7 * q^9	\( q - \zeta_{50}^{13} q^{2} - \zeta_{50} q^{4} - \zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} - \zeta_{50}^{7} q^{9} + \zeta_{50}^{22} q^{10} + ( - \zeta_{50}^{11} - \zeta_{50}^{3}) q^{13} + \zeta_{50}^{2} q^{16} + ( - \zeta_{50}^{17} + \zeta_{50}^{6}) q^{17} + \zeta_{50}^{20} q^{18} + \zeta_{50}^{10} q^{20} + \zeta_{50}^{18} q^{25} + (\zeta_{50}^{24} + \zeta_{50}^{16}) q^{26} + (\zeta_{50}^{8} + \zeta_{50}^{4}) q^{29} - \zeta_{50}^{15} q^{32} + ( - \zeta_{50}^{19} - \zeta_{50}^{5}) q^{34} + \zeta_{50}^{8} q^{36} + (\zeta_{50}^{24} - \zeta_{50}^{5}) q^{37} - \zeta_{50}^{23} q^{40} + ( - \zeta_{50}^{23} - \zeta_{50}^{21}) q^{41} + \zeta_{50}^{16} q^{45} + \zeta_{50}^{10} q^{49} + \zeta_{50}^{6} q^{50} + (\zeta_{50}^{12} + \zeta_{50}^{4}) q^{52} + ( - \zeta_{50}^{15} + \zeta_{50}^{2}) q^{53} + ( - \zeta_{50}^{21} - \zeta_{50}^{17}) q^{58} + ( - \zeta_{50}^{19} + \zeta_{50}^{12}) q^{61} - \zeta_{50}^{3} q^{64} + (\zeta_{50}^{20} + \zeta_{50}^{12}) q^{65} + (\zeta_{50}^{18} - \zeta_{50}^{7}) q^{68} - \zeta_{50}^{21} q^{72} + (\zeta_{50}^{22} - \zeta_{50}^{19}) q^{73} + (\zeta_{50}^{18} + \zeta_{50}^{12}) q^{74} - \zeta_{50}^{11} q^{80} + \zeta_{50}^{14} q^{81} + ( - \zeta_{50}^{11} - \zeta_{50}^{9}) q^{82} + ( - \zeta_{50}^{15} - \zeta_{50}) q^{85} + ( - \zeta_{50}^{21} + \zeta_{50}^{20}) q^{89} + \zeta_{50}^{4} q^{90} + ( - \zeta_{50}^{23} + \zeta_{50}^{14}) q^{97} - \zeta_{50}^{23} q^{98} +O(q^{100}) \) q - z^13 * q^2 - z * q^4 - z^9 * q^5 + z^14 * q^8 - z^7 * q^9 + z^22 * q^10 + (-z^11 - z^3) * q^13 + z^2 * q^16 + (-z^17 + z^6) * q^17 + z^20 * q^18 + z^10 * q^20 + z^18 * q^25 + (z^24 + z^16) * q^26 + (z^8 + z^4) * q^29 - z^15 * q^32 + (-z^19 - z^5) * q^34 + z^8 * q^36 + (z^24 - z^5) * q^37 - z^23 * q^40 + (-z^23 - z^21) * q^41 + z^16 * q^45 + z^10 * q^49 + z^6 * q^50 + (z^12 + z^4) * q^52 + (-z^15 + z^2) * q^53 + (-z^21 - z^17) * q^58 + (-z^19 + z^12) * q^61 - z^3 * q^64 + (z^20 + z^12) * q^65 + (z^18 - z^7) * q^68 - z^21 * q^72 + (z^22 - z^19) * q^73 + (z^18 + z^12) * q^74 - z^11 * q^80 + z^14 * q^81 + (-z^11 - z^9) * q^82 + (-z^15 - z) * q^85 + (-z^21 + z^20) * q^89 + z^4 * q^90 + (-z^23 + z^14) * q^97 - z^23 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 5 q^{18} - 5 q^{20} - 5 q^{32} - 5 q^{34} - 5 q^{37} - 5 q^{49} - 5 q^{53} - 5 q^{65} - 5 q^{85} - 5 q^{89}+O(q^{100}) \) 20 * q - 5 * q^18 - 5 * q^20 - 5 * q^32 - 5 * q^34 - 5 * q^37 - 5 * q^49 - 5 * q^53 - 5 * q^65 - 5 * q^85 - 5 * q^89

Introduction

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Modular forms

Varieties

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Database

Properties

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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	500.1.p.a

Level	$500$
Weight	$1$
Character orbit	500.p
Analytic conductor	$0.250$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{25}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.249532506317\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{50})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{25}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
125.g	even	25	1	inner
500.p	odd	50	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} + T^{15} + \cdots + 1 \) T^20 + T^15 + T^10 + T^5 + 1
$3$	\( T^{20} \) T^20
$5$	\( T^{20} + T^{15} + \cdots + 1 \) T^20 + T^15 + T^10 + T^5 + 1
$7$	\( T^{20} \) T^20
$11$	\( T^{20} \) T^20
$13$	\( T^{20} - 20 T^{17} + \cdots + 1 \) T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$17$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 - 75T^13 + 20T^12 + 145T^11 - 246T^10 - 225T^9 + 675T^8 - 615T^7 + 385T^6 - 122T^5 - 25T^4 + 200T^3 + 85T^2 + 10*T + 1
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	\( T^{20} \) T^20
$37$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$41$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$43$	\( T^{20} \) T^20
$47$	\( T^{20} \) T^20
$53$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$59$	\( T^{20} \) T^20
$61$	\( T^{20} - 20 T^{17} + \cdots + 1 \) T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 - 75T^13 + 20T^12 + 145T^11 - 246T^10 - 225T^9 + 675T^8 - 615T^7 + 385T^6 - 122T^5 - 25T^4 + 200T^3 + 85T^2 + 10*T + 1
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\(n\)	\(251\)	\(377\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{50}\)