LMFDB - Newform orbit 2020.1.bt.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2020,1,Mod(279,2020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2020, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 25, 11]))

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

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

chi := DirichletCharacter("2020.279");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2020 = 2^{2} \cdot 5 \cdot 101 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2020.bt (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:	$1.00811132552$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$2$ over $\Q(\zeta_{50})$
Coefficient field:	$\Q(\zeta_{100})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{30} + x^{20} - x^{10} + 1 $ x^40 - x^30 + x^20 - x^10 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

$q$-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_{100} q^{2} + ( - \zeta_{100}^{23} - \zeta_{100}^{15}) q^{3} + \zeta_{100}^{2} q^{4} - \zeta_{100}^{24} q^{5} + (\zeta_{100}^{24} + \zeta_{100}^{16}) q^{6} + ( - \zeta_{100}^{35} + \zeta_{100}^{33}) q^{7} - \zeta_{100}^{3} q^{8} + (\zeta_{100}^{46} + \cdots + \zeta_{100}^{30}) q^{9} +O(q^{10}) $ q - z * q^2 + (-z^23 - z^15) * q^3 + z^2 * q^4 - z^24 * q^5 + (z^24 + z^16) * q^6 + (-z^35 + z^33) * q^7 - z^3 * q^8 + (z^46 + z^38 + z^30) * q^9	$ q - \zeta_{100} q^{2} + ( - \zeta_{100}^{23} - \zeta_{100}^{15}) q^{3} + \zeta_{100}^{2} q^{4} - \zeta_{100}^{24} q^{5} + (\zeta_{100}^{24} + \zeta_{100}^{16}) q^{6} + ( - \zeta_{100}^{35} + \zeta_{100}^{33}) q^{7} - \zeta_{100}^{3} q^{8} + (\zeta_{100}^{46} + \cdots + \zeta_{100}^{30}) q^{9} + \cdots + (\zeta_{100}^{21} + \cdots + \zeta_{100}^{17}) q^{98} +O(q^{100}) $ q - z * q^2 + (-z^23 - z^15) * q^3 + z^2 * q^4 - z^24 * q^5 + (z^24 + z^16) * q^6 + (-z^35 + z^33) * q^7 - z^3 * q^8 + (z^46 + z^38 + z^30) * q^9 + z^25 * q^10 + (-z^25 - z^17) * q^12 + (z^36 - z^34) * q^14 + (z^47 + z^39) * q^15 + z^4 * q^16 + (-z^47 - z^39 - z^31) * q^18 - z^26 * q^20 + (-z^48 - z^8 + z^6 - 1) * q^21 + (z^43 + z^29) * q^23 + (z^26 + z^18) * q^24 + z^48 * q^25 + (-z^45 + z^19 - z^11 - z^3) * q^27 + (-z^37 + z^35) * q^28 + (z^42 + z^40) * q^29 + (-z^48 - z^40) * q^30 - z^5 * q^32 + (-z^9 + z^7) * q^35 + (z^48 + z^40 + z^32) * q^36 + z^27 * q^40 + (z^36 - z^4) * q^41 + (z^49 + z^9 - z^7 + z) * q^42 + (z^23 - z^11) * q^43 + (z^20 + z^12 + z^4) * q^45 + (-z^44 - z^30) * q^46 + (z^37 - z^29) * q^47 + (-z^27 - z^19) * q^48 + (-z^20 + z^18 - z^16) * q^49 - z^49 * q^50 + (z^46 - z^20 - z^12 - z^4) * q^54 + (z^38 - z^36) * q^56 + (-z^43 - z^41) * q^58 + (z^49 + z^41) * q^60 + (-z^40 - z^14) * q^61 + (z^31 - z^29 + z^23 - z^21 + z^15 - z^13) * q^63 + z^6 * q^64 + (z^45 + z^17) * q^67 + (-z^44 + z^16 + z^8 + z^2) * q^69 + (z^10 - z^8) * q^70 + (-z^49 - z^41 - z^33) * q^72 + (z^21 + z^13) * q^75 - z^28 * q^80 + (-z^42 + z^34 + z^26 + z^18 - z^10) * q^81 + (-z^37 + z^5) * q^82 + (-z^45 - z^33) * q^83 + (-z^10 + z^8 - z^2 + 1) * q^84 + (-z^24 + z^12) * q^86 + (z^15 + z^13 + z^7 + z^5) * q^87 + (-z^36 - z^6) * q^89 + (-z^21 - z^13 - z^5) * q^90 + (z^45 + z^31) * q^92 + (-z^38 + z^30) * q^94 + (z^28 + z^20) * q^96 + (z^21 - z^19 + z^17) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 10 q^{9}+O(q^{10}) $ 40 * q + 10 * q^9	$ 40 q + 10 q^{9} - 40 q^{21} - 10 q^{29} + 10 q^{30} - 10 q^{36} - 10 q^{45} - 10 q^{46} + 10 q^{49} + 10 q^{54} + 10 q^{61} + 10 q^{70} - 10 q^{81} + 30 q^{84} + 10 q^{94} - 10 q^{96}+O(q^{100}) $ 40 * q + 10 * q^9 - 40 * q^21 - 10 * q^29 + 10 * q^30 - 10 * q^36 - 10 * q^45 - 10 * q^46 + 10 * q^49 + 10 * q^54 + 10 * q^61 + 10 * q^70 - 10 * q^81 + 30 * q^84 + 10 * q^94 - 10 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2020\mathbb{Z}\right)^\times$.

$n$	$1011$	$1617$	$1921$
$\chi(n)$	$-1$	$-1$	$\zeta_{100}^{2}$

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} $

279.1

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

−0.770513

−

0.637424i

1.57007

0.996398i

0.187381

0.982287i

0.637424

0.770513i

−0.574633

−

1.76854i

−1.27233

0.0800484i

0.481754

−

0.876307i

1.04654

2.22401i

−

1.00000i

279.2

0.770513

0.637424i

−1.57007

−

0.996398i

0.187381

0.982287i

0.637424

0.770513i

−0.574633

−

1.76854i

1.27233

−

0.0800484i

−0.481754

0.876307i

1.04654

2.22401i

1.00000i

379.1

−0.998027

0.0627905i

−0.713118

1.80113i

0.992115

−

0.125333i

−0.0627905

0.998027i

0.598617

−

1.84235i

0.106032

−

0.0672897i

−0.982287

0.187381i

−2.00657

−

1.88429i

−

1.00000i

379.2

0.998027

−

0.0627905i

0.713118

−

1.80113i

0.992115

−

0.125333i

−0.0627905

0.998027i

0.598617

−

1.84235i

−0.106032

0.0672897i

0.982287

−

0.187381i

−2.00657

−

1.88429i

1.00000i

399.1

−0.684547

0.728969i

−1.94908

−

0.371808i

−0.0627905

−

0.998027i

−0.728969

0.684547i

1.60528

−

1.16630i

0.702367

−

1.27760i

0.770513

0.637424i

2.73091

1.08124i

−

1.00000i

399.2

0.684547

−

0.728969i

1.94908

0.371808i

−0.0627905

−

0.998027i

−0.728969

0.684547i

1.60528

−

1.16630i

−0.702367

1.27760i

−0.770513

−

0.637424i

2.73091

1.08124i

1.00000i

619.1

−0.481754

0.876307i

0.106729

−

0.844844i

−0.535827

−

0.844328i

−0.876307

0.481754i

0.688925

0.500534i

−1.72157

0.328407i

0.998027

−

0.0627905i

0.266213

0.0683519i

−

1.00000i

619.2

0.481754

−

0.876307i

−0.106729

0.844844i

−0.535827

−

0.844328i

−0.876307

0.481754i

0.688925

0.500534i

1.72157

−

0.328407i

−0.998027

0.0627905i

0.266213

0.0683519i

1.00000i

639.1

−0.844328

0.535827i

−0.317042

1.23480i

0.425779

−

0.904827i

−0.535827

0.844328i

−0.393950

−

1.21245i

0.394502

0.996398i

0.125333

0.992115i

−0.547900

−

0.301210i

−

1.00000i

639.2

0.844328

−

0.535827i

0.317042

−

1.23480i

0.425779

−

0.904827i

−0.535827

0.844328i

−0.393950

−

1.21245i

−0.394502

−

0.996398i

−0.125333

−

0.992115i

−0.547900

−

0.301210i

1.00000i

939.1

−0.982287

0.187381i

1.31918

−

0.620759i

0.929776

−

0.368125i

0.187381

−

0.982287i

−1.17950

0.856954i

0.0469702

0.371808i

−0.844328

0.535827i

0.717472

−

0.867275i

1.00000i

939.2

0.982287

−

0.187381i

−1.31918

0.620759i

0.929776

−

0.368125i

0.187381

−

0.982287i

−1.17950

0.856954i

−0.0469702

−

0.371808i

0.844328

−

0.535827i

0.717472

−

0.867275i

−

1.00000i

979.1

−0.481754

−

0.876307i

0.106729

0.844844i

−0.535827

0.844328i

−0.876307

−

0.481754i

0.688925

−

0.500534i

−1.72157

−

0.328407i

0.998027

0.0627905i

0.266213

−

0.0683519i

1.00000i

979.2

0.481754

0.876307i

−0.106729

−

0.844844i

−0.535827

0.844328i

−0.876307

−

0.481754i

0.688925

−

0.500534i

1.72157

0.328407i

−0.998027

−

0.0627905i

0.266213

−

0.0683519i

−

1.00000i

1019.1

−0.368125

−

0.929776i

0.0967619

0.0800484i

−0.728969

0.684547i

0.929776

0.368125i

0.0388067

−

0.119435i

0.462452

1.80113i

0.904827

0.425779i

−0.184426

−

0.966796i

−

1.00000i

1019.2

0.368125

0.929776i

−0.0967619

−

0.0800484i

−0.728969

0.684547i

0.929776

0.368125i

0.0388067

−

0.119435i

−0.462452

−

1.80113i

−0.904827

−

0.425779i

−0.184426

−

0.966796i

1.00000i

1059.1

−0.844328

−

0.535827i

−0.317042

−

1.23480i

0.425779

0.904827i

−0.535827

−

0.844328i

−0.393950

1.21245i

0.394502

−

0.996398i

0.125333

−

0.992115i

−0.547900

0.301210i

1.00000i

1059.2

0.844328

0.535827i

0.317042

1.23480i

0.425779

0.904827i

−0.535827

−

0.844328i

−0.393950

1.21245i

−0.394502

0.996398i

−0.125333

0.992115i

−0.547900

0.301210i

−

1.00000i

1259.1

−0.248690

0.968583i

−1.06954

−

0.0672897i

−0.876307

−

0.481754i

−0.968583

0.248690i

0.331159

−

1.01920i

1.49261

1.23480i

0.684547

−

0.728969i

0.147271

0.0186046i

−

1.00000i

1259.2

0.248690

−

0.968583i

1.06954

0.0672897i

−0.876307

−

0.481754i

−0.968583

0.248690i

0.331159

−

1.01920i

−1.49261

−

1.23480i

−0.684547

0.728969i

0.147271

0.0186046i

1.00000i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
4.b	odd	2	1	inner
5.b	even	2	1	inner
101.h	even	50	1	inner
404.n	odd	50	1	inner
505.w	even	50	1	inner
2020.bt	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2020.1.bt.c	✓	40
4.b	odd	2	1	inner	2020.1.bt.c	✓	40
5.b	even	2	1	inner	2020.1.bt.c	✓	40
20.d	odd	2	1	CM	2020.1.bt.c	✓	40
101.h	even	50	1	inner	2020.1.bt.c	✓	40
404.n	odd	50	1	inner	2020.1.bt.c	✓	40
505.w	even	50	1	inner	2020.1.bt.c	✓	40
2020.bt	odd	50	1	inner	2020.1.bt.c	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2020.1.bt.c	✓	40	1.a	even	1	1	trivial
2020.1.bt.c	✓	40	4.b	odd	2	1	inner
2020.1.bt.c	✓	40	5.b	even	2	1	inner
2020.1.bt.c	✓	40	20.d	odd	2	1	CM
2020.1.bt.c	✓	40	101.h	even	50	1	inner
2020.1.bt.c	✓	40	404.n	odd	50	1	inner
2020.1.bt.c	✓	40	505.w	even	50	1	inner
2020.1.bt.c	✓	40	2020.bt	odd	50	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(2020, [\chi])$:

$ T_{3}^{40} - 5 T_{3}^{38} + 15 T_{3}^{36} - 45 T_{3}^{34} + 190 T_{3}^{32} - 874 T_{3}^{30} + 3615 T_{3}^{28} + \cdots + 1 $

$ T_{13} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{40} - T^{30} + \cdots + 1 $ T^40 - T^30 + T^20 - T^10 + 1
$3$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 - 45T^34 + 190T^32 - 874T^30 + 3615T^28 - 6720T^26 + 17960T^24 + 12780T^22 - 51369T^20 + 42185T^18 + 290645T^16 - 285085T^14 - 411505T^12 + 171246T^10 + 230205T^8 + 33910T^6 + 3870T^4 - 15*T^2 + 1
$5$	$ (T^{20} - T^{15} + T^{10} + \cdots + 1)^{2} $ (T^20 - T^15 + T^10 - T^5 + 1)^2
$7$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 + 5T^34 - 60T^32 - 124T^30 + 2340T^28 - 9245T^26 + 10985T^24 - 1545T^22 + 118381T^20 - 152165T^18 - 144655T^16 - 131060T^14 + 286545T^12 + 121246T^10 + 219730T^8 + 53335T^6 + 3320T^4 - 40*T^2 + 1
$11$	$ T^{40} $ T^40
$13$	$ T^{40} $ T^40
$17$	$ T^{40} $ T^40
$19$	$ T^{40} $ T^40
$23$	$ T^{40} - 10 T^{36} + \cdots + 25 $ T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 6250T^26 - 15625T^24 + 47300T^22 + 348135T^20 + 800625T^18 + 927075T^16 + 680375T^14 + 357000T^12 + 78775T^10 + 70000T^8 - 21875T^6 + 8125T^4 - 750T^2 + 25
$29$	$ (T^{20} + 5 T^{19} + \cdots + 5)^{2} $ (T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 120T^15 + 200T^14 + 325T^13 + 475T^12 + 525T^11 + 135T^10 - 525T^9 - 925T^8 - 650T^7 + 75T^6 + 240T^5 + 400T^4 + 375T^3 + 200T^2 + 50*T + 5)^2
$31$	$ T^{40} $ T^40
$37$	$ T^{40} $ T^40
$41$	$ (T^{20} - 5 T^{18} + \cdots + 5)^{2} $ (T^20 - 5T^18 + 20T^16 - 75T^14 + 275T^12 + 100T^11 - 370T^10 - 125T^9 + 500T^8 + 125T^7 - 650T^6 + 750T^4 - 650T^3 + 250T^2 - 50T + 5)^2
$43$	$ T^{40} - 10 T^{36} + \cdots + 25 $ T^40 - 10T^36 - 25T^34 + 125T^32 + 500T^30 - 375T^28 + 3225T^26 - 11000T^24 - 35375T^22 + 77510T^20 - 46125T^18 + 390575T^16 - 563375T^14 + 435875T^12 - 43100T^10 - 14375T^8 + 25000T^6 + 6875T^4 + 250T^2 + 25
$47$	$ T^{40} - 10 T^{36} + \cdots + 25 $ T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 6250T^26 - 15625T^24 + 47300T^22 + 348135T^20 + 800625T^18 + 927075T^16 + 680375T^14 + 357000T^12 + 78775T^10 + 70000T^8 - 21875T^6 + 8125T^4 - 750T^2 + 25
$53$	$ T^{40} $ T^40
$59$	$ T^{40} $ T^40
$61$	$ (T^{20} - 5 T^{19} + \cdots + 5)^{2} $ (T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 275T^13 + 475T^12 - 775T^11 + 1010T^10 - 675T^9 - 75T^8 + 900T^7 - 1025T^6 + 260T^5 + 350T^4 - 200T^3 + 25T^2 + 5)^2
$67$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 - 45T^34 + 190T^32 - 874T^30 + 315T^28 + 6205T^26 - 13715T^24 - 13645T^22 + 89506T^20 + 98385T^18 + 32295T^16 - 25010T^14 + 236695T^12 - 402754T^10 + 189980T^8 + 910T^6 + 3945T^4 + 110*T^2 + 1
$71$	$ T^{40} $ T^40
$73$	$ T^{40} $ T^40
$79$	$ T^{40} $ T^40
$83$	$ T^{40} - 5 T^{38} + \cdots + 1 $ T^40 - 5T^38 + 15T^36 - 45T^34 - 160T^32 + 876T^30 - 1635T^28 + 1880T^26 + 23435T^24 - 67445T^22 + 37256T^20 + 54735T^18 - 64380T^16 - 9985T^14 + 367645T^12 + 274996T^10 + 157755T^8 - 26390T^6 + 3370T^4 + 85*T^2 + 1
$89$	$ (T^{20} - 25 T^{15} + \cdots + 3125)^{2} $ (T^20 - 25T^15 + 500T^10 + 2500*T^5 + 3125)^2
$97$	$ T^{40} $ T^40
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2020 = 2^{2} \cdot 5 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2020.bt (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{100} q^{2} + ( - \zeta_{100}^{23} - \zeta_{100}^{15}) q^{3} + \zeta_{100}^{2} q^{4} - \zeta_{100}^{24} q^{5} + (\zeta_{100}^{24} + \zeta_{100}^{16}) q^{6} + ( - \zeta_{100}^{35} + \zeta_{100}^{33}) q^{7} - \zeta_{100}^{3} q^{8} + (\zeta_{100}^{46} + \cdots + \zeta_{100}^{30}) q^{9} +O(q^{10}) \) q - z * q^2 + (-z^23 - z^15) * q^3 + z^2 * q^4 - z^24 * q^5 + (z^24 + z^16) * q^6 + (-z^35 + z^33) * q^7 - z^3 * q^8 + (z^46 + z^38 + z^30) * q^9	\( q - \zeta_{100} q^{2} + ( - \zeta_{100}^{23} - \zeta_{100}^{15}) q^{3} + \zeta_{100}^{2} q^{4} - \zeta_{100}^{24} q^{5} + (\zeta_{100}^{24} + \zeta_{100}^{16}) q^{6} + ( - \zeta_{100}^{35} + \zeta_{100}^{33}) q^{7} - \zeta_{100}^{3} q^{8} + (\zeta_{100}^{46} + \cdots + \zeta_{100}^{30}) q^{9} + \cdots + (\zeta_{100}^{21} + \cdots + \zeta_{100}^{17}) q^{98} +O(q^{100}) \) q - z * q^2 + (-z^23 - z^15) * q^3 + z^2 * q^4 - z^24 * q^5 + (z^24 + z^16) * q^6 + (-z^35 + z^33) * q^7 - z^3 * q^8 + (z^46 + z^38 + z^30) * q^9 + z^25 * q^10 + (-z^25 - z^17) * q^12 + (z^36 - z^34) * q^14 + (z^47 + z^39) * q^15 + z^4 * q^16 + (-z^47 - z^39 - z^31) * q^18 - z^26 * q^20 + (-z^48 - z^8 + z^6 - 1) * q^21 + (z^43 + z^29) * q^23 + (z^26 + z^18) * q^24 + z^48 * q^25 + (-z^45 + z^19 - z^11 - z^3) * q^27 + (-z^37 + z^35) * q^28 + (z^42 + z^40) * q^29 + (-z^48 - z^40) * q^30 - z^5 * q^32 + (-z^9 + z^7) * q^35 + (z^48 + z^40 + z^32) * q^36 + z^27 * q^40 + (z^36 - z^4) * q^41 + (z^49 + z^9 - z^7 + z) * q^42 + (z^23 - z^11) * q^43 + (z^20 + z^12 + z^4) * q^45 + (-z^44 - z^30) * q^46 + (z^37 - z^29) * q^47 + (-z^27 - z^19) * q^48 + (-z^20 + z^18 - z^16) * q^49 - z^49 * q^50 + (z^46 - z^20 - z^12 - z^4) * q^54 + (z^38 - z^36) * q^56 + (-z^43 - z^41) * q^58 + (z^49 + z^41) * q^60 + (-z^40 - z^14) * q^61 + (z^31 - z^29 + z^23 - z^21 + z^15 - z^13) * q^63 + z^6 * q^64 + (z^45 + z^17) * q^67 + (-z^44 + z^16 + z^8 + z^2) * q^69 + (z^10 - z^8) * q^70 + (-z^49 - z^41 - z^33) * q^72 + (z^21 + z^13) * q^75 - z^28 * q^80 + (-z^42 + z^34 + z^26 + z^18 - z^10) * q^81 + (-z^37 + z^5) * q^82 + (-z^45 - z^33) * q^83 + (-z^10 + z^8 - z^2 + 1) * q^84 + (-z^24 + z^12) * q^86 + (z^15 + z^13 + z^7 + z^5) * q^87 + (-z^36 - z^6) * q^89 + (-z^21 - z^13 - z^5) * q^90 + (z^45 + z^31) * q^92 + (-z^38 + z^30) * q^94 + (z^28 + z^20) * q^96 + (z^21 - z^19 + z^17) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 10 q^{9}+O(q^{10}) \) 40 * q + 10 * q^9	\( 40 q + 10 q^{9} - 40 q^{21} - 10 q^{29} + 10 q^{30} - 10 q^{36} - 10 q^{45} - 10 q^{46} + 10 q^{49} + 10 q^{54} + 10 q^{61} + 10 q^{70} - 10 q^{81} + 30 q^{84} + 10 q^{94} - 10 q^{96}+O(q^{100}) \) 40 * q + 10 * q^9 - 40 * q^21 - 10 * q^29 + 10 * q^30 - 10 * q^36 - 10 * q^45 - 10 * q^46 + 10 * q^49 + 10 * q^54 + 10 * q^61 + 10 * q^70 - 10 * q^81 + 30 * q^84 + 10 * q^94 - 10 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2020.1.bt.c

Level	$2020$
Weight	$1$
Character orbit	2020.bt
Analytic conductor	$1.008$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{50}$
CM discriminant	-20
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.00811132552\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(2\) over \(\Q(\zeta_{50})\)
Coefficient field:	\(\Q(\zeta_{100})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{30} + x^{20} - x^{10} + 1 \) x^40 - x^30 + x^20 - x^10 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{50}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

\(n\)	\(1011\)	\(1617\)	\(1921\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{100}^{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 279.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{40} - T^{30} + \cdots + 1 \) T^40 - T^30 + T^20 - T^10 + 1
$3$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 - 45T^34 + 190T^32 - 874T^30 + 3615T^28 - 6720T^26 + 17960T^24 + 12780T^22 - 51369T^20 + 42185T^18 + 290645T^16 - 285085T^14 - 411505T^12 + 171246T^10 + 230205T^8 + 33910T^6 + 3870T^4 - 15*T^2 + 1
$5$	\( (T^{20} - T^{15} + T^{10} + \cdots + 1)^{2} \) (T^20 - T^15 + T^10 - T^5 + 1)^2
$7$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 + 5T^34 - 60T^32 - 124T^30 + 2340T^28 - 9245T^26 + 10985T^24 - 1545T^22 + 118381T^20 - 152165T^18 - 144655T^16 - 131060T^14 + 286545T^12 + 121246T^10 + 219730T^8 + 53335T^6 + 3320T^4 - 40*T^2 + 1
$11$	\( T^{40} \) T^40
$13$	\( T^{40} \) T^40
$17$	\( T^{40} \) T^40
$19$	\( T^{40} \) T^40
$23$	\( T^{40} - 10 T^{36} + \cdots + 25 \) T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 6250T^26 - 15625T^24 + 47300T^22 + 348135T^20 + 800625T^18 + 927075T^16 + 680375T^14 + 357000T^12 + 78775T^10 + 70000T^8 - 21875T^6 + 8125T^4 - 750T^2 + 25
$29$	\( (T^{20} + 5 T^{19} + \cdots + 5)^{2} \) (T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 120T^15 + 200T^14 + 325T^13 + 475T^12 + 525T^11 + 135T^10 - 525T^9 - 925T^8 - 650T^7 + 75T^6 + 240T^5 + 400T^4 + 375T^3 + 200T^2 + 50*T + 5)^2
$31$	\( T^{40} \) T^40
$37$	\( T^{40} \) T^40
$41$	\( (T^{20} - 5 T^{18} + \cdots + 5)^{2} \) (T^20 - 5T^18 + 20T^16 - 75T^14 + 275T^12 + 100T^11 - 370T^10 - 125T^9 + 500T^8 + 125T^7 - 650T^6 + 750T^4 - 650T^3 + 250T^2 - 50T + 5)^2
$43$	\( T^{40} - 10 T^{36} + \cdots + 25 \) T^40 - 10T^36 - 25T^34 + 125T^32 + 500T^30 - 375T^28 + 3225T^26 - 11000T^24 - 35375T^22 + 77510T^20 - 46125T^18 + 390575T^16 - 563375T^14 + 435875T^12 - 43100T^10 - 14375T^8 + 25000T^6 + 6875T^4 + 250T^2 + 25
$47$	\( T^{40} - 10 T^{36} + \cdots + 25 \) T^40 - 10T^36 - 25T^34 + 75T^32 + 500T^30 + 125T^28 - 6250T^26 - 15625T^24 + 47300T^22 + 348135T^20 + 800625T^18 + 927075T^16 + 680375T^14 + 357000T^12 + 78775T^10 + 70000T^8 - 21875T^6 + 8125T^4 - 750T^2 + 25
$53$	\( T^{40} \) T^40
$59$	\( T^{40} \) T^40
$61$	\( (T^{20} - 5 T^{19} + \cdots + 5)^{2} \) (T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 275T^13 + 475T^12 - 775T^11 + 1010T^10 - 675T^9 - 75T^8 + 900T^7 - 1025T^6 + 260T^5 + 350T^4 - 200T^3 + 25T^2 + 5)^2
$67$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 - 45T^34 + 190T^32 - 874T^30 + 315T^28 + 6205T^26 - 13715T^24 - 13645T^22 + 89506T^20 + 98385T^18 + 32295T^16 - 25010T^14 + 236695T^12 - 402754T^10 + 189980T^8 + 910T^6 + 3945T^4 + 110*T^2 + 1
$71$	\( T^{40} \) T^40
$73$	\( T^{40} \) T^40
$79$	\( T^{40} \) T^40
$83$	\( T^{40} - 5 T^{38} + \cdots + 1 \) T^40 - 5T^38 + 15T^36 - 45T^34 - 160T^32 + 876T^30 - 1635T^28 + 1880T^26 + 23435T^24 - 67445T^22 + 37256T^20 + 54735T^18 - 64380T^16 - 9985T^14 + 367645T^12 + 274996T^10 + 157755T^8 - 26390T^6 + 3370T^4 + 85*T^2 + 1
$89$	\( (T^{20} - 25 T^{15} + \cdots + 3125)^{2} \) (T^20 - 25T^15 + 500T^10 + 2500*T^5 + 3125)^2
$97$	\( T^{40} \) T^40
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