LMFDB - Newform orbit 10000.2.a.bj

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

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:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9}+O(q^{10}) $ q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9	$ q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9} - 2 q^{11} - \beta_{5} q^{13} + ( - 2 \beta_{7} + \beta_{6} + \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{19} + (\beta_{3} - \beta_{2} + 1) q^{21} + (\beta_{6} + \beta_{5}) q^{23} + ( - 2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{27}+ \cdots + 2 \beta_{3} q^{99}+O(q^{100}) $ q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9 - 2 * q^11 - b5 * q^13 + (-2b7 + b6 + b1) q^17 + (-2b4 + b2 - 2) q^19 + (b3 - b2 + 1) * q^21 + (b6 + b5) * q^23 + (-2b7 + b6 - b5 + 2b1) * q^27 + (3b4 + b3 - 2b2 + 4) * q^29 + (b3 + 2b2 - 1) q^31 + (2b6 - 2b5) * q^33 + (4b7 + b5 - b1) q^37 + (2b4 + b3 - b2 - 1) q^39 + (-b4 - 2b3 + b2 + 2) q^41 + (3b7 + 2b6 - b5 - b1) * q^43 + (-b7 + b6 - 2b5 + 3b1) * q^47 + (2b3 - b2 - 1) q^49 + (-2b3 + 2b2) * q^51 + (-3b7 + 2b6 - 2b5 - 3b1) * q^53 + (b7 + 4b6 - b5 - 3b1) * q^57 + (2b4 - 2b3 - b2 - 4) * q^59 + (b4 + 3b3 + b2 + 3) q^61 + (2b6 + 2b1) * q^63 + (-2b7 + 2b6 + 4b1) q^67 + (-4b4 - b3 + 2b2 - 1) * q^69 + (-2b4 + b3 - 3b2 - 7) * q^71 + (3b7 - 2b6 + 2b5 - b1) q^73 + (-2b7 + 2b6 + 2b1) q^77 + (4b4 + 4b3 - 3b2 + 2) q^79 + (4b4 + 2b3 - 1) * q^81 + (-4b7 + 5b6 - b5 + 4b1) q^83 + (b7 - 6b6 + b5 + 3b1) * q^87 + (-7b4 - b3 + b2) q^89 + (b2 + 2) * q^91 + (7b6 - 5b5 - 4b1) q^93 + (-5b7 - 4b6 - 4b5 + 4b1) * q^97 + 2b3 q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11*x^6 + 36*x^4 - 31*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{4} - 5\nu^{2} + 1 $ v^4 - 5*v^2 + 1
$\beta_{4}$	$=$	$ ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4 $ (v^6 - 8v^4 + 16v^2 - 7) / 4
$\beta_{5}$	$=$	$ ( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4 $ (v^7 - 8v^5 + 16v^3 - 7*v) / 4
$\beta_{6}$	$=$	$ ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4 $ (-v^7 + 12v^5 - 40v^3 + 27*v) / 4
$\beta_{7}$	$=$	$ ( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4 $ (v^7 - 12v^5 + 44v^3 - 43*v) / 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{7} + \beta_{6} + 4\beta_1 $ b7 + b6 + 4*b1
$\nu^{4}$	$=$	$ \beta_{3} + 5\beta_{2} + 14 $ b3 + 5*b2 + 14
$\nu^{5}$	$=$	$ 6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1 $ 6b7 + 7b6 + b5 + 19*b1
$\nu^{6}$	$=$	$ 4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71 $ 4b4 + 8b3 + 24*b2 + 71
$\nu^{7}$	$=$	$ 32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1 $ 32b7 + 40b6 + 12b5 + 95b1

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

−1.13370
2.08529
0.183172
2.30927
−2.30927
−0.183172
−2.08529
1.13370

−2.60278

0.407162

3.77447

1.2

−2.19849

0.992398

1.83337

1.3

−1.47195

−3.26086

−0.833366

1.4

−0.474903

−3.03582

−2.77447

1.5

0.474903

3.03582

−2.77447

1.6

1.47195

3.26086

−0.833366

1.7

2.19849

−0.992398

1.83337

1.8

2.60278

−0.407162

3.77447

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$5$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	10000.2.a.bj		8
4.b	odd	2	1		625.2.a.f		8
5.b	even	2	1	inner	10000.2.a.bj		8
12.b	even	2	1		5625.2.a.x		8
20.d	odd	2	1		625.2.a.f		8
20.e	even	4	2		625.2.b.c		8
25.f	odd	20	2		400.2.y.c		8
60.h	even	2	1		5625.2.a.x		8
100.h	odd	10	2		125.2.d.b		16
100.h	odd	10	2		625.2.d.o		16
100.j	odd	10	2		125.2.d.b		16
100.j	odd	10	2		625.2.d.o		16
100.l	even	20	2		25.2.e.a	✓	8
100.l	even	20	2		125.2.e.b		8
100.l	even	20	2		625.2.e.a		8
100.l	even	20	2		625.2.e.i		8
300.u	odd	20	2		225.2.m.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
25.2.e.a	✓	8	100.l	even	20	2
125.2.d.b		16	100.h	odd	10	2
125.2.d.b		16	100.j	odd	10	2
125.2.e.b		8	100.l	even	20	2
225.2.m.a		8	300.u	odd	20	2
400.2.y.c		8	25.f	odd	20	2
625.2.a.f		8	4.b	odd	2	1
625.2.a.f		8	20.d	odd	2	1
625.2.b.c		8	20.e	even	4	2
625.2.d.o		16	100.h	odd	10	2
625.2.d.o		16	100.j	odd	10	2
625.2.e.a		8	100.l	even	20	2
625.2.e.i		8	100.l	even	20	2
5625.2.a.x		8	12.b	even	2	1
5625.2.a.x		8	60.h	even	2	1
10000.2.a.bj		8	1.a	even	1	1	trivial
10000.2.a.bj		8	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(10000))$:

$ T_{3}^{8} - 14T_{3}^{6} + 61T_{3}^{4} - 84T_{3}^{2} + 16 $

$ T_{7}^{8} - 21T_{7}^{6} + 121T_{7}^{4} - 116T_{7}^{2} + 16 $

$ T_{11} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 14 T^{6} + \cdots + 16 $ T^8 - 14T^6 + 61T^4 - 84*T^2 + 16
$5$	$ T^{8} $ T^8
$7$	$ T^{8} - 21 T^{6} + \cdots + 16 $ T^8 - 21T^6 + 121T^4 - 116*T^2 + 16
$11$	$ (T + 2)^{8} $ (T + 2)^8
$13$	$ T^{8} - 14 T^{6} + \cdots + 1 $ T^8 - 14T^6 + 31T^4 - 14*T^2 + 1
$17$	$ T^{8} - 41 T^{6} + \cdots + 1936 $ T^8 - 41T^6 + 441T^4 - 1636*T^2 + 1936
$19$	$ (T^{4} + 5 T^{3} - 5 T^{2} + \cdots - 20)^{2} $ (T^4 + 5T^3 - 5T^2 - 30*T - 20)^2
$23$	$ T^{8} - 34 T^{6} + \cdots + 256 $ T^8 - 34T^6 + 301T^4 - 544*T^2 + 256
$29$	$ (T^{4} - 10 T^{3} + \cdots - 695)^{2} $ (T^4 - 10T^3 - 5T^2 + 290*T - 695)^2
$31$	$ (T^{4} + 8 T^{3} - 41 T^{2} + \cdots - 44)^{2} $ (T^4 + 8T^3 - 41T^2 - 328*T - 44)^2
$37$	$ T^{8} - 111 T^{6} + \cdots + 116281 $ T^8 - 111T^6 + 3556T^4 - 39331*T^2 + 116281
$41$	$ (T^{4} - 13 T^{3} + \cdots + 116)^{2} $ (T^4 - 13T^3 + 19T^2 + 148*T + 116)^2
$43$	$ T^{8} - 129 T^{6} + \cdots + 246016 $ T^8 - 129T^6 + 4421T^4 - 56784*T^2 + 246016
$47$	$ T^{8} - 141 T^{6} + \cdots + 65536 $ T^8 - 141T^6 + 4661T^4 - 47616*T^2 + 65536
$53$	$ T^{8} - 239 T^{6} + \cdots + 8755681 $ T^8 - 239T^6 + 20356T^4 - 722619*T^2 + 8755681
$59$	$ (T^{4} + 15 T^{3} + \cdots - 2020)^{2} $ (T^4 + 15T^3 + 5T^2 - 630*T - 2020)^2
$61$	$ (T^{4} - 3 T^{3} + \cdots + 341)^{2} $ (T^4 - 3T^3 - 146T^2 - 237*T + 341)^2
$67$	$ T^{8} - 176 T^{6} + \cdots + 246016 $ T^8 - 176T^6 + 7776T^4 - 84736*T^2 + 246016
$71$	$ (T^{4} + 23 T^{3} + \cdots - 4924)^{2} $ (T^4 + 23T^3 + 99T^2 - 798*T - 4924)^2
$73$	$ T^{8} - 79 T^{6} + \cdots + 1 $ T^8 - 79T^6 + 76T^4 - 19*T^2 + 1
$79$	$ (T^{4} + 5 T^{3} + \cdots + 5780)^{2} $ (T^4 + 5T^3 - 195T^2 + 5780)^2
$83$	$ T^{8} - 374 T^{6} + \cdots + 99856 $ T^8 - 374T^6 + 35061T^4 - 138164*T^2 + 99856
$89$	$ (T^{4} - 15 T^{3} + \cdots + 1180)^{2} $ (T^4 - 15T^3 - 35T^2 + 530*T + 1180)^2
$97$	$ T^{8} - 666 T^{6} + \cdots + 301334881 $ T^8 - 666T^6 + 146971T^4 - 12038986*T^2 + 301334881
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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9}+O(q^{10}) \) q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9	\( q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9} - 2 q^{11} - \beta_{5} q^{13} + ( - 2 \beta_{7} + \beta_{6} + \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{19} + (\beta_{3} - \beta_{2} + 1) q^{21} + (\beta_{6} + \beta_{5}) q^{23} + ( - 2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{27}+ \cdots + 2 \beta_{3} q^{99}+O(q^{100}) \) q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9 - 2 * q^11 - b5 * q^13 + (-2b7 + b6 + b1) q^17 + (-2b4 + b2 - 2) q^19 + (b3 - b2 + 1) * q^21 + (b6 + b5) * q^23 + (-2b7 + b6 - b5 + 2b1) * q^27 + (3b4 + b3 - 2b2 + 4) * q^29 + (b3 + 2b2 - 1) q^31 + (2b6 - 2b5) * q^33 + (4b7 + b5 - b1) q^37 + (2b4 + b3 - b2 - 1) q^39 + (-b4 - 2b3 + b2 + 2) q^41 + (3b7 + 2b6 - b5 - b1) * q^43 + (-b7 + b6 - 2b5 + 3b1) * q^47 + (2b3 - b2 - 1) q^49 + (-2b3 + 2b2) * q^51 + (-3b7 + 2b6 - 2b5 - 3b1) * q^53 + (b7 + 4b6 - b5 - 3b1) * q^57 + (2b4 - 2b3 - b2 - 4) * q^59 + (b4 + 3b3 + b2 + 3) q^61 + (2b6 + 2b1) * q^63 + (-2b7 + 2b6 + 4b1) q^67 + (-4b4 - b3 + 2b2 - 1) * q^69 + (-2b4 + b3 - 3b2 - 7) * q^71 + (3b7 - 2b6 + 2b5 - b1) q^73 + (-2b7 + 2b6 + 2b1) q^77 + (4b4 + 4b3 - 3b2 + 2) q^79 + (4b4 + 2b3 - 1) * q^81 + (-4b7 + 5b6 - b5 + 4b1) q^83 + (b7 - 6b6 + b5 + 3b1) * q^87 + (-7b4 - b3 + b2) q^89 + (b2 + 2) * q^91 + (7b6 - 5b5 - 4b1) q^93 + (-5b7 - 4b6 - 4b5 + 4b1) * q^97 + 2b3 q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more