↑

Properties

Label 2000.2.a.o
Level $2000$
Weight $2$
Character orbit 2000.a
Self dual yes
Analytic conductor $15.970$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2000,2,Mod(1,2000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2000, 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("2000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2000.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: \(15.9700804043\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 125)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{2} + 1) q^{9} - 2 q^{11} + 2 \beta_{3} q^{13} - 2 \beta_1 q^{17} + ( - 2 \beta_{2} - 6) q^{19} + ( - 4 \beta_{2} - 5) q^{21} + (\beta_{3} - \beta_1) q^{23} + (\beta_{3} - 2 \beta_1) q^{27} + (3 \beta_{2} - 1) q^{29} - 2 q^{31} - 2 \beta_1 q^{33} + ( - 2 \beta_{3} + 2 \beta_1) q^{37} + (6 \beta_{2} + 2) q^{39} + ( - 5 \beta_{2} - 3) q^{41} + (2 \beta_{3} - 3 \beta_1) q^{43} + ( - 2 \beta_{3} + \beta_1) q^{47} + (5 \beta_{2} + 2) q^{49} + ( - 2 \beta_{2} - 8) q^{51} + 4 \beta_1 q^{53} + ( - 2 \beta_{3} - 6 \beta_1) q^{57} + (4 \beta_{2} + 2) q^{59} + (5 \beta_{2} + 2) q^{61} + ( - \beta_{3} - 2 \beta_1) q^{63} + 2 \beta_1 q^{67} + (2 \beta_{2} - 3) q^{69} + ( - 10 \beta_{2} - 2) q^{71} + ( - 6 \beta_{3} - 4 \beta_1) q^{73} + (2 \beta_{3} + 2 \beta_1) q^{77} + (2 \beta_{2} - 4) q^{79} + ( - 2 \beta_{2} - 10) q^{81} + ( - \beta_{3} + 3 \beta_1) q^{83} + (3 \beta_{3} - \beta_1) q^{87} + ( - \beta_{2} - 8) q^{89} + ( - 2 \beta_{2} - 8) q^{91} - 2 \beta_1 q^{93} + (2 \beta_{3} + 4 \beta_1) q^{97} + ( - 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} - 8 q^{11} - 20 q^{19} - 12 q^{21} - 10 q^{29} - 8 q^{31} - 4 q^{39} - 2 q^{41} - 2 q^{49} - 28 q^{51} - 2 q^{61} - 16 q^{69} + 12 q^{71} - 20 q^{79} - 36 q^{81} - 30 q^{89} - 28 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display

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
−2.14896
−1.54336
1.54336
2.14896
0 −2.14896 0 0 0 3.47709 0 1.61803 0
1.2 0 −1.54336 0 0 0 −0.953850 0 −0.618034 0
1.3 0 1.54336 0 0 0 0.953850 0 −0.618034 0
1.4 0 2.14896 0 0 0 −3.47709 0 1.61803 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2000.2.a.o 4
4.b odd 2 1 125.2.a.c ✓ 4
5.b even 2 1 inner 2000.2.a.o 4
5.c odd 4 2 2000.2.c.c 4
8.b even 2 1 8000.2.a.bk 4
8.d odd 2 1 8000.2.a.bj 4
12.b even 2 1 1125.2.a.k 4
20.d odd 2 1 125.2.a.c ✓ 4
20.e even 4 2 125.2.b.a 4
28.d even 2 1 6125.2.a.o 4
40.e odd 2 1 8000.2.a.bj 4
40.f even 2 1 8000.2.a.bk 4
60.h even 2 1 1125.2.a.k 4
60.l odd 4 2 1125.2.b.a 4
100.h odd 10 2 625.2.d.k 8
100.h odd 10 2 625.2.d.l 8
100.j odd 10 2 625.2.d.k 8
100.j odd 10 2 625.2.d.l 8
100.l even 20 4 625.2.e.b 8
100.l even 20 4 625.2.e.h 8
140.c even 2 1 6125.2.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.c ✓ 4 4.b odd 2 1
125.2.a.c ✓ 4 20.d odd 2 1
125.2.b.a 4 20.e even 4 2
625.2.d.k 8 100.h odd 10 2
625.2.d.k 8 100.j odd 10 2
625.2.d.l 8 100.h odd 10 2
625.2.d.l 8 100.j odd 10 2
625.2.e.b 8 100.l even 20 4
625.2.e.h 8 100.l even 20 4
1125.2.a.k 4 12.b even 2 1
1125.2.a.k 4 60.h even 2 1
1125.2.b.a 4 60.l odd 4 2
2000.2.a.o 4 1.a even 1 1 trivial
2000.2.a.o 4 5.b even 2 1 inner
2000.2.c.c 4 5.c odd 4 2
6125.2.a.o 4 28.d even 2 1
6125.2.a.o 4 140.c even 2 1
8000.2.a.bj 4 8.d odd 2 1
8000.2.a.bj 4 40.e odd 2 1
8000.2.a.bk 4 8.b even 2 1
8000.2.a.bk 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2000))\):

\( T_{3}^{4} - 7T_{3}^{2} + 11 \) Copy content Toggle raw display
\( T_{7}^{4} - 13T_{7}^{2} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 7T^{2} + 11 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 11 \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 32T^{2} + 176 \) Copy content Toggle raw display
$17$ \( T^{4} - 28T^{2} + 176 \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 17T^{2} + 11 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 68T^{2} + 176 \) Copy content Toggle raw display
$41$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 107T^{2} + 1331 \) Copy content Toggle raw display
$47$ \( T^{4} - 43T^{2} + 11 \) Copy content Toggle raw display
$53$ \( T^{4} - 112T^{2} + 2816 \) Copy content Toggle raw display
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 28T^{2} + 176 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 116)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 352 T^{2} + 21296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 77T^{2} + 1331 \) Copy content Toggle raw display
$89$ \( (T^{2} + 15 T + 55)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 128T^{2} + 176 \) Copy content Toggle raw display
show more
show less