Properties

Label 800.2.c.g
Level $800$
Weight $2$
Character orbit 800.c
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - 2 \beta_{2} q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - 2 \beta_{2} q^{7} - 2 q^{9} + \beta_{3} q^{11} + 4 \beta_1 q^{13} + 7 \beta_1 q^{17} + 3 \beta_{3} q^{19} + 10 q^{21} - 2 \beta_{2} q^{23} + \beta_{2} q^{27} + 2 \beta_{3} q^{31} + 5 \beta_1 q^{33} - 2 \beta_1 q^{37} - 4 \beta_{3} q^{39} + 5 q^{41} + 4 \beta_{2} q^{47} - 13 q^{49} - 7 \beta_{3} q^{51} + 6 \beta_1 q^{53} + 15 \beta_1 q^{57} - 4 \beta_{3} q^{59} + 10 q^{61} + 4 \beta_{2} q^{63} + \beta_{2} q^{67} + 10 q^{69} - 4 \beta_{3} q^{71} - 9 \beta_1 q^{73} - 10 \beta_1 q^{77} + 2 \beta_{3} q^{79} - 11 q^{81} - 5 \beta_{2} q^{83} + 5 q^{89} + 8 \beta_{3} q^{91} + 10 \beta_1 q^{93} - 2 \beta_1 q^{97} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 40 q^{21} + 20 q^{41} - 52 q^{49} + 40 q^{61} + 40 q^{69} - 44 q^{81} + 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.61803i
0.618034i
1.61803i
0.618034i
0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.2 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.3 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.4 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.c.g 4
3.b odd 2 1 7200.2.f.bg 4
4.b odd 2 1 inner 800.2.c.g 4
5.b even 2 1 inner 800.2.c.g 4
5.c odd 4 1 800.2.a.k 2
5.c odd 4 1 800.2.a.l yes 2
8.b even 2 1 1600.2.c.o 4
8.d odd 2 1 1600.2.c.o 4
12.b even 2 1 7200.2.f.bg 4
15.d odd 2 1 7200.2.f.bg 4
15.e even 4 1 7200.2.a.cf 2
15.e even 4 1 7200.2.a.cn 2
20.d odd 2 1 inner 800.2.c.g 4
20.e even 4 1 800.2.a.k 2
20.e even 4 1 800.2.a.l yes 2
40.e odd 2 1 1600.2.c.o 4
40.f even 2 1 1600.2.c.o 4
40.i odd 4 1 1600.2.a.ba 2
40.i odd 4 1 1600.2.a.bb 2
40.k even 4 1 1600.2.a.ba 2
40.k even 4 1 1600.2.a.bb 2
60.h even 2 1 7200.2.f.bg 4
60.l odd 4 1 7200.2.a.cf 2
60.l odd 4 1 7200.2.a.cn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 5.c odd 4 1
800.2.a.k 2 20.e even 4 1
800.2.a.l yes 2 5.c odd 4 1
800.2.a.l yes 2 20.e even 4 1
800.2.c.g 4 1.a even 1 1 trivial
800.2.c.g 4 4.b odd 2 1 inner
800.2.c.g 4 5.b even 2 1 inner
800.2.c.g 4 20.d odd 2 1 inner
1600.2.a.ba 2 40.i odd 4 1
1600.2.a.ba 2 40.k even 4 1
1600.2.a.bb 2 40.i odd 4 1
1600.2.a.bb 2 40.k even 4 1
1600.2.c.o 4 8.b even 2 1
1600.2.c.o 4 8.d odd 2 1
1600.2.c.o 4 40.e odd 2 1
1600.2.c.o 4 40.f even 2 1
7200.2.a.cf 2 15.e even 4 1
7200.2.a.cf 2 60.l odd 4 1
7200.2.a.cn 2 15.e even 4 1
7200.2.a.cn 2 60.l odd 4 1
7200.2.f.bg 4 3.b odd 2 1
7200.2.f.bg 4 12.b even 2 1
7200.2.f.bg 4 15.d odd 2 1
7200.2.f.bg 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{11}^{2} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T - 5)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$89$ \( (T - 5)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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