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Properties

Label 1600.2.d.g
Level $1600$
Weight $2$
Character orbit 1600.d
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(801,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.801");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\), degree \(1\), 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: \(12.7760643234\)
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: \( 2^{5} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{7} + 3 q^{9} - \beta_1 q^{11} + \beta_{2} q^{13} - 3 \beta_1 q^{19} + \beta_{3} q^{23} - \beta_{2} q^{37} + 2 q^{41} + 3 \beta_{3} q^{47} + 13 q^{49} - 3 \beta_{2} q^{53} - 7 \beta_1 q^{59} - 3 \beta_{3} q^{63} + 2 \beta_{2} q^{77} + 9 q^{81} + 14 q^{89} - 10 \beta_1 q^{91} - 3 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 8 q^{41} + 52 q^{49} + 36 q^{81} + 56 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
801.1
0.618034i
− 0.618034i
− 1.61803i
1.61803i
0 0 0 0 0 −4.47214 0 3.00000 0
801.2 0 0 0 0 0 −4.47214 0 3.00000 0
801.3 0 0 0 0 0 4.47214 0 3.00000 0
801.4 0 0 0 0 0 4.47214 0 3.00000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.d.g 4
4.b odd 2 1 inner 1600.2.d.g 4
5.b even 2 1 inner 1600.2.d.g 4
5.c odd 4 2 320.2.f.a ✓ 4
8.b even 2 1 inner 1600.2.d.g 4
8.d odd 2 1 inner 1600.2.d.g 4
15.e even 4 2 2880.2.d.e 4
16.e even 4 1 6400.2.a.bi 2
16.e even 4 1 6400.2.a.bj 2
16.f odd 4 1 6400.2.a.bi 2
16.f odd 4 1 6400.2.a.bj 2
20.d odd 2 1 inner 1600.2.d.g 4
20.e even 4 2 320.2.f.a ✓ 4
40.e odd 2 1 CM 1600.2.d.g 4
40.f even 2 1 inner 1600.2.d.g 4
40.i odd 4 2 320.2.f.a ✓ 4
40.k even 4 2 320.2.f.a ✓ 4
60.l odd 4 2 2880.2.d.e 4
80.i odd 4 1 1280.2.c.b 2
80.i odd 4 1 1280.2.c.c 2
80.j even 4 1 1280.2.c.b 2
80.j even 4 1 1280.2.c.c 2
80.k odd 4 1 6400.2.a.bi 2
80.k odd 4 1 6400.2.a.bj 2
80.q even 4 1 6400.2.a.bi 2
80.q even 4 1 6400.2.a.bj 2
80.s even 4 1 1280.2.c.b 2
80.s even 4 1 1280.2.c.c 2
80.t odd 4 1 1280.2.c.b 2
80.t odd 4 1 1280.2.c.c 2
120.q odd 4 2 2880.2.d.e 4
120.w even 4 2 2880.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.a ✓ 4 5.c odd 4 2
320.2.f.a ✓ 4 20.e even 4 2
320.2.f.a ✓ 4 40.i odd 4 2
320.2.f.a ✓ 4 40.k even 4 2
1280.2.c.b 2 80.i odd 4 1
1280.2.c.b 2 80.j even 4 1
1280.2.c.b 2 80.s even 4 1
1280.2.c.b 2 80.t odd 4 1
1280.2.c.c 2 80.i odd 4 1
1280.2.c.c 2 80.j even 4 1
1280.2.c.c 2 80.s even 4 1
1280.2.c.c 2 80.t odd 4 1
1600.2.d.g 4 1.a even 1 1 trivial
1600.2.d.g 4 4.b odd 2 1 inner
1600.2.d.g 4 5.b even 2 1 inner
1600.2.d.g 4 8.b even 2 1 inner
1600.2.d.g 4 8.d odd 2 1 inner
1600.2.d.g 4 20.d odd 2 1 inner
1600.2.d.g 4 40.e odd 2 1 CM
1600.2.d.g 4 40.f even 2 1 inner
2880.2.d.e 4 15.e even 4 2
2880.2.d.e 4 60.l odd 4 2
2880.2.d.e 4 120.q odd 4 2
2880.2.d.e 4 120.w even 4 2
6400.2.a.bi 2 16.e even 4 1
6400.2.a.bi 2 16.f odd 4 1
6400.2.a.bi 2 80.k odd 4 1
6400.2.a.bi 2 80.q even 4 1
6400.2.a.bj 2 16.e even 4 1
6400.2.a.bj 2 16.f odd 4 1
6400.2.a.bj 2 80.k odd 4 1
6400.2.a.bj 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 20 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T - 14)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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