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Properties

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

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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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{3} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{3} - 4) q^{9} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} + 3) q^{17} + ( - 3 \beta_{2} + 2 \beta_1) q^{19} + ( - 3 \beta_{2} - 5 \beta_1) q^{27} + (2 \beta_{3} + 9) q^{33} + ( - 2 \beta_{3} + 3) q^{41} - 5 \beta_1 q^{43} - 7 q^{49} + (\beta_{2} - 5 \beta_1) q^{51} + (\beta_{3} + 17) q^{57} + 3 \beta_1 q^{59} + (3 \beta_{2} - 5 \beta_1) q^{67} + ( - 3 \beta_{3} - 1) q^{73} + (5 \beta_{3} + 19) q^{81} + ( - \beta_{2} + 5 \beta_1) q^{83} + (\beta_{3} - 9) q^{89} + 10 q^{97} + (10 \beta_{2} + 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{9} + 12 q^{17} + 36 q^{33} + 12 q^{41} - 28 q^{49} + 68 q^{57} - 4 q^{73} + 76 q^{81} - 36 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 6\beta_{2} - 3\beta_1 ) / 4 \) 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
1.22474 − 1.22474i
−1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 − 3.44949i 0 0 0 0 0 −8.89898 0
801.2 0 − 1.44949i 0 0 0 0 0 0.898979 0
801.3 0 1.44949i 0 0 0 0 0 0.898979 0
801.4 0 3.44949i 0 0 0 0 0 −8.89898 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.d yes 4
4.b odd 2 1 inner 1600.2.d.d yes 4
5.b even 2 1 1600.2.d.c ✓ 4
5.c odd 4 1 1600.2.f.f 4
5.c odd 4 1 1600.2.f.j 4
8.b even 2 1 inner 1600.2.d.d yes 4
8.d odd 2 1 CM 1600.2.d.d yes 4
16.e even 4 1 6400.2.a.bc 2
16.e even 4 1 6400.2.a.ci 2
16.f odd 4 1 6400.2.a.bc 2
16.f odd 4 1 6400.2.a.ci 2
20.d odd 2 1 1600.2.d.c ✓ 4
20.e even 4 1 1600.2.f.f 4
20.e even 4 1 1600.2.f.j 4
40.e odd 2 1 1600.2.d.c ✓ 4
40.f even 2 1 1600.2.d.c ✓ 4
40.i odd 4 1 1600.2.f.f 4
40.i odd 4 1 1600.2.f.j 4
40.k even 4 1 1600.2.f.f 4
40.k even 4 1 1600.2.f.j 4
80.k odd 4 1 6400.2.a.bd 2
80.k odd 4 1 6400.2.a.ch 2
80.q even 4 1 6400.2.a.bd 2
80.q even 4 1 6400.2.a.ch 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c ✓ 4 5.b even 2 1
1600.2.d.c ✓ 4 20.d odd 2 1
1600.2.d.c ✓ 4 40.e odd 2 1
1600.2.d.c ✓ 4 40.f even 2 1
1600.2.d.d yes 4 1.a even 1 1 trivial
1600.2.d.d yes 4 4.b odd 2 1 inner
1600.2.d.d yes 4 8.b even 2 1 inner
1600.2.d.d yes 4 8.d odd 2 1 CM
1600.2.f.f 4 5.c odd 4 1
1600.2.f.f 4 20.e even 4 1
1600.2.f.f 4 40.i odd 4 1
1600.2.f.f 4 40.k even 4 1
1600.2.f.j 4 5.c odd 4 1
1600.2.f.j 4 20.e even 4 1
1600.2.f.j 4 40.i odd 4 1
1600.2.f.j 4 40.k even 4 1
6400.2.a.bc 2 16.e even 4 1
6400.2.a.bc 2 16.f odd 4 1
6400.2.a.bd 2 80.k odd 4 1
6400.2.a.bd 2 80.q even 4 1
6400.2.a.ch 2 80.k odd 4 1
6400.2.a.ch 2 80.q even 4 1
6400.2.a.ci 2 16.e even 4 1
6400.2.a.ci 2 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 14T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 30T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 15)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 110T^{2} + 2809 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 87)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 206T^{2} + 25 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 174T^{2} + 5625 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 57)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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