Properties

Label 1600.2.l.d
Level $1600$
Weight $2$
Character orbit 1600.l
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(401,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{9} + (\beta_{2} - 2 \beta_1 - 1) q^{11} + (2 \beta_{3} - 2) q^{13} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{3} + 2 \beta_1 - 1) q^{19} + (5 \beta_1 + 5) q^{21} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{23} + ( - \beta_{2} + 6 \beta_1 + 5) q^{27} + ( - 2 \beta_1 + 2) q^{29} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 7) q^{33} + (2 \beta_{2} + 3 \beta_1 + 5) q^{37} + ( - 2 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 2) q^{39} + 5 \beta_1 q^{41} + ( - 4 \beta_{2} + 3 \beta_1 - 1) q^{43} + 8 q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 5) q^{49} + ( - \beta_{3} + 5 \beta_1 - 4) q^{51} + (2 \beta_{2} - \beta_1 + 1) q^{53} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{57} + ( - 2 \beta_{2} - \beta_1 - 3) q^{59} + ( - 4 \beta_{3} - \beta_1 + 5) q^{61} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 8) q^{63} + (\beta_{3} + 7 \beta_1 - 8) q^{67} + (2 \beta_{2} + 3 \beta_1 + 5) q^{69} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{71} + ( - \beta_{3} + \beta_{2} + 10 \beta_1 + 1) q^{73} + (2 \beta_{3} + 3 \beta_1 - 5) q^{77} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{79} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 2) q^{81} + ( - \beta_{3} - 5 \beta_1 + 6) q^{83} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{87} + (3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 3) q^{89} + (10 \beta_1 + 10) q^{91} + (5 \beta_1 - 5) q^{93} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{97} + (6 \beta_{3} + 7 \beta_1 - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 6 q^{11} - 4 q^{13} - 8 q^{17} - 6 q^{19} + 20 q^{21} + 22 q^{27} + 8 q^{29} - 4 q^{31} + 28 q^{33} + 16 q^{37} + 4 q^{43} + 32 q^{47} - 20 q^{49} - 18 q^{51} - 8 q^{59} + 12 q^{61} - 32 q^{63} - 30 q^{67} + 16 q^{69} - 16 q^{77} - 4 q^{79} - 8 q^{81} + 22 q^{83} + 40 q^{91} - 20 q^{93} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.65831 0.500000i
1.65831 0.500000i
−1.65831 + 0.500000i
1.65831 + 0.500000i
0 −2.15831 2.15831i 0 0 0 2.31662i 0 6.31662i 0
401.2 0 1.15831 + 1.15831i 0 0 0 4.31662i 0 0.316625i 0
1201.1 0 −2.15831 + 2.15831i 0 0 0 2.31662i 0 6.31662i 0
1201.2 0 1.15831 1.15831i 0 0 0 4.31662i 0 0.316625i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.d 4
4.b odd 2 1 400.2.l.e yes 4
5.b even 2 1 1600.2.l.e 4
5.c odd 4 1 1600.2.q.c 4
5.c odd 4 1 1600.2.q.d 4
16.e even 4 1 inner 1600.2.l.d 4
16.f odd 4 1 400.2.l.e yes 4
20.d odd 2 1 400.2.l.d 4
20.e even 4 1 400.2.q.c 4
20.e even 4 1 400.2.q.d 4
80.i odd 4 1 1600.2.q.d 4
80.j even 4 1 400.2.q.c 4
80.k odd 4 1 400.2.l.d 4
80.q even 4 1 1600.2.l.e 4
80.s even 4 1 400.2.q.d 4
80.t odd 4 1 1600.2.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.d 4 20.d odd 2 1
400.2.l.d 4 80.k odd 4 1
400.2.l.e yes 4 4.b odd 2 1
400.2.l.e yes 4 16.f odd 4 1
400.2.q.c 4 20.e even 4 1
400.2.q.c 4 80.j even 4 1
400.2.q.d 4 20.e even 4 1
400.2.q.d 4 80.s even 4 1
1600.2.l.d 4 1.a even 1 1 trivial
1600.2.l.d 4 16.e even 4 1 inner
1600.2.l.e 4 5.b even 2 1
1600.2.l.e 4 80.q even 4 1
1600.2.q.c 4 5.c odd 4 1
1600.2.q.c 4 80.t odd 4 1
1600.2.q.d 4 5.c odd 4 1
1600.2.q.d 4 80.i odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 10T_{3} + 25 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} - 10 T + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 100 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 18 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 8 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 18 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$23$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$41$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + 8 T^{2} + 344 T + 7396 \) Copy content Toggle raw display
$47$ \( (T - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 484 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + 32 T^{2} - 112 T + 196 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 4900 \) Copy content Toggle raw display
$67$ \( T^{4} + 30 T^{3} + 450 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$71$ \( T^{4} + 96T^{2} + 1600 \) Copy content Toggle raw display
$73$ \( T^{4} + 222T^{2} + 7921 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$89$ \( T^{4} + 270T^{2} + 3969 \) Copy content Toggle raw display
$97$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
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