Properties

Label 7200.2.f.bj
Level $7200$
Weight $2$
Character orbit 7200.f
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(6049,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7200.6049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.f (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: \(57.4922894553\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1440)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{7} + \beta_{3} q^{11} - 2 \beta_1 q^{13} + \beta_1 q^{17} + 2 \beta_{2} q^{23} + 6 q^{29} + 2 \beta_{3} q^{31} + 4 \beta_1 q^{37} + 8 q^{41} + 2 \beta_{2} q^{47} - 13 q^{49} + 3 \beta_1 q^{53} - \beta_{3} q^{59} + 10 q^{61} + 2 \beta_{2} q^{67} + 2 \beta_{3} q^{71} - 3 \beta_1 q^{73} - 10 \beta_1 q^{77} + 2 \beta_{3} q^{79} + 2 \beta_{2} q^{83} - 4 q^{89} - 4 \beta_{3} q^{91} + \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{29} + 32 q^{41} - 52 q^{49} + 40 q^{61} - 16 q^{89}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 4\nu \) 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
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) 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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
1.61803i
0.618034i
1.61803i
0.618034i
0 0 0 0 0 4.47214i 0 0 0
6049.2 0 0 0 0 0 4.47214i 0 0 0
6049.3 0 0 0 0 0 4.47214i 0 0 0
6049.4 0 0 0 0 0 4.47214i 0 0 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 7200.2.f.bj 4
3.b odd 2 1 7200.2.f.be 4
4.b odd 2 1 inner 7200.2.f.bj 4
5.b even 2 1 inner 7200.2.f.bj 4
5.c odd 4 1 1440.2.a.q yes 2
5.c odd 4 1 7200.2.a.cg 2
12.b even 2 1 7200.2.f.be 4
15.d odd 2 1 7200.2.f.be 4
15.e even 4 1 1440.2.a.p 2
15.e even 4 1 7200.2.a.ch 2
20.d odd 2 1 inner 7200.2.f.bj 4
20.e even 4 1 1440.2.a.q yes 2
20.e even 4 1 7200.2.a.cg 2
40.i odd 4 1 2880.2.a.bi 2
40.k even 4 1 2880.2.a.bi 2
60.h even 2 1 7200.2.f.be 4
60.l odd 4 1 1440.2.a.p 2
60.l odd 4 1 7200.2.a.ch 2
120.q odd 4 1 2880.2.a.bj 2
120.w even 4 1 2880.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.a.p 2 15.e even 4 1
1440.2.a.p 2 60.l odd 4 1
1440.2.a.q yes 2 5.c odd 4 1
1440.2.a.q yes 2 20.e even 4 1
2880.2.a.bi 2 40.i odd 4 1
2880.2.a.bi 2 40.k even 4 1
2880.2.a.bj 2 120.q odd 4 1
2880.2.a.bj 2 120.w even 4 1
7200.2.a.cg 2 5.c odd 4 1
7200.2.a.cg 2 20.e even 4 1
7200.2.a.ch 2 15.e even 4 1
7200.2.a.ch 2 60.l odd 4 1
7200.2.f.be 4 3.b odd 2 1
7200.2.f.be 4 12.b even 2 1
7200.2.f.be 4 15.d odd 2 1
7200.2.f.be 4 60.h even 2 1
7200.2.f.bj 4 1.a even 1 1 trivial
7200.2.f.bj 4 4.b odd 2 1 inner
7200.2.f.bj 4 5.b even 2 1 inner
7200.2.f.bj 4 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{2} + 20 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display
\( T_{31}^{2} - 80 \) 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
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T - 8)^{4} \) 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
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$89$ \( (T + 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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