Properties

Label 4200.2.a.bg
Level $4200$
Weight $2$
Character orbit 4200.a
Self dual yes
Analytic conductor $33.537$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4200,2,Mod(1,4200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)

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: yes
Analytic conductor: \(33.5371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{7} + q^{9} + \beta q^{11} - 2 q^{13} + (\beta - 2) q^{17} - \beta q^{19} + q^{21} + \beta q^{23} - q^{27} + (\beta - 2) q^{29} + 4 q^{31} - \beta q^{33} + ( - \beta - 6) q^{37} + 2 q^{39} + 2 q^{41} + ( - \beta + 4) q^{43} + ( - \beta + 8) q^{47} + q^{49} + ( - \beta + 2) q^{51} + (\beta - 2) q^{53} + \beta q^{57} - 4 q^{59} + ( - \beta + 6) q^{61} - q^{63} + (\beta + 4) q^{67} - \beta q^{69} + ( - 2 \beta + 4) q^{71} - 6 q^{73} - \beta q^{77} + 2 \beta q^{79} + q^{81} + 4 q^{83} + ( - \beta + 2) q^{87} - 14 q^{89} + 2 q^{91} - 4 q^{93} - 6 q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{21} - 2 q^{27} - 4 q^{29} + 8 q^{31} - 12 q^{37} + 4 q^{39} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} - 8 q^{59} + 12 q^{61} - 2 q^{63} + 8 q^{67} + 8 q^{71} - 12 q^{73} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 28 q^{89} + 4 q^{91} - 8 q^{93} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.00000 0 0 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4200.2.a.bg 2
4.b odd 2 1 8400.2.a.de 2
5.b even 2 1 840.2.a.k 2
5.c odd 4 2 4200.2.t.u 4
15.d odd 2 1 2520.2.a.v 2
20.d odd 2 1 1680.2.a.u 2
35.c odd 2 1 5880.2.a.bl 2
40.e odd 2 1 6720.2.a.cu 2
40.f even 2 1 6720.2.a.co 2
60.h even 2 1 5040.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.k 2 5.b even 2 1
1680.2.a.u 2 20.d odd 2 1
2520.2.a.v 2 15.d odd 2 1
4200.2.a.bg 2 1.a even 1 1 trivial
4200.2.t.u 4 5.c odd 4 2
5040.2.a.br 2 60.h even 2 1
5880.2.a.bl 2 35.c odd 2 1
6720.2.a.co 2 40.f even 2 1
6720.2.a.cu 2 40.e odd 2 1
8400.2.a.de 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\):

\( T_{11}^{2} - 32 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} - 32 \) Copy content Toggle raw display
\( T_{23}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 32 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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