Properties

Label 4050.2.a.bw
Level $4050$
Weight $2$
Character orbit 4050.a
Self dual yes
Analytic conductor $32.339$
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] = [4050,2,Mod(1,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta q^{7} + q^{8} + ( - \beta - 1) q^{11} + 2 \beta q^{13} - \beta q^{14} + q^{16} + ( - \beta + 5) q^{17} + ( - \beta + 1) q^{19} + ( - \beta - 1) q^{22} + ( - \beta + 2) q^{23} + \cdots + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} - 3 q^{11} + 2 q^{13} - q^{14} + 2 q^{16} + 9 q^{17} + q^{19} - 3 q^{22} + 3 q^{23} + 2 q^{26} - q^{28} - 3 q^{29} - 2 q^{31} + 2 q^{32} + 9 q^{34} + 8 q^{37}+ \cdots + 3 q^{98}+O(q^{100}) \) 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
3.37228
−2.37228
1.00000 0 1.00000 0 0 −3.37228 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 2.37228 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4050.2.a.bw 2
3.b odd 2 1 4050.2.a.bo 2
5.b even 2 1 810.2.a.i 2
5.c odd 4 2 4050.2.c.v 4
9.c even 3 2 450.2.e.j 4
9.d odd 6 2 1350.2.e.l 4
15.d odd 2 1 810.2.a.k 2
15.e even 4 2 4050.2.c.ba 4
20.d odd 2 1 6480.2.a.be 2
45.h odd 6 2 270.2.e.c 4
45.j even 6 2 90.2.e.c 4
45.k odd 12 4 450.2.j.g 8
45.l even 12 4 1350.2.j.f 8
60.h even 2 1 6480.2.a.bn 2
180.n even 6 2 2160.2.q.f 4
180.p odd 6 2 720.2.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 45.j even 6 2
270.2.e.c 4 45.h odd 6 2
450.2.e.j 4 9.c even 3 2
450.2.j.g 8 45.k odd 12 4
720.2.q.f 4 180.p odd 6 2
810.2.a.i 2 5.b even 2 1
810.2.a.k 2 15.d odd 2 1
1350.2.e.l 4 9.d odd 6 2
1350.2.j.f 8 45.l even 12 4
2160.2.q.f 4 180.n even 6 2
4050.2.a.bo 2 3.b odd 2 1
4050.2.a.bw 2 1.a even 1 1 trivial
4050.2.c.v 4 5.c odd 4 2
4050.2.c.ba 4 15.e even 4 2
6480.2.a.be 2 20.d odd 2 1
6480.2.a.bn 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 32 \) Copy content Toggle raw display
\( T_{17}^{2} - 9T_{17} + 12 \) Copy content Toggle raw display
\( T_{23}^{2} - 3T_{23} - 6 \) Copy content Toggle raw display
\( T_{41} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 17T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 132 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$67$ \( (T - 7)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11T - 44 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T - 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 11T + 22 \) Copy content Toggle raw display
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