↑

Properties

Label 5200.2.a.bx
Level $5200$
Weight $2$
Character orbit 5200.a
Self dual yes
Analytic conductor $41.522$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,-1,0,11,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.5222090511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1300)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta q^{3} + (\beta - 1) q^{7} + (\beta + 5) q^{9} + (\beta + 1) q^{11} + q^{13} + ( - \beta - 1) q^{17} + 4 q^{19} + 8 q^{21} + (\beta - 2) q^{23} + (3 \beta + 8) q^{27} + (2 \beta - 1) q^{29} + (\beta - 1) q^{31}+ \cdots + (7 \beta + 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{7} + 11 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{17} + 8 q^{19} + 16 q^{21} - 3 q^{23} + 19 q^{27} - q^{31} + 18 q^{33} - 20 q^{37} + q^{39} + 6 q^{41} - 13 q^{43} - 15 q^{47} + 3 q^{49}+ \cdots + 33 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.37228
3.37228
0 −2.37228 0 0 0 −3.37228 0 2.62772 0
1.2 0 3.37228 0 0 0 2.37228 0 8.37228 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( -1 \)

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 5200.2.a.bx 2
4.b odd 2 1 1300.2.a.g ✓ 2
5.b even 2 1 5200.2.a.bp 2
20.d odd 2 1 1300.2.a.h yes 2
20.e even 4 2 1300.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1300.2.a.g ✓ 2 4.b odd 2 1
1300.2.a.h yes 2 20.d odd 2 1
1300.2.c.e 4 20.e even 4 2
5200.2.a.bp 2 5.b even 2 1
5200.2.a.bx 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\):

\( T_{3}^{2} - T_{3} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 8 \) 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
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 33 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$47$ \( T^{2} + 15T + 48 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 3 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 17 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 74 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 48 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 19T + 82 \) Copy content Toggle raw display
$83$ \( T^{2} + 21T + 102 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 16T - 68 \) Copy content Toggle raw display
show more
show less