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Properties

Label 2128.2.a.l
Level $2128$
Weight $2$
Character orbit 2128.a
Self dual yes
Analytic conductor $16.992$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2128,2,Mod(1,2128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2128.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2128.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,-6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.9921655501\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 133)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - 3 q^{5} - q^{7} + (3 \beta + 1) q^{9} + (\beta + 2) q^{11} + (2 \beta - 3) q^{13} + ( - 3 \beta - 3) q^{15} + (\beta - 4) q^{17} - q^{19} + ( - \beta - 1) q^{21} + 3 q^{23} + 4 q^{25}+ \cdots + (10 \beta + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} - 9 q^{15} - 7 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 8 q^{25} + 18 q^{27} + 9 q^{29} + q^{31} + 14 q^{33} + 6 q^{35} + 7 q^{39} + 5 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
0 −0.302776 0 −3.00000 0 −1.00000 0 −2.90833 0
1.2 0 3.30278 0 −3.00000 0 −1.00000 0 7.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(19\) \( +1 \)

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 2128.2.a.l 2
4.b odd 2 1 133.2.a.b ✓ 2
8.b even 2 1 8512.2.a.l 2
8.d odd 2 1 8512.2.a.bh 2
12.b even 2 1 1197.2.a.h 2
20.d odd 2 1 3325.2.a.n 2
28.d even 2 1 931.2.a.g 2
28.f even 6 2 931.2.f.g 4
28.g odd 6 2 931.2.f.h 4
76.d even 2 1 2527.2.a.d 2
84.h odd 2 1 8379.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.a.b ✓ 2 4.b odd 2 1
931.2.a.g 2 28.d even 2 1
931.2.f.g 4 28.f even 6 2
931.2.f.h 4 28.g odd 6 2
1197.2.a.h 2 12.b even 2 1
2128.2.a.l 2 1.a even 1 1 trivial
2527.2.a.d 2 76.d even 2 1
3325.2.a.n 2 20.d odd 2 1
8379.2.a.bf 2 84.h odd 2 1
8512.2.a.l 2 8.b even 2 1
8512.2.a.bh 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2128))\):

\( T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 9 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 9T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$37$ \( T^{2} - 13 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 51 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 43 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T - 17 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$73$ \( T^{2} + 15T - 25 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 15T + 27 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 23 \) Copy content Toggle raw display
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