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Properties

Label 20.10.a.b
Level $20$
Weight $10$
Character orbit 20.a
Self dual yes
Analytic conductor $10.301$
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] = [20,10,Mod(1,20)] 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("20.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(\beta = 16\sqrt{79}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 130) q^{3} - 625 q^{5} + (69 \beta - 190) q^{7} + ( - 260 \beta + 17441) q^{9} + (30 \beta + 51360) q^{11} + (564 \beta + 89570) q^{13} + ( - 625 \beta + 81250) q^{15} + (348 \beta + 158010) q^{17}+ \cdots + ( - 12830370 \beta + 738022560) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 260 q^{3} - 1250 q^{5} - 380 q^{7} + 34882 q^{9} + 102720 q^{11} + 179140 q^{13} + 162500 q^{15} + 316020 q^{17} + 137272 q^{19} + 2840312 q^{21} - 665460 q^{23} + 781250 q^{25} - 9933560 q^{27} - 6893748 q^{29}+ \cdots + 1476045120 q^{99}+O(q^{100}) \) 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
−8.88819
8.88819
0 −272.211 0 −625.000 0 −10002.6 0 54415.9 0
1.2 0 12.2111 0 −625.000 0 9622.57 0 −19533.9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 20.10.a.b ✓ 2
3.b odd 2 1 180.10.a.e 2
4.b odd 2 1 80.10.a.j 2
5.b even 2 1 100.10.a.c 2
5.c odd 4 2 100.10.c.c 4
8.b even 2 1 320.10.a.t 2
8.d odd 2 1 320.10.a.l 2
20.d odd 2 1 400.10.a.l 2
20.e even 4 2 400.10.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.b ✓ 2 1.a even 1 1 trivial
80.10.a.j 2 4.b odd 2 1
100.10.a.c 2 5.b even 2 1
100.10.c.c 4 5.c odd 4 2
180.10.a.e 2 3.b odd 2 1
320.10.a.l 2 8.d odd 2 1
320.10.a.t 2 8.b even 2 1
400.10.a.l 2 20.d odd 2 1
400.10.c.l 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 260T_{3} - 3324 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(20))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 260T - 3324 \) Copy content Toggle raw display
$5$ \( (T + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 380 T - 96250364 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 2619648000 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 1589611396 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 22517952804 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 533765233904 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 715854707676 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 11181707706276 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 13914308520944 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 41043496652924 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 116248533661476 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 166415379974300 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 939697938528804 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 891341422077276 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 669513681826416 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 954028889496956 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 15\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 82\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 73\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 54\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
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