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Properties

Label 207.2.a.b
Level $207$
Weight $2$
Character orbit 207.a
Self dual yes
Analytic conductor $1.653$
Analytic rank $1$
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] = [207,2,Mod(1,207)] 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("207.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.65290332184\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + ( - \beta - 2) q^{5} + (\beta - 2) q^{7} + (\beta - 3) q^{8} - \beta q^{10} - 2 \beta q^{11} + ( - 3 \beta + 4) q^{14} + 3 q^{16} + (\beta - 6) q^{17} + (3 \beta - 2) q^{19} + \cdots + (3 \beta - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 4 q^{7} - 6 q^{8} + 8 q^{14} + 6 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 2 q^{23} + 2 q^{25} - 12 q^{28} + 6 q^{32} + 16 q^{34} + 4 q^{35} - 4 q^{37} + 16 q^{38}+ \cdots - 14 q^{98}+O(q^{100}) \) 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.

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
−1.41421
1.41421
−2.41421 0 3.82843 −0.585786 0 −3.41421 −4.41421 0 1.41421
1.2 0.414214 0 −1.82843 −3.41421 0 −0.585786 −1.58579 0 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(23\) \( +1 \)

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 207.2.a.b ✓ 2
3.b odd 2 1 207.2.a.e yes 2
4.b odd 2 1 3312.2.a.u 2
5.b even 2 1 5175.2.a.bo 2
12.b even 2 1 3312.2.a.be 2
15.d odd 2 1 5175.2.a.bc 2
23.b odd 2 1 4761.2.a.k 2
69.c even 2 1 4761.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.a.b ✓ 2 1.a even 1 1 trivial
207.2.a.e yes 2 3.b odd 2 1
3312.2.a.u 2 4.b odd 2 1
3312.2.a.be 2 12.b even 2 1
4761.2.a.k 2 23.b odd 2 1
4761.2.a.z 2 69.c even 2 1
5175.2.a.bc 2 15.d odd 2 1
5175.2.a.bo 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 72 \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 98 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 94 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 14 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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