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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3698,2,Mod(1,3698)] 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("3698.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3698, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3698 = 2 \cdot 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3698.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-1,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.5286786675\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 86)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} + (\beta + 1) q^{5} + \beta q^{6} + (4 \beta - 2) q^{7} - q^{8} + (\beta - 2) q^{9} + ( - \beta - 1) q^{10} + (4 \beta - 4) q^{11} - \beta q^{12} + (4 \beta - 2) q^{13} + \cdots + ( - 8 \beta + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} + q^{6} - 2 q^{8} - 3 q^{9} - 3 q^{10} - 4 q^{11} - q^{12} - 4 q^{15} + 2 q^{16} - q^{17} + 3 q^{18} - 11 q^{19} + 3 q^{20} - 10 q^{21} + 4 q^{22} + 3 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−1.00000 −1.61803 1.00000 2.61803 1.61803 4.47214 −1.00000 −0.381966 −2.61803
1.2 −1.00000 0.618034 1.00000 0.381966 −0.618034 −4.47214 −1.00000 −2.61803 −0.381966
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(43\) \( -1 \)

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 3698.2.a.e 2
43.b odd 2 1 86.2.a.b ✓ 2
129.d even 2 1 774.2.a.k 2
172.d even 2 1 688.2.a.e 2
215.d odd 2 1 2150.2.a.t 2
215.g even 4 2 2150.2.b.q 4
301.c even 2 1 4214.2.a.j 2
344.e even 2 1 2752.2.a.p 2
344.h odd 2 1 2752.2.a.k 2
516.h odd 2 1 6192.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
86.2.a.b ✓ 2 43.b odd 2 1
688.2.a.e 2 172.d even 2 1
774.2.a.k 2 129.d even 2 1
2150.2.a.t 2 215.d odd 2 1
2150.2.b.q 4 215.g even 4 2
2752.2.a.k 2 344.h odd 2 1
2752.2.a.p 2 344.e even 2 1
3698.2.a.e 2 1.a even 1 1 trivial
4214.2.a.j 2 301.c even 2 1
6192.2.a.bp 2 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3698))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 20 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 11T + 29 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$37$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$59$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$79$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$97$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
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