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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3184,2,Mod(1,3184)] 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("3184.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3184 = 2^{4} \cdot 199 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3184.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,-2] 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: \(25.4243680036\)
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 398)
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (2 \beta - 2) q^{5} + (\beta + 3) q^{7} + (3 \beta - 1) q^{9} + ( - \beta + 4) q^{11} - 2 q^{13} + 2 \beta q^{15} + ( - 4 \beta + 4) q^{17} + (5 \beta + 4) q^{21} + (7 \beta - 2) q^{23}+ \cdots + (10 \beta - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{5} + 7 q^{7} + q^{9} + 7 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} + 13 q^{21} + 3 q^{23} + 2 q^{25} - 6 q^{29} + 5 q^{31} + 8 q^{33} - 2 q^{35} - 9 q^{37} - 6 q^{39} + 6 q^{41} - 2 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
0 0.381966 0 −3.23607 0 2.38197 0 −2.85410 0
1.2 0 2.61803 0 1.23607 0 4.61803 0 3.85410 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(199\) \( +1 \)

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 3184.2.a.f 2
4.b odd 2 1 398.2.a.c ✓ 2
12.b even 2 1 3582.2.a.g 2
20.d odd 2 1 9950.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
398.2.a.c ✓ 2 4.b odd 2 1
3184.2.a.f 2 1.a even 1 1 trivial
3582.2.a.g 2 12.b even 2 1
9950.2.a.r 2 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3184))\):

\( T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 11 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 11 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 11 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 17T + 61 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T - 55 \) Copy content Toggle raw display
$83$ \( T^{2} + 7T - 199 \) Copy content Toggle raw display
$89$ \( T^{2} - 33T + 271 \) Copy content Toggle raw display
$97$ \( T^{2} + 22T + 76 \) Copy content Toggle raw display
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