Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,3,Mod(689,736)] 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("736.689"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 736.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.0545474569\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{23}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2\sqrt{23}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} + 9 q^{9} - \beta q^{11} - \beta q^{19} - 23 q^{23} + 67 q^{25} - 30 q^{31} + 7 \beta q^{37} + 10 q^{41} + 7 \beta q^{43} - 9 \beta q^{45} + 90 q^{47} + 49 q^{49} + 7 \beta q^{53} + 92 q^{55} + \cdots - 9 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{9} - 46 q^{23} + 134 q^{25} - 60 q^{31} + 20 q^{41} + 180 q^{47} + 98 q^{49} + 184 q^{55} + 84 q^{71} - 108 q^{73} + 162 q^{81} + 184 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1
4.79583
−4.79583
0 0 0 −9.59166 0 0 0 9.00000 0
689.2 0 0 0 9.59166 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
184.e odd 2 1 CM by \(\Q(\sqrt{-46}) \)
8.b even 2 1 inner
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 736.3.e.a 2
4.b odd 2 1 184.3.e.a ✓ 2
8.b even 2 1 inner 736.3.e.a 2
8.d odd 2 1 184.3.e.a ✓ 2
23.b odd 2 1 inner 736.3.e.a 2
92.b even 2 1 184.3.e.a ✓ 2
184.e odd 2 1 CM 736.3.e.a 2
184.h even 2 1 184.3.e.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.3.e.a ✓ 2 4.b odd 2 1
184.3.e.a ✓ 2 8.d odd 2 1
184.3.e.a ✓ 2 92.b even 2 1
184.3.e.a ✓ 2 184.h even 2 1
736.3.e.a 2 1.a even 1 1 trivial
736.3.e.a 2 8.b even 2 1 inner
736.3.e.a 2 23.b odd 2 1 inner
736.3.e.a 2 184.e odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(736, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 92 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 92 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 92 \) Copy content Toggle raw display
$23$ \( (T + 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 30)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4508 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4508 \) Copy content Toggle raw display
$47$ \( (T - 90)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 4508 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 92 \) Copy content Toggle raw display
$67$ \( T^{2} - 4508 \) Copy content Toggle raw display
$71$ \( (T - 42)^{2} \) Copy content Toggle raw display
$73$ \( (T + 54)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 26588 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
show more
show less