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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,6,Mod(37,72)] 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("72.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(11.5476350265\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} - 32 q^{4} - 11 \beta q^{5} + 242 q^{7} - 64 \beta q^{8} + 176 q^{10} + 262 \beta q^{11} + 484 \beta q^{14} + 1024 q^{16} + 352 \beta q^{20} - 4192 q^{22} + 2157 q^{25} - 7744 q^{28} + \cdots + 83514 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} + 484 q^{7} + 352 q^{10} + 2048 q^{16} - 8384 q^{22} + 4314 q^{25} - 15488 q^{28} + 13900 q^{31} - 11264 q^{40} + 83514 q^{49} + 46112 q^{55} - 99488 q^{58} - 65536 q^{64} + 85184 q^{70}+ \cdots + 180484 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
− 1.41421i
1.41421i
− 5.65685i 0 −32.0000 31.1127i 0 242.000 181.019i 0 176.000
37.2 5.65685i 0 −32.0000 − 31.1127i 0 242.000 − 181.019i 0 176.000
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.6.d.a ✓ 2
3.b odd 2 1 inner 72.6.d.a ✓ 2
4.b odd 2 1 288.6.d.a 2
8.b even 2 1 inner 72.6.d.a ✓ 2
8.d odd 2 1 288.6.d.a 2
12.b even 2 1 288.6.d.a 2
24.f even 2 1 288.6.d.a 2
24.h odd 2 1 CM 72.6.d.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.6.d.a ✓ 2 1.a even 1 1 trivial
72.6.d.a ✓ 2 3.b odd 2 1 inner
72.6.d.a ✓ 2 8.b even 2 1 inner
72.6.d.a ✓ 2 24.h odd 2 1 CM
288.6.d.a 2 4.b odd 2 1
288.6.d.a 2 8.d odd 2 1
288.6.d.a 2 12.b even 2 1
288.6.d.a 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 968 \) acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 32 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 968 \) Copy content Toggle raw display
$7$ \( (T - 242)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 549152 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 77327048 \) Copy content Toggle raw display
$31$ \( (T - 6950)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 218405000 \) Copy content Toggle raw display
$59$ \( T^{2} + 2018411648 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 90706)^{2} \) Copy content Toggle raw display
$79$ \( (T + 17050)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 15755255072 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 90242)^{2} \) Copy content Toggle raw display
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