Properties

Label 72.4.d.c
Level $72$
Weight $4$
Character orbit 72.d
Analytic conductor $4.248$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,4,Mod(37,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 4, names="a")
 
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, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] 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: no
Analytic conductor: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-10}, \sqrt{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + 10 q^{7} + (4 \beta_{2} + 2 \beta_1) q^{8} + (2 \beta_{3} - 10) q^{10} + (6 \beta_{2} + 6 \beta_1) q^{11} - 8 \beta_{3} q^{13}+ \cdots - 243 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 40 q^{7} - 40 q^{10} - 184 q^{16} - 240 q^{22} + 340 q^{25} + 120 q^{28} + 248 q^{31} + 704 q^{34} - 560 q^{40} - 1408 q^{46} - 972 q^{49} + 1760 q^{52} - 960 q^{55} + 1560 q^{58} - 1872 q^{64}+ \cdots + 520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} + 2\beta_1 \) 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
−2.34521 1.58114i
−2.34521 + 1.58114i
2.34521 1.58114i
2.34521 + 1.58114i
−2.34521 1.58114i 0 3.00000 + 7.41620i 6.32456i 0 10.0000 4.69042 22.1359i 0 −10.0000 + 14.8324i
37.2 −2.34521 + 1.58114i 0 3.00000 7.41620i 6.32456i 0 10.0000 4.69042 + 22.1359i 0 −10.0000 14.8324i
37.3 2.34521 1.58114i 0 3.00000 7.41620i 6.32456i 0 10.0000 −4.69042 22.1359i 0 −10.0000 14.8324i
37.4 2.34521 + 1.58114i 0 3.00000 + 7.41620i 6.32456i 0 10.0000 −4.69042 + 22.1359i 0 −10.0000 + 14.8324i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.d.c 4
3.b odd 2 1 inner 72.4.d.c 4
4.b odd 2 1 288.4.d.c 4
8.b even 2 1 inner 72.4.d.c 4
8.d odd 2 1 288.4.d.c 4
12.b even 2 1 288.4.d.c 4
16.e even 4 2 2304.4.a.bx 4
16.f odd 4 2 2304.4.a.cc 4
24.f even 2 1 288.4.d.c 4
24.h odd 2 1 inner 72.4.d.c 4
48.i odd 4 2 2304.4.a.bx 4
48.k even 4 2 2304.4.a.cc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.d.c 4 1.a even 1 1 trivial
72.4.d.c 4 3.b odd 2 1 inner
72.4.d.c 4 8.b even 2 1 inner
72.4.d.c 4 24.h odd 2 1 inner
288.4.d.c 4 4.b odd 2 1
288.4.d.c 4 8.d odd 2 1
288.4.d.c 4 12.b even 2 1
288.4.d.c 4 24.f even 2 1
2304.4.a.bx 4 16.e even 4 2
2304.4.a.bx 4 48.i odd 4 2
2304.4.a.cc 4 16.f odd 4 2
2304.4.a.cc 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 40 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$7$ \( (T - 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1440)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3520)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5632)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 14080)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 22528)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 60840)^{2} \) Copy content Toggle raw display
$31$ \( (T - 62)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3520)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 140800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 14080)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 202752)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 17640)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 538240)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 285120)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 506880)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 30)^{4} \) Copy content Toggle raw display
$79$ \( (T - 94)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 449440)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 563200)^{2} \) Copy content Toggle raw display
$97$ \( (T - 130)^{4} \) Copy content Toggle raw display
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