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Properties

Label 72.5.p.a
Level $72$
Weight $5$
Character orbit 72.p
Analytic conductor $7.443$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,5,Mod(43,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + ( - \beta_{3} + 7 \beta_{2} + \beta_1) q^{3} + (16 \beta_{2} - 16) q^{4} + ( - 28 \beta_{2} - 4 \beta_1 + 28) q^{6} + 64 q^{8} + (17 \beta_{2} + 14 \beta_1 - 17) q^{9} + (21 \beta_{3} - 23 \beta_{2} + 21 \beta_1) q^{11}+ \cdots + (35 \beta_{3} + 18816 \beta_{2} + \cdots - 9017) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 14 q^{3} - 32 q^{4} + 56 q^{6} + 256 q^{8} - 34 q^{9} - 46 q^{11} - 448 q^{12} - 512 q^{16} + 1148 q^{17} + 272 q^{18} - 868 q^{19} - 184 q^{22} + 896 q^{24} - 1250 q^{25} + 1316 q^{27}+ \cdots + 1564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
−2.00000 + 3.46410i −1.39898 − 8.89060i −8.00000 − 13.8564i 0 33.5959 + 12.9350i 0 64.0000 −77.0857 + 24.8755i 0
43.2 −2.00000 + 3.46410i 8.39898 − 3.23375i −8.00000 − 13.8564i 0 −5.59592 + 35.5624i 0 64.0000 60.0857 − 54.3204i 0
67.1 −2.00000 − 3.46410i −1.39898 + 8.89060i −8.00000 + 13.8564i 0 33.5959 − 12.9350i 0 64.0000 −77.0857 − 24.8755i 0
67.2 −2.00000 − 3.46410i 8.39898 + 3.23375i −8.00000 + 13.8564i 0 −5.59592 − 35.5624i 0 64.0000 60.0857 + 54.3204i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.5.p.a ✓ 4
3.b odd 2 1 216.5.p.a 4
4.b odd 2 1 288.5.t.a 4
8.b even 2 1 288.5.t.a 4
8.d odd 2 1 CM 72.5.p.a ✓ 4
9.c even 3 1 inner 72.5.p.a ✓ 4
9.d odd 6 1 216.5.p.a 4
12.b even 2 1 864.5.t.a 4
24.f even 2 1 216.5.p.a 4
24.h odd 2 1 864.5.t.a 4
36.f odd 6 1 288.5.t.a 4
36.h even 6 1 864.5.t.a 4
72.j odd 6 1 864.5.t.a 4
72.l even 6 1 216.5.p.a 4
72.n even 6 1 288.5.t.a 4
72.p odd 6 1 inner 72.5.p.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.p.a ✓ 4 1.a even 1 1 trivial
72.5.p.a ✓ 4 8.d odd 2 1 CM
72.5.p.a ✓ 4 9.c even 3 1 inner
72.5.p.a ✓ 4 72.p odd 6 1 inner
216.5.p.a 4 3.b odd 2 1
216.5.p.a 4 9.d odd 6 1
216.5.p.a 4 24.f even 2 1
216.5.p.a 4 72.l even 6 1
288.5.t.a 4 4.b odd 2 1
288.5.t.a 4 8.b even 2 1
288.5.t.a 4 36.f odd 6 1
288.5.t.a 4 72.n even 6 1
864.5.t.a 4 12.b even 2 1
864.5.t.a 4 24.h odd 2 1
864.5.t.a 4 36.h even 6 1
864.5.t.a 4 72.j odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 14 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 1747825249 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 574 T + 78913)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 434 T - 202607)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 47952398004289 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 4030461775201 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9506 T + 5169313)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 11186 T + 125126596)^{2} \) Copy content Toggle raw display
$89$ \( (T - 5474)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
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