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Properties

Label 216.2.l.a
Level $216$
Weight $2$
Character orbit 216.l
Analytic conductor $1.725$
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] = [216,2,Mod(35,216)] 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("216.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.l (of order \(6\), degree \(2\), not 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: \(1.72476868366\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} + (3 \beta_{2} - \beta_1 + 3) q^{11} + (4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{3} - 6 \beta_{2} + 3) q^{17} + (3 \beta_{3} - 6 \beta_1 - 1) q^{19}+ \cdots - 7 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 18 q^{11} - 8 q^{16} - 4 q^{19} - 4 q^{22} + 10 q^{25} + 8 q^{34} - 36 q^{38} - 18 q^{41} - 10 q^{43} - 14 q^{49} - 18 q^{59} - 32 q^{64} + 14 q^{67} + 36 q^{68} - 4 q^{73} - 4 q^{76} + 32 q^{82}+ \cdots - 10 q^{97}+O(q^{100}) \) 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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i 0 1.00000 + 1.73205i 0 0 0 − 2.82843i 0 0
35.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
179.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 0 0 0 2.82843i 0 0
179.2 1.22474 − 0.707107i 0 1.00000 − 1.73205i 0 0 0 − 2.82843i 0 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.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.l.a 4
3.b odd 2 1 72.2.l.a ✓ 4
4.b odd 2 1 864.2.p.a 4
8.b even 2 1 864.2.p.a 4
8.d odd 2 1 CM 216.2.l.a 4
9.c even 3 1 72.2.l.a ✓ 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 216.2.l.a 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 288.2.p.a 4
24.f even 2 1 72.2.l.a ✓ 4
24.h odd 2 1 288.2.p.a 4
36.f odd 6 1 288.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 864.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 864.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 216.2.l.a 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 288.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 72.2.l.a ✓ 4
72.p odd 6 1 648.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a ✓ 4 3.b odd 2 1
72.2.l.a ✓ 4 9.c even 3 1
72.2.l.a ✓ 4 24.f even 2 1
72.2.l.a ✓ 4 72.p odd 6 1
216.2.l.a 4 1.a even 1 1 trivial
216.2.l.a 4 8.d odd 2 1 CM
216.2.l.a 4 9.d odd 6 1 inner
216.2.l.a 4 72.l even 6 1 inner
288.2.p.a 4 12.b even 2 1
288.2.p.a 4 24.h odd 2 1
288.2.p.a 4 36.f odd 6 1
288.2.p.a 4 72.n even 6 1
648.2.f.a 4 9.c even 3 1
648.2.f.a 4 9.d odd 6 1
648.2.f.a 4 72.l even 6 1
648.2.f.a 4 72.p odd 6 1
864.2.p.a 4 4.b odd 2 1
864.2.p.a 4 8.b even 2 1
864.2.p.a 4 36.h even 6 1
864.2.p.a 4 72.j odd 6 1
2592.2.f.a 4 36.f odd 6 1
2592.2.f.a 4 36.h even 6 1
2592.2.f.a 4 72.j odd 6 1
2592.2.f.a 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 70T^{2} + 361 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 53)^{2} \) 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
$41$ \( T^{4} + 18 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + \cdots + 36481 \) Copy content Toggle raw display
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