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Properties

Label 216.3.p.a
Level $216$
Weight $3$
Character orbit 216.p
Analytic conductor $5.886$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(19,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
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: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + 8 q^{8} + (\beta_{2} - 7 \beta_1) q^{11} - 16 \beta_1 q^{16} + (2 \beta_{3} + 1) q^{17} + (\beta_{3} + 17) q^{19} + (2 \beta_{3} - 2 \beta_{2} + 14 \beta_1 - 14) q^{22} - 25 \beta_1 q^{25} + (32 \beta_1 - 32) q^{32} + ( - 4 \beta_{2} - 2 \beta_1) q^{34} + ( - 2 \beta_{2} - 34 \beta_1) q^{38} + (4 \beta_{3} - 4 \beta_{2} - 23 \beta_1 + 23) q^{41} + (5 \beta_{2} + 7 \beta_1) q^{43} + ( - 4 \beta_{3} + 28) q^{44} + (49 \beta_1 - 49) q^{49} + (50 \beta_1 - 50) q^{50} + ( - 5 \beta_{3} + 5 \beta_{2} + \cdots + 41) q^{59}+ \cdots + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 14 q^{11} - 32 q^{16} + 4 q^{17} + 68 q^{19} - 28 q^{22} - 50 q^{25} - 64 q^{32} - 4 q^{34} - 68 q^{38} + 46 q^{41} + 14 q^{43} + 112 q^{44} - 98 q^{49} - 100 q^{50} + 82 q^{59} + 256 q^{64} + 62 q^{67} - 8 q^{68} + 284 q^{73} - 136 q^{76} - 184 q^{82} + 316 q^{83} + 28 q^{86} - 112 q^{88} - 584 q^{89} - 94 q^{97} + 392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} + 12\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.00000 + 1.73205i 0 −2.00000 − 3.46410i 0 0 0 8.00000 0 0
19.2 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i 0 0 0 8.00000 0 0
91.1 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 0 0 0 8.00000 0 0
91.2 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 0 0 0 8.00000 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.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 216.3.p.a 4
3.b odd 2 1 72.3.p.a ✓ 4
4.b odd 2 1 864.3.t.a 4
8.b even 2 1 864.3.t.a 4
8.d odd 2 1 CM 216.3.p.a 4
9.c even 3 1 inner 216.3.p.a 4
9.c even 3 1 648.3.b.b 2
9.d odd 6 1 72.3.p.a ✓ 4
9.d odd 6 1 648.3.b.a 2
12.b even 2 1 288.3.t.a 4
24.f even 2 1 72.3.p.a ✓ 4
24.h odd 2 1 288.3.t.a 4
36.f odd 6 1 864.3.t.a 4
36.f odd 6 1 2592.3.b.a 2
36.h even 6 1 288.3.t.a 4
36.h even 6 1 2592.3.b.b 2
72.j odd 6 1 288.3.t.a 4
72.j odd 6 1 2592.3.b.b 2
72.l even 6 1 72.3.p.a ✓ 4
72.l even 6 1 648.3.b.a 2
72.n even 6 1 864.3.t.a 4
72.n even 6 1 2592.3.b.a 2
72.p odd 6 1 inner 216.3.p.a 4
72.p odd 6 1 648.3.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.p.a ✓ 4 3.b odd 2 1
72.3.p.a ✓ 4 9.d odd 6 1
72.3.p.a ✓ 4 24.f even 2 1
72.3.p.a ✓ 4 72.l even 6 1
216.3.p.a 4 1.a even 1 1 trivial
216.3.p.a 4 8.d odd 2 1 CM
216.3.p.a 4 9.c even 3 1 inner
216.3.p.a 4 72.p odd 6 1 inner
288.3.t.a 4 12.b even 2 1
288.3.t.a 4 24.h odd 2 1
288.3.t.a 4 36.h even 6 1
288.3.t.a 4 72.j odd 6 1
648.3.b.a 2 9.d odd 6 1
648.3.b.a 2 72.l even 6 1
648.3.b.b 2 9.c even 3 1
648.3.b.b 2 72.p odd 6 1
864.3.t.a 4 4.b odd 2 1
864.3.t.a 4 8.b even 2 1
864.3.t.a 4 36.f odd 6 1
864.3.t.a 4 72.n even 6 1
2592.3.b.a 2 36.f odd 6 1
2592.3.b.a 2 72.n even 6 1
2592.3.b.b 2 36.h even 6 1
2592.3.b.b 2 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) 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
$11$ \( T^{4} + 14 T^{3} + \cdots + 27889 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 863)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 34 T + 73)^{2} \) 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
$41$ \( T^{4} - 46 T^{3} + \cdots + 8567329 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + \cdots + 28633201 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 82 T^{3} + \cdots + 13830961 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 62 T^{3} + \cdots + 92602129 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 142 T + 4177)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 158 T + 24964)^{2} \) Copy content Toggle raw display
$89$ \( (T + 146)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 94 T^{3} + \cdots + 376010881 \) Copy content Toggle raw display
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