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Properties

Label 216.3.e.a
Level $216$
Weight $3$
Character orbit 216.e
Analytic conductor $5.886$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,3,Mod(161,216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("216.161"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.88557371018\)
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: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 3 q^{7} + 7 \beta q^{11} + 7 q^{13} + 5 \beta q^{17} - 19 q^{19} + \beta q^{23} + 17 q^{25} + 18 \beta q^{29} - 10 q^{31} - 3 \beta q^{35} + 63 q^{37} - 18 \beta q^{41} - 50 q^{43} + 15 \beta q^{47} + \cdots - 97 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7} + 14 q^{13} - 38 q^{19} + 34 q^{25} - 20 q^{31} + 126 q^{37} - 100 q^{43} - 80 q^{49} - 112 q^{55} + 158 q^{61} + 154 q^{67} - 34 q^{73} - 22 q^{79} - 80 q^{85} - 42 q^{91} - 194 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 1.41421i
1.41421i
0 0 0 − 2.82843i 0 −3.00000 0 0 0
161.2 0 0 0 2.82843i 0 −3.00000 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.e.a ✓ 2
3.b odd 2 1 inner 216.3.e.a ✓ 2
4.b odd 2 1 432.3.e.f 2
8.b even 2 1 1728.3.e.h 2
8.d odd 2 1 1728.3.e.j 2
9.c even 3 2 648.3.m.c 4
9.d odd 6 2 648.3.m.c 4
12.b even 2 1 432.3.e.f 2
24.f even 2 1 1728.3.e.j 2
24.h odd 2 1 1728.3.e.h 2
36.f odd 6 2 1296.3.q.g 4
36.h even 6 2 1296.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.a ✓ 2 1.a even 1 1 trivial
216.3.e.a ✓ 2 3.b odd 2 1 inner
432.3.e.f 2 4.b odd 2 1
432.3.e.f 2 12.b even 2 1
648.3.m.c 4 9.c even 3 2
648.3.m.c 4 9.d odd 6 2
1296.3.q.g 4 36.f odd 6 2
1296.3.q.g 4 36.h even 6 2
1728.3.e.h 2 8.b even 2 1
1728.3.e.h 2 24.h odd 2 1
1728.3.e.j 2 8.d odd 2 1
1728.3.e.j 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 392 \) Copy content Toggle raw display
$13$ \( (T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 200 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 2592 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T - 63)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2592 \) Copy content Toggle raw display
$43$ \( (T + 50)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1800 \) Copy content Toggle raw display
$53$ \( T^{2} + 5408 \) Copy content Toggle raw display
$59$ \( T^{2} + 9800 \) Copy content Toggle raw display
$61$ \( (T - 79)^{2} \) Copy content Toggle raw display
$67$ \( (T - 77)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 6272 \) Copy content Toggle raw display
$73$ \( (T + 17)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1568 \) Copy content Toggle raw display
$89$ \( T^{2} + 1800 \) Copy content Toggle raw display
$97$ \( (T + 97)^{2} \) Copy content Toggle raw display
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