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Properties

Label 216.3.m.a
Level $216$
Weight $3$
Character orbit 216.m
Analytic conductor $5.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(17,216)]
 
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([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.m (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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{11} + ( - 2 \beta_{3} + 7 \beta_{2} + \beta_1 - 7) q^{13} + ( - 2 \beta_{3} + 16 \beta_{2} - 8) q^{17} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{19} + ( - 5 \beta_{2} + \beta_1 - 5) q^{23} + ( - 2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{25} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{29} + (2 \beta_{3} - 37 \beta_{2} - \beta_1 + 37) q^{31} + ( - 58 \beta_{2} + 29) q^{35} + ( - 2 \beta_{3} + 4 \beta_1 - 30) q^{37} + (23 \beta_{2} + 23) q^{41} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{43} + (3 \beta_{3} + 29 \beta_{2} - 3 \beta_1 - 58) q^{47} + (12 \beta_{3} + 56 \beta_{2} - 6 \beta_1 - 56) q^{49} + (8 \beta_{3} + 64 \beta_{2} - 32) q^{53} + ( - \beta_{3} + 2 \beta_1 + 55) q^{55} + (3 \beta_{2} - 2 \beta_1 + 3) q^{59} + ( - \beta_{3} - 31 \beta_{2} - \beta_1) q^{61} + (10 \beta_{3} - 39 \beta_{2} - 10 \beta_1 + 78) q^{65} + (20 \beta_{3} + 11 \beta_{2} - 10 \beta_1 - 11) q^{67} + ( - 2 \beta_{3} - 48 \beta_{2} + 24) q^{71} + ( - 6 \beta_{3} + 12 \beta_1 + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1 - 73) q^{77} + (7 \beta_{3} - 43 \beta_{2} + 7 \beta_1) q^{79} + (14 \beta_{3} - 11 \beta_{2} - 14 \beta_1 + 22) q^{83} + (20 \beta_{3} - 88 \beta_{2} - 10 \beta_1 + 88) q^{85} + ( - 6 \beta_{3} + 48 \beta_{2} - 24) q^{89} + ( - 4 \beta_{3} + 8 \beta_1 - 75) q^{91} + (62 \beta_{2} - 4 \beta_1 + 62) q^{95} + ( - 2 \beta_{3} + 121 \beta_{2} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 6 q^{7} + 18 q^{11} - 14 q^{13} + 8 q^{19} - 30 q^{23} + 20 q^{25} - 6 q^{29} + 74 q^{31} - 120 q^{37} + 138 q^{41} + 10 q^{43} - 174 q^{47} - 112 q^{49} + 220 q^{55} + 18 q^{59} - 62 q^{61} + 234 q^{65} - 22 q^{67} + 40 q^{73} - 438 q^{77} - 86 q^{79} + 66 q^{83} + 176 q^{85} - 300 q^{91} + 372 q^{95} + 242 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 4\nu \) 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 \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
0 0 0 −6.39898 − 3.69445i 0 3.39898 + 5.88721i 0 0 0
17.2 0 0 0 3.39898 + 1.96240i 0 −6.39898 − 11.0834i 0 0 0
89.1 0 0 0 −6.39898 + 3.69445i 0 3.39898 − 5.88721i 0 0 0
89.2 0 0 0 3.39898 − 1.96240i 0 −6.39898 + 11.0834i 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
9.d 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.m.a 4
3.b odd 2 1 72.3.m.a ✓ 4
4.b odd 2 1 432.3.q.c 4
8.b even 2 1 1728.3.q.e 4
8.d odd 2 1 1728.3.q.f 4
9.c even 3 1 72.3.m.a ✓ 4
9.c even 3 1 648.3.e.b 4
9.d odd 6 1 inner 216.3.m.a 4
9.d odd 6 1 648.3.e.b 4
12.b even 2 1 144.3.q.d 4
24.f even 2 1 576.3.q.c 4
24.h odd 2 1 576.3.q.h 4
36.f odd 6 1 144.3.q.d 4
36.f odd 6 1 1296.3.e.c 4
36.h even 6 1 432.3.q.c 4
36.h even 6 1 1296.3.e.c 4
72.j odd 6 1 1728.3.q.e 4
72.l even 6 1 1728.3.q.f 4
72.n even 6 1 576.3.q.h 4
72.p odd 6 1 576.3.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a ✓ 4 3.b odd 2 1
72.3.m.a ✓ 4 9.c even 3 1
144.3.q.d 4 12.b even 2 1
144.3.q.d 4 36.f odd 6 1
216.3.m.a 4 1.a even 1 1 trivial
216.3.m.a 4 9.d odd 6 1 inner
432.3.q.c 4 4.b odd 2 1
432.3.q.c 4 36.h even 6 1
576.3.q.c 4 24.f even 2 1
576.3.q.c 4 72.p odd 6 1
576.3.q.h 4 24.h odd 2 1
576.3.q.h 4 72.n even 6 1
648.3.e.b 4 9.c even 3 1
648.3.e.b 4 9.d odd 6 1
1296.3.e.c 4 36.f odd 6 1
1296.3.e.c 4 36.h even 6 1
1728.3.q.e 4 8.b even 2 1
1728.3.q.e 4 72.j odd 6 1
1728.3.q.f 4 8.d odd 2 1
1728.3.q.f 4 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6T_{5}^{3} - 17T_{5}^{2} - 174T_{5} + 841 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} - 17 T^{2} - 174 T + 841 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 123 T^{2} + \cdots + 7569 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + 7 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$13$ \( T^{4} + 14 T^{3} + 243 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$17$ \( T^{4} + 640T^{2} + 4096 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 380)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 30 T^{3} + 343 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} - 273 T^{2} + \cdots + 81225 \) Copy content Toggle raw display
$31$ \( T^{4} - 74 T^{3} + 4203 T^{2} + \cdots + 1620529 \) Copy content Toggle raw display
$37$ \( (T^{2} + 60 T + 516)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 69 T + 1587)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + 1611 T^{2} + \cdots + 2283121 \) Copy content Toggle raw display
$47$ \( T^{4} + 174 T^{3} + 12327 T^{2} + \cdots + 4995225 \) Copy content Toggle raw display
$53$ \( T^{4} + 10240 T^{2} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + 7 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$61$ \( T^{4} + 62 T^{3} + 2979 T^{2} + \cdots + 748225 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + 9963 T^{2} + \cdots + 89851441 \) Copy content Toggle raw display
$71$ \( T^{4} + 3712 T^{2} + \cdots + 2560000 \) Copy content Toggle raw display
$73$ \( (T^{2} - 20 T - 3356)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 86 T^{3} + 10251 T^{2} + \cdots + 8151025 \) Copy content Toggle raw display
$83$ \( T^{4} - 66 T^{3} - 4457 T^{2} + \cdots + 34916281 \) Copy content Toggle raw display
$89$ \( T^{4} + 5760 T^{2} + 331776 \) Copy content Toggle raw display
$97$ \( T^{4} - 242 T^{3} + \cdots + 203262049 \) Copy content Toggle raw display
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