Properties

Label 54.2.e.a
Level $54$
Weight $2$
Character orbit 54.e
Analytic conductor $0.431$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,2,Mod(7,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 54.e (of order \(9\), degree \(6\), 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: \(0.431192170915\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{18} q^{2} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{5}+ \cdots - 3 \zeta_{18}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18} q^{2} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{5}+ \cdots + ( - 3 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 9 q^{6} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{5} - 9 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 12 q^{13} - 3 q^{14} - 6 q^{17} + 9 q^{19} + 6 q^{20} + 3 q^{22} - 6 q^{23} - 9 q^{25} + 18 q^{26} + 27 q^{27} - 12 q^{28} + 15 q^{29} + 9 q^{30} - 18 q^{31} - 9 q^{33} + 6 q^{34} + 3 q^{35} + 9 q^{36} + 15 q^{37} - 15 q^{38} + 3 q^{40} - 3 q^{41} - 9 q^{42} - 18 q^{43} - 3 q^{44} - 18 q^{45} - 3 q^{46} + 9 q^{47} - 9 q^{48} + 21 q^{49} - 9 q^{50} + 27 q^{51} - 12 q^{52} - 12 q^{53} - 18 q^{55} - 3 q^{56} - 27 q^{57} + 3 q^{58} - 6 q^{59} + 18 q^{61} - 12 q^{62} - 9 q^{63} - 3 q^{64} + 21 q^{65} + 18 q^{66} - 9 q^{67} - 15 q^{68} - 3 q^{70} + 12 q^{71} + 3 q^{73} - 3 q^{74} - 18 q^{75} + 15 q^{76} + 39 q^{77} + 18 q^{78} + 33 q^{79} + 6 q^{80} - 6 q^{82} - 18 q^{83} + 9 q^{84} + 27 q^{85} + 18 q^{86} + 9 q^{87} + 12 q^{88} - 15 q^{89} - 9 q^{90} - 12 q^{91} + 12 q^{92} + 27 q^{93} + 21 q^{95} - 12 q^{97} - 3 q^{98} - 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.939693 0.342020i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i −1.11334 1.32683i 0.766044 0.642788i 0.439693 + 2.49362i −1.50000 0.866025i −1.79813 1.50881i 0.500000 0.866025i −0.520945 + 2.95442i 1.26604 + 2.19285i
13.1 −0.173648 0.984808i −0.592396 1.62760i −0.939693 + 0.342020i −0.673648 0.565258i −1.50000 + 0.866025i 3.31908 + 1.20805i 0.500000 + 0.866025i −2.29813 + 1.92836i −0.439693 + 0.761570i
25.1 −0.173648 + 0.984808i −0.592396 + 1.62760i −0.939693 0.342020i −0.673648 + 0.565258i −1.50000 0.866025i 3.31908 1.20805i 0.500000 0.866025i −2.29813 1.92836i −0.439693 0.761570i
31.1 0.939693 + 0.342020i −1.11334 + 1.32683i 0.766044 + 0.642788i 0.439693 2.49362i −1.50000 + 0.866025i −1.79813 + 1.50881i 0.500000 + 0.866025i −0.520945 2.95442i 1.26604 2.19285i
43.1 −0.766044 0.642788i 1.70574 0.300767i 0.173648 + 0.984808i −1.26604 0.460802i −1.50000 0.866025i −0.0209445 + 0.118782i 0.500000 0.866025i 2.81908 1.02606i 0.673648 + 1.16679i
49.1 −0.766044 + 0.642788i 1.70574 + 0.300767i 0.173648 0.984808i −1.26604 + 0.460802i −1.50000 + 0.866025i −0.0209445 0.118782i 0.500000 + 0.866025i 2.81908 + 1.02606i 0.673648 1.16679i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.2.e.a 6
3.b odd 2 1 162.2.e.a 6
4.b odd 2 1 432.2.u.a 6
9.c even 3 1 486.2.e.b 6
9.c even 3 1 486.2.e.d 6
9.d odd 6 1 486.2.e.a 6
9.d odd 6 1 486.2.e.c 6
27.e even 9 1 inner 54.2.e.a 6
27.e even 9 1 486.2.e.b 6
27.e even 9 1 486.2.e.d 6
27.e even 9 1 1458.2.a.a 3
27.e even 9 2 1458.2.c.d 6
27.f odd 18 1 162.2.e.a 6
27.f odd 18 1 486.2.e.a 6
27.f odd 18 1 486.2.e.c 6
27.f odd 18 1 1458.2.a.d 3
27.f odd 18 2 1458.2.c.a 6
108.j odd 18 1 432.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a 6 1.a even 1 1 trivial
54.2.e.a 6 27.e even 9 1 inner
162.2.e.a 6 3.b odd 2 1
162.2.e.a 6 27.f odd 18 1
432.2.u.a 6 4.b odd 2 1
432.2.u.a 6 108.j odd 18 1
486.2.e.a 6 9.d odd 6 1
486.2.e.a 6 27.f odd 18 1
486.2.e.b 6 9.c even 3 1
486.2.e.b 6 27.e even 9 1
486.2.e.c 6 9.d odd 6 1
486.2.e.c 6 27.f odd 18 1
486.2.e.d 6 9.c even 3 1
486.2.e.d 6 27.e even 9 1
1458.2.a.a 3 27.e even 9 1
1458.2.a.d 3 27.f odd 18 1
1458.2.c.a 6 27.f odd 18 2
1458.2.c.d 6 27.e even 9 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 24T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 25281 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 32041 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} - 15 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( T^{6} + 18 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$37$ \( T^{6} - 15 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 47961 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$61$ \( T^{6} - 18 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + \cdots + 106929 \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + \cdots + 72361 \) Copy content Toggle raw display
$79$ \( T^{6} - 33 T^{5} + \cdots + 466489 \) Copy content Toggle raw display
$83$ \( T^{6} + 18 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( T^{6} + 15 T^{5} + \cdots + 25281 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
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