Properties

Label 27.3.d.a
Level $27$
Weight $3$
Character orbit 27.d
Analytic conductor $0.736$
Analytic rank $0$
Dimension $2$
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] = [27,3,Mod(8,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 27.d (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: \(0.735696713773\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + (2 \zeta_{6} - 4) q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 10 \zeta_{6} + 5) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + (2 \zeta_{6} - 4) q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 10 \zeta_{6} + 5) q^{8} - 6 q^{10} + (\zeta_{6} + 1) q^{11} + 4 \zeta_{6} q^{13} + (2 \zeta_{6} - 4) q^{14} + ( - 11 \zeta_{6} + 11) q^{16} + (18 \zeta_{6} - 9) q^{17} + 11 q^{19} + (2 \zeta_{6} + 2) q^{20} + 3 \zeta_{6} q^{22} + ( - 16 \zeta_{6} + 32) q^{23} + (13 \zeta_{6} - 13) q^{25} + (8 \zeta_{6} - 4) q^{26} + 2 q^{28} + ( - 26 \zeta_{6} - 26) q^{29} - 32 \zeta_{6} q^{31} + (9 \zeta_{6} - 18) q^{32} + (27 \zeta_{6} - 27) q^{34} + ( - 8 \zeta_{6} + 4) q^{35} - 34 q^{37} + (11 \zeta_{6} + 11) q^{38} + 30 \zeta_{6} q^{40} + ( - 7 \zeta_{6} + 14) q^{41} + ( - 61 \zeta_{6} + 61) q^{43} + ( - 2 \zeta_{6} + 1) q^{44} + 48 q^{46} + (28 \zeta_{6} + 28) q^{47} + 45 \zeta_{6} q^{49} + (13 \zeta_{6} - 26) q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 6 q^{55} + (10 \zeta_{6} + 10) q^{56} - 78 \zeta_{6} q^{58} + (29 \zeta_{6} - 58) q^{59} + (56 \zeta_{6} - 56) q^{61} + ( - 64 \zeta_{6} + 32) q^{62} - 71 q^{64} + ( - 8 \zeta_{6} - 8) q^{65} + 31 \zeta_{6} q^{67} + ( - 9 \zeta_{6} + 18) q^{68} + ( - 12 \zeta_{6} + 12) q^{70} + ( - 36 \zeta_{6} + 18) q^{71} + 65 q^{73} + ( - 34 \zeta_{6} - 34) q^{74} - 11 \zeta_{6} q^{76} + (2 \zeta_{6} - 4) q^{77} + (38 \zeta_{6} - 38) q^{79} + (44 \zeta_{6} - 22) q^{80} + 21 q^{82} + (28 \zeta_{6} + 28) q^{83} - 54 \zeta_{6} q^{85} + ( - 61 \zeta_{6} + 122) q^{86} + ( - 15 \zeta_{6} + 15) q^{88} + (144 \zeta_{6} - 72) q^{89} - 8 q^{91} + ( - 16 \zeta_{6} - 16) q^{92} + 84 \zeta_{6} q^{94} + (22 \zeta_{6} - 44) q^{95} + ( - 115 \zeta_{6} + 115) q^{97} + (90 \zeta_{6} - 45) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - q^{4} - 6 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - q^{4} - 6 q^{5} - 2 q^{7} - 12 q^{10} + 3 q^{11} + 4 q^{13} - 6 q^{14} + 11 q^{16} + 22 q^{19} + 6 q^{20} + 3 q^{22} + 48 q^{23} - 13 q^{25} + 4 q^{28} - 78 q^{29} - 32 q^{31} - 27 q^{32} - 27 q^{34} - 68 q^{37} + 33 q^{38} + 30 q^{40} + 21 q^{41} + 61 q^{43} + 96 q^{46} + 84 q^{47} + 45 q^{49} - 39 q^{50} + 4 q^{52} - 12 q^{55} + 30 q^{56} - 78 q^{58} - 87 q^{59} - 56 q^{61} - 142 q^{64} - 24 q^{65} + 31 q^{67} + 27 q^{68} + 12 q^{70} + 130 q^{73} - 102 q^{74} - 11 q^{76} - 6 q^{77} - 38 q^{79} + 42 q^{82} + 84 q^{83} - 54 q^{85} + 183 q^{86} + 15 q^{88} - 16 q^{91} - 48 q^{92} + 84 q^{94} - 66 q^{95} + 115 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i 0 −0.500000 0.866025i −3.00000 + 1.73205i 0 −1.00000 + 1.73205i 8.66025i 0 −6.00000
17.1 1.50000 0.866025i 0 −0.500000 + 0.866025i −3.00000 1.73205i 0 −1.00000 1.73205i 8.66025i 0 −6.00000
\(n\): 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 27.3.d.a 2
3.b odd 2 1 9.3.d.a 2
4.b odd 2 1 432.3.q.a 2
5.b even 2 1 675.3.j.a 2
5.c odd 4 2 675.3.i.a 4
8.b even 2 1 1728.3.q.a 2
8.d odd 2 1 1728.3.q.b 2
9.c even 3 1 9.3.d.a 2
9.c even 3 1 81.3.b.a 2
9.d odd 6 1 inner 27.3.d.a 2
9.d odd 6 1 81.3.b.a 2
12.b even 2 1 144.3.q.a 2
15.d odd 2 1 225.3.j.a 2
15.e even 4 2 225.3.i.a 4
21.c even 2 1 441.3.r.a 2
21.g even 6 1 441.3.j.b 2
21.g even 6 1 441.3.n.a 2
21.h odd 6 1 441.3.j.a 2
21.h odd 6 1 441.3.n.b 2
24.f even 2 1 576.3.q.a 2
24.h odd 2 1 576.3.q.b 2
36.f odd 6 1 144.3.q.a 2
36.f odd 6 1 1296.3.e.a 2
36.h even 6 1 432.3.q.a 2
36.h even 6 1 1296.3.e.a 2
45.h odd 6 1 675.3.j.a 2
45.j even 6 1 225.3.j.a 2
45.k odd 12 2 225.3.i.a 4
45.l even 12 2 675.3.i.a 4
63.g even 3 1 441.3.j.a 2
63.h even 3 1 441.3.n.b 2
63.k odd 6 1 441.3.j.b 2
63.l odd 6 1 441.3.r.a 2
63.t odd 6 1 441.3.n.a 2
72.j odd 6 1 1728.3.q.a 2
72.l even 6 1 1728.3.q.b 2
72.n even 6 1 576.3.q.b 2
72.p odd 6 1 576.3.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 3.b odd 2 1
9.3.d.a 2 9.c even 3 1
27.3.d.a 2 1.a even 1 1 trivial
27.3.d.a 2 9.d odd 6 1 inner
81.3.b.a 2 9.c even 3 1
81.3.b.a 2 9.d odd 6 1
144.3.q.a 2 12.b even 2 1
144.3.q.a 2 36.f odd 6 1
225.3.i.a 4 15.e even 4 2
225.3.i.a 4 45.k odd 12 2
225.3.j.a 2 15.d odd 2 1
225.3.j.a 2 45.j even 6 1
432.3.q.a 2 4.b odd 2 1
432.3.q.a 2 36.h even 6 1
441.3.j.a 2 21.h odd 6 1
441.3.j.a 2 63.g even 3 1
441.3.j.b 2 21.g even 6 1
441.3.j.b 2 63.k odd 6 1
441.3.n.a 2 21.g even 6 1
441.3.n.a 2 63.t odd 6 1
441.3.n.b 2 21.h odd 6 1
441.3.n.b 2 63.h even 3 1
441.3.r.a 2 21.c even 2 1
441.3.r.a 2 63.l odd 6 1
576.3.q.a 2 24.f even 2 1
576.3.q.a 2 72.p odd 6 1
576.3.q.b 2 24.h odd 2 1
576.3.q.b 2 72.n even 6 1
675.3.i.a 4 5.c odd 4 2
675.3.i.a 4 45.l even 12 2
675.3.j.a 2 5.b even 2 1
675.3.j.a 2 45.h odd 6 1
1296.3.e.a 2 36.f odd 6 1
1296.3.e.a 2 36.h even 6 1
1728.3.q.a 2 8.b even 2 1
1728.3.q.a 2 72.j odd 6 1
1728.3.q.b 2 8.d odd 2 1
1728.3.q.b 2 72.l even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 243 \) Copy content Toggle raw display
$19$ \( (T - 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$29$ \( T^{2} + 78T + 2028 \) Copy content Toggle raw display
$31$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$37$ \( (T + 34)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$43$ \( T^{2} - 61T + 3721 \) Copy content Toggle raw display
$47$ \( T^{2} - 84T + 2352 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 87T + 2523 \) Copy content Toggle raw display
$61$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$67$ \( T^{2} - 31T + 961 \) Copy content Toggle raw display
$71$ \( T^{2} + 972 \) Copy content Toggle raw display
$73$ \( (T - 65)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 38T + 1444 \) Copy content Toggle raw display
$83$ \( T^{2} - 84T + 2352 \) Copy content Toggle raw display
$89$ \( T^{2} + 15552 \) Copy content Toggle raw display
$97$ \( T^{2} - 115T + 13225 \) Copy content Toggle raw display
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